Ibm spss categories

i
IBM SPSS Categories 20
Jacqueline J. Meulman
Willem J. Heiser

Note: Before using this information and the product it supports, read the general information
under Notices on p. 301.
This edition applies to IBM® SPSS® Statistics 20 and to all subsequent releases and modifications
until otherwise indicated in new editions.
Adobe product screenshot(s) reprinted with permission from Adobe Systems Incorporated.
Microsoft product screenshot(s) reprinted with permission from Microsoft Corporation.
Licensed Materials - Property of IBM
© Copyright IBM Corporation 1989, 2011.
U.S. Government Users Restricted Rights - Use, duplication or disclosure restricted by GSA ADP
Schedule Contract with IBM Corp.

Preface
IBM® SPSS® Statistics is a comprehensive system for analyzing data. The Categories optional
add-on module provides the additional analytic techniques described in this manual. The
Categories add-on module must be used with the SPSS Statistics Core system and is completely
integrated into that system.
About IBM Business Analytics
IBM Business Analytics software delivers complete, consistent and accurate information that
decision-makers trust to improve business performance. A comprehensive portfolio of business
intelligence, predictive analytics, financial performance and strategy management, and analytic
applications provides clear, immediate and actionable insights into current performance and the
ability to predict future outcomes. Combined with rich industry solutions, proven practices and
professional services, organizations of every size can drive the highest productivity, confidently
automate decisions and deliver better results.
As part of this portfolio, IBM SPSS Predictive Analytics software helps organizations predict
future events and proactively act upon that insight to drive better business outcomes. Commercial,
government and academic customers worldwide rely on IBM SPSS technology as a competitive
advantage in attracting, retaining and growing customers, while reducing fraud and mitigating
risk. By incorporating IBM SPSS software into their daily operations, organizations become
predictive enterprises – able to direct and automate decisions to meet business goals and achieve
measurable competitive advantage. For further information or to reach a representative visit
http://www.ibm.com/spss.
Technical support
Technical support is available to maintenance customers. Customers may contact Technical
Support for assistance in using IBM Corp. products or for installation help for one of the
supported hardware environments. To reach Technical Support, see the IBM Corp. web site
at http://www.ibm.com/support. Be prepared to identify yourself, your organization, and your
support agreement when requesting assistance.
Technical Support for Students
If you’re a student using a student, academic or grad pack version of any IBM
SPSS software product, please see our special online Solutions for Education
(http://www.ibm.com/spss/rd/students/) pages for students. If you’re a student using a
university-supplied copy of the IBM SPSS software, please contact the IBM SPSS product
coordinator at your university.
Customer Service
If you have any questions concerning your shipment or account, contact your local office. Please
have your serial number ready for identification.
© Copyright IBM Corporation 1989, 2011. iii

Training Seminars
IBM Corp. provides both public and onsite training seminars. All seminars feature hands-on
workshops. Seminars will be offered in major cities on a regular basis. For more information on
these seminars, go to http://www.ibm.com/software/analytics/spss/training.
Additional Publications
The SPSS Statistics: Guide to Data Analysis, SPSS Statistics: Statistical Procedures Companion,
and SPSS Statistics: Advanced Statistical Procedures Companion, written by Marija Norušis and
published by Prentice Hall, are available as suggested supplemental material. These publications
cover statistical procedures in the SPSS Statistics Base module, Advanced Statistics module
and Regression module. Whether you are just getting starting in data analysis or are ready for
advanced applications, these books will help you make best use of the capabilities found within
the IBM® SPSS® Statistics offering. For additional information including publication contents
and sample chapters, please see the author’s website: http://www.norusis.com
Acknowledgements
The optimal scaling procedures and their implementation in IBM® SPSS® Statistics were
developed by the Data Theory Scaling System Group (DTSS), consisting of members of the
departments of Education and Psychology of the Faculty of Social and Behavioral Sciences
at Leiden University.
Willem Heiser, Jacqueline Meulman, Gerda van den Berg, and Patrick Groenen were involved
with the original 1990 procedures. Jacqueline Meulman and Peter Neufeglise participated in
the development of procedures for categorical regression, correspondence analysis, categorical
principal components analysis, and multidimensional scaling. In addition, Anita van der Kooij
contributed especially to CATREG, CORRESPONDENCE, and CATPCA. Willem Heiser,
Jacques Commandeur, Frank Busing, Gerda van den Berg, and Patrick Groenen participated in the
development of the PROXSCAL procedure. Frank Busing, Willem Heiser, Patrick Groenen, and
Peter Neufeglise participated in the development of the PREFSCAL procedure.
iv

Contents
Part I: User’s Guide
1 Introduction to Optimal Scaling Procedures for Categorical
Data 1
What Is Optimal Scaling? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Why Use Optimal Scaling? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Optimal Scaling Level and Measurement Level . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
Selecting the Optimal Scaling Level . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
Transformation Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
Category Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
Which Procedure Is Best for Your Application? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
Categorical Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
Categorical Principal Components Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
Nonlinear Canonical Correlation Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
Correspondence Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
Multiple Correspondence Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
Multidimensional Scaling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
Multidimensional Unfolding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
Aspect Ratio in Optimal Scaling Charts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
Recommended Readings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2 Categorical Regression (CATREG) 14
Define Scale in Categorical Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
Categorical Regression Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
Categorical Regression Missing Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
Categorical Regression Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
Categorical Regression Regularization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
Categorical Regression Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
Categorical Regression Save . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
Categorical Regression Transformation Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
CATREG Command Additional Features. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
v

3 Categorical Principal Components Analysis (CATPCA) 26
Define Scale and Weight in CATPCA. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
Categorical Principal Components Analysis Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
Categorical Principal Components Analysis Missing Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
Categorical Principal Components Analysis Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
Categorical Principal Components Analysis Output. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
Categorical Principal Components Analysis Save . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
Categorical Principal Components Analysis Object Plots. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
Categorical Principal Components Analysis Category Plots. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
Categorical Principal Components Analysis Loading Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
CATPCA Command Additional Features. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
4 Nonlinear Canonical Correlation Analysis (OVERALS) 40
Define Range and Scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
Define Range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
Nonlinear Canonical Correlation Analysis Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
OVERALS Command Additional Features. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
5 Correspondence Analysis 46
Define Row Range in Correspondence Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
Define Column Range in Correspondence Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
Correspondence Analysis Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
Correspondence Analysis Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
Correspondence Analysis Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
CORRESPONDENCE Command Additional Features. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
6 Multiple Correspondence Analysis 55
Define Variable Weight in Multiple Correspondence Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
Multiple Correspondence Analysis Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
Multiple Correspondence Analysis Missing Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
Multiple Correspondence Analysis Options. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
vi

Multiple Correspondence Analysis Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
Multiple Correspondence Analysis Save. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
Multiple Correspondence Analysis Object Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
Multiple Correspondence Analysis Variable Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
MULTIPLE CORRESPONDENCE Command Additional Features . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
7 Multidimensional Scaling (PROXSCAL) 67
Proximities in Matrices across Columns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
Proximities in Columns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
Proximities in One Column . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
Create Proximities from Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
Create Measure from Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
Define a Multidimensional Scaling Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
Multidimensional Scaling Restrictions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
Multidimensional Scaling Options. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
Multidimensional Scaling Plots, Version 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
Multidimensional Scaling Plots, Version 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
Multidimensional Scaling Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
PROXSCAL Command Additional Features . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
8 Multidimensional Unfolding (PREFSCAL) 82
Define a Multidimensional Unfolding Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
Multidimensional Unfolding Restrictions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
Multidimensional Unfolding Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
Multidimensional Unfolding Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
Multidimensional Unfolding Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
PREFSCAL Command Additional Features . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
vii

Part II: Examples
9 Categorical Regression 94
Example: Carpet Cleaner Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
A Standard Linear Regression Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
A Categorical Regression Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
Example: Ozone Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
Discretizing Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
Selection of Transformation Type. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
Optimality of the Quantifications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
Effects of Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
Recommended Readings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
10 Categorical Principal Components Analysis 137
Example: Examining Interrelations of Social Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
Running the Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
Number of Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
Quantifications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
Object Scores . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
Component Loadings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
Additional Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
Example: Symptomatology of Eating Disorders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
Transformation Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
Model Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
Object Scores . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
Examining the Structure of the Course of Illness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
11 Nonlinear Canonical Correlation Analysis 186
Example: An Analysis of Survey Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
Examining the Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
Accounting for Similarity between Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
viii

Transformation Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
Single Category versus Multiple Category Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200
Centroids and Projected Centroids. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
An Alternative Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204
General Suggestions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
12 Correspondence analysis 211
Normalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212
Example: Perceptions of Coffee Brands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212
Dimensionality. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
Contributions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218
Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
Symmetrical Normalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220
Recommended readings. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222
13 Multiple Correspondence Analysis 223
Example: Characteristics of Hardware . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
Model Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226
Object Scores . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
Discrimination Measures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228
Category Quantifications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229
A More Detailed Look at Object Scores . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231
Omission of Outliers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
14 Multidimensional Scaling 239
Example: An Examination of Kinship Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239
Choosing the Number of Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239
A Three-Dimensional Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246
A Three-Dimensional Solution with Nondefault Transformations . . . . . . . . . . . . . . . . . . . . . 252
Discussion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256
ix

15 Multidimensional Unfolding 258
Example: Breakfast Item Preferences. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258
Producing a Degenerate Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258
Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261
Common Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262
Running a Nondegenerate Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263
Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264
Common Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265
Example: Three-Way Unfolding of Breakfast Item Preferences . . . . . . . . . . . . . . . . . . . . . . . . . . 265
Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270
Common Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271
Individual Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272
Using a Different Initial Configuration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274
Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276
Common Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277
Individual Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278
Example: Examining Behavior-Situation Appropriateness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280
Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285
Common Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286
Proximity Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287
Changing the Proximities Transformation (Ordinal). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287
Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289
Common Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290
Proximity Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291
x

Appendices
A Sample Files 292
B Notices 301
Bibliography 304
Index 310
xi

Chapter
1
Introduction to Optimal Scaling
Procedures for Categorical Data
Categories procedures use optimal scaling to analyze data that are difficult or impossible for
standard statistical procedures to analyze. This chapter describes what each procedure does, the
situations in which each procedure is most appropriate, the relationships between the procedures,
and the relationships of these procedures to their standard statistical counterparts.
Note: These procedures and their implementation in IBM® SPSS® Statistics were developed by
the Data Theory Scaling System Group (DTSS), consisting of members of the departments of
Education and Psychology, Faculty of Social and Behavioral Sciences, Leiden University.
What Is Optimal Scaling?
The idea behind optimal scaling is to assign numerical quantifications to the categories of each
variable, thus allowing standard procedures to be used to obtain a solution on the quantified
variables.
The optimal scale values are assigned to categories of each variable based on the optimizing
criterion of the procedure in use. Unlike the original labels of the nominal or ordinal variables in
the analysis, these scale values have metric properties.
In most Categories procedures, the optimal quantification for each scaled variable is obtained
through an iterative method called alternating least squares in which, after the current
quantifications are used to find a solution, the quantifications are updated using that solution.
The updated quantifications are then used to find a new solution, which is used to update the
quantifications, and so on, until some criterion is reached that signals the process to stop.
Why Use Optimal Scaling?
Categorical data are often found in marketing research, survey research, and research in the social
and behavioral sciences. In fact, many researchers deal almost exclusively with categorical data.
While adaptations of most standard models exist specifically to analyze categorical data, they
often do not perform well for datasets that feature:
 Too few observations
 Too many variables
 Too many values per variable
By quantifying categories, optimal scaling techniques avoid problems in these situations.
Moreover, they are useful even when specialized techniques are appropriate.
Rather than interpreting parameter estimates, the interpretation of optimal scaling output
is often based on graphical displays. Optimal scaling techniques offer excellent exploratory
analyses, which complement other IBM® SPSS® Statistics models well. By narrowing the
© Copyright IBM Corporation 1989, 2011. 1

2
Chapter 1
focus of your investigation, visualizing your data through optimal scaling can form the basis of
an analysis that centers on interpretation of model parameters.
Optimal Scaling Level and Measurement Level
This can be a very confusing concept when you first use Categories procedures. When specifying
the level, you specify not the level at which variables are measured but the level at which they
are scaled. The idea is that the variables to be quantified may have nonlinear relations regardless
of how they are measured.
For Categories purposes, there are three basic levels of measurement:
 The nominal level implies that a variable’s values represent unordered categories. Examples
of variables that might be nominal are region, zip code area, religious affiliation, and multiple
choice categories.
 The ordinal level implies that a variable’s values represent ordered categories. Examples
include attitude scales representing degree of satisfaction or confidence and preference rating
scores.
 The numerical level implies that a variable’s values represent ordered categories with a
meaningful metric so that distance comparisons between categories are appropriate. Examples
include age in years and income in thousands of dollars.
For example, suppose that the variables region, job, and age are coded as shown in the following
table.
Table 1-1
Coding scheme for region, job, and age
Region code Region value Job code Job value Age
1 North 1 intern 20
2 South 2 sales rep 22
3 East 3 manager 25
4 West 27
The values shown represent the categories of each variable. Region would be a nominal variable.
There are four categories of region, with no intrinsic ordering. Values 1 through 4 simply represent
the four categories; the coding scheme is completely arbitrary. Job, on the other hand, could be
assumed to be an ordinal variable. The original categories form a progression from intern to
manager. Larger codes represent a job higher on the corporate ladder. However, only the order
information is known—nothing can be said about the distance between adjacent categories. In
contrast, age could be assumed to be a numerical variable. In the case of age, the distances
between the values are intrinsically meaningful. The distance between 20 and 22 is the same as the
distance between 25 and 27, while the distance between 22 and 25 is greater than either of these.

3
Introduction to Optimal Scaling Procedures for Categorical Data
Selecting the Optimal Scaling Level
It is important to understand that there are no intrinsic properties of a variable that automatically
predefine what optimal scaling level you should specify for it. You can explore your data in any
way that makes sense and makes interpretation easier. By analyzing a numerical-level variable
at the ordinal level, for example, the use of a nonlinear transformation may allow a solution
in fewer dimensions.
The following two examples illustrate how the “obvious” level of measurement might not be
the best optimal scaling level. Suppose that a variable sorts objects into age groups. Although age
can be scaled as a numerical variable, it may be true that for people younger than 25 safety has a
positive relation with age, whereas for people older than 60 safety has a negative relation with
age. In this case, it might be better to treat age as a nominal variable.
As another example, a variable that sorts persons by political preference appears to be
essentially nominal. However, if you order the parties from political left to political right, you
might want the quantification of parties to respect this order by using an ordinal level of analysis.
Even though there are no predefined properties of a variable that make it exclusively one
level or another, there are some general guidelines to help the novice user. With single-nominal
quantification, you don’t usually know the order of the categories but you want the analysis to
impose one. If the order of the categories is known, you should try ordinal quantification. If the
categories are unorderable, you might try multiple-nominal quantification.
Transformation Plots
The different levels at which each variable can be scaled impose different restrictions on the
quantifications. Transformation plots illustrate the relationship between the quantifications and
the original categories resulting from the selected optimal scaling level. For example, a linear
transformation plot results when a variable is treated as numerical. Variables treated as ordinal
result in a nondecreasing transformation plot. Transformation plots for variables treated nominally
that are U-shaped (or the inverse) display a quadratic relationship. Nominal variables could
also yield transformation plots without apparent trends by changing the order of the categories
completely. The following figure displays a sample transformation plot.
Transformation plots are particularly suited to determining how well the selected optimal
scaling level performs. If several categories receive similar quantifications, collapsing these
categories into one category may be warranted. Alternatively, if a variable treated as nominal
receives quantifications that display an increasing trend, an ordinal transformation may result in a
similar fit. If that trend is linear, numerical treatment may be appropriate. However, if collapsing
categories or changing scaling levels is warranted, the analysis will not change significantly.

4
Chapter 1
Figure 1-1
Transformation plot of price (numerical)
Category Codes
Some care should be taken when coding categorical variables because some coding schemes may
yield unwanted output or incomplete analyses. Possible coding schemes for job are displayed in
the following table.
Table 1-2
Alternative coding schemes for job
Category A B C D
intern 1 1 5 1
sales rep 2 2 6 5
manager 3 7 7 3
Some Categories procedures require that the range of every variable used be defined. Any value
outside this range is treated as a missing value. The minimum category value is always 1. The
maximum category value is supplied by the user. This value is not the number of categories for a
variable—it is the largest category value. For example, in the table, scheme A has a maximum
category value of 3 and scheme B has a maximum category value of 7, yet both schemes code
the same three categories.
The variable range determines which categories will be omitted from the analysis. Any
categories with codes outside the defined range are omitted from the analysis. This is a simple
method for omitting categories but can result in unwanted analyses. An incorrectly defined
maximum category can omit valid categories from the analysis. For example, for scheme B,
defining the maximum category value to be 3 indicates that job has categories coded from 1 to 3;
the manager category is treated as missing. Because no category has actually been coded 3, the
third category in the analysis contains no cases. If you wanted to omit all manager categories, this
analysis would be appropriate. However, if managers are to be included, the maximum category
must be defined as 7, and missing values must be coded with values above 7 or below 1.

5
For variables treated as nominal or ordinal, the range of the categories does not affect the
results. For nominal variables, only the label and not the value associated with that label is
important. For ordinal variables, the order of the categories is preserved in the quantifications;
the category values themselves are not important. All coding schemes resulting in the same
category ordering will have identical results. For example, the first three schemes in the table
are functionally equivalent if job is analyzed at an ordinal level. The order of the categories is
identical in these schemes. Scheme D, on the other hand, inverts the second and third categories
and will yield different results than the other schemes.
Although many coding schemes for a variable are functionally equivalent, schemes with small
differences between codes are preferred because the codes have an impact on the amount of
output produced by a procedure. All categories coded with values between 1 and the user-defined
maximum are valid. If any of these categories are empty, the corresponding quantifications will be
either system-missing or 0, depending on the procedure. Although neither of these assignments
affect the analyses, output is produced for these categories. Thus, for scheme B, job has four
categories that receive system-missing values. For scheme C, there are also four categories
receiving system-missing indicators. In contrast, for scheme A there are no system-missing
quantifications. Using consecutive integers as codes for variables treated as nominal or ordinal
results in much less output without affecting the results.
Coding schemes for variables treated as numerical are more restricted than the ordinal case.
For these variables, the differences between consecutive categories are important. The following
table displays three coding schemes for age.
Table 1-3
Alternative coding schemes for age
Category A B C
20 20 1 1
22 22 3 2
25 25 6 3
27 27 8 4
Any recoding of numerical variables must preserve the differences between the categories. Using
the original values is one method for ensuring preservation of differences. However, this can
result in many categories having system-missing indicators. For example, scheme A employs
the original observed values. For all Categories procedures except for Correspondence Analysis,
the maximum category value is 27 and the minimum category value is set to 1. The first 19
categories are empty and receive system-missing indicators. The output can quickly become
rather cumbersome if the maximum category is much greater than 1 and there are many empty
categories between 1 and the maximum.
To reduce the amount of output, recoding can be done. However, in the numerical case,
the Automatic Recode facility should not be used. Coding to consecutive integers results in
differences of 1 between all consecutive categories, and, as a result, all quantifications will be
equally spaced. The metric characteristics deemed important when treating a variable as numerical
are destroyed by recoding to consecutive integers. For example, scheme C in the table corresponds
to automatically recoding age. The difference between categories 22 and 25 has changed from
three to one, and the quantifications will reflect the latter difference.

6
Chapter 1
An alternative recoding scheme that preserves the differences between categories is to subtract
the smallest category value from every category and add 1 to each difference. Scheme B
results from this transformation. The smallest category value, 20, has been subtracted from
each category, and 1 was added to each result. The transformed codes have a minimum of 1,
and all differences are identical to the original data. The maximum category value is now 8,
and the zero quantifications before the first nonzero quantification are all eliminated. Yet, the
nonzero quantifications corresponding to each category resulting from scheme B are identical to
the quantifications from scheme A.
Which Procedure Is Best for Your Application?
The techniques embodied in four of these procedures (Correspondence Analysis, Multiple
Correspondence Analysis, Categorical Principal Components Analysis, and Nonlinear Canonical
Correlation Analysis) fall into the general area of multivariate data analysis known as dimension
reduction. That is, relationships between variables are represented in a few dimensions—say
two or three—as often as possible. This enables you to describe structures or patterns in the
relationships that would be too difficult to fathom in their original richness and complexity. In
market research applications, these techniques can be a form of perceptual mapping. A major
advantage of these procedures is that they accommodate data with different levels of optimal
scaling.
Categorical Regression describes the relationship between a categorical response variable and a
combination of categorical predictor variables. The influence of each predictor variable on the
response variable is described by the corresponding regression weight. As in the other procedures,
data can be analyzed with different levels of optimal scaling.
Multidimensional Scaling and Multidimensional Unfolding describe relationships between
objects in a low-dimensional space, using the proximities between the objects.
Following are brief guidelines for each of the procedures:
 Use Categorical Regression to predict the values of a categorical dependent variable from a
combination of categorical independent variables.
 Use Categorical Principal Components Analysis to account for patterns of variation in a single
set of variables of mixed optimal scaling levels.
 Use Nonlinear Canonical Correlation Analysis to assess the extent to which two or more sets
of variables of mixed optimal scaling levels are correlated.
 Use Correspondence Analysis to analyze two-way contingency tables or data that can be
expressed as a two-way table, such as brand preference or sociometric choice data.
 Use Multiple Correspondence Analysis to analyze a categorical multivariate data matrix
when you are willing to make no stronger assumption that all variables are analyzed at the
nominal level.
 Use Multidimensional Scaling to analyze proximity data to find a least-squares representation
of a single set of objects in a low-dimensional space.
 Use Multidimensional Unfolding to analyze proximity data to find a least-squares
representation of two sets of objects in a low-dimensional space.

7
Categorical Regression
The use of Categorical Regression is most appropriate when the goal of your analysis is to
predict a dependent (response) variable from a set of independent (predictor) variables. As with
all optimal scaling procedures, scale values are assigned to each category of every variable
such that these values are optimal with respect to the regression. The solution of a categorical
regression maximizes the squared correlation between the transformed response and the weighted
combination of transformed predictors.
Relation to other Categories procedures. Categorical regression with optimal scaling is comparable
to optimal scaling canonical correlation analysis with two sets, one of which contains only the
dependent variable. In the latter technique, similarity of sets is derived by comparing each set
to an unknown variable that lies somewhere between all of the sets. In categorical regression,
similarity of the transformed response and the linear combination of transformed predictors
is assessed directly.
Relation to standard techniques. In standard linear regression, categorical variables can either be
recoded as indicator variables or be treated in the same fashion as interval level variables. In the
first approach, the model contains a separate intercept and slope for each combination of the
levels of the categorical variables. This results in a large number of parameters to interpret. In
the second approach, only one parameter is estimated for each variable. However, the arbitrary
nature of the category codings makes generalizations impossible.
If some of the variables are not continuous, alternative analyses are available. If the response
is continuous and the predictors are categorical, analysis of variance is often employed. If the
response is categorical and the predictors are continuous, logistic regression or discriminant
analysis may be appropriate. If the response and the predictors are both categorical, loglinear
models are often used.
Regression with optimal scaling offers three scaling levels for each variable. Combinations
of these levels can account for a wide range of nonlinear relationships for which any single
“standard” method is ill-suited. Consequently, optimal scaling offers greater flexibility than the
standard approaches with minimal added complexity.
In addition, nonlinear transformations of the predictors usually reduce the dependencies among
the predictors. If you compare the eigenvalues of the correlation matrix for the predictors with
the eigenvalues of the correlation matrix for the optimally scaled predictors, the latter set will
usually be less variable than the former. In other words, in categorical regression, optimal
scaling makes the larger eigenvalues of the predictor correlation matrix smaller and the smaller
eigenvalues larger.
Categorical Principal Components Analysis
The use of Categorical Principal Components Analysis is most appropriate when you want to
account for patterns of variation in a single set of variables of mixed optimal scaling levels. This
technique attempts to reduce the dimensionality of a set of variables while accounting for as much
of the variation as possible. Scale values are assigned to each category of every variable so that
these values are optimal with respect to the principal components solution. Objects in the analysis
receive component scores based on the quantified data. Plots of the component scores reveal
patterns among the objects in the analysis and can reveal unusual objects in the data. The solution

8
Chapter 1
of a categorical principal components analysis maximizes the correlations of the object scores
with each of the quantified variables for the number of components (dimensions) specified.
An important application of categorical principal components is to examine preference data,
in which respondents rank or rate a number of items with respect to preference. In the usual
IBM® SPSS® Statistics data configuration, rows are individuals, columns are measurements for
the items, and the scores across rows are preference scores (on a 0 to 10 scale, for example),
making the data row-conditional. For preference data, you may want to treat the individuals as
variables. Using the Transpose procedure, you can transpose the data. The raters become the
variables, and all variables are declared ordinal. There is no objection to using more variables
than objects in CATPCA.
Relation to other Categories procedures. If all variables are declared multiple nominal, categorical
principal components analysis produces an analysis equivalent to a multiple correspondence
analysis run on the same variables. Thus, categorical principal components analysis can be
seen as a type of multiple correspondence analysis in which some of the variables are declared
ordinal or numerical.
Relation to standard techniques. If all variables are scaled on the numerical level, categorical
principal components analysis is equivalent to standard principal components analysis.
More generally, categorical principal components analysis is an alternative to computing
the correlations between non-numerical scales and analyzing them using a standard principal
components or factor-analysis approach. Naive use of the usual Pearson correlation coefficient as a
measure of association for ordinal data can lead to nontrivial bias in estimation of the correlations.
Nonlinear Canonical Correlation Analysis
Nonlinear Canonical Correlation Analysis is a very general procedure with many different
applications. The goal of nonlinear canonical correlation analysis is to analyze the relationships
between two or more sets of variables instead of between the variables themselves, as in principal
components analysis. For example, you may have two sets of variables, where one set of
variables might be demographic background items on a set of respondents and a second set
might be responses to a set of attitude items. The scaling levels in the analysis can be any mix of
nominal, ordinal, and numerical. Optimal scaling canonical correlation analysis determines the
similarity among the sets by simultaneously comparing the canonical variables from each set to a
compromise set of scores assigned to the objects.
Relation to other Categories procedures. If there are two or more sets of variables with only one
variable per set, optimal scaling canonical correlation analysis is equivalent to optimal scaling
principal components analysis. If all variables in a one-variable-per-set analysis are multiple
nominal, optimal scaling canonical correlation analysis is equivalent to multiple correspondence
analysis. If there are two sets of variables, one of which contains only one variable, optimal
scaling canonical correlation analysis is equivalent to categorical regression with optimal scaling.
Relation to standard techniques. Standard canonical correlation analysis is a statistical technique
that finds a linear combination of one set of variables and a linear combination of a second set
of variables that are maximally correlated. Given this set of linear combinations, canonical
correlation analysis can find subsequent independent sets of linear combinations, referred to as
canonical variables, up to a maximum number equal to the number of variables in the smaller set.

9
If there are two sets of variables in the analysis and all variables are defined to be numerical,
optimal scaling canonical correlation analysis is equivalent to a standard canonical correlation
analysis. Although IBM® SPSS® Statistics does not have a canonical correlation analysis
procedure, many of the relevant statistics can be obtained from multivariate analysis of variance.
Optimal scaling canonical correlation analysis has various other applications. If you have two
sets of variables and one of the sets contains a nominal variable declared as single nominal,
optimal scaling canonical correlation analysis results can be interpreted in a similar fashion to
regression analysis. If you consider the variable to be multiple nominal, the optimal scaling
analysis is an alternative to discriminant analysis. Grouping the variables in more than two sets
provides a variety of ways to analyze your data.
Correspondence Analysis
The goal of correspondence analysis is to make biplots for correspondence tables. In a
correspondence table, the row and column variables are assumed to represent unordered
categories; therefore, the nominal optimal scaling level is always used. Both variables are
inspected for their nominal information only. That is, the only consideration is the fact that some
objects are in the same category while others are not. Nothing is assumed about the distance or
order between categories of the same variable.
One specific use of correspondence analysis is the analysis of two-way contingency tables. If a
table has r active rows and c active columns, the number of dimensions in the correspondence
analysis solution is the minimum of r minus 1 or c minus 1, whichever is less. In other words, you
could perfectly represent the row categories or the column categories of a contingency table in
a space of dimensions. Practically speaking, however, you would like to represent the row and
column categories of a two-way table in a low-dimensional space, say two dimensions, for the
reason that two-dimensional plots are more easily comprehensible than multidimensional spatial
representations.
When fewer than the maximum number of possible dimensions is used, the statistics
produced in the analysis describe how well the row and column categories are represented in the
low-dimensional representation. Provided that the quality of representation of the two-dimensional
solution is good, you can examine plots of the row points and the column points to learn which
categories of the row variable are similar, which categories of the column variable are similar,
and which row and column categories are similar to each other.
Relation to other Categories procedures. Simple correspondence analysis is limited to two-way
tables. If there are more than two variables of interest, you can combine variables to create
interaction variables. For example, for the variables region, job, and age, you can combine region
and job to create a new variable rejob with the 12 categories shown in the following table. This
new variable forms a two-way table with age (12 rows, 4 columns), which can be analyzed
in correspondence analysis.
Table 1-4
Combinations of region and job
Category code Category definition Category code Category definition
1 North, intern 7 East, intern
2 North, sales rep 8 East, sales rep
3 North, manager 9 East, manager

10
Chapter 1
Category code Category definition Category code Category definition
4 South, intern 10 West, intern
5 South, sales rep 11 West, sales rep
6 South, manager 12 West, manager
One shortcoming of this approach is that any pair of variables can be combined. We can combine
job and age, yielding another 12-category variable. Or we can combine region and age, which
results in a new 16-category variable. Each of these interaction variables forms a two-way table
with the remaining variable. Correspondence analyses of these three tables will not yield identical
results, yet each is a valid approach. Furthermore, if there are four or more variables, two-way
tables comparing an interaction variable with another interaction variable can be constructed.
The number of possible tables to analyze can get quite large, even for a few variables. You can
select one of these tables to analyze, or you can analyze all of them. Alternatively, the Multiple
Correspondence Analysis procedure can be used to examine all of the variables simultaneously
without the need to construct interaction variables.
Relation to standard techniques. The Crosstabs procedure can also be used to analyze contingency
tables, with independence as a common focus in the analyses. However, even in small
tables, detecting the cause of departures from independence may be difficult. The utility of
correspondence analysis lies in displaying such patterns for two-way tables of any size. If there
is an association between the row and column variables—that is, if the chi-square value is
significant—correspondence analysis may help reveal the nature of the relationship.
Multiple Correspondence Analysis
Multiple Correspondence Analysis tries to produce a solution in which objects within the same
category are plotted close together and objects in different categories are plotted far apart. Each
object is as close as possible to the category points of categories that apply to the object. In this
way, the categories divide the objects into homogeneous subgroups. Variables are considered
homogeneous when they classify objects in the same categories into the same subgroups.
For a one-dimensional solution, multiple correspondence analysis assigns optimal scale values
(category quantifications) to each category of each variable in such a way that overall, on average,
the categories have maximum spread. For a two-dimensional solution, multiple correspondence
analysis finds a second set of quantifications of the categories of each variable unrelated to the
first set, attempting again to maximize spread, and so on. Because categories of a variable
receive as many scorings as there are dimensions, the variables in the analysis are assumed to
be multiple nominal in optimal scaling level.
Multiple correspondence analysis also assigns scores to the objects in the analysis in such a
way that the category quantifications are the averages, or centroids, of the object scores of objects
in that category.
Relation to other Categories procedures. Multiple correspondence analysis is also known
as homogeneity analysis or dual scaling. It gives comparable, but not identical, results to
correspondence analysis when there are only two variables. Correspondence analysis produces
unique output summarizing the fit and quality of representation of the solution, including stability
information. Thus, correspondence analysis is usually preferable to multiple correspondence
analysis in the two-variable case. Another difference between the two procedures is that the input

11
to multiple correspondence analysis is a data matrix, where the rows are objects and the columns
are variables, while the input to correspondence analysis can be the same data matrix, a general
proximity matrix, or a joint contingency table, which is an aggregated matrix in which both the
rows and columns represent categories of variables. Multiple correspondence analysis can also be
thought of as principal components analysis of data scaled at the multiple nominal level.
Relation to standard techniques. Multiple correspondence analysis can be thought of as the analysis
of a multiway contingency table. Multiway contingency tables can also be analyzed with the
Crosstabs procedure, but Crosstabs gives separate summary statistics for each category of each
control variable. With multiple correspondence analysis, it is often possible to summarize the
relationship between all of the variables with a single two-dimensional plot. An advanced use of
multiple correspondence analysis is to replace the original category values with the optimal scale
values from the first dimension and perform a secondary multivariate analysis. Since multiple
correspondence analysis replaces category labels with numerical scale values, many different
procedures that require numerical data can be applied after the multiple correspondence analysis.
For example, the Factor Analysis procedure produces a first principal component that is equivalent
to the first dimension of multiple correspondence analysis. The component scores in the first
dimension are equal to the object scores, and the squared component loadings are equal to the
discrimination measures. The second multiple correspondence analysis dimension, however, is
not equal to the second dimension of factor analysis.
Multidimensional Scaling
The use of Multidimensional Scaling is most appropriate when the goal of your analysis is to
find the structure in a set of distance measures between a single set of objects or cases. This is
accomplished by assigning observations to specific locations in a conceptual low-dimensional
space so that the distances between points in the space match the given (dis)similarities as closely
as possible. The result is a least-squares representation of the objects in that low-dimensional
space, which, in many cases, will help you further understand your data.
Relation to other Categories procedures. When you have multivariate data from which you create
distances and which you then analyze with multidimensional scaling, the results are similar
to analyzing the data using categorical principal components analysis with object principal
normalization. This kind of PCA is also known as principal coordinates analysis.
Relation to standard techniques. The Categories Multidimensional Scaling procedure
(PROXSCAL) offers several improvements upon the scaling procedure available in the Statistics
Base option (ALSCAL). PROXSCAL offers an accelerated algorithm for certain models and
allows you to put restrictions on the common space. Moreover, PROXSCAL attempts to minimize
normalized raw stress rather than S-stress (also referred to as strain). The normalized raw stress is
generally preferred because it is a measure based on the distances, while the S-stress is based on
the squared distances.
Multidimensional Unfolding
The use of Multidimensional Unfolding is most appropriate when the goal of your analysis is to
find the structure in a set of distance measures between two sets of objects (referred to as the
row and column objects). This is accomplished by assigning observations to specific locations

12
Chapter 1
in a conceptual low-dimensional space so that the distances between points in the space match
the given (dis)similarities as closely as possible. The result is a least-squares representation
of the row and column objects in that low-dimensional space, which, in many cases, will help
you further understand your data.
Relation to other Categories procedures. If your data consist of distances between a single set of
objects (a square, symmetrical matrix), use Multidimensional Scaling.
Relation to standard techniques. The Categories Multidimensional Unfolding procedure
(PREFSCAL) offers several improvements upon the unfolding functionality available in the
Statistics Base option (through ALSCAL). PREFSCAL allows you to put restrictions on the
common space; moreover, PREFSCAL attempts to minimize a penalized stress measure that helps
it to avoid degenerate solutions (to which older algorithms are prone).
Aspect Ratio in Optimal Scaling Charts
Aspect ratio in optimal scaling plots is isotropic. In a two-dimensional plot, the distance
representing one unit in dimension 1 is equal to the distance representing one unit in dimension 2.
If you change the range of a dimension in a two-dimensional plot, the system changes the size
of the other dimension to keep the physical distances equal. Isotropic aspect ratio cannot be
overridden for the optimal scaling procedures.
Recommended Readings
See the following texts for general information on optimal scaling techniques:
Barlow, R. E., D. J. Bartholomew, D. J. Bremner, and H. D. Brunk. 1972. Statistical inference
under order restrictions. New York: John Wiley and Sons.
Benzécri, J. P. 1969. Statistical analysis as a tool to make patterns emerge from data. In:
Methodologies of Pattern Recognition, S. Watanabe, ed. New York: Academic Press, 35–74.
Bishop, Y. M., S. E. Feinberg, and P. W. Holland. 1975. Discrete multivariate analysis: Theory
and practice. Cambridge, Mass.: MIT Press.
De Leeuw, J. 1984. The Gifi system of nonlinear multivariate analysis. In: Data Analysis and
Informatics III, E. Diday, et al., ed., 415–424.
De Leeuw, J. 1990. Multivariate analysis with optimal scaling. In: Progress in Multivariate
Analysis, S. Das Gupta, and J. Sethuraman, eds. Calcutta: Indian Statistical Institute.
De Leeuw, J., and J. Van Rijckevorsel. 1980. HOMALS and PRINCALS—Some generalizations
of principal components analysis. In: Data Analysis and Informatics, E. Diday,et al., ed.
Amsterdam: North-Holland, 231–242.
De Leeuw, J., F. W. Young, and Y. Takane. 1976. Additive structure in qualitative data: An
alternating least squares method with optimal scaling features. Psychometrika, 41, 471–503.
Gifi, A. 1990. Nonlinear multivariate analysis. Chichester: John Wiley and Sons.

13
Heiser, W. J., and J. J. Meulman. 1995. Nonlinear methods for the analysis of homogeneity and
heterogeneity. In: Recent Advances in Descriptive Multivariate Analysis, W. J. Krzanowski,
ed. Oxford: Oxford UniversityPress, 51–89.
Israëls, A. 1987. Eigenvalue techniques for qualitative data. Leiden: DSWO Press.
Krzanowski, W. J., and F. H. C. Marriott. 1994. Multivariate analysis: Part I, distributions,
ordination and inference. London: Edward Arnold.
Lebart, L., A. Morineau, and K. M. Warwick. 1984. Multivariate descriptive statistical analysis.
New York: John Wiley and Sons.
Max, J. 1960. Quantizing for minimum distortion. Proceedings IEEE (Information Theory), 6,
7–12.
Meulman, J. J. 1986. A distance approach to nonlinear multivariate analysis. Leiden: DSWO
Press.
Meulman, J. J. 1992. The integration of multidimensional scaling and multivariate analysis with
optimal transformations of the variables. Psychometrika, 57, 539–565.
Nishisato, S. 1980. Analysis of categorical data: Dual scaling and its applications. Toronto:
University of Toronto Press.
Nishisato, S. 1994. Elements of dual scaling: An introduction to practical data analysis. Hillsdale,
N.J.: Lawrence Erlbaum Associates, Inc.
Rao, C. R. 1973. Linear statistical inference and its applications, 2nd ed. New York: John
Wiley and Sons.
Rao, C. R. 1980. Matrix approximations and reduction of dimensionality in multivariate statistical
analysis. In: Multivariate Analysis, Vol. 5, P. R. Krishnaiah, ed. Amsterdam: North-Holland,
3–22.
Roskam, E. E. 1968. Metric analysis of ordinal data in psychology. Voorschoten: VAM.
Shepard, R. N. 1966. Metric structures in ordinal data. Journal of Mathematical Psychology, 3,
287–315.
Wolter, K. M. 1985. Introduction to variance estimation. Berlin: Springer-Verlag.
Young, F. W. 1981. Quantitative analysis of qualitative data. Psychometrika, 46, 357–387.

Chapter
2
Categorical Regression (CATREG)
Categorical regression quantifies categorical data by assigning numerical values to the
categories, resulting in an optimal linear regression equation for the transformed variables.
Categorical regression is also known by the acronym CATREG, for categorical regression.
Standard linear regression analysis involves minimizing the sum of squared differences
between a response (dependent) variable and a weighted combination of predictor (independent)
variables. Variables are typically quantitative, with (nominal) categorical data recoded to binary
or contrast variables. As a result, categorical variables serve to separate groups of cases, and the
technique estimates separate sets of parameters for each group. The estimated coefficients reflect
how changes in the predictors affect the response. Prediction of the response is possible for
any combination of predictor values.
An alternative approach involves regressing the response on the categorical predictor values
themselves. Consequently, one coefficient is estimated for each variable. However, for categorical
variables, the category values are arbitrary. Coding the categories in different ways yield different
coefficients, making comparisons across analyses of the same variables difficult.
CATREG extends the standard approach by simultaneously scaling nominal, ordinal, and
numerical variables. The procedure quantifies categorical variables so that the quantifications
reflect characteristics of the original categories. The procedure treats quantified categorical
variables in the same way as numerical variables. Using nonlinear transformations allow variables
to be analyzed at a variety of levels to find the best-fitting model.
Example. Categorical regression could be used to describe how job satisfaction depends on job
category, geographic region, and amount of travel. You might find that high levels of satisfaction
correspond to managers and low travel. The resulting regression equation could be used to predict
job satisfaction for any combination of the three independent variables.
Statistics and plots. Frequencies, regression coefficients, ANOVA table, iteration history, category
quantifications, correlations between untransformed predictors, correlations between transformed
predictors, residual plots, and transformation plots.
Data. CATREG operates on category indicator variables. The category indicators should be
positive integers. You can use the Discretization dialog box to convert fractional-value variables
and string variables into positive integers.
Assumptions. Only one response variable is allowed, but the maximum number of predictor
variables is 200. The data must contain at least three valid cases, and the number of valid cases
must exceed the number of predictor variables plus one.
Related procedures. CATREG is equivalent to categorical canonical correlation analysis with
optimal scaling (OVERALS) with two sets, one of which contains only one variable. Scaling all
variables at the numerical level corresponds to standard multiple regression analysis.

15
To Obtain a Categorical Regression
E From the menus choose:
Analyze > Regression > Optimal Scaling (CATREG)...
Figure 2-1
Categorical Regression dialog box
E Select the dependent variable and independent variable(s).
E Click OK.
Optionally, change the scaling level for each variable.
Define Scale in Categorical Regression
You can set the optimal scaling level for the dependent and independent variables. By default, they
are scaled as second-degree monotonic splines (ordinal) with two interior knots. Additionally,
you can set the weight for analysis variables.

16
Chapter 2
Figure 2-2
Define Scale dialog box
Optimal Scaling Level. You can also select the scaling level for quantifying each variable.
 Spline Ordinal. The order of the categories of the observed variable is preserved in the
optimally scaled variable. Category points will be on a straight line (vector) through the origin.
The resulting transformation is a smooth monotonic piecewise polynomial of the chosen
degree. The pieces are specified by the user-specified number and procedure-determined
placement of the interior knots.
 Spline Nominal. The only information in the observed variable that is preserved in the
optimally scaled variable is the grouping of objects in categories. The order of the categories
of the observed variable is not preserved. Category points will be on a straight line (vector)
through the origin. The resulting transformation is a smooth, possibly nonmonotonic,
piecewise polynomial of the chosen degree. The pieces are specified by the user-specified
number and procedure-determined placement of the interior knots.
 Ordinal. The order of the categories of the observed variable is preserved in the optimally
scaled variable. Category points will be on a straight line (vector) through the origin. The
resulting transformation fits better than the spline ordinal transformation but is less smooth.
 Nominal. The only information in the observed variable that is preserved in the optimally
scaled variable is the grouping of objects in categories. The order of the categories of the
observed variable is not preserved. Category points will be on a straight line (vector) through
the origin. The resulting transformation fits better than the spline nominal transformation
but is less smooth.
 Numeric. Categories are treated as ordered and equally spaced (interval level). The order of
the categories and the equal distances between category numbers of the observed variable are
preserved in the optimally scaled variable. Category points will be on a straight line (vector)
through the origin. When all variables are at the numeric level, the analysis is analogous
to standard principal components analysis.
Categorical Regression Discretization
The Discretization dialog box allows you to select a method of recoding your variables.
Fractional-value variables are grouped into seven categories (or into the number of distinct values
of the variable if this number is less than seven) with an approximately normal distribution unless

17
otherwise specified. String variables are always converted into positive integers by assigning
category indicators according to ascending alphanumeric order. Discretization for string variables
applies to these integers. Other variables are left alone by default. The discretized variables are
then used in the analysis.
Figure 2-3
Discretization dialog box
Method. Choose between grouping, ranking, and multiplying.
 Grouping. Recode into a specified number of categories or recode by interval.
 Ranking. The variable is discretized by ranking the cases.
 Multiplying. The current values of the variable are standardized, multiplied by 10, rounded,
and have a constant added so that the lowest discretized value is 1.
Grouping. The following options are available when discretizing variables by grouping:
 Number of categories. Specify a number of categories and whether the values of the variable
should follow an approximately normal or uniform distribution across those categories.
 Equal intervals. Variables are recoded into categories defined by these equally sized intervals.
You must specify the length of the intervals.
Categorical Regression Missing Values
The Missing Values dialog box allows you to choose the strategy for handling missing values in
analysis variables and supplementary variables.

18
Chapter 2
Figure 2-4
Missing Values dialog box
Strategy. Choose to exclude objects with missing values (listwise deletion) or impute missing
values (active treatment).
 Exclude objects with missing values on this variable. Objects with missing values on
the selected variable are excluded from the analysis. This strategy is not available for
supplementary variables.
 Impute missing values. Objects with missing values on the selected variable have those values
imputed. You can choose the method of imputation. Select Mode to replace missing values
with the most frequent category. When there are multiple modes, the one with the smallest
category indicator is used. Select Extra category to replace missing values with the same
quantification of an extra category. This implies that objects with a missing value on this
variable are considered to belong to the same (extra) category.
Categorical Regression Options
The Options dialog box allows you to select the initial configuration style, specify iteration and
convergence criteria, select supplementary objects, and set the labeling of plots.

19
Figure 2-5
Options dialog box
Supplementary Objects. This allows you to specify the objects that you want to treat as
supplementary. Simply type the number of a supplementary object (or specify a range of cases)
and click Add. You cannot weight supplementary objects (specified weights are ignored).
Initial Configuration. If no variables are treated as nominal, select the Numerical configuration. If at
least one variable is treated as nominal, select the Random configuration.
Alternatively, if at least one variable has an ordinal or spline ordinal scaling level, the usual
model-fitting algorithm can result in a suboptimal solution. Choosing Multiple systematic starts
with all possible sign patterns to test will always find the optimal solution, but the necessary
processing time rapidly increases as the number of ordinal and spline ordinal variables in the
dataset increase. You can reduce the number of test patterns by specifying a percentage of loss
of variance threshold, where the higher the threshold, the more sign patterns will be excluded.
With this option, obtaining the optimal solution is not garantueed, but the chance of obtaining
a suboptimal solution is diminished. Also, if the optimal solution is not found, the chance that
the suboptimal solution is very different from the optimal solution is diminished. When multiple
systematic starts are requested, the signs of the regression coefficients for each start are written
to an external IBM® SPSS® Statistics data file or dataset in the current session. For more
information, see the topic Categorical Regression Save on p. 23.

20
Chapter 2
The results of a previous run with multiple systematic starts allows you to Use fixed signs for the
regression coefficients. The signs (indicated by 1 and −1) need to be in a row of the specified
dataset or file. The integer-valued starting number is the case number of the row in this file
that contains the signs to be used.
Criteria. You can specify the maximum number of iterations that the regression may go through in
its computations. You can also select a convergence criterion value. The regression stops iterating
if the difference in total fit between the last two iterations is less than the convergence value or if
the maximum number of iterations is reached.
Label Plots By. Allows you to specify whether variables and value labels or variable names and
values will be used in the plots. You can also specify a maximum length for labels.
Categorical Regression Regularization
Figure 2-6
Regularization dialog box
Method. Regularization methods can improve the predictive error of the model by reducing the
variability in the estimates of regression coefficient by shrinking the estimates toward 0. The
Lasso and Elastic Net will shrink some coefficient estimates to exactly 0, thus providing a form
of variable selection. When a regularization method is requested, the regularized model and
coefficients for each penalty coefficient value are written to an external IBM® SPSS® Statistics
data file or dataset in the current session. For more information, see the topic Categorical
Regression Save on p. 23.

21
 Ridge regression. Ridge regression shrinks coefficients by introducing a penalty term equal
to the sum of squared coefficients times a penalty coefficient. This coefficient can range
from 0 (no penalty) to 1; the procedure will search for the “best” value of the penalty if
you specify a range and increment.
 Lasso. The Lasso’s penalty term is based on the sum of absolute coefficients, and the
specification of a penalty coefficient is similar to that of Ridge regression; however, the
Lasso is more computationally intensive.
 Elastic net. The Elastic Net simply combines the Lasso and Ridge regression penalties, and
will search over the grid of values specified to find the “best” Lasso and Ridge regression
penalty coefficients. For a given pair of Lasso and Ridge regression penalties, the Elastic Net
is not much more computationally expensive than the Lasso.
Display regularization plots. These are plots of the regression coefficients versus the regularization
penalty. When searching a range of values for the “best” penalty coefficient, it provides a view of
how the regression coefficients change over that range.
Elastic Net Plots. For the Elastic Net method, separate regularization plots are produced by values
of the Ridge regression penalty. All possible plots uses every value in the range determined
by the minimum and maximum Ridge regression penalty values specified. For some Ridge
penalties allows you to specify a subset of the values in the range determined by the minimum and
maximum. Simply type the number of a penalty value (or specify a range of values) and click Add.
Categorical Regression Output
The Output dialog box allows you to select the statistics to display in the output.

22
Chapter 2
Figure 2-7
Output dialog box
Tables. Produces tables for:
 Multiple R. Includes R2, adjusted R2, and adjusted R2 taking the optimal scaling into account.
 ANOVA. This option includes regression and residual sums of squares, mean squares, and F.
Two ANOVA tables are displayed: one with degrees of freedom for the regression equal to
the number of predictor variables and one with degrees of freedom for the regression taking
the optimal scaling into account.
 Coefficients. This option gives three tables: a Coefficients table that includes betas, standard
error of the betas, t values, and significance; a Coefficients-Optimal Scaling table with the
standard error of the betas taking the optimal scaling degrees of freedom into account; and a
table with the zero-order, part, and partial correlation, Pratt’s relative importance measure for
the transformed predictors, and the tolerance before and after transformation.
 Iteration history. For each iteration, including the starting values for the algorithm, the multiple
R and regression error are shown. The increase in multiple R is listed starting from the first
iteration.
 Correlations of original variables. A matrix showing the correlations between the untransformed
variables is displayed.

23
 Correlations of transformed variables. A matrix showing the correlations between the
transformed variables is displayed.
 Regularized models and coefficients. Displays penalty values, R-square, and the regression
coefficients for each regularized model. If a resampling method is specified or if
supplementary objects (test cases) are specified, it also displays the prediction error or test
MSE.
Resampling. Resampling methods give you an estimate of the prediction error of the model.
 Crossvalidation. Crossvalidation divides the sample into a number of subsamples, or folds.
Categorical regression models are then generated, excluding the data from each subsample
in turn. The first model is based on all of the cases except those in the first sample fold, the
second model is based on all of the cases except those in the second sample fold, and so on.
For each model, the prediction error is estimated by applying the model to the subsample
excluded in generating it.
 .632 Bootstrap. With the bootstrap, observations are drawn randomly from the data with
replacement, repeating this process a number of times to obtain a number bootstrap samples.
A model is fit for each bootstrap sample, and the prediction error for each model is estimated
by this fitted model is then applied to the cases not in the bootstrap sample.
Category Quantifications. Tables showing the transformed values of the selected variables are
displayed.
Descriptive Statistics. Tables showing the frequencies, missing values, and modes of the selected
variables are displayed.
Categorical Regression Save
The Save dialog box allows you to save predicted values, residuals, and transformed values
to the active dataset and/or save discretized data, transformed values, regularized models and
coefficients, and signs of regression coefficients to an external IBM® SPSS® Statistics data
file or dataset in the current session.
 Datasets are available during the current session but are not available in subsequent sessions
unless you explicitly save them as data files. Dataset names must adhere to variable naming
rules.
 Filenames or dataset names must be different for each type of data saved.

24
Chapter 2
Figure 2-8
Save dialog box
Regularized models and coefficients are saved whenever a regularization method is selected on
the Regularization dialog. By default, the procedure creates a new dataset with a unique name, but
you can of course specify a name of your own choosing or write to an external file.
Signs of regression coefficients are saved whenever multiple systematic starts are used as the initial
configuration on the Options dialog. By default, the procedure creates a new dataset with a unique
name, but you can of course specify a name of your own choosing or write to an external file.
Categorical Regression Transformation Plots
The Plots dialog box allows you to specify the variables that will produce transformation and
residual plots.

25
Figure 2-9
Plots dialog box
Transformation Plots. For each of these variables, the category quantifications are plotted against
the original category values. Empty categories appear on the horizontal axis but do not affect the
computations. These categories are identified by breaks in the line connecting the quantifications.
Residual Plots. For each of these variables, residuals (computed for the dependent variable
predicted from all predictor variables except the predictor variable in question) are plotted
against category indicators and the optimal category quantifications multiplied with beta against
category indicators.
CATREG Command Additional Features
You can customize your categorical regression if you paste your selections into a syntax window
and edit the resulting CATREG command syntax. The command syntax language also allows you to:
 Specify rootnames for the transformed variables when saving them to the active dataset (with
the SAVE subcommand).
See the Command Syntax Reference for complete syntax information.

Chapter
3
Categorical Principal Components
Analysis (CATPCA)
This procedure simultaneously quantifies categorical variables while reducing the dimensionality
of the data. Categorical principal components analysis is also known by the acronym CATPCA,
for categorical principal components analysis.
The goal of principal components analysis is to reduce an original set of variables into a
smaller set of uncorrelated components that represent most of the information found in the
original variables. The technique is most useful when a large number of variables prohibits
effective interpretation of the relationships between objects (subjects and units). By reducing the
dimensionality, you interpret a few components rather than a large number of variables.
Standard principal components analysis assumes linear relationships between numeric
variables. On the other hand, the optimal-scaling approach allows variables to be scaled at
different levels. Categorical variables are optimally quantified in the specified dimensionality. As
a result, nonlinear relationships between variables can be modeled.
Example. Categorical principal components analysis could be used to graphically display the
relationship between job category, job division, region, amount of travel (high, medium, and low),
and job satisfaction. You might find that two dimensions account for a large amount of variance.
The first dimension might separate job category from region, whereas the second dimension might
separate job division from amount of travel. You also might find that high job satisfaction is
related to a medium amount of travel.
Statistics and plots. Frequencies, missing values, optimal scaling level, mode, variance accounted
for by centroid coordinates, vector coordinates, total per variable and per dimension, component
loadings for vector-quantified variables, category quantifications and coordinates, iteration history,
correlations of the transformed variables and eigenvalues of the correlation matrix, correlations
of the original variables and eigenvalues of the correlation matrix, object scores, category plots,
joint category plots, transformation plots, residual plots, projected centroid plots, object plots,
biplots, triplots, and component loadings plots.
Data. String variable values are always converted into positive integers by ascending alphanumeric
order. User-defined missing values, system-missing values, and values less than 1 are considered
missing; you can recode or add a constant to variables with values less than 1 to make them
nonmissing.
Assumptions. The data must contain at least three valid cases. The analysis is based on positive
integer data. The discretization option will automatically categorize a fractional-valued variable
by grouping its values into categories with a close to “normal” distribution and will automatically
convert values of string variables into positive integers. You can specify other discretization
schemes.
Related procedures. Scaling all variables at the numeric level corresponds to standard principal
components analysis. Alternate plotting features are available by using the transformed variables
in a standard linear principal components analysis. If all variables have multiple nominal scaling

27
Categorical Principal Components Analysis (CATPCA)
levels, categorical principal components analysis is identical to multiple correspondence analysis.
If sets of variables are of interest, categorical (nonlinear) canonical correlation analysis should be
used.
To Obtain a Categorical Principal Components Analysis
Analyze > Dimension Reduction > Optimal Scaling...
Figure 3-1
Optimal Scaling dialog box
E Select Some variable(s) not multiple nominal.
E Select One set.
E Click Define.

28
Chapter 3
Figure 3-2
Categorical Principal Components dialog box
E Select at least two analysis variables and specify the number of dimensions in the solution.
E Click OK.
You may optionally specify supplementary variables, which are fitted into the solution found,
or labeling variables for the plots.
Define Scale and Weight in CATPCA
You can set the optimal scaling level for analysis variables and supplementary variables. By
default, they are scaled as second-degree monotonic splines (ordinal) with two interior knots.
Additionally, you can set the weight for analysis variables.

29
Figure 3-3
Define Scale and Weight dialog box
Variable weight. You can choose to define a weight for each variable. The value specified must
be a positive integer. The default value is 1.
Optimal Scaling Level. You can also select the scaling level to be used to quantify each variable.
 Spline ordinal. The order of the categories of the observed variable is preserved in the optimally
resulting transformation is a smooth monotonic piecewise polynomial of the chosen degree.
The pieces are specified by the user-specified number and procedure-determined placement
of the interior knots.
 Spline nominal. The only information in the observed variable that is preserved in the optimally
the origin. The resulting transformation is a smooth, possibly nonmonotonic, piecewise
polynomial of the chosen degree. The pieces are specified by the user-specified number and
procedure-determined placement of the interior knots.
 Multiple nominal. The only information in the observed variable that is preserved in the
optimally scaled variable is the grouping of objects in categories. The order of the categories
of the observed variable is not preserved. Category points will be in the centroid of the
objects in the particular categories. Multiple indicates that different sets of quantifications
are obtained for each dimension.
 Ordinal. The order of the categories of the observed variable is preserved in the optimally
resulting transformation fits better than the spline ordinal transformation but is less smooth.
 Nominal. The only information in the observed variable that is preserved in the optimally

30
Chapter 3
the origin. The resulting transformation fits better than the spline nominal transformation
but is less smooth.
 Numeric. Categories are treated as ordered and equally spaced (interval level). The order of
the categories and the equal distances between category numbers of the observed variable are
preserved in the optimally scaled variable. Category points will be on a straight line (vector)
through the origin. When all variables are at the numeric level, the analysis is analogous
to standard principal components analysis.
Categorical Principal Components Analysis Discretization
Fractional-valued variables are grouped into seven categories (or into the number of distinct
values of the variable if this number is less than seven) with an approximately normal distribution,
unless specified otherwise. String variables are always converted into positive integers by
assigning category indicators according to ascending alphanumeric order. Discretization for string
variables applies to these integers. Other variables are left alone by default. The discretized
variables are then used in the analysis.
Figure 3-4
and have a constant added such that the lowest discretized value is 1.

31
Grouping. The following options are available when you are discretizing variables by grouping:
Categorical Principal Components Analysis Missing Values
Figure 3-5
Strategy. Choose to exclude missing values (passive treatment), impute missing values (active
treatment), or exclude objects with missing values (listwise deletion).
 Exclude missing values; for correlations impute after quantification. Objects with missing values
on the selected variable do not contribute to the analysis for this variable. If all variables
are given passive treatment, then objects with missing values on all variables are treated as
supplementary. If correlations are specified in the Output dialog box, then (after analysis)
missing values are imputed with the most frequent category, or mode, of the variable for the
correlations of the original variables. For the correlations of the optimally scaled variables,
you can choose the method of imputation. Select Mode to replace missing values with the
mode of the optimally scaled variable. Select Extra category to replace missing values with

32
Chapter 3
the quantification of an extra category. This implies that objects with a missing value on this
Categorical Principal Components Analysis Options
The Options dialog box allows you to select the initial configuration, specify iteration and
convergence criteria, select a normalization method, choose the method for labeling plots, and
specify supplementary objects.
Figure 3-6
Options dialog box

33
Supplementary Objects. Specify the case number of the object, or the first and last case numbers
of a range of objects, that you want to make supplementary and then click Add. Continue until
you have specified all of your supplementary objects. If an object is specified as supplementary,
then case weights are ignored for that object.
Normalization Method. You can specify one of five options for normalizing the object scores and
the variables. Only one normalization method can be used in a given analysis.
 Variable Principal. This option optimizes the association between variables. The coordinates
of the variables in the object space are the component loadings (correlations with principal
components, such as dimensions and object scores). This is useful when you are primarily
interested in the correlation between the variables.
 Object Principal. This option optimizes distances between objects. This is useful when you are
primarily interested in differences or similarities between the objects.
 Symmetrical. Use this normalization option if you are primarily interested in the relation
between objects and variables.
 Independent. Use this normalization option if you want to examine distances between objects
and correlations between variables separately.
 Custom. You can specify any real value in the closed interval [–1, 1]. A value of 1 is equal to
the Object Principal method, a value of 0 is equal to the Symmetrical method, and a value of
–1 is equal to the Variable Principal method. By specifying a value greater than –1 and less
than 1, you can spread the eigenvalue over both objects and variables. This method is useful
for making a tailor-made biplot or triplot.
Criteria. You can specify the maximum number of iterations the procedure can go through in its
computations. You can also select a convergence criterion value. The algorithm stops iterating if
the difference in total fit between the last two iterations is less than the convergence value or if
Plot Dimensions. Allows you to control the dimensions displayed in the output.
 Display all dimensions in the solution. All dimensions in the solution are displayed in a
scatterplot matrix.
 Restrict the number of dimensions. The displayed dimensions are restricted to plotted pairs. If
you restrict the dimensions, you must select the lowest and highest dimensions to be plotted.
The lowest dimension can range from 1 to the number of dimensions in the solution minus 1
and is plotted against higher dimensions. The highest dimension value can range from 2 to the
number of dimensions in the solution and indicates the highest dimension to be used in plotting
the dimension pairs. This specification applies to all requested multidimensional plots.
Configuration. You can read data from a file containing the coordinates of a configuration. The
first variable in the file should contain the coordinates for the first dimension, the second variable
should contain the coordinates for the second dimension, and so on.

34
Chapter 3
 Initial. The configuration in the file specified will be used as the starting point of the analysis.
 Fixed. The configuration in the file specified will be used to fit in the variables. The variables
that are fitted in must be selected as analysis variables, but because the configuration is
fixed, they are treated as supplementary variables (so they do not need to be selected as
supplementary variables).
Categorical Principal Components Analysis Output
The Output dialog box allows you to produce tables for object scores, component loadings,
iteration history, correlations of original and transformed variables, the variance accounted for
per variable and per dimension, category quantifications for selected variables, and descriptive
statistics for selected variables.
Figure 3-7
Output dialog box
Object scores. Displays the object scores and has the following options:
 Include Categories Of. Displays the category indicators of the analysis variables selected.
 Label Object Scores By. From the list of variables specified as labeling variables, you can
select one to label the objects.
Component loadings. Displays the component loadings for all variables that were not given
multiple nominal scaling levels.

35
Iteration history. For each iteration, the variance accounted for, loss, and increase in variance
accounted for are shown.
Correlations of original variables. Shows the correlation matrix of the original variables and the
eigenvalues of that matrix.
Correlations of transformed variables. Shows the correlation matrix of the transformed (optimally
scaled) variables and the eigenvalues of that matrix.
Variance accounted for. Displays the amount of variance accounted for by centroid coordinates,
vector coordinates, and total (centroid and vector coordinates combined) per variable and per
dimension.
Category Quantifications. Gives the category quantifications and coordinates for each dimension
of the variable(s) selected.
Descriptive Statistics. Displays frequencies, number of missing values, and mode of the variable(s)
selected.
Categorical Principal Components Analysis Save
The Save dialog box allows you to save discretized data, object scores, transformed values, and
approximations to an external IBM® SPSS® Statistics data file or dataset in the current session.
You can also save transformed values, object scores, and approximations to the active dataset.
rules.
 If you save object scores or transformed values to the active dataset, you can specify the
number of multiple nominal dimensions.

36
Chapter 3
Figure 3-8
Save dialog box
Categorical Principal Components Analysis Object Plots
The Object and Variable Plots dialog box allows you to specify the types of plots desired and the
variables for which plots will be produced.

37
Figure 3-9
Object and Variable Plots dialog box
Object points. A plot of the object points is displayed.
Objects and variables (biplot). The object points are plotted with your choice of the variable
coordinates—component loadings or variable centroids.
Objects, loadings, and centroids (triplot). The object points are plotted with the centroids of multiple
nominal-scaling-level variables and the component loadings of other variables.
Biplot and Triplot Variables. You can choose to use all variables for the biplots and triplots or
select a subset.
Label Objects. You can choose to have objects labeled with the categories of selected variables
(you may choose category indicator values or value labels in the Options dialog box) or with their
case numbers. One plot is produced per variable if Variable is selected.
Categorical Principal Components Analysis Category Plots
The Category Plots dialog box allows you to specify the types of plots desired and the variables
for which plots will be produced.

38
Chapter 3
Figure 3-10
Category Plots dialog box
Category Plots. For each variable selected, a plot of the centroid and vector coordinates is plotted.
For variables with multiple nominal scaling levels, categories are in the centroids of the objects in
the particular categories. For all other scaling levels, categories are on a vector through the origin.
Joint Category Plots. This is a single plot of the centroid and vector coordinates of each selected
variable.
Transformation Plots. Displays a plot of the optimal category quantifications versus the category
indicators. You can specify the number of dimensions desired for variables with multiple nominal
scaling levels; one plot will be generated for each dimension. You can also choose to display
residual plots for each variable selected.
Project Centroids Of. You may choose a variable and project its centroids onto selected variables.
Variables with multiple nominal scaling levels cannot be selected to project on. When this plot is
requested, a table with the coordinates of the projected centroids is also displayed.
Categorical Principal Components Analysis Loading Plots
The Loading Plots dialog box allows you to specify the variables that will be included in the plot,
and whether or not to include centroids in the plot.

39
Figure 3-11
Loading Plots dialog box
Display component loadings. If selected, a plot of the component loadings is displayed.
Loading Variables. You can choose to use all variables for the component loadings plot or select a
subset.
Include centroids. Variables with multiple nominal scaling levels do not have component loadings,
but you may choose to include the centroids of those variables in the plot. You can choose to use
all multiple nominal variables or select a subset.
CATPCA Command Additional Features
You can customize your categorical principal components analysis if you paste your selections
into a syntax window and edit the resulting CATPCA command syntax. The command syntax
language also allows you to:
 Specify rootnames for the transformed variables, object scores, and approximations when
saving them to the active dataset (with the SAVE subcommand).
 Specify a maximum length for labels for each plot separately (with the PLOT subcommand).
 Specify a separate variable list for residual plots (with the PLOT subcommand).

Chapter
4
Nonlinear Canonical Correlation
Analysis (OVERALS)
Nonlinear canonical correlation analysis corresponds to categorical canonical correlation analysis
with optimal scaling. The purpose of this procedure is to determine how similar sets of categorical
variables are to one another. Nonlinear canonical correlation analysis is also known by the
acronym OVERALS.
Standard canonical correlation analysis is an extension of multiple regression, where the
second set does not contain a single response variable but instead contain multiple response
variables. The goal is to explain as much as possible of the variance in the relationships among
two sets of numerical variables in a low dimensional space. Initially, the variables in each set
are linearly combined such that the linear combinations have a maximal correlation. Given these
combinations, subsequent linear combinations are determined that are uncorrelated with the
previous combinations and that have the largest correlation possible.
The optimal scaling approach expands the standard analysis in three crucial ways. First,
OVERALS allows more than two sets of variables. Second, variables can be scaled as either
nominal, ordinal, or numerical. As a result, nonlinear relationships between variables can be
analyzed. Finally, instead of maximizing correlations between the variable sets, the sets are
compared to an unknown compromise set that is defined by the object scores.
Example. Categorical canonical correlation analysis with optimal scaling could be used to
graphically display the relationship between one set of variables containing job category and
years of education and another set of variables containing region of residence and gender. You
might find that years of education and region of residence discriminate better than the remaining
variables. You might also find that years of education discriminates best on the first dimension.
Statistics and plots. Frequencies, centroids, iteration history, object scores, category
quantifications, weights, component loadings, single and multiple fit, object scores plots, category
coordinates plots, component loadings plots, category centroids plots, transformation plots.
Data. Use integers to code categorical variables (nominal or ordinal scaling level). To minimize
output, use consecutive integers beginning with 1 to code each variable. Variables that are scaled
at the numerical level should not be recoded to consecutive integers. To minimize output, for each
variable that is scaled at the numerical level, subtract the smallest observed value from every value
and add 1. Fractional values are truncated after the decimal.
Assumptions. Variables can be classified into two or more sets. Variables in the analysis are scaled
as multiple nominal, single nominal, ordinal, or numerical. The maximum number of dimensions
that are used in the procedure depends on the optimal scaling level of the variables. If all variables
are specified as ordinal, single nominal, or numerical, the maximum number of dimensions is the
lesser of the following two values: the number of observations minus 1 or the total number of
variables. However, if only two sets of variables are defined, the maximum number of dimensions
is the number of variables in the smaller set. If some variables are multiple nominal, the maximum
number of dimensions is the total number of multiple nominal categories plus the number of

41
Nonlinear Canonical Correlation Analysis (OVERALS)
nonmultiple nominal variables minus the number of multiple nominal variables. For example, if
the analysis involves five variables, one of which is multiple nominal with four categories, the
maximum number of dimensions is (4 + 4 – 1), or 7. If you specify a number that is greater than
the maximum, the maximum value is used.
Related procedures. If each set contains one variable, nonlinear canonical correlation analysis is
equivalent to principal components analysis with optimal scaling. If each of these variables is
multiple nominal, the analysis corresponds to multiple correspondence analysis. If two sets of
variables are involved, and one of the sets contains only one variable, the analysis is identical to
categorical regression with optimal scaling.
To Obtain a Nonlinear Canonical Correlation Analysis
Figure 4-1
E Select either All variables multiple nominal or Some variable(s) not multiple nominal.
E Select Multiple sets.
E Click Define.

42
Chapter 4
Figure 4-2
Nonlinear Canonical Correlation Analysis (OVERALS) dialog box
E Define at least two sets of variables. Select the variable(s) that you want to include in the first
set. To move to the next set, click Next, and select the variables that you want to include in the
second set. You can add additional sets as desired. Click Previous to return to the previously
defined variable set.
E Define the value range and measurement scale (optimal scaling level) for each selected variable.
E Click OK.
E Optionally:
 Select one or more variables to provide point labels for object scores plots. Each variable
produces a separate plot, with the points labeled by the values of that variable. You must
define a range for each of these plot label variables. When you are using the dialog box, a
single variable cannot be used both in the analysis and as a labeling variable. If labeling the
object scores plot with a variable that is used in the analysis is desired, use the Compute
facility (available from the Transform menu) to create a copy of that variable. Use the new
variable to label the plot. Alternatively, command syntax can be used.
 Specify the number of dimensions that you want in the solution. In general, choose as few
dimensions as needed to explain most of the variation. If the analysis involves more than
two dimensions, three-dimensional plots of the first three dimensions are produced. Other
dimensions can be displayed by editing the chart.

43
Define Range and Scale
Figure 4-3
Define Range and Scale dialog box
You must define a range for each variable. The maximum value that is specified must be an integer.
Fractional data values are truncated in the analysis. A category value that is outside of the specified
range is ignored in the analysis. To minimize output, use the Automatic Recode facility (available
from the Transform menu) to create consecutive categories beginning with 1 for variables that
are treated as nominal or ordinal. Recoding to consecutive integers is not recommended for
variables that are scaled at the numerical level. To minimize output for variables that are treated as
numerical, for each variable, subtract the minimum value from every value and add 1.
You must also select the scaling to be used to quantify each variable.
 Ordinal. The order of the categories of the observed variable is preserved in the quantified
variable.
 Single nominal. In the quantified variable, objects in the same category receive the same score.
 Multiple nominal. The quantifications can be different for each dimension.
 Discrete numeric. Categories are treated as ordered and equally spaced. The differences
between category numbers and the order of the categories of the observed variable are
preserved in the quantified variable.
Define Range
Figure 4-4
Define Range dialog box
You must define a range for each variable. The maximum value that is specified must be an
integer. Fractional data values are truncated in the analysis. A category value that is outside of the
specified range is ignored in the analysis. To minimize output, use the Automatic Recode facility
(available from the Transform menu) to create consecutive categories beginning with 1.

44
Chapter 4
You must also define a range for each variable that is used to label the object scores plots.
However, labels for categories with data values that are outside of the defined range for the
variable do appear on the plots.
Nonlinear Canonical Correlation Analysis Options
The Options dialog box allows you to select optional statistics and plots, save object scores as
new variables in the active dataset, specify iteration and convergence criteria, and specify an
initial configuration for the analysis.
Figure 4-5
Options dialog box
Display. Available statistics include marginal frequencies (counts), centroids, iteration history,
weights and component loadings, category quantifications, object scores, and single and multiple
fit statistics.
 Centroids. Category quantifications, and the projected and the actual averages of the object
scores for the objects (cases) included in each set for those belonging to the same category
of the variable.
 Weights and component loadings. The regression coefficients in each dimension for every
quantified variable in a set, where the object scores are regressed on the quantified variables,
and the projection of the quantified variable in the object space. Provides an indication of the
contribution each variable makes to the dimension within each set.
 Single and multiple fit. Measures of goodness of fit of the single- and multiple-category
coordinates/category quantifications with respect to the objects.
 Category quantifications. Optimal scale values assigned to the categories of a variable.
 Object scores. Optimal score assigned to an object (case) in a particular dimension.

45
Plot. You can produce plots of category coordinates, object scores, component loadings, category
centroids, and transformations.
Save object scores. You can save the object scores as new variables in the active dataset. Object
scores are saved for the number of dimensions that are specified in the main dialog box.
Use random initial configuration. A random initial configuration should be used if some or all of the
variables are single nominal. If this option is not selected, a nested initial configuration is used.
Criteria. You can specify the maximum number of iterations that the nonlinear canonical
correlation analysis can go through in its computations. You can also select a convergence
criterion value. The analysis stops iterating if the difference in total fit between the last two
iterations is less than the convergence value or if the maximum number of iterations is reached.
OVERALS Command Additional Features
You can customize your nonlinear canonical correlation analysis if you paste your selections
into a syntax window and edit the resulting OVERALS command syntax. The command syntax
 Specify the dimension pairs to be plotted, rather than plotting all extracted dimensions (using
theNDIM keyword on the PLOT subcommand).
 Specify the number of value label characters that are used to label points on the plots (with
thePLOT subcommand).
 Designate more than five variables as labeling variables for object scores plots (with thePLOT
subcommand).
 Select variables that are used in the analysis as labeling variables for the object scores plots
(with the PLOT subcommand).
 Select variables to provide point labels for the quantification score plot (with the PLOT
subcommand).
 Specify the number of cases to be included in the analysis if you do not want to use all cases in
the active dataset (with the NOBSERVATIONS subcommand).
 Specify rootnames for variables created by saving object scores (with the SAVE subcommand).
 Specify the number of dimensions to be saved, rather than saving all extracted dimensions
(with the SAVE subcommand).
 Write category quantifications to a matrix file (using the MATRIX subcommand).
 Produce low-resolution plots that may be easier to read than the usual high-resolution plots
(using the SET command).
 Produce centroid and transformation plots for specified variables only (with the PLOT
subcommand).

Chapter
5
One of the goals of correspondence analysis is to describe the relationships between two nominal
variables in a correspondence table in a low-dimensional space, while simultaneously describing
the relationships between the categories for each variable. For each variable, the distances
between category points in a plot reflect the relationships between the categories with similar
categories plotted close to each other. Projecting points for one variable on the vector from the
origin to a category point for the other variable describe the relationship between the variables.
An analysis of contingency tables often includes examining row and column profiles and
testing for independence via the chi-square statistic. However, the number of profiles can be quite
large, and the chi-square test does not reveal the dependence structure. The Crosstabs procedure
offers several measures of association and tests of association but cannot graphically represent any
relationships between the variables.
Factor analysis is a standard technique for describing relationships between variables in a
low-dimensional space. However, factor analysis requires interval data, and the number of
observations should be five times the number of variables. Correspondence analysis, on the other
hand, assumes nominal variables and can describe the relationships between categories of each
variable, as well as the relationship between the variables. In addition, correspondence analysis
can be used to analyze any table of positive correspondence measures.
Example. Correspondence analysis could be used to graphically display the relationship between
staff category and smoking habits. You might find that with regard to smoking, junior managers
differ from secretaries, but secretaries do not differ from senior managers. You might also find
that heavy smoking is associated with junior managers, whereas light smoking is associated
with secretaries.
Statistics and plots. Correspondence measures, row and column profiles, singular values, row
and column scores, inertia, mass, row and column score confidence statistics, singular value
confidence statistics, transformation plots, row point plots, column point plots, and biplots.
Data. Categorical variables to be analyzed are scaled nominally. For aggregated data or for a
correspondence measure other than frequencies, use a weighting variable with positive similarity
values. Alternatively, for table data, use syntax to read the table.
Assumptions. The maximum number of dimensions used in the procedure depends on the number
of active rows and column categories and the number of equality constraints. If no equality
constraints are used and all categories are active, the maximum dimensionality is one fewer
than the number of categories for the variable with the fewest categories. For example, if one
variable has five categories and the other has four, the maximum number of dimensions is three.
Supplementary categories are not active. For example, if one variable has five categories, two of
which are supplementary, and the other variable has four categories, the maximum number of
dimensions is two. Treat all sets of categories that are constrained to be equal as one category. For
example, if a variable has five categories, three of which are constrained to be equal, that variable
should be treated as having three categories when determining the maximum dimensionality. Two
of the categories are unconstrained, and the third category corresponds to the three constrained

47
categories. If you specify a number of dimensions greater than the maximum, the maximum
value is used.
Related procedures. If more than two variables are involved, use multiple correspondence analysis.
If the variables should be scaled ordinally, use categorical principal components analysis.
To Obtain a Correspondence Analysis
Analyze > Dimension Reduction > Correspondence Analysis...
Figure 5-1
Correspondence Analysis dialog box
E Select a row variable.
E Select a column variable.
E Define the ranges for the variables.
E Click OK.
Define Row Range in Correspondence Analysis
You must define a range for the row variable. The minimum and maximum values specified must
be integers. Fractional data values are truncated in the analysis. A category value that is outside of
the specified range is ignored in the analysis.

48
Chapter 5
Figure 5-2
Define Row Range dialog box
All categories are initially unconstrained and active. You can constrain row categories to equal
other row categories, or you can define a row category as supplementary.
 Categories must be equal. Categories must have equal scores. Use equality constraints if
the obtained order for the categories is undesirable or counterintuitive. The maximum
number of row categories that can be constrained to be equal is the total number of active
row categories minus 1. To impose different equality constraints on sets of categories, use
syntax. For example, use syntax to constrain categories 1 and 2 to be equal and categories
3 and 4 to be equal.
 Category is supplemental. Supplementary categories do not influence the analysis but are
represented in the space defined by the active categories. Supplementary categories play no
role in defining the dimensions. The maximum number of supplementary row categories is
the total number of row categories minus 2.
Define Column Range in Correspondence Analysis
You must define a range for the column variable. The minimum and maximum values specified
must be integers. Fractional data values are truncated in the analysis. A category value that is
outside of the specified range is ignored in the analysis.

49
Figure 5-3
Define Column Range dialog box
All categories are initially unconstrained and active. You can constrain column categories to equal
other column categories, or you can define a column category as supplementary.
 Categories must be equal. Categories must have equal scores. Use equality constraints if the
obtained order for the categories is undesirable or counterintuitive. The maximum number of
column categories that can be constrained to be equal is the total number of active column
categories minus 1. To impose different equality constraints on sets of categories, use syntax.
For example, use syntax to constrain categories 1 and 2 to be equal and categories 3 and
4 to be equal.
 Category is supplemental. Supplementary categories do not influence the analysis but are
represented in the space defined by the active categories. Supplementary categories play no
role in defining the dimensions. The maximum number of supplementary column categories
is the total number of column categories minus 2.
Correspondence Analysis Model
The Model dialog box allows you to specify the number of dimensions, the distance measure, the
standardization method, and the normalization method.

50
Chapter 5
Figure 5-4
Model dialog box
Dimensions in solution. Specify the number of dimensions. In general, choose as few dimensions
as needed to explain most of the variation. The maximum number of dimensions depends on the
number of active categories used in the analysis and on the equality constraints. The maximum
number of dimensions is the smaller of:
 The number of active row categories minus the number of row categories constrained to be
equal, plus the number of constrained row category sets.
 The number of active column categories minus the number of column categories constrained
to be equal, plus the number of constrained column category sets.
Distance Measure. You can select the measure of distance among the rows and columns of the
correspondence table. Choose one of the following alternatives:
 Chi-square. Use a weighted profile distance, where the weight is the mass of the rows or
columns. This measure is required for standard correspondence analysis.
 Euclidean. Use the square root of the sum of squared differences between pairs of rows and
pairs of columns.
Standardization Method. Choose one of the following alternatives:
 Row and column means are removed. Both the rows and columns are centered. This method is
required for standard correspondence analysis.
 Row means are removed. Only the rows are centered.
 Column means are removed. Only the columns are centered.

51
 Row totals are equalized and means are removed. Before centering the rows, the row margins
are equalized.
 Column totals are equalized and means are removed. Before centering the columns, the column
margins are equalized.
Normalization Method. Choose one of the following alternatives:
 Symmetrical. For each dimension, the row scores are the weighted average of the column
scores divided by the matching singular value, and the column scores are the weighted
average of row scores divided by the matching singular value. Use this method if you want to
examine the differences or similarities between the categories of the two variables.
 Principal. The distances between row points and column points are approximations of the
distances in the correspondence table according to the selected distance measure. Use this
method if you want to examine differences between categories of either or both variables
instead of differences between the two variables.
 Row principal. The distances between row points are approximations of the distances in the
correspondence table according to the selected distance measure. The row scores are the
weighted average of the column scores. Use this method if you want to examine differences
or similarities between categories of the row variable.
 Column principal. The distances between column points are approximations of the distances in
the correspondence table according to the selected distance measure. The column scores are
the weighted average of the row scores. Use this method if you want to examine differences
or similarities between categories of the column variable.
 Custom. You must specify a value between –1 and 1. A value of –1 corresponds to column
principal. A value of 1 corresponds to row principal. A value of 0 corresponds to symmetrical.
All other values spread the inertia over both the row and column scores to varying degrees.
This method is useful for making tailor-made biplots.
Correspondence Analysis Statistics
The Statistics dialog box allows you to specify the numerical output produced.

52
Chapter 5
Figure 5-5
Statistics dialog box
Correspondence table. A crosstabulation of the input variables with row and column marginal
totals.
Overview of row points. For each row category, the scores, mass, inertia, contribution to the inertia
of the dimension, and the contribution of the dimension to the inertia of the point.
Overview of column points. For each column category, the scores, mass, inertia, contribution to the
inertia of the dimension, and the contribution of the dimension to the inertia of the point.
Row profiles. For each row category, the distribution across the categories of the column variable.
Column profiles. For each column category, the distribution across the categories of the row
variable.
Permutations of the correspondence table. The correspondence table reorganized such that the rows
and columns are in increasing order according to the scores on the first dimension. Optionally,
you can specify the maximum dimension number for which permuted tables will be produced. A
permuted table for each dimension from 1 to the number specified is produced.
Confidence Statistics for Row points. Includes standard deviation and correlations for all
nonsupplementary row points.
Confidence Statistics for Column points. Includes standard deviation and correlations for all
nonsupplementary column points.
Correspondence Analysis Plots
The Plots dialog box allows you to specify which plots are produced.

53
Figure 5-6
Plots dialog box
Scatterplots. Produces a matrix of all pairwise plots of the dimensions. Available scatterplots
include:
 Biplot. Produces a matrix of joint plots of the row and column points. If principal
normalization is selected, the biplot is not available.
 Row points. Produces a matrix of plots of the row points.
 Column points. Produces a matrix of plots of the column points.
Optionally, you can specify how many value label characters to use when labeling the points. This
value must be a non-negative integer less than or equal to 20.
Line Plots. Produces a plot for every dimension of the selected variable. Available line plots
include:
 Transformed row categories. Produces a plot of the original row category values against their
corresponding row scores.
 Transformed column categories. Produces a plot of the original column category values against
their corresponding column scores.
Optionally, you can specify how many value label characters to use when labeling the category
axis. This value must be a non-negative integer less than or equal to 20.

54
Chapter 5
scatterplot matrix.
The lowest dimension can range from 1 to the number of dimensions in the solution minus
1, and is plotted against higher dimensions. The highest dimension value can range from 2
to the number of dimensions in the solution, and indicates the highest dimension to be used
in plotting the dimension pairs. This specification applies to all requested multidimensional
plots.
CORRESPONDENCE Command Additional Features
You can customize your correspondence analysis if you paste your selections into a syntax
window and edit the resulting CORRESPONDENCE command syntax. The command syntax
 Specify table data as input instead of using casewise data (using the TABLE = ALL
subcommand).
 Specify the number of value-label characters used to label points for each type of scatterplot
matrix or biplot matrix (with the PLOT subcommand).
 Specify the number of value-label characters used to label points for each type of line plot
(with the PLOT subcommand).
 Write a matrix of row and column scores to a matrix data file (with the OUTFILE
subcommand).
 Write a matrix of confidence statistics (variances and covariances) for the singular values and
the scores to a matrix data file (with the OUTFILE subcommand).
 Specify multiple sets of categories to be equal (with the EQUAL subcommand).

Chapter
6
Multiple Correspondence Analysis quantifies nominal (categorical) data by assigning numerical
values to the cases (objects) and categories so that objects within the same category are close
together and objects in different categories are far apart. Each object is as close as possible to the
category points of categories that apply to the object. In this way, the categories divide the objects
into homogeneous subgroups. Variables are considered homogeneous when they classify objects
in the same categories into the same subgroups.
Example. Multiple Correspondence Analysis could be used to graphically display the relationship
between job category, minority classification, and gender. You might find that minority
classification and gender discriminate between people but that job category does not. You might
also find that the Latino and African-American categories are similar to each other.
Statistics and plots. Object scores, discrimination measures, iteration history, correlations of
original and transformed variables, category quantifications, descriptive statistics, object points
plots, biplots, category plots, joint category plots, transformation plots, and discrimination
measures plots.
Data. String variable values are always converted into positive integers by ascending alphanumeric
order. User-defined missing values, system-missing values, and values less than 1 are considered
missing; you can recode or add a constant to variables with values less than 1 to make them
nonmissing.
Assumptions. All variables have the multiple nominal scaling level. The data must contain at least
three valid cases. The analysis is based on positive integer data. The discretization option will
automatically categorize a fractional-valued variable by grouping its values into categories with a
close-to-normal distribution and will automatically convert values of string variables into positive
integers. You can specify other discretization schemes.
Related procedures. For two variables, Multiple Correspondence Analysis is analogous to
Correspondence Analysis. If you believe that variables possess ordinal or numerical properties,
Categorical Principal Components Analysis should be used. If sets of variables are of interest,
Nonlinear Canonical Correlation Analysis should be used.
To Obtain a Multiple Correspondence Analysis

56
Chapter 6
Figure 6-1
E Select All variables multiple nominal.
E Select One set.
E Click Define.
Figure 6-2
Multiple Correspondence Analysis dialog box
E Select at least two analysis variables and specify the number of dimensions in the solution.

57
E Click OK.
You may optionally specify supplementary variables, which are fitted into the solution found,
or labeling variables for the plots.
Define Variable Weight in Multiple Correspondence Analysis
You can set the weight for analysis variables.
Figure 6-3
Define Variable Weight dialog box
Variable weight. You can choose to define a weight for each variable. The value specified must
be a positive integer. The default value is 1.
Multiple Correspondence Analysis Discretization
Fractional-valued variables are grouped into seven categories (or into the number of distinct
values of the variable if this number is less than seven) with an approximately normal distribution
unless otherwise specified. String variables are always converted into positive integers by
assigning category indicators according to ascending alphanumeric order. Discretization for string
variables applies to these integers. Other variables are left alone by default. The discretized
variables are then used in the analysis.

58
Chapter 6
Figure 6-4
and have a constant added so that the lowest discretized value is 1.
Grouping. The following options are available when discretizing variables by grouping:
Multiple Correspondence Analysis Missing Values

59
Figure 6-5
Missing Value Strategy. Choose to exclude missing values (passive treatment), impute missing
values (active treatment), or exclude objects with missing values (listwise deletion).
 Exclude missing values; for correlations impute after quantification. Objects with missing values
on the selected variable do not contribute to the analysis for this variable. If all variables
are given passive treatment, then objects with missing values on all variables are treated as
supplementary. If correlations are specified in the Output dialog box, then (after analysis)
missing values are imputed with the most frequent category, or mode, of the variable for the
correlations of the original variables. For the correlations of the optimally scaled variables,
you can choose the method of imputation. Select Mode to replace missing values with the
mode of the optimally scaled variable. Select Extra category to replace missing values with
the quantification of an extra category. This implies that objects with a missing value on this

60
Chapter 6
Multiple Correspondence Analysis Options
The Options dialog box allows you to select the initial configuration, specify iteration and
convergence criteria, select a normalization method, choose the method for labeling plots, and
specify supplementary objects.
Figure 6-6
Options dialog box
Supplementary Objects. Specify the case number of the object (or the first and last case numbers
of a range of objects) that you want to make supplementary, and then click Add. Continue until
you have specified all of your supplementary objects. If an object is specified as supplementary,
then case weights are ignored for that object.
Normalization Method. You can specify one of five options for normalizing the object scores and
the variables. Only one normalization method can be used in a given analysis.
 Variable Principal. This option optimizes the association between variables. The coordinates
of the variables in the object space are the component loadings (correlations with principal
components, such as dimensions and object scores). This is useful when you are interested
primarily in the correlation between the variables.
 Object Principal. This option optimizes distances between objects. This is useful when you are
interested primarily in differences or similarities between the objects.
 Symmetrical. Use this normalization option if you are interested primarily in the relation
between objects and variables.

61
 Independent. Use this normalization option if you want to examine distances between objects
and correlations between variables separately.
 Custom. You can specify any real value in the closed interval [–1, 1]. A value of 1 is equal to
the Object Principal method, a value of 0 is equal to the Symmetrical method, and a value of
–1 is equal to the Variable Principal method. By specifying a value greater than –1 and less
than 1, you can spread the eigenvalue over both objects and variables. This method is useful
for making a tailor-made biplot or triplot.
Criteria. You can specify the maximum number of iterations the procedure can go through in its
computations. You can also select a convergence criterion value. The algorithm stops iterating if
the difference in total fit between the last two iterations is less than the convergence value or if
scatterplot matrix.
The lowest dimension can range from 1 to the number of dimensions in the solution minus 1
and is plotted against higher dimensions. The highest dimension value can range from 2 to the
number of dimensions in the solution and indicates the highest dimension to be used in plotting
the dimension pairs. This specification applies to all requested multidimensional plots.
Configuration. You can read data from a file containing the coordinates of a configuration. The
first variable in the file should contain the coordinates for the first dimension, the second variable
should contain the coordinates for the second dimension, and so on.
 Initial. The configuration in the file specified will be used as the starting point of the analysis.
 Fixed. The configuration in the file specified will be used to fit in the variables. The variables
that are fitted in must be selected as analysis variables, but, because the configuration is
fixed, they are treated as supplementary variables (so they do not need to be selected as
supplementary variables).
Multiple Correspondence Analysis Output
The Output dialog box allows you to produce tables for object scores, discrimination measures,
iteration history, correlations of original and transformed variables, category quantifications for
selected variables, and descriptive statistics for selected variables.

62
Chapter 6
Figure 6-7
Output dialog box
Object scores. Displays the object scores, including mass, inertia, and contributions, and has
the following options:
 Include Categories Of. Displays the category indicators of the analysis variables selected.
 Label Object Scores By. From the list of variables specified as labeling variables, you can
select one to label the objects.
Discrimination measures. Displays the discrimination measures per variable and per dimension.
Iteration history. For each iteration, the variance accounted for, loss, and increase in variance
accounted for are shown.
Correlations of original variables. Shows the correlation matrix of the original variables and the
eigenvalues of that matrix.
Correlations of transformed variables. Shows the correlation matrix of the transformed (optimally
scaled) variables and the eigenvalues of that matrix.
Category Quantifications and Contributions. Gives the category quantifications (coordinates),
including mass, inertia, and contributions, for each dimension of the variable(s) selected.
Note: the coordinates and contributions (including the mass and inertia) are displayed in
separate layers of the pivot table output, with the coordinates shown by default. To display
the contributions, activate (double-click) on the table and select Contributions from the Layer
dropdown list.

63
Descriptive Statistics. Displays frequencies, number of missing values, and mode of the variable(s)
selected.
Multiple Correspondence Analysis Save
The Save dialog box allows you to save discretized data, object scores, and transformed values
to an external IBM® SPSS® Statistics data file or dataset in the current session. You can also
save transformed values and object scores to the active dataset.
rules.
 If you save object scores or transformed values to the active dataset, you can specify the
number of multiple nominal dimensions.
Figure 6-8
Save dialog box
Multiple Correspondence Analysis Object Plots
The Object Plots dialog box allows you to specify the types of plots desired and the variables to
be plotted

64
Chapter 6
Figure 6-9
Object Plots dialog box
Object points. A plot of the object points is displayed.
Objects and centroids (biplot). The object points are plotted with the variable centroids.
Biplot Variables. You can choose to use all variables for the biplots or select a subset.
Label Objects. You can choose to have objects labeled with the categories of selected variables
(you may choose category indicator values or value labels in the Options dialog box) or with their
case numbers. One plot is produced per variable if Variable is selected.
Multiple Correspondence Analysis Variable Plots
The Variable Plots dialog box allows you to specify the types of plots desired and the variables to
be plotted.

65
Figure 6-10
Variable Plots dialog box
Category Plots. For each variable selected, a plot of the centroid coordinates is plotted. Categories
are in the centroids of the objects in the particular categories.
Joint Category Plots. This is a single plot of the centroid coordinates of each selected variable.
Transformation Plots. Displays a plot of the optimal category quantifications versus the category
indicators. You can specify the number of dimensions desired; one plot will be generated for each
dimension. You can also choose to display residual plots for each variable selected.
Discrimination Measures. Produces a single plot of the discrimination measures for the selected
variables.
MULTIPLE CORRESPONDENCE Command Additional Features
You can customize your Multiple Correspondence Analysis if you paste your selections into a
syntax window and edit the resulting MULTIPLE CORRESPONDENCE command syntax. The
command syntax language also allows you to:
 Specify rootnames for the transformed variables, object scores, and approximations when
saving them to the active dataset (with the SAVE subcommand).

66
Chapter 6
 Specify a maximum length for labels for each plot separately (with the PLOT subcommand).
 Specify a separate variable list for residual plots (with the PLOT subcommand).

Chapter
7
Multidimensional Scaling (PROXSCAL)
Multidimensional scaling attempts to find the structure in a set of proximity measures between
objects. This process is accomplished by assigning observations to specific locations in a
conceptual low-dimensional space such that the distances between points in the space match the
given (dis)similarities as closely as possible. The result is a least-squares representation of the
objects in that low-dimensional space, which, in many cases, will help you to further understand
your data.
Example. Multidimensional scaling can be very useful in determining perceptual relationships.
For example, when considering your product image, you can conduct a survey to obtain a dataset
that describes the perceived similarity (or proximity) of your product to those of your competitors.
Using these proximities and independent variables (such as price), you can try to determine
which variables are important to how people view these products, and you can adjust your image
accordingly.
Statistics and plots. Iteration history, stress measures, stress decomposition, coordinates of
the common space, object distances within the final configuration, individual space weights,
individual spaces, transformed proximities, transformed independent variables, stress plots,
common space scatterplots, individual space weight scatterplots, individual spaces scatterplots,
transformation plots, Shepard residual plots, and independent variables transformation plots.
Data. Data can be supplied in the form of proximity matrices or variables that are converted into
proximity matrices. The matrices can be formatted in columns or across columns. The proximities
can be treated on the ratio, interval, ordinal, or spline scaling levels.
Assumptions. At least three variables must be specified. The number of dimensions cannot exceed
the number of objects minus one. Dimensionality reduction is omitted if combined with multiple
random starts. If only one source is specified, all models are equivalent to the identity model;
therefore, the analysis defaults to the identity model.
Related procedures. Scaling all variables at the numerical level corresponds to standard
multidimensional scaling analysis.

68
Chapter 7
To Obtain a Multidimensional Scaling
Analyze > Scale > Multidimensional Scaling (PROXSCAL)...
This opens the Data Format dialog box.
Figure 7-1
Data Format dialog box
E Specify the format of your data:
Data Format. Specify whether your data consist of proximity measures or you want to create
proximities from the data.
Number of Sources. If your data are proximities, specify whether you have a single source or
multiple sources of proximity measures.
One Source. If there is one source of proximities, specify whether your dataset is formatted with
the proximities in a matrix across the columns or in a single column with two separate variables to
identify the row and column of each proximity.
 The proximities are in a matrix across columns. The proximity matrix is spread across a number
of columns equal to the number of objects. This leads to the Proximities in Matrices across
Columns dialog box.
 The proximities are in a single column. The proximity matrix is collapsed into a single column,
or variable. Two additional variables, identifying the row and column for each cell, are
necessary. This leads to the Proximities in One Column dialog box.
Multiple Sources. If there are multiple sources of proximities, specify whether the dataset is
formatted with the proximities in stacked matrices across columns, in multiple columns with
one source per column, or in a single column.

69
 The proximities are in stacked matrices across columns. The proximity matrices are spread
across a number of columns equal to the number of objects and are stacked above one another
across a number of rows equal to the number of objects times the number of sources. This
leads to the Proximities in Matrices across Columns dialog box.
 The proximities are in columns, one source per column. The proximity matrices are collapsed
into multiple columns, or variables. Two additional variables, identifying the row and column
for each cell, are necessary. This leads to the Proximities in Columns dialog box.
 The proximites are stacked in a single column. The proximity matrices are collapsed into
a single column, or variable. Three additional variables, identifying the row, column, and
source for each cell, are necessary. This leads to the Proximities in One Column dialog box.
E Click Define.
Proximities in Matrices across Columns
If you select the proximities in matrices data model for either one source or multiple sources in the
Data Format dialog box, the main dialog box will appear as follows:
Figure 7-2
Proximities in Matrices across Columns dialog box
E Select three or more proximities variables. (Be sure that the order of the variables in the list
matches the order of the columns of the proximities.)
E Optionally, select a number of weights variables equal to the number of proximities variables. (Be
sure that the order of the weights matches the order of the proximities that they weight.)
E Optionally, if there are multiple sources, select a sources variable. (The number of cases in each
proximities variable should equal the number of proximities variables times the number of
sources.)
Additionally, you can define a model for the multidimensional scaling, place restrictions on the
common space, set convergence criteria, specify the initial configuration to be used, and choose
plots and output.

70
Chapter 7
Proximities in Columns
If you select the multiple columns model for multiple sources in the Data Format dialog box,
the main dialog box will appear as follows:
Figure 7-3
Proximities in Columns dialog box
E Select two or more proximities variables. (Each variable is assumed to be a matrix of proximities
from a separate source.)
E Select a rows variable to define the row locations for the proximities in each proximities variable.
E Select a columns variable to define the column locations for the proximities in each proximities
variable. (Cells of the proximity matrix that are not given a row/column designation are treated as
missing.)
E Optionally, select a number of weights variables equal to the number of proximities variables.
plots and output.

71
Proximities in One Column
If you select the one column model for either one source or multiple sources in the Data Format
dialog box, the main dialog box will appear as follows:
Figure 7-4
Proximities in One Column dialog box
E Select a proximities variable. (t is assumed to be one or more matrices of proximities.)
E Select a rows variable to define the row locations for the proximities in the proximities variable.
E Select a columns variable to define the column locations for the proximities in the proximities
variable.
E If there are multiple sources, select a sources variable. (For each source, cells of the proximity
matrix that are not given a row/column designation are treated as missing.)
E Optionally, select a weights variable.
plots and output.

72
Chapter 7
Create Proximities from Data
If you choose to create proximities from the data in the Data Format dialog box, the main dialog
box will appear as follows:
Figure 7-5
Create Proximities from Data dialog box
E If you create distances between variables (see the Create Measure from Data dialog box), select at
least three variables. These variables will be used to create the proximity matrix (or matrices, if
there are multiple sources). If you create distances between cases, only one variable is needed.
E If there are multiple sources, select a sources variable.
E Optionally, choose a measure for creating proximities.
plots and output.

73
Create Measure from Data
Figure 7-6
Create Measure from Data dialog box
Multidimensional scaling uses dissimilarity data to create a scaling solution. If your data are
multivariate data (values of measured variables), you must create dissimilarity data in order to
compute a multidimensional scaling solution. You can specify the details of creating dissimilarity
measures from your data.
Measure. Allows you to specify the dissimilarity measure for your analysis. Select one alternative
from the Measure group corresponding to your type of data, and then select one of the measures
from the drop-down list corresponding to that type of measure. Available alternatives are:
 Interval. Euclidean distance, Squared Euclidean distance, Chebychev, Block, Minkowski,
or Customized.
 Counts. Chi-square measure or Phi-square measure.
 Binary. Euclidean distance, Squared Euclidean distance, Size difference, Pattern difference,
Variance, or Lance and Williams.
Create Distance Matrix. Allows you to choose the unit of analysis. Alternatives are Between
variables or Between cases.
Transform Values. In certain cases, such as when variables are measured on very different scales,
you want to standardize values before computing proximities (not applicable to binary data).
Select a standardization method from the Standardize drop-down list (if no standardization is
required, select None).

74
Chapter 7
Define a Multidimensional Scaling Model
The Model dialog box allows you to specify a scaling model, its minimum and maximum number
of dimensions, the structure of the proximity matrix, the transformation to use on the proximities,
and whether proximities are transformed within each source separately or unconditionally on
the source.
Figure 7-7
Model dialog box
Scaling Model. Choose from the following alternatives:
 Identity. All sources have the same configuration.
 Weighted Euclidean. This model is an individual differences model. Each source has an
individual space in which every dimension of the common space is weighted differentially.
 Generalized Euclidean. This model is an individual differences model. Each source has an
individual space that is equal to a rotation of the common space, followed by a differential
weighting of the dimensions.
 Reduced rank. This model is a generalized Euclidean model for which you can specify the
rank of the individual space. You must specify a rank that is greater than or equal to 1 and less
than the maximum number of dimensions.
Shape. Specify whether the proximities should be taken from the lower-triangular part or the
upper-triangular part of the proximity matrix. You can specify that the full matrix be used, in
which case the weighted sum of the upper-triangular part and the lower-triangular part will be
analyzed. In any case, the complete matrix should be specified, including the diagonal, though
only the specified parts will be used.

75
Proximities. Specify whether your proximity matrix contains measures of similarity or
dissimilarity.
Proximity Transformations. Choose from the following alternatives:
 Ratio. The transformed proximities are proportional to the original proximities. This is
allowed only for positively valued proximities.
 Interval. The transformed proximities are proportional to the original proximities, plus an
intercept term. The intercept assures all transformed proximities to be positive.
 Ordinal. The transformed proximities have the same order as the original proximities. You
specify whether tied proximities should be kept tied or allowed to become untied.
 Spline. The transformed proximities are a smooth nondecreasing piecewise polynomial
transformation of the original proximities. You specify the degree of the polynomial and the
number of interior knots.
Apply Transformations. Specify whether only proximities within each source are compared with
each other or whether the comparisons are unconditional on the source.
Dimensions. By default, a solution is computed in two dimensions (Minimum = 2, Maximum = 2).
You choose an integer minimum and maximum from 1 to the number of objects minus 1 (as long
as the minimum is less than or equal to the maximum). The procedure computes a solution in the
maximum dimensions and then reduces the dimensionality in steps until the lowest is reached.
Multidimensional Scaling Restrictions
The Restrictions dialog box allows you to place restrictions on the common space.

76
Chapter 7
Figure 7-8
Restrictions dialog box
Restrictions on Common Space. Specify the type of restriction desired.
 No restrictions. No restrictions are placed on the common space.
 Some coordinates fixed. The first variable selected contains the coordinates of the objects on
the first dimension, the second variable corresponds to coordinates on the second dimension,
and so on. A missing value indicates that a coordinate on a dimension is free. The number of
variables selected must equal the maximum number of dimensions requested.
 Linear combination of independent variables. The common space is restricted to be a linear
combination of the variables selected.
Restriction Variables. Select the variables that define the restrictions on the common space. If you
specified a linear combination, you specify an interval, nominal, ordinal, or spline transformation
for the restriction variables. In either case, the number of cases for each variable must equal
the number of objects.
Multidimensional Scaling Options
convergence criteria, and select standard or relaxed updates.

77
Figure 7-9
Options dialog box
Initial Configuration. Choose one of the following alternatives:
 Simplex. Objects are placed at the same distance from each other in the maximum dimension.
One iteration is taken to improve this high-dimensional configuration, followed by a
dimension reduction operation to obtain an initial configuration that has the maximum number
of dimensions that you specified in the Model dialog box.
 Torgerson. A classical scaling solution is used as the initial configuration.
 Single random start. A configuration is chosen at random.
 Multiple random starts. Several configurations are chosen at random, and the configuration
with the lowest normalized raw stress is used as the initial configuration.
 Custom. You select variables that contain the coordinates of your own initial configuration.
The number of variables selected should equal the maximum number of dimensions specified,
with the first variable corresponding to coordinates on dimension 1, the second variable
corresponding to coordinates on dimension 2, and so on. The number of cases in each variable
should equal the number of objects.
Iteration Criteria. Specify the iteration criteria values.
 Stress convergence. The algorithm will stop iterating when the difference in consecutive
normalized raw stress values is less than the number that is specified here, which must lie
between 0.0 and 1.0.
 Minimum stress. The algorithm will stop when the normalized raw stress falls below the
number that is specified here, which must lie between 0.0 and 1.0.

78
Chapter 7
 Maximum iterations. The algorithm will perform the number of specified iterations, unless
one of the above criteria is satisfied first.
 Use relaxed updates. Relaxed updates will speed up the algorithm; these updates cannot be
used with models other than the identity model or used with restrictions.
Multidimensional Scaling Plots, Version 1
The Plots dialog box allows you to specify which plots will be produced. If you have the
Proximities in Columns data format, the following Plots dialog box is displayed. For Individual
space weights, Original vs. transformed proximities, and Transformed proximities vs. distances plots,
you specify the sources for which the plots should be produced. The list of available sources is the
list of proximities variables in the main dialog box.
Figure 7-10
Plots dialog box, version 1
Stress. A plot is produced of normalized raw stress versus dimensions. This plot is produced only
if the maximum number of dimensions is larger than the minimum number of dimensions.
Common space. A scatterplot matrix of coordinates of the common space is displayed.
Individual spaces. For each source, the coordinates of the individual spaces are displayed in
scatterplot matrices. This is possible only if one of the individual differences models is specified
in the Model dialog box.
Individual space weights. A scatterplot is produced of the individual space weights. This is
possible only if one of the individual differences models is specified in the Model dialog box.
For the weighted Euclidean model, the weights are printed in plots, with one dimension on each
axis. For the generalized Euclidean model, one plot is produced per dimension, indicating both
rotation and weighting of that dimension. The reduced rank model produces the same plot as the
generalized Euclidean model but reduces the number of dimensions for the individual spaces.

79
Original vs. transformed proximities. Plots are produced of the original proximities versus the
transformed proximities.
Transformed proximities vs. distances. The transformed proximities versus the distances are plotted.
Transformed independent variables. Transformation plots are produced for the independent
variables.
Variable and dimension correlations. A plot of correlations between the independent variables and
the dimensions of the common space is displayed.
Multidimensional Scaling Plots, Version 2
The Plots dialog box allows you to specify which plots will be produced. If your data format
is anything other than Proximities in Columns, the following Plots dialog box is displayed.
For Individual space weights, Original vs. transformed proximities, and Transformed proximities vs.
distances plots, you specify the sources for which the plots should be produced. The source
numbers entered must be values of the sources variable that is specified in the main dialog box
and must range from 1 to the number of sources.
Figure 7-11
Plots dialog box, version 2
Multidimensional Scaling Output
The Output dialog box allows you to control the amount of displayed output and save some
of it to separate files.

80
Chapter 7
Figure 7-12
Output dialog box
Display. Select one or more of the following items for display:
 Common space coordinates. Displays the coordinates of the common space.
 Individual space coordinates. The coordinates of the individual spaces are displayed only if the
model is not the identity model.
 Individual space weights. Displays the individual space weights only if one of the individual
differences models is specified. Depending on the model, the space weights are decomposed
in rotation weights and dimension weights, which are also displayed.
 Distances. Displays the distances between the objects in the configuration.
 Transformed proximities. Displays the transformed proximities between the objects in the
configuration.
 Input data. Includes the original proximities and, if present, the data weights, the initial
configuration, and the fixed coordinates of the independent variables.
 Stress for random starts. Displays the random number seed and normalized raw stress value of
each random start.
 Iteration history. Displays the history of iterations of the main algorithm.
 Multiple stress measures. Displays different stress values. The table contains values for
normalized raw stress, Stress-I, Stress-II, S-Stress, Dispersion Accounted For (DAF), and
Tucker’s Coefficient of Congruence.
 Stress decomposition. Displays an objects and sources decomposition of final normalized raw
stress, including the average per object and the average per source.

81
 Transformed independent variables. If a linear combination restriction was selected, the
transformed independent variables and the corresponding regression weights are displayed.
 Variable and dimension correlations. If a linear combination restriction was selected, the
correlations between the independent variables and the dimensions of the common space
are displayed.
Save to New File. You can save the common space coordinates, individual space weights,
distances, transformed proximities, and transformed independent variables to separate IBM®
SPSS® Statistics data files.
PROXSCAL Command Additional Features
You can customize your multidimensional scaling of proximities analysis if you paste your
selections into a syntax window and edit the resulting PROXSCAL command syntax. The command
syntax language also allows you to:
 Specify separate variable lists for transformations and residuals plots (with the PLOT
subcommand).
 Specify separate source lists for individual space weights, transformations, and residuals
plots (with the PLOT subcommand).
 Specify a subset of the independent variables transformation plots to be displayed (with
the PLOT subcommand).

Chapter
8
(PREFSCAL)
The Multidimensional Unfolding procedure attempts to find a common quantitative scale that
allows you to visually examine the relationships between two sets of objects.
Examples. You have asked 21 individuals to rank 15 breakfast items in order of preference, 1 to
15. Using Multidimensional Unfolding, you can determine that the individuals discriminate
between breakfast items in two primary ways: between soft and hard breads, and between
fattening and non-fattening items.
Alternatively, you have asked a group of drivers to rate 26 models of cars on 10 attributes on a
6-point scale ranging from 1=“not true at all” to 6=“very true.” Averaged over individuals, the
values are taken as similarities. Using Multidimensional Unfolding, you find clusterings of similar
models and the attributes with which they are most closely associated.
Statistics and plots. The Multidimensional Unfolding procedure can produce an iteration history,
stress measures, stress decomposition, coordinates of the common space, object distances within
the final configuration, individual space weights, individual spaces, transformed proximities,
stress plots, common space scatterplots, individual space weight scatterplots, individual spaces
scatterplots, transformation plots, and Shepard residual plots.
Data. Data are supplied in the form of rectangular proximity matrices. Each column is considered
a separate column object. Each row of a proximity matrix is considered a separate row object.
When there are multiple sources of proximities, the matrices are stacked.
Assumptions. At least two variables must be specified. The number of dimensions in the solution
may not exceed the number of objects minus one. If only one source is specified, all models are
equivalent to the identity model; therefore, the analysis defaults to the identity model.
To Obtain a Multidimensional Unfolding
Analyze > Scale > Multidimensional Unfolding (PREFSCAL)...

83
Multidimensional Unfolding (PREFSCAL)
Figure 8-1
Multidimensional Unfolding main dialog box
E Select two or more variables that identify the columns in the rectangular proximity matrix. Each
variable represents a separate column object.
E Optionally, select a number of weights variables equal to the number of column object variables.
The order of the weights variables should match the order of the column objects they weight.
E Optionally, select a rows variable. The values (or value labels) of this variable are used to label
row objects in the output.
E If there are multiple sources, optionally select a sources variable. The number of cases in the data
file should equal the number of row objects times the number of sources.
Additionally, you can define a model for the multidimensional unfolding, place restrictions on
the common space, set convergence criteria, specify the initial configuration to be used, and
choose plots and output.
Define a Multidimensional Unfolding Model
The Model dialog box allows you to specify a scaling model, its minimum and maximum number
of dimensions, the structure of the proximity matrix, the transformation to use on the proximities,
and whether proximities are transformed conditonal upon the row, conditional upon the source, or
unconditionally on the source.

84
Chapter 8
Figure 8-2
Model dialog box
Scaling Model. Choose from the following alternatives:
 Identity. All sources have the same configuration.
 Weighted Euclidean. This model is an individual differences model. Each source has an
individual space in which every dimension of the common space is weighted differentially.
 Generalized Euclidean. This model is an individual differences model. Each source has an
individual space that is equal to a rotation of the common space, followed by a differential
weighting of the dimensions.
Proximities. Specify whether your proximity matrix contains measures of similarity or
dissimilarity.
Dimensions. By default, a solution is computed in two dimensions (Minimum = 2, Maximum =
2). You can choose an integer minimum and maximum from 1 to the number of objects minus
1 as long as the minimum is less than or equal to the maximum. The procedure computes a
solution in the maximum dimensionality and then reduces the dimensionality in steps until the
lowest is reached.
Proximity Transformations. Choose from the following alternatives:
 None. The proximities are not transformed. You can optionally select Include intercept, in
which case the proximities can be shifted by a constant term.

85
 Linear. The transformed proximities are proportional to the original proximities; that is, the
transformation function estimates a slope and the intercept is fixed at 0. This is also called a
ratio transformation. You can optionally select Include intercept, in which case the proximities
can also be shifted by a constant term. This is also called an interval transformation.
 Spline. The transformed proximities are a smooth nondecreasing piecewise polynomial
transformation of the original proximities. You can specify the degree of the polynomial and
the number of interior knots. You can optionally select Include intercept, in which case the
proximities can also be shifted by a constant term.
 Smooth. The transformed proximities have the same order as the original proximities,
including a restriction that takes the differences between subsequent values into account. The
result is a “smooth ordinal” transformation. You can specify whether tied proximities should
be kept tied or allowed to become untied.
 Ordinal. The transformed proximities have the same order as the original proximities. You can
specify whether tied proximities should be kept tied or allowed to become untied.
Apply Transformations. Specify whether only proximities within each row are compared with each
other, or only proximities within each source are compared with each other, or the comparisons
are unconditional on the row or source; that is, whether the transformations are performed per
row, per source, or over all proximities at once.
Multidimensional Unfolding Restrictions
The Restrictions dialog box allows you to place restrictions on the common space.

86
Chapter 8
Figure 8-3
Restrictions on Common Space. You can choose to fix the coordinates of row and/or column
objects in the common space.
Row/Column Restriction Variables. Choose the file containing the restrictions and select the
variables that define the restrictions on the common space. The first variable selected contains the
coordinates of the objects on the first dimension, the second variable corresponds to coordinates
on the second dimension, and so on. A missing value indicates that a coordinate on a dimension is
free. The number of variables selected must equal the maximum number of dimensions requested.
The number of cases for each variable must equal the number of objects.
Multidimensional Unfolding Options
convergence criteria, and set the penalty term for stress.

87
Figure 8-4
Options dialog box
Initial Configuration. Choose one of the following alternatives:
 Classical. The rectangular proximity matrix is used to supplement the intra-blocks (values
between rows and between columns) of the complete symmetrical MDS matrix. Once the
complete matrix is formed, a classical scaling solution is used as the initial configuration. The
intra-blocks can be filled via imputation using the triangle inequality or Spearman distances.
 Ross-Cliff. The Ross-Cliff start uses the results of a singular value decomposition on the double
centered and squared proximity matrix as the initial values for the row and column objects.
 Correspondence. The correspondence start uses the results of a correspondence analysis
on the reversed data (similarities instead of dissimilarities), with symmetric normalization
of row and column scores.
 Centroids. The procedure starts by positioning the row objects in the configuration using an
eigenvalue decomposition. Then the column objects are positioned at the centroid of the
specified choices. For the number of choices, specify a positive integer between 1 and the
number of proximities variables.

88
Chapter 8
 Multiple random starts. Solutions are computed for several initial configurations chosen at
random, and the one with the lowest penalized stress is shown as the best solution.
 Custom. You can select variables that contain the coordinates of your own initial configuration.
The number of variables selected should equal the maximum number of dimensions specified,
with the first variable corresponding to coordinates on dimension 1, the second variable
corresponding to coordinates on dimension 2, and so on. The number of cases in each
variable should equal the combined number of row and column objects. The row and column
coordinates should be stacked, with the column coordinates following the row coordinates.
Iteration Criteria. Specify the iteration criteria values.
 Stress convergence. The algorithm will stop iterating when the relative difference in
consecutive penalized stress values is less than the number specified here, which must be
non-negative.
 Minimum stress. The algorithm will stop when the penalized stress falls below the number
specified here, which must be non-negative.
 Maximum iterations. The algorithm will perform the number of iterations specified here unless
one of the above criteria is satisfied first.
Penalty Term. The algorithm attempts to minimize penalized stress, a goodness-of-fit measure
equal to the product of Kruskal’s Stress-I and a penalty term based on the coefficient of variation
of the transformed proximities. These controls allow you to set the strength and range of the
penalty term.
 Strength. The smaller the value of the strength parameter, the stronger the penalty. Specify a
value between 0.0 and 1.0.
 Range. This parameter sets the moment at which the penalty becomes active. If set to 0.0, the
penalty is inactive. Increasing the value causes the algorithm to search for a solution with
greater variation among the transformed proximities. Specify a non-negative value.
Multidimensional Unfolding Plots
The Plots dialog box allows you to specify which plots will be produced.

89
Figure 8-5
Plots dialog box
Plots. The following plots are available:
 Multiple starts. Displays a stacked histogram of penalized stress displaying both stress and
penalty.
 Initial common space. Displays a scatterplot matrix of the coordinates of the initial common
space.
 Stress per dimension. Produces a lineplot of penalized stress versus dimensionality. This plot
is produced only if the maximum number of dimensions is larger than the minimum number
of dimensions.
 Final common space. A scatterplot matrix of coordinates of the common space is displayed.

90
Chapter 8
 Space weights. A scatterplot is produced of the individual space weights. This is possible
only if one of the individual differences models is specified in the Model dialog box. For
the weighted Euclidean model, the weights for all sources are displayed in a plot, with one
dimension on each axis. For the generalized Euclidean model, one plot is produced per
dimension, indicating both rotation and weighting of that dimension for each source.
 Individual spaces. A scatterplot matrix of coordinates of the individual space of each source
is displayed. This is possible only if one of the individual differences models is specified
in the Model dialog box.
 Transformation plots. A scatterplot is produced of the original proximities versus the
transformed proximities. Depending on how transformations are applied, a separate color is
assigned to each row or source. An unconditional transformation produces a single color.
 Shepard plots. The original proximities versus both transformed proximities and distances.
The distances are indicated by points, and the transformed proximities are indicated by a line.
Depending on how transformations are applied, a separate line is produced for each row or
source. An unconditional transformation produces one line.
 Scatterplot of fit. A scatterplot of the transformed proximities versus the distances is displayed.
A separate color is assigned to each source if multiple sources are specified.
 Residuals plots. A scatterplot of the transformed proximities versus the residuals (transformed
proximities minus distances) is displayed. A separate color is assigned to each source if
multiple sources are specified.
Row Object Styles. These give you further control of the display of row objects in plots. The values
of the optional colors variable are used to cycle through all colors. The values of the optional
markers variable are used to cycle through all possible markers.
Source Plots. For Individual spaces, Scatterplot of fit, and Residuals plots—and if transformations
are applied by source, for Transformation plots and Shepard plots—you can specify the sources for
which the plots should be produced. The source numbers entered must be values of the sources
variable specified in the main dialog box and range from 1 to the number of sources.
Row Plots. If transformations are applied by row, for Transformation plots and Shepard plots, you
can specify the row for which the plots should be produced. The row numbers entered must
range from 1 to the number of rows.
Multidimensional Unfolding Output
The Output dialog box allows you to control the amount of displayed output and save some
of it to separate files.

91
Figure 8-6
Output dialog box
Display. Select one or more of the following for display:
 Input data. Includes the original proximities and, if present, the data weights, the initial
configuration, and the fixed coordinates.
 Multiple starts. Displays the random number seed and penalized stress value of each random
start.
 Initial data. Displays the coordinates of the initial common space.
 Iteration history. Displays the history of iterations of the main algorithm.
 Fit measures. Displays different measures. The table contains several goodness-of-fit,
badness-of-fit, correlation, variation, and nondegeneracy measures.
 Stress decomposition. Displays an objects, rows, and sources decomposition of penalized
stress, including row, column, and source means and standard deviations.
 Transformed proximities. Displays the transformed proximities.
 Final common space. Displays the coordinates of the common space.
 Space weights. Displays the individual space weights. This option is available only if one of
the individual differences models is specified. Depending on the model, the space weights are
decomposed in rotation weights and dimension weights, which are also displayed.
 Individual spaces. The coordinates of the individual spaces are displayed. This option is
available only if one of the individual differences models is specified.
 Fitted distances. Displays the distances between the objects in the configuration.
Save to New File. You can save the common space coordinates, individual space weights,
distances, and transformed proximities to separate IBM® SPSS® Statistics data files.

92
Chapter 8
PREFSCAL Command Additional Features
You can customize your Multidimensional Unfolding of proximities analysis if you paste your
selections into a syntax window and edit the resulting PREFSCAL command syntax. The command
syntax language also allows you to:
 Specify multiple source lists for Individual spaces, Scatterplots of fit, and Residuals
plots—and in the case of matrix conditional transformations, for Transformation plots and
Shepard plots—when multiple sources are available (with the PLOT subcommand).
 Specify multiple row lists for Transformation plots and Shepard plots in the case of row
conditional transformations (with the PLOT subcommand).
 Specify a number of rows instead of a row ID variable (with the INPUT subcommand).
 Specify a number of sources instead of a source ID variable (with the INPUT subcommand).

Chapter
9
The goal of categorical regression with optimal scaling is to describe the relationship between a
response variable and a set of predictors. By quantifying this relationship, values of the response
can be predicted for any combination of predictors.
In this chapter, two examples serve to illustrate the analyses involved in optimal scaling
regression. The first example uses a small data set to illustrate the basic concepts. The second
example uses a much larger set of variables and observations in a practical example.
Example: Carpet Cleaner Data
In a popular example (Green and Wind, 1973), a company interested in marketing a new carpet
cleaner wants to examine the influence of five factors on consumer preference—package design,
brand name, price, a Good Housekeeping seal, and a money-back guarantee. There are three
factor levels for package design, each one differing in the location of the applicator brush; three
brand names (K2R, Glory, and Bissell); three price levels; and two levels (either no or yes) for
each of the last two factors. The following table displays the variables used in the carpet-cleaner
study, with their variable labels and values.
Table 9-1
Explanatory variables in the carpet-cleaner study
Variable name Variable label Value label
package Package design A*, B*, C*
brand Brand name K2R, Glory, Bissell
price Price $1.19, $1.39, $1.59
seal Good Housekeeping
seal
No, yes
money Money-back
guarantee
No, yes
Ten consumers rank 22 profiles defined by these factors. The variable Preference contains the
rank of the average rankings for each profile. Low rankings correspond to high preference. This
variable reflects an overall measure of preference for each profile. Using categorical regression,
you will explore how the five factors are related to preference. This data set can be found in
carpet.sav. For more information, see the topic Sample Files in Appendix A in IBM SPSS
Categories 20.
A Standard Linear Regression Analysis
E To produce standard linear regression output, from the menus choose:
Analyze > Regression > Linear...
Note: This feature requires the Statistics Base option.

95
Figure 9-1
Linear Regression dialog box
E Select Preference as the dependent variable.
E Select Package design through Money-back guarantee as independent variables.
E Click Plots.
Figure 9-2
Plots dialog box
E Select *ZRESID as the y-axis variable.
E Select *ZPRED as the x-axis variable.
E Click Continue.

96
Chapter 9
E Click Save in the Linear Regression dialog box.
Figure 9-3
Save dialog box
E Select Standardized in the Residuals group.
E Click Continue.
E Click OK in the Linear Regression dialog box.
Model Summary
Figure 9-4
Model summary for standard linear regression
The standard approach for describing the relationships in this problem is linear regression. The
most common measure of how well a regression model fits the data is R2. This statistic represents
how much of the variance in the response is explained by the weighted combination of predictors.

97
The closer R2 is to 1, the better the model fits. Regressing Preference on the five predictors results
in an R2 of 0.707, indicating that approximately 71% of the variance in the preference rankings is
explained by the predictor variables in the linear regression.
Coefficients
The standardized coefficients are shown in the table. The sign of the coefficient indicates
whether the predicted response increases or decreases when the predictor increases, all other
predictors being constant. For categorical data, the category coding determines the meaning of an
increase in a predictor. For instance, an increase in Money-back guarantee, Package design, or
Good Housekeeping seal will result in a decrease in predicted preference ranking. Money-back
guarantee is coded 1 for no money-back guarantee and 2 for money-back guarantee. An increase
in Money-back guarantee corresponds to the addition of a money-back guarantee. Thus, adding
a money-back guarantee reduces the predicted preference ranking, which corresponds to an
increased predicted preference.
Figure 9-5
Regression coefficients
The value of the coefficient reflects the amount of change in the predicted preference ranking.
Using standardized coefficients, interpretations are based on the standard deviations of the
variables. Each coefficient indicates the number of standard deviations that the predicted response
changes for a one standard deviation change in a predictor, all other predictors remaining
constant. For example, a one standard deviation change in Brand name yields an increase in
predicted preference of 0.056 standard deviations. The standard deviation of Preference is 6.44,
so Preference increases by . Changes in Package design yield the greatest
changes in predicted preference.

98
Chapter 9
Residual Scatterplots
Figure 9-6
Residuals versus predicted values
The standardized residuals are plotted against the standardized predicted values. No patterns
should be present if the model fits well. Here you see a U-shape in which both low and high
standardized predicted values have positive residuals. Standardized predicted values near 0 tend
to have negative residuals.
E To produce a scatterplot of the residuals by the predictor Package design, from the menus choose:
Graphs > Chart Builder...

99
Figure 9-7
Chart Builder
E Select the Scatter/Dot gallery and choose Simple Scatter.
E Select Standardized Residual as the y-axis variable and Package design as the x-axis variable.
E Click OK.

100
Chapter 9
Figure 9-8
Residuals versus package design
The U-shape is more pronounced in the plot of the standardized residuals against package. Every
residual for Design B* is negative, whereas all but one of the residuals is positive for the other two
designs. Because the linear regression model fits one parameter for each variable, the relationship
cannot be captured by the standard approach.
A Categorical Regression Analysis
The categorical nature of the variables and the nonlinear relationship between Preference and
Package design suggest that regression on optimal scores may perform better than standard
regression. The U-shape of the residual plots indicates that a nominal treatment of Package design
should be used. All other predictors will be treated at the numerical scaling level.
The response variable warrants special consideration. You want to predict the values of
Preference. Thus, recovering as many properties of its categories as possible in the quantifications
is desirable. Using an ordinal or nominal scaling level ignores the differences between the response
categories. However, linearly transforming the response categories preserves category differences.
Consequently, scaling the response numerically is generally preferred and will be employed here.
Running the Analysis
E To run a Categorical Regression analysis, from the menus choose:

101
Figure 9-9
E Select Preference as the dependent variable.
E Select Package design through Money-back guarantee as independent variables.
E Select Preference and click Define Scale.
Figure 9-10
E Select Numeric as the optimal scaling level.
E Click Continue.
E Select Package design and click Define Scale in the Categorical Regression dialog box.

102
Chapter 9
Figure 9-11
E Select Nominal as the optimal scaling level.
E Click Continue.
E Select Brand name through Money-back guarantee and click Define Scale in the Categorical
Regression dialog box.
Figure 9-12
E Click Continue.
E Click Output in the Categorical Regression dialog box.

103
Figure 9-13
Output dialog box
E Select Correlations of original variables and Correlations of transformed variables.
E Deselect ANOVA.
E Click Continue.
E Click Save in the Categorical Regression dialog box.

104
Chapter 9
Figure 9-14
Save dialog box
E Select Save residuals to the active dataset.
E Select Save transformed variables to the active dataset in the Transformed Variables group.
E Click Continue.
E Click Plots in the Categorical Regression dialog box.

105
Figure 9-15
Plots dialog box
E Choose to create transformation plots for Package design and Price.
E Click Continue.
E Click OK in the Categorical Regression dialog box.
Intercorrelations
The intercorrelations among the predictors are useful for identifying multicollinearity in the
regression. Variables that are highly correlated will lead to unstable regression estimates.
However, due to their high correlation, omitting one of them from the model only minimally
affects prediction. The variance in the response that can be explained by the omitted variable
is still explained by the remaining correlated variable. However, zero-order correlations are
sensitive to outliers and also cannot identify multicollinearity due to a high correlation between a
predictor and a combination of other predictors.
Figure 9-16
Original predictor correlations

106
Chapter 9
Figure 9-17
Transformed predictor correlations
The intercorrelations of the predictors for both the untransformed and transformed predictors are
displayed. All values are near 0, indicating that multicollinearity between individual variables is
not a concern.
Notice that the only correlations that change involve Package design. Because all other
predictors are treated numerically, the differences between the categories and the order of the
categories are preserved for these variables. Consequently, the correlations cannot change.
Model Fit and Coefficients
The Categorical Regression procedure yields an R2 of 0.948, indicating that almost 95% of the
variance in the transformed preference rankings is explained by the regression on the optimally
transformed predictors. Transforming the predictors improves the fit over the standard approach.
Figure 9-18
Model summary for categorical regression
The following table shows the standardized regression coefficients. Categorical regression
standardizes the variables, so only standardized coefficients are reported. These values are divided
by their corresponding standard errors, yielding an F test for each variable. However, the test for
each variable is contingent upon the other predictors being in the model. In other words, the test
determines if omission of a predictor variable from the model with all other predictors present
significantly worsens the predictive capabilities of the model. These values should not be used to

107
omit several variables at one time for a subsequent model. Moreover, alternating least squares
optimizes the quantifications, implying that these tests must be interpreted conservatively.
Figure 9-19
Standardized coefficients for transformed predictors
The largest coefficient occurs for Package design. A one standard deviation increase in Package
design yields a 0.748 standard deviation decrease in predicted preference ranking. However,
Package design is treated nominally, so an increase in the quantifications need not correspond
to an increase in the original category codes.
Standardized coefficients are often interpreted as reflecting the importance of each predictor.
However, regression coefficients cannot fully describe the impact of a predictor or the relationships
between the predictors. Alternative statistics must be used in conjunction with the standardized
coefficients to fully explore predictor effects.
Correlations and Importance
To interpret the contributions of the predictors to the regression, it is not sufficient to only inspect
the regression coefficients. In addition, the correlations, partial correlations, and part correlations
should be inspected. The following table contains these correlational measures for each variable.
The zero-order correlation is the correlation between the transformed predictor and the
transformed response. For this data, the largest correlation occurs for Package design. However, if
you can explain some of the variation in either the predictor or the response, you will get a better
representation of how well the predictor is doing.
Figure 9-20
Zero-order, part, and partial correlations (transformed variables)
Other variables in the model can confound the performance of a given predictor in predicting the
response. The partial correlation coefficient removes the linear effects of other predictors from
both the predictor and the response. This measure equals the correlation between the residuals
from regressing the predictor on the other predictors and the residuals from regressing the

108
Chapter 9
response on the other predictors. The squared partial correlation corresponds to the proportion of
the variance explained relative to the residual variance of the response remaining after removing
the effects of the other variables. For example, Package design has a partial correlation of –0.955.
Removing the effects of the other variables, Package design explains (–0.955)2 = 0.91 = 91% of
the variation in the preference rankings. Both Price and Good Housekeeping seal also explain a
large portion of variance if the effects of the other variables are removed.
As an alternative to removing the effects of variables from both the response and a predictor,
you can remove the effects from just the predictor. The correlation between the response and the
residuals from regressing a predictor on the other predictors is the part correlation. Squaring
this value yields a measure of the proportion of variance explained relative to the total variance
of response. If you remove the effects of Brand name, Good Housekeeping seal, Money back
guarantee, and Price from Package design, the remaining part of Package design explains
(–0.733)2 = 0.54 = 54% of the variation in preference rankings.
Importance
In addition to the regression coefficients and the correlations, Pratt’s measure of relative
importance (Pratt, 1987) aids in interpreting predictor contributions to the regression. Large
individual importances relative to the other importances correspond to predictors that are crucial
to the regression. Also, the presence of suppressor variables is signaled by a low importance for a
variable that has a coefficient of similar size to the important predictors.
In contrast to the regression coefficients, this measure defines the importance of the predictors
additively—that is, the importance of a set of predictors is the sum of the individual importances of
the predictors. Pratt’s measure equals the product of the regression coefficient and the zero-order
correlation for a predictor. These products add to R2, so they are divided by R2, yielding a sum of
1. The set of predictors Package design and Brand name, for example, have an importance of
0.654. The largest importance corresponds to Package design, with Package design, Price, and
Good Housekeeping seal accounting for 95% of the importance for this combination of predictors.
Multicollinearity
Large correlations between predictors will dramatically reduce a regression model’s stability.
Correlated predictors result in unstable parameter estimates. Tolerance reflects how much the
independent variables are linearly related to one another. This measure is the proportion of a
variable’s variance not accounted for by other independent variables in the equation. If the other
predictors can explain a large amount of a predictor’s variance, that predictor is not needed in the
model. A tolerance value near 1 indicates that the variable cannot be predicted very well from the
other predictors. In contrast, a variable with a very low tolerance contributes little information
to a model, and can cause computational problems. Moreover, large negative values of Pratt’s
importance measure indicate multicollinearity.
All of the tolerance measures are very high. None of the predictors are predicted very well by
the other predictors and multicollinearity is not present.

109
Plotting the original category values against their corresponding quantifications can reveal trends
that might not be noticed in a list of the quantifications. Such plots are commonly referred to as
transformation plots. Attention should be given to categories that receive similar quantifications.
These categories affect the predicted response in the same manner. However, the transformation
type dictates the basic appearance of the plot.
Variables treated as numerical result in a linear relationship between the quantifications and the
original categories, corresponding to a straight line in the transformation plot. The order and the
difference between the original categories is preserved in the quantifications.
The order of the quantifications for variables treated as ordinal correspond to the order of
the original categories. However, the differences between the categories are not preserved. As
a result, the transformation plot is nondecreasing but need not be a straight line. If consecutive
categories correspond to similar quantifications, the category distinction may be unnecessary and
the categories could be combined. Such categories result in a plateau on the transformation plot.
However, this pattern can also result from imposing an ordinal structure on a variable that should
be treated as nominal. If a subsequent nominal treatment of the variable reveals the same pattern,
combining categories is warranted. Moreover, if the quantifications for a variable treated as
ordinal fall along a straight line, a numerical transformation may be more appropriate.
For variables treated as nominal, the order of the categories along the horizontal axis
corresponds to the order of the codes used to represent the categories. Interpretations of category
order or of the distance between the categories is unfounded. The plot can assume any nonlinear
or linear form. If an increasing trend is present, an ordinal treatment should be attempted. If the
nominal transformation plot displays a linear trend, a numerical transformation may be more
appropriate.
The following figure displays the transformation plot for Price, which was treated as numerical.
Notice that the order of the categories along the straight line correspond to the order of the original
categories. Also, the difference between the quantifications for $1.19 and $1.39 (–1.173 and 0) is
the same as the difference between the quantifications for $1.39 and $1.59 (0 and 1.173). The fact
that categories 1 and 3 are the same distance from category 2 is preserved in the quantifications.

110
Chapter 9
Figure 9-21
Transformation plot of Price (numerical)
The nominal transformation of Package design yields the following transformation plot. Notice
the distinct nonlinear shape in which the second category has the largest quantification. In terms
of the regression, the second category decreases predicted preference ranking, whereas the first
and third categories have the opposite effect.
Figure 9-22
Transformation plot of Package design (nominal)

111
Residual Analysis
Using the transformed data and residuals that you saved to the active dataset allows you to create a
scatterplot of the predicted values by the transformed values of Package design.
To obtain such a scatterplot, recall the Chart Builder and click Reset to clear your previous
selections and restore the default options.
Figure 9-23
Chart Builder
E Select Residual as the y-axis variable.
E Select Package design Quantification as the x-axis variable.
E Click OK.
The scatterplot shows the standardized residuals plotted against the optimal scores for Package
design. All of the residuals are within two standard deviations of 0. A random scatter of points
replaces the U-shape present in the scatterplot from the standard linear regression. Predictive
abilities are improved by optimally quantifying the categories.

112
Chapter 9
Figure 9-24
Residuals for Categorical Regression
Example: Ozone Data
In this example, you will use a larger set of data to illustrate the selection and effects of optimal
scaling transformations. The data include 330 observations on six meteorological variables
previously analyzed by Breiman and Friedman (Breiman and Friedman, 1985), and Hastie
and Tibshirani (Hastie and Tibshirani, 1990), among others. The following table describes the
original variables. Your categorical regression attempts to predict the ozone concentration from
the remaining variables. Previous researchers found nonlinearities among these variables, which
hinder standard regression approaches.
Table 9-2
Original variables
Variable Description
ozon daily ozone level; categorized into one of 38
categories
ibh inversion base height
dpg pressure gradient (mm Hg)
vis visibility (miles)
temp temperature (degrees F)
doy day of the year

113
This dataset can be found in ozone.sav.For more information, see the topic Sample Files in
Appendix A in IBM SPSS Categories 20.
Discretizing Variables
If a variable has more categories than is practically interpretable, you should modify the categories
using the Discretization dialog box to reduce the category range to a more manageable number.
The variable Day of the year has a minimum value of 3 and a maximum value of 365. Using
this variable in a categorical regression corresponds to using a variable with 365 categories.
Similarly, Visibility (miles) ranges from 0 to 350. To simplify interpretation of analyses, discretize
these variables into equal intervals of length 10.
The variable Inversion base height ranges from 111 to 5000. A variable with this many
categories results in very complex relationships. However, discretizing this variable into equal
intervals of length 100 yields roughly 50 categories. Using a 50-category variable rather than a
5000-category variable simplifies interpretations significantly.
Pressure gradient (mm Hg) ranges from –69 to 107. The procedure omits any categories coded
with negative numbers from the analysis, but discretizing this variable into equal intervals of
length 10 yields roughly 19 categories.
Temperature (degrees F) ranges from 25 to 93 on the Fahrenheit scale. In order to analyze the
data as if it were on the Celsius scale, discretize this variable into equal intervals of length 1.8.
Different discretizations for variables may be desired. The choices used here are purely
subjective. If you desire fewer categories, choose larger intervals. For example, Day of the year
could have been divided into months of the year or seasons.
Selection of Transformation Type
Each variable can be analyzed at one of several different levels. However, because prediction of
the response is the goal, you should scale the response “as is” by employing the numerical optimal
scaling level. Consequently, the order and the differences between categories will be preserved
in the transformed variable.
E To run a Categorical Regression analysis, from the menus choose:

114
Chapter 9
Figure 9-25
E Select Daily ozone level as the dependent variable.
E Select Inversion base height through Day of the year as independent variables.
E Select Daily ozone level and click Define Scale.
Figure 9-26
E Click Continue.
E Select Inversion base height through Day of the year, and click Define Scale in the Categorical

115
Figure 9-27
E Click Continue.
E Click Discretize in the Categorical Regression dialog box.
Figure 9-28
E Select ibh.
E Select Equal intervals and type 100 as the interval length.
E Click Change.
E Select dpg, vis, and doy.

116
Chapter 9
E Type 10 as the interval length.
E Click Change.
E Select temp.
E Type 1.8 as the interval length.
E Click Change.
E Click Continue.
Figure 9-29
Plots dialog box
E Select transformation plots for Inversion base height through Day of the year.
E Click Continue.
Figure 9-30
Model summary
Treating all predictors as nominal yields an R2 of 0.880. This large amount of variance accounted
for is not surprising because nominal treatment imposes no restrictions on the quantifications.
However, interpreting the results can be quite difficult.

117
Figure 9-31
Regression coefficients (all predictors nominal)
This table shows the standardized regression coefficients of the predictors. A common mistake
made when interpreting these values involves focusing on the coefficients while neglecting the
quantifications. You cannot simply assert that a positive value of Inversion base height, for
example, implies that as the predictor increases, predicted Ozone increases. All interpretations
must be relative to the transformed variables, so that as the quantifications for Inversion base
height increase, predicted Ozone increases. To examine the effects of the original variables, you
must relate the categories to the quantifications.
Figure 9-32
Transformation plot of Inversion base height (nominal)
The transformation plot of Inversion base height shows no apparent pattern. As evidenced by the
jagged nature of the plot, moving from low categories to high categories yields fluctuations in the
quantifications in both directions. Thus, describing the effects of this variable requires focusing
on the individual categories. Imposing ordinal or linear restrictions on the quantifications for
this variable might significantly reduce the fit.

118
Chapter 9
Figure 9-33
Transformation plot of Pressure gradient (nominal)
This figure displays the transformation plot of Pressure gradient. The initial discretized categories
(1 through 6) receive small quantifications and thus have minimal contributions to the predicted
response. The next three categories receive somewhat higher, positive values, resulting in a
moderate increase in predicted ozone.
The quantifications decrease up to category 16, where Pressure gradient has its greatest
decreasing effect on predicted ozone. Although the line increases after this category, using an
ordinal scaling level for Pressure gradient may not significantly reduce the fit, while simplifying
the interpretations of the effects. However, the importance measure of 0.04 and the regression
coefficient for Pressure gradient indicates that this variable is not very useful in the regression.

119
Figure 9-34
Transformation plot of Visibility (nominal)
The transformation plot of Visibility, like that for Inversion base height, shows no apparent
pattern. Imposing ordinal or linear restrictions on the quantifications for this variable might
significantly reduce the fit.

120
Chapter 9
Figure 9-35
Transformation plot of Temperature (nominal)
The transformation plot of Temperature displays an alternative pattern. As the categories increase,
the quantifications tend to increase. As a result, as Temperature increases, predicted ozone tends
to increase. This pattern suggests scaling Temperature at the ordinal level.

121
Figure 9-36
Transformation plot of Day of the year (nominal)
This figure shows the transformation plot of Day of the year. The quantifications tend to
increase up to the midpoint of the graph, at which point they tend to decrease, yielding an
inverted U-shape. Considering the sign of the regression coefficient for Day of the year, the
initial categories receive quantifications that have a decreasing effect on predicted ozone. For
the middle categories, the effect of the quantifications on predicted ozone increases, reaching a
maximum around the midpoint of the graph.
Beyond that point, the quantifications tend to decrease the predicted ozone. Although the line is
quite jagged, the general shape is still identifiable. Thus, the transformation plots suggest scaling
Temperature at the ordinal level while keeping all other predictors nominally scaled.
To recompute the regression, scaling Temperature at the ordinal level, recall the Categorical
Figure 9-37

122
Chapter 9
E Select Temperature and click Define Scale.
E Select Ordinal as the optimal scaling level.
E Click Continue.
E Click Save in the Categorical Regression dialog box.
Figure 9-38
Save dialog box
E Select Save transformed variables to the active dataset in the Transformed Variables group.
E Click Continue.
Figure 9-39
Model summary for regression with Temperature (ordinal)
This model results in an R2 of 0.872, so the variance accounted for decreases negligibly when
the quantifications for Temperature are restricted to be ordered.

123
Figure 9-40
Regression coefficients with Temperature (ordinal)
This table displays the coefficients for the model in which Temperature is scaled as ordinal.
Comparing the coefficients to those for the model in which Temperature is scaled as nominal, no
large changes occur.
Figure 9-41
Correlations, importance, and tolerance
Moreover, the importance measures suggest that Temperature is still much more important to the
regression than the other variables. Now, however, as a result of the ordinal scaling level of
Temperature and the positive regression coefficient, you can assert that as Temperature increases,
predicted ozone increases.

124
Chapter 9
Figure 9-42
Transformation plot of Temperature (ordinal)
The transformation plot illustrates the ordinal restriction on the quantifications for Temperature.
The jagged line from the nominal transformation is replaced here by a smooth ascending line.
Moreover, no long plateaus are present, indicating that collapsing categories is not needed.
Optimality of the Quantifications
The transformed variables from a categorical regression can be used in a standard linear
regression, yielding identical results. However, the quantifications are optimal only for the model
that produced them. Using a subset of the predictors in linear regression does not correspond to an
optimal scaling regression on the same subset.
For example, the categorical regression that you have computed has an R2 of 0.875. You have
saved the transformed variables, so in order to fit a linear regression using only Temperature,
Pressure gradient, and Inversion base height as predictors, from the menus choose:
Analyze > Regression > Linear...

125
Figure 9-43
Linear Regression dialog box
E Select Daily ozone level Quantification as the dependent variable.
E Select Inversion base height Quantification, Pressure gradient (mm Hg) Quantification, and
Temperature (degrees F) Quantification as independent variables.
E Click OK.
Figure 9-44
Model summary for regression with subset of optimally scaled predictors
Using the quantifications for the response, Temperature, Pressure gradient, and Inversion base
height in a standard linear regression results in a fit of 0.732. To compare this to the fit of a
categorical regression using just those three predictors, recall the Categorical Regression dialog
box.

126
Chapter 9
Figure 9-45
E Deselect Visibility (miles) and Day of the year as independent variables.
E Click OK.
Figure 9-46
Model summary for categorical regression on three predictors
The categorical regression analysis has a fit of 0.796, which is better than the fit of 0.732.
This demonstrates the property of the scalings that the quantifications obtained in the original
regression are only optimal when all five variables are included in the model.
Effects of Transformations
Transforming the variables makes a nonlinear relationship between the original response and the
original set of predictors linear for the transformed variables. However, when there are multiple
predictors, pairwise relationships are confounded by the other variables in the model.
To focus your analysis on the relationship between Daily ozone level and Day of the year, begin
by looking at a scatterplot. From the menus choose:

127
Figure 9-47
Chart Builder dialog box
E Select Daily ozone level as the y-axis variable and Day of the year as the x-axis variable.
E Click OK.

128
Chapter 9
Figure 9-48
Scatterplot of Daily ozone level and Day of the year
This figure illustrates the relationship between Daily ozone level and Day of the year. As Day of
the year increases to approximately 200, Daily ozone level increases. However, for Day of the year
values greater than 200, Daily ozone level decreases. This inverted U pattern suggests a quadratic
relationship between the two variables. A linear regression cannot capture this relationship.
E To see a best-fit line overlaid on the points in the scatterplot, activate the graph by double-clicking
on it.
E Select a point in the Chart Editor.
E Click the Add Fit Line at Total tool, and close the Chart Editor.

129
Figure 9-49
Scatterplot showing best-fit line
A linear regression of Daily ozone level on Day of the year yields an R2 of 0.004. This fit suggests
that Day of the year has no predictive value for Daily ozone level. This is not surprising, given
the pattern in the figure. By using optimal scaling, however, you can linearize the quadratic
relationship and use the transformed Day of the year to predict the response.

130
Chapter 9
Figure 9-50
To obtain a categorical regression of Daily ozone level on Day of the year, recall the Categorical
E Deselect Inversion base height through Temperature (degrees F) as independent variables.
E Select Day of the year as an independent variable.
E Click Define Scale.
Figure 9-51
E Click Continue.
E Click Discretize in the Categorical Regression dialog box.

131
Figure 9-52
E Select doy.
E Select Equal intervals.
E Type 10 as the interval length.
E Click Change.
E Click Continue.

132
Chapter 9
Figure 9-53
Plots dialog box
E Select doy for transformation plots.
E Click Continue.
Figure 9-54
Model summary for categorical regression of Daily ozone level on Day of the year
The optimal scaling regression treats Daily ozone level as numerical and Day of the year as
nominal. This results in an R2 of 0.549. Although only 55% of the variation in Daily ozone level
is accounted for by the categorical regression, this is a substantial improvement over the original
regression. Transforming Day of the year allows for the prediction of Daily ozone level.

133
Figure 9-55
Transformation plot of Day of the year (nominal)
This figure displays the transformation plot of Day of the year. The extremes of Day of the year
both receive negative quantifications, whereas the central values have positive quantifications.
By applying this transformation, the low and high Day of the year values have similar effects on
predicted Daily ozone level.

134
Chapter 9
Figure 9-56
Chart Builder
To see a scatterplot of the transformed variables, recall the Chart Builder, and click Reset to
clear your previous selections.
E Select Daily ozone level Quantification [TRA1_3] as the y-axis variable and Day of the year
Quantification [TRA2_3] as the x-axis variable.
E Click OK.

135
Figure 9-57
Scatterplot of the transformed variables
This figure depicts the relationship between the transformed variables. An increasing trend
replaces the inverted U. The regression line has a positive slope, indicating that as transformed
Day of the year increases, predicted Daily ozone level increases. Using optimal scaling linearizes
the relationship and allows interpretations that would otherwise go unnoticed.
See the following texts for more information on categorical regression:
Buja, A. 1990. Remarks on functional canonical variates, alternating least squares methods
and ACE. Annals of Statistics, 18, 1032–1069.
Hastie, T., R. Tibshirani, and A. Buja. 1994. Flexible discriminant analysis. Journal of the
American Statistical Association, 89, 1255–1270.
Hayashi, C. 1952. On the prediction of phenomena from qualitative data and the quantification
of qualitative data from the mathematico-statistical point of view. Annals of the Institute of
Statitical Mathematics, 2, 93–96.
Kruskal, J. B. 1965. Analysis of factorial experiments by estimating monotone transformations of
the data. Journal of the Royal Statistical Society Series B, 27, 251–263.
Meulman, J. J. 2003. Prediction and classification in nonlinear data analysis: Something old,
something new, something borrowed, something blue. Psychometrika, 4, 493–517.

136
Chapter 9
Ramsay, J. O. 1989. Monotone regression splines in action. Statistical Science, 4, 425–441.
Van der Kooij, A. J., and J. J. Meulman. 1997. MURALS: Multiple regression and optimal
scaling using alternating least squares. In: Softstat ’97, F. Faulbaum, and W. Bandilla, eds.
Stuttgart: Gustav Fisher, 99–106.
Winsberg, S., and J. O. Ramsay. 1980. Monotonic transformations to additivity using splines.
Biometrika, 67, 669–674.
Winsberg, S., and J. O. Ramsay. 1983. Monotone spline transformations for dimension reduction.
Psychometrika, 48, 575–595.
Young, F. W., J. De Leeuw, and Y. Takane. 1976. Regression with qualitative and quantitative
variables: An alternating least squares method with optimal scaling features. Psychometrika, 41,
505–528.

Chapter
10
Categorical Principal Components
Analysis
Categorical principal components analysis can be thought of as a method of dimension reduction.
A set of variables is analyzed to reveal major dimensions of variation. The original data set can
then be replaced by a new, smaller data set with minimal loss of information. The method reveals
relationships among variables, among cases, and among variables and cases.
The criterion used by categorical principal components analysis for quantifying the observed
data is that the object scores (component scores) should have large correlations with each of the
quantified variables. A solution is good to the extent that this criterion is satisfied.
Two examples of categorical principal components analysis will be presented. The first
employs a rather small data set useful for illustrating the basic concepts and interpretations
associated with the procedure. The second example examines a practical application.
Example: Examining Interrelations of Social Systems
This example examines Guttman’s (Guttman, 1968) adaptation of a table by Bell (Bell, 1961).
The data are also discussed by Lingoes (Lingoes, 1968).
Bell presented a table to illustrate possible social groups. Guttman used a portion of this
table, in which five variables describing such things as social interaction, feelings of belonging
to a group, physical proximity of members, and formality of the relationship were crossed with
seven theoretical social groups, including crowds (for example, people at a football game),
audiences (for example, people at a theater or classroom lecture), public (for example, newspaper
or television audiences), mobs (like a crowd but with much more intense interaction), primary
groups (intimate), secondary groups (voluntary), and the modern community (loose confederation
resulting from close physical proximity and a need for specialized services).
The following table shows the variables in the dataset resulting from the classification into
seven social groups used in the Guttman-Bell data, with their variable labels and the value labels
(categories) associated with the levels of each variable. This dataset can be found in guttman.sav.
For more information, see the topic Sample Files in Appendix A in IBM SPSS Categories 20.
In addition to selecting variables to be included in the computation of the categorical principal
components analysis, you can select variables that are used to label objects in plots. In this
example, the first five variables in the data are included in the analysis, while cluster is used
exclusively as a labeling variable. When you specify a categorical principal components analysis,
you must specify the optimal scaling level for each analysis variable. In this example, an ordinal
level is specified for all analysis variables.
Table 10-1
Variables in the Guttman-Bell dataset
intnsity Intensity of interaction Slight, low, moderate, high

138
Chapter 10
frquency Frequency of interaction Slight, nonrecurring, infrequent, frequent
blonging Feeling of belonging None, slight, variable, high
proxmity Physical proximity Distant, close
formlity Formality of relationship No relationship, formal, informal
cluster Crowds, audiences, public, mobs,
primary groups, secondary groups,
modern community
E To produce categorical principal components output for this dataset, from the menus choose:
Figure 10-1
E Select Some variable(s) not multiple nominal in the Optimal Scaling Level group.
E Click Define.

139
Figure 10-2
E Select Intensity of interaction through Formality of relationship as analysis variables.
E Click Define Scale and Weight.
Figure 10-3
E Select Ordinal in the Optimal Scaling Level group.
E Click Continue.
E Select cluster as a labeling variable in the Categorical Principal Components dialog box.

140
Chapter 10
E Click Output.
Figure 10-4
Output dialog box
E Select Object scores and deselect Correlations of transformed variables in the Tables group.
E Choose to produce category quantifications for intnsity (Intensity of interaction) through formlity
(Formality of relationship).
E Choose to label object scores by cluster.
E Click Continue.
E Click Object in the Plots group of the Categorical Principal Components dialog box.

141
Figure 10-5
E Select Objects and variables (biplot) in the Plots group.
E Choose to label objects by Variable in the Label Objects group, and then select cluster as the
variable to label objects by.
E Click Continue.
E Click Category in the Plots group of the Categorical Principal Components dialog box.

142
Chapter 10
Figure 10-6
E Choose to produce joint category plots for intnsity (Intensity of interaction) through formlity
(Formality of relationship).
E Click Continue.
E Click OK in the Categorical Principal Components dialog box.
Number of Dimensions
These figures show some of the initial output for the categorical principal components analysis.
After the iteration history of the algorithm, the model summary, including the eigenvalues of
each dimension, is displayed. These eigenvalues are equivalent to those of classical principal
components analysis. They are measures of how much variance is accounted for by each
dimension.

143
Figure 10-7
Iteration history
Figure 10-8
Model summary
The eigenvalues can be used as an indication of how many dimensions are needed. In this
example, the default number of dimensions, 2, was used. Is this the right number? As a general
rule, when all variables are either single nominal, ordinal, or numerical, the eigenvalue for a
dimension should be larger than 1. Since the two-dimensional solution accounts for 94.52% of the
variance, a third dimension probably would not add much more information.
For multiple nominal variables, there is no easy rule of thumb to determine the appropriate
number of dimensions. If the number of variables is replaced by the total number of categories
minus the number of variables, the above rule still holds. But this rule alone would probably allow
more dimensions than are needed. When choosing the number of dimensions, the most useful
guideline is to keep the number small enough so that meaningful interpretations are possible. The
model summary table also shows Cronbach’s alpha (a measure of reliability), which is maximized
by the procedure.
Quantifications
For each variable, the quantifications, the vector coordinates, and the centroid coordinates for
each dimension are presented. The quantifications are the values assigned to each category.
The centroid coordinates are the average of the object scores of objects in the same category.
The vector coordinates are the coordinates of the categories when they are required to be on a
line, representing the variable in the object space. This is required for variables with the ordinal
and numerical scaling level.

144
Chapter 10
Figure 10-9
Quantifications for Intensity of interaction
Glancing at the quantifications in the joint plot of the category points, you can see that some of the
categories of some variables were not clearly separated by the categorical principal components
analysis as cleanly as would have been expected if the level had been truly ordinal. Variables
Intensity of interaction and Frequency of interaction, for example, have equal or almost equal
quantifications for their two middle categories. This kind of result might suggest trying alternative
categorical principal components analyses, perhaps with some categories collapsed, or perhaps
with a different level of analysis, such as (multiple) nominal.
Figure 10-10
Joint plot category points
The joint plot of category points resembles the plot for the component loadings, but it also shows
where the endpoints are located that correspond to the lowest quantifications (for example, slight
for Intensity of interaction and none for Feeling of belonging). The two variables measuring
interaction, Intensity of interaction and Frequency of interaction, appear very close together and
account for much of the variance in dimension 1. Formality of Relationship also appears close to
Physical Proximity.
By focusing on the category points, you can see the relationships even more clearly. Not only
are Intensity of interaction and Frequency of interaction close, but the directions of their scales are
similar; that is, slight intensity is close to slight frequency, and frequent interaction is near high

145
intensity of interaction. You also see that close physical proximity seems to go hand-in-hand with
an informal type of relationship, and physical distance is related to no relationship.
Object Scores
You can also request a listing and plot of object scores. The plot of the object scores can be useful
for detecting outliers, detecting typical groups of objects, or revealing some special patterns.
The object scores table shows the listing of object scores labeled by social group for the
Guttman-Bell data. By examining the values for the object points, you can identify specific
objects in the plot.
Figure 10-11
Object scores
The first dimension appears to separate CROWDS and PUBLIC, which have relatively large
negative scores, from MOBS and PRIMARY GROUPS, which have relatively large positive
scores. The second dimension has three clumps: PUBLIC and SECONDARY GROUPS with large
negative values, CROWDS with large positive values, and the other social groups in between.
This is easier to see by inspecting the plot of the object scores.
Figure 10-12
Object scores plot

146
Chapter 10
In the plot, you see PUBLIC and SECONDARY GROUPS at the bottom, CROWDS at the top, and
the other social groups in the middle. Examining patterns among individual objects depends on the
additional information available for the units of analysis. In this case, you know the classification
of the objects. In other cases, you can use supplementary variables to label the objects. You
can also see that the categorical principal components analysis does not separate MOBS from
PRIMARY GROUPS. Although most people usually don’t think of their families as mobs, on the
variables used, these two groups received the same score on four of the five variables! Obviously,
you might want to explore possible shortcomings of the variables and categories used. For
example, high intensity of interaction and informal relationships probably mean different things to
these two groups. Alternatively, you might consider a higher dimensional solution.
Component Loadings
This figure shows the plot of component loadings. The vectors (lines) are relatively long,
indicating again that the first two dimensions account for most of the variance of all of the
quantified variables. On the first dimension, all variables have high (positive) component loadings.
The second dimension is correlated mainly with the quantified variables Feeling of belonging and
Physical Proximity, in opposite directions. This means that objects with a large negative score in
dimension 2 will have a high score in feeling of belonging and a low score in physical proximity.
The second dimension, therefore, reveals a contrast between these two variables while having
little relation with the quantified variables Intensity of interaction and Frequency of interaction.
Figure 10-13
Component loadings
To examine the relation between the objects and the variables, look at the biplot of objects and
component loadings. The vector of a variable points into the direction of the highest category
of the variable. For example, for Physical Proximity and Feeling of belonging the highest
categories are close and high, respectively. Therefore, CROWDS are characterized by close

147
physical proximity and no feeling of belonging, and SECONDARY GROUPS, by distant physical
proximity and a high feeling of belonging.
Figure 10-14
Biplot
Additional Dimensions
Increasing the number of dimensions will increase the amount of variation accounted for and
may reveal differences concealed in lower dimensional solutions. As noted previously, in two
dimensions MOBS and PRIMARY GROUPS cannot be separated. However, increasing the
dimensionality may allow the two groups to be differentiated.
E To obtain a three-dimensional solution, recall the Categorical Principal Components dialog box.
E Type 3 as the number of dimensions in the solution.

148
Chapter 10
Model Summary
Figure 10-15
Model summary
A three-dimensional solution has eigenvalues of 3.424, 0.844, and 0.732, accounting for nearly all
of the variance.
Object Scores
The object scores for the three-dimensional solution are plotted in a scatterplot matrix. In a
scatterplot matrix, every dimension is plotted against every other dimension in a series of
two-dimensional scatterplots. Note that the first two eigenvalues in three dimensions are not equal
to the eigenvalues in the two-dimensional solution; in other words, the solutions are not nested.
Because the eigenvalues in dimensions 2 and 3 are now smaller than 1 (giving a Cronbach’s
alpha that is negative), you should prefer the two-dimensional solution. The three-dimensional
solution is included for purposes of illustration.
Figure 10-16
Three-dimensional object scores scatterplot matrix
The top row of plots reveals that the first dimension separates PRIMARY GROUPS and MOBS
from the other groups. Notice that the order of the objects along the vertical axis does not change
in any of the plots in the top row; each of these plots employs dimension 1 as the y axis.

149
The middle row of plots allows for interpretation of dimension 2. The second dimension has
changed slightly from the two-dimensional solution. Previously, the second dimension had three
distinct clumps, but now the objects are more spread out along the axis.
The third dimension helps to separate MOBS from PRIMARY GROUPS, which did not occur in
the two-dimensional solution.
Look more closely at the dimension 2 versus dimension 3 and dimension 1 versus dimension
2 plots. On the plane defined by dimensions 2 and 3, the objects form a rough rectangle,
with CROWDS, MODERN COMMUNITY, SECONDARY GROUPS, and PUBLIC at the
vertices. On this plane, MOBS and PRIMARY GROUPS appear to be convex combinations
of PUBLIC-CROWDS and SECONDARY GROUPS-MODERN COMMUNITY, respectively.
However, as previously mentioned, they are separated from the other groups along dimension
1. AUDIENCES is not separated from the other groups along dimension 1 and appears to be a
combination of CROWDS and MODERN COMMUNITY.
Component Loadings
Figure 10-17
Three-dimensional component loadings
Knowing how the objects are separated does not reveal which variables correspond to which
dimensions. This is accomplished using the component loadings. The first dimension corresponds
primarily to Feeling of belonging, Intensity of interaction, and Formality of Relationship; the
second dimension separates Frequency of interaction and Physical Proximity; and the third
dimension separates these from the others.
Example: Symptomatology of Eating Disorders
Eating disorders are debilitating illnesses associated with disturbances in eating behavior,
severe body image distortion, and an obsession with weight that affects the mind and body
simultaneously. Millions of people are affected each year, with adolescents particularly at risk.
Treatments are available and most are helpful when the condition is identified early.
A health professional can attempt to diagnose an eating disorder through a psychological and
medical evaluation. However, it can be difficult to assign a patient to one of several different
classes of eating disorders because there is no standardized symptomatology of anorectic/bulimic
behavior. Are there symptoms that clearly differentiate patients into the four groups? Which
symptoms do they have in common?
In order to try to answer these questions, researchers (Van der Ham, Meulman, Van Strien, and
Van Engeland, 1997) made a study of 55 adolescents with known eating disorders, as shown in

150
Chapter 10
Table 10-2
Patient diagnoses
Diagnosis Number of Patients
Anorexia nervosa 25
Anorexia with bulimia nervosa 9
Bulimia nervosa after anorexia 14
Atypical eating disorder 7
Total 55
Each patient was seen four times over four years, for a total of 220 observations. At each
observation, the patients were scored for each of the 16 symptoms outlined in the following table.
Symptom scores are missing for patient 71 at time 2, patient 76 at time 2, and patient 47 at time 3,
leaving 217 valid observations. The data can be found in anorectic.sav.For more information, see
the topic Sample Files in Appendix A in IBM SPSS Categories 20.
Table 10-3
Modified Morgan-Russell subscales measuring well-being
Variable
name
Variable label Lower end (score1) Upper end (score 3 or 4)
weight Body weight Outside normal range Normal
mens Menstruation Amenorrhea Regular periods
fast Restriction of food intake
(fasting)
Less than 1200 calories Normal/regular meals
binge Binge eating Greater than once a week No bingeing
vomit Vomiting Greater than once a week No vomiting
purge Purging Greater than once a week No purging
hyper Hyperactivity Not able to be at rest No hyperactivity
fami Family relations Poor Good
eman Emancipation from
family
Very dependent Adequate
frie Friends No good friends Two or more good friends
school School/employment
record
Stopped school/work Moderate to good record
satt Sexual attitude Inadequate Adequate
sbeh Sexual behavior Inadequate Can enjoy sex
mood Mental state (mood) Very depressed Normal
preo Preoccupation with food
and weight
Complete No preoccupation
body Body perception Disturbed Normal
Principal components analysis is ideal for this situation, since the purpose of the study is to
ascertain the relationships between symptoms and the different classes of eating disorders.
Moreover, categorical principal components analysis is likely to be more useful than classical
principal components analysis because the symptoms are scored on an ordinal scale.

151
In order to properly examine the structure of the course of illness for each diagnosis, you will
want to make the results of the projected centroids table available as data for scatterplots. You can
accomplish this using the Output Management System.
E To begin an OMS request, from the menus choose:
Utilities > OMS Control Panel...
Figure 10-18
Output Management System Control Panel
E Select Tables as the output type.
E Select CATPCA as the command.
E Select Projected Centroids as the table type.
E Select File in the Output Destinations group and type projected_centroids.sav as the filename.
E Click Options.

152
Chapter 10
Figure 10-19
Options dialog box
E Select IBM® SPSS® Statistics Data File as the output format.
E Type TableNumber_1 as the table number variable.
E Click Continue.

153
Figure 10-20
E Click Add.
E Click OK, and then click OK to confirm the OMS session.
The Output Management System is now set to write the results of the projected centroids table
to the file projected_centroids.sav.
E To produce categorical principal components output for this dataset, from the menus choose:

154
Chapter 10
Figure 10-21
E Click Define.
Figure 10-22
E Select Body weight through Body perception as analysis variables.
E Click Define Scale and Weight.

155
Figure 10-23
E Select Ordinal as the optimal scaling level.
E Click Continue.
E Select Time/diagnosis interaction as a supplementary variable and click Define Scale in the
Categorical Principal Components dialog box.
Figure 10-24
E Select Multiple nominal as the optimal scaling level.
E Click Continue.

156
Chapter 10
Figure 10-25
E Select Time of interview through Patient number as labeling variables.
E Click Options.

157
Figure 10-26
Options dialog box
E Choose to label plots by Variable names or values.
E Click Continue.
E Click Output in the Categorical Principal Components dialog box.

158
Chapter 10
Figure 10-27
Output dialog box
E Select Object scores in the Tables group.
E Request category quantifications for tidi.
E Choose to include categories of time, diag, and number.
E Click Continue.
E Click Save in the Categorical Principal Components dialog box.

159
Figure 10-28
Save dialog box
E In the Transformed Variables group, select Save to the active dataset.
E Click Continue.
E Click Object in the Categorical Principal Components dialog box.

160
Chapter 10
Figure 10-29
E Choose to label objects by Variable.
E Select time and diag as the variables to label objects by.
E Click Continue.
E Click Category in the Categorical Principal Components dialog box.

161
Figure 10-30
E Request category plots for tidi.
E Request transformation plots for weight through body.
E Choose to project centroids of tidi onto binge, satt, and preo.
E Click Continue.
The procedure results in scores for the subjects (with mean 0 and unit variance) and quantifications
of the categories that maximize the mean squared correlation of the subject scores and the
transformed variables. In the present analysis, the category quantifications were constrained
to reflect the ordinal information.

162
Chapter 10
Finally, to write the projected centroids table information to projected_centroids.sav, you need
to end the OMS request. Recall the OMS Control Panel.
Figure 10-31
E Click End.
E Click OK, and then click OK to confirm.
The transformation plots display the original category number on the horizontal axes; the vertical
axes give the optimal quantifications.

163
Figure 10-32
Transformation plot for menstruation
Some variables, like Menstruation, obtained nearly linear transformations, so in this analysis
you may interpret them as numerical.
Figure 10-33
Transformation plot for School/employment record
The quantifications for other variables like School/employment record did not obtain linear
transformations and should be interpreted at the ordinal scaling level. The difference between
the second and third categories is much more important than that between the first and second
categories.

164
Chapter 10
Figure 10-34
Transformation plot for Binge eating
An interesting case arises in the quantifications for Binge eating. The transformation obtained is
linear for categories 1 through 3, but the quantified values for categories 3 and 4 are equal. This
result shows that scores of 3 and 4 do not differentiate between patients and suggests that you
could use the numerical scaling level in a two-component solution by recoding 4’s as 3’s.
Model Summary
Figure 10-35
Model summary
To see how well your model fits the data, look at the model summary. About 47% of the total
variance is explained by the two-component model, 35% by the first dimension and 12% by
the second. So, almost half of the variability on the individual objects level is explained by
the two-component model.

165
Component Loadings
To begin to interpret the two dimensions of your solution, look at the component loadings. All
variables have a positive component loading in the first dimension, which means that there is a
common factor that correlates positively with all of the variables.
Figure 10-36
Component loadings plot
The second dimension separates the variables. The variables Binge eating, Vomiting, and Purging
form a bundle having large positive loadings in the second dimension. These symptoms are
typically considered to be representative of bulimic behavior.
The variables Emancipation from family, School/employment record, Sexual attitude, Body
weight, and Menstruation form another bundle, and you can include Restriction of food intake
(fasting) and Family relations in this bundle, because their vectors are close to the main cluster,
and these variables are considered to be anorectic symptoms (fasting, weight, menstruation) or are
psychosocial in nature (emancipation, school/work record, sexual attitude, family relations). The
vectors of this bundle are orthogonal (perpendicular) to the vectors of binge, vomit, and purge,
which means that this set of variables is uncorrelated with the set of bulimic variables.
The variables Friends, Mental state (mood), and Hyperactivity do not appear to fit very well
into the solution. You can see this in the plot by observing the lengths of each vector. The length
of a given variable’s vector corresponds to its fit, and these variables have the shortest vectors.
Based on a two-component solution, you would probably drop these variables from a proposed
symptomatology for eating disorders. They may, however, fit better in a higher dimensional
solution.
The variables Sexual behavior, Preoccupation with food and weight, and Body perception form
another theoretic group of symptoms, pertaining to how the patient experiences his or her body.
While correlated with the two orthogonal bundles of variables, these variables have fairly long
vectors and are strongly associated with the first dimension and therefore may provide some
useful information about the “common” factor.

166
Chapter 10
Object Scores
The following figure shows a plot of the object scores, in which the subjects are labeled with
their diagnosis category.
Figure 10-37
Object scores plot labeled by diagnosis
This plot does not help to interpret the first dimension, because patients are not separated by
diagnosis along it. However, there is some information about the second dimension. Anorexia
subjects (1) and patients with atypical eating disorder (4) form a group, located above subjects
with some form of bulimia (2 and 3). Thus, the second dimension separates bulimic patients from
others, as you have also seen in the previous section (the variables in the bulimic bundle have
large positive component loadings in the second dimension). This makes sense, given that the
component loadings of the symptoms that are traditionally associated with bulimia have large
values in the second dimension.

167
This figure shows a plot of the object scores, in which the subjects are labeled with their
time of diagnosis.
Figure 10-38
Object scores labeled by time of interview
Labeling the object scores by time reveals that the first dimension has a relation to time because
there seems to be a progression of times of diagnosis from the 1’s mostly to the left and others to
the right. Note that you can connect the time points in this plot by saving the object scores and
creating a scatterplot using the dimension 1 scores on the x axis, the dimension 2 scores on the y
axis, and setting the markers using the patient numbers.
Comparing the object scores plot labeled by time with the one labeled by diagnosis can give
you some insight into unusual objects. For example, in the plot labeled by time, there is a patient
whose diagnosis at time 4 lies to the left of all other points in the plot. This is unusual because the
general trend of the points is for the later times to lie further to the right. Interestingly, this point
that seems out of place in time also has an unusual diagnosis, in that the patient is an anorectic
whose scores place the patient in the bulimic cluster. By looking in the table of object scores, you
find that this is patient 43, diagnosed with anorexia nervosa, whose object scores are shown in
Table 10-4
Object scores for patient 43
Time Dimension 1 Dimension 2
1 –2.031 1.250
2 –2.067 0.131
3 –1.575 –1.467
4 –2.405 –1.807
The patient’s scores at time 1 are prototypical for anorectics, with the large negative score
in dimension 1 corresponding to poor body image and the positive score in dimension 2
corresponding to anorectic symptoms or poor psychosocial behavior. However, unlike the
majority of patients, there is little or no progress in dimension 1. In dimension 2, there is

168
Chapter 10
apparently some progress toward “normal” (around 0, between anorectic and bulimic behavior),
but then the patient shifts to exhibit bulimic symptoms.
Examining the Structure of the Course of Illness
To find out more about how the two dimensions were related to the four diagnosis categories
and the four time points, a supplementary variable Time/diagnosis interaction was created by a
cross-classification of the four categories of Patient diagnosis and the four categories of Time
of interview. Thus, Time/diagnosis interaction has 16 categories, where the first category
indicates the anorexia nervosa patients at their first visit. The fifth category indicates the anorexia
nervosa patients at time point 2, and so on, with the sixteenth category indicating the atypical
eating disorder patients at time point 4. The use of the supplementary variable Time/diagnosis
interaction allows for the study of the courses of illness for the different groups over time. The
variable was given a multiple nominal scaling level, and the category points are displayed in
the following figure.
Figure 10-39
Category points for time/diagnosis interaction
Some of the structure is apparent from this plot: the diagnosis categories at time point 1 clearly
separate anorexia nervosa and atypical eating disorder from anorexia nervosa with bulimia
nervosa and bulimia nervosa after anorexia nervosa in the second dimension. After that, it’s a
little more difficult to see the patterns.
However, you can make the patterns more easily visible by creating a scatterplot based on the
quantifications. To do this, from the menus choose:

169
Figure 10-40
Scatter/Dot gallery
E Select the Scatter/Dot gallery and choose Grouped Scatter.

170
Chapter 10
Figure 10-41
Chart Builder
E Select Time/diagnosis interaction Quantification dimension 2 as the y-axis variable and
Time/diagnosis interaction Quantification dimension 1 as the x-axis variable.
E Choose to set color by Patient Diagnosis.
E Click OK.

171
Figure 10-42
Structures of the courses of illness
E Then, to connect the points, double-click on the graph, and click the Add interpolation line tool
in the Chart Editor.
E Close the Chart Editor.

172
Chapter 10
Figure 10-43
Structures of the courses of illness
By connecting the category points for each diagnostic category across time, the patterns
immediately suggest that the first dimension is related to time and the second, to diagnosis, as you
previously determined from the object scores plots.
However, this plot further shows that, over time, the illnesses tend to become more alike.
Moreover, for all groups, the progress is greatest between time points 1 and 2; the anorectic
patients show some more progress from 2 to 3, but the other groups show little progress.
Differential Development for Selected Variables
One variable from each bundle of symptoms identified by the component loadings was selected as
“representative” of the bundle. Binge eating was selected from the bulimic bundle; sexual attitude,
from the anorectic/psychosocial bundle; and body preoccupation, from the third bundle.

173
In order to examine the possible differential courses of illness, the projections of Time/diagnosis
interaction on Binge eating, Sexual attitude, and Preoccupation with food and weight were
computed and plotted in the following figure.
Figure 10-44
Projected centroids of Time/diagnosis interaction on Binge eating, Sexual attitude, and Preoccupation
with food and weight
This plot shows that at the first time point, the symptom binge eating separates bulimic patients
(2 and 3) from others (1 and 4); sexual attitude separates anorectic and atypical patients (1 and
4) from others (2 and 3); and body preoccupation does not really separate the patients. In many
applications, this plot would be sufficient to describe the relationship between the symptoms and
diagnosis, but because of the complication of multiple time points, the picture becomes muddled.

174
Chapter 10
In order to view these projections over time, you need to be able to plot the contents of the
projected centroids table. This is made possible by the OMS request that saved this information
to projected_centroids.sav.
Figure 10-45
Projected_centroids.sav
The variables Bingeeating, Sexualattitude, and Preoccupationwithfoodandweight contain the
values of the centroids projected on each of the symptoms of interest. The case number (1 through
16) corresponds to the time/diagnosis interaction. You will need to compute new variables that
separate out the Time and Diagnosis values.
Transform > Compute Variable...

175
Figure 10-46
Compute Variable dialog box
E Type time as the target variable.
E Type trunc(($casenum-1)/4) + 1 as the numeric expression.
E Click OK.

176
Chapter 10
Figure 10-47
Compute Variable dialog box
E Recall the Compute Variable dialog box.
E Type diagnosis as the target variable.
E Type mod($casenum-1, 4) + 1 as the numeric expression.
E Click OK.

177
Figure 10-48
Projected_centroids.sav
In the Variable View, change the measure for diagnosis from Scale to Nominal.

178
Chapter 10
Figure 10-49
Chart Builder
E Finally, to view the projected centroids of time of diagnosis on binging over time, recall the Chart
Builder and click Reset to clear your previous selections.
E Select the Scatter/Dot gallery and choose Grouped Scatter.
E Select Centroids Projected on Binge eating as the y-axis variable and time as the x-axis variable.
E Choose to set colors by diagnosis.
E Click OK.

179
Figure 10-50
Projected centroids of Time of diagnosis on Binge eating over time
With respect to binge eating, it is clear that the anorectic groups have different starting values
from the bulimic groups. This difference shrinks over time, as the anorectic groups hardly change,
while the bulimic groups show progress.

180
Chapter 10
Figure 10-51
Chart Builder
E Recall the Chart Builder.
E Deselect Centroids Projected on Binge eating as the y-axis variable and select Centroids Projected
on Sexual attitude as the y-axis variable.
E Click OK.

181
Figure 10-52
Projected centroids of Time of diagnosis on Sexual attitude over time
With respect to sexual attitude, the four trajectories are more or less parallel over time, and
all groups show progress. The bulimic groups, however, have higher (better) scores than the
anorectic group.

182
Chapter 10
Figure 10-53
Chart Builder
E Recall the Chart Builder.
E Deselect Centroids Projected on Sexual attitude as the y-axis variable and select Centroids
Projected on Preoccupation with food and weight as the y-axis variable.
E Click OK.

183
Figure 10-54
Projected centroids of Time of diagnosis on Body preoccupation over time
Body preoccupation is a variable that represents the core symptoms, which are shared by the four
different groups. Apart from the atypical eating disorder patients, the anorectic group and the two
bulimic groups have very similar levels both at the beginning and at the end.
See the following texts for more information on categorical principal components analysis:
De Haas, M., J. A. Algera, H. F. J. M. Van Tuijl, and J. J. Meulman. 2000. Macro and micro goal
setting: In search of coherence. Applied Psychology, 49, 579–595.

184
Chapter 10
De Leeuw, J. 1982. Nonlinear principal components analysis. In: COMPSTAT Proceedings in
Computational Statistics, Vienna: Physica Verlag, 77–89.
Eckart, C., and G. Young. 1936. The approximation of one matrix by another one of lower
rank. Psychometrika, 1, 211–218.
Gabriel, K. R. 1971. The biplot graphic display of matrices with application to principal
components analysis. Biometrika, 58, 453–467.
Gifi, A. 1985. PRINCALS. Research Report UG-85-02. Leiden: Department of Data Theory,
University of Leiden.
Gower, J. C., and J. J. Meulman. 1993. The treatment of categorical information in physical
anthropology. International Journal of Anthropology, 8, 43–51.
Heiser, W. J., and J. J. Meulman. 1994. Homogeneity analysis: Exploring the distribution of
variables and their nonlinear relationships. In: Correspondence Analysis in the Social Sciences:
Recent Developments and Applications, M. Greenacre, and J. Blasius, eds. New York: Academic
Press, 179–209.
Kruskal, J. B. 1978. Factor analysis and principal components analysis: Bilinear methods. In:
International Encyclopedia of Statistics, W. H. Kruskal, and J. M. Tanur, eds. New York: The
Free Press, 307–330.
Kruskal, J. B., and R. N. Shepard. 1974. A nonmetric variety of linear factor analysis.
Meulman, J. J. 1993. Principal coordinates analysis with optimal transformations of the variables:
Minimizing the sum of squares of the smallest eigenvalues. British Journal of Mathematical and
Statistical Psychology, 46, 287–300.
Meulman, J. J., and P. Verboon. 1993. Points of view analysis revisited: Fitting multidimensional
structures to optimal distance components with cluster restrictions on the variables.
Meulman, J. J., A. J. Van der Kooij, and A. Babinec. 2000. New features of categorical principal
components analysis for complicated data sets, including data mining. In: Classification,
Automation and New Media, W. Gaul, and G. Ritter, eds. Berlin: Springer-Verlag, 207–217.
Meulman, J. J., A. J. Van der Kooij, and W. J. Heiser. 2004. Principal components analysis
with nonlinear optimal scaling transformations for ordinal and nominal data. In: Handbook of
Quantitative Methodology for the Social Sciences, D. Kaplan, ed. Thousand Oaks, Calif.: Sage
Publications, Inc., 49–70.
Theunissen, N. C. M., J. J. Meulman, A. L. Den Ouden, H. M. Koopman, G. H. Verrips, S.
P. Verloove-Vanhorick, and J. M. Wit. 2003. Changes can be studied when the measurement
instrument is different at different time points. Health Services and Outcomes Research
Methodology, 4, 109–126.
Tucker, L. R. 1960. Intra-individual and inter-individual multidimensionality. In: Psychological
Scaling: Theory & Applications, H. Gulliksen, and S. Messick, eds. NewYork: John Wiley
and Sons, 155–167.
Vlek, C., and P. J. Stallen. 1981. Judging risks and benefits in the small and in the large.
Organizational Behavior and Human Performance, 28, 235–271.

185
Wagenaar, W. A. 1988. Paradoxes of gambling behaviour. London: Lawrence Erlbaum
Associates, Inc.
Young, F. W., Y. Takane, and J. De Leeuw. 1978. The principal components of mixed
measurement level multivariate data: An alternating least squares method with optimal scaling
features. Psychometrika, 43, 279–281.
Zeijl, E., Y. te Poel, M. du Bois-Reymond, J. Ravesloot, and J. J. Meulman. 2000. The role of
parents and peers in the leisure activities of young adolescents. Journal of Leisure Research, 32,
281–302.

Chapter
11
Nonlinear Canonical Correlation
Analysis
The purpose of nonlinear canonical correlation analysis is to determine how similar two or
more sets of variables are to one another. As in linear canonical correlation analysis, the aim
is to account for as much of the variance in the relationships among the sets as possible in
a low-dimensional space. Unlike linear canonical correlation analysis, however, nonlinear
canonical correlation analysis does not assume an interval level of measurement or assume
that the relationships are linear. Another important difference is that nonlinear canonical
correlation analysis establishes the similarity between the sets by simultaneously comparing linear
combinations of the variables in each set to an unknown set—the object scores.
Example: An Analysis of Survey Results
The example in this chapter is from a survey (Verdegaal, 1985). The responses of 15 subjects
to 8 variables were recorded. The variables, variable labels, and value labels (categories) in
the dataset are shown in the following table.
Table 11-1
Survey data
age Age in years 20–25, 26–30, 31–35, 36–40,
41–45, 46–50, 51–55, 56–60,
61–65, 66–70
marital Marital status Single, Married, Other
pet Pets owned No, Cat(s), Dog(s), Other than cat
or dog, Various domestic animals
news Newspaper read most
often
None, Telegraaf, Volkskrant,
NRC, Other
music Music preferred Classical, New wave, Popular,
Variety, Don’t like music
live Neighborhood
preference
Town, Village, Countryside
math Math test score 0–5, 6–10, 11–15
language Language test score 0–5, 6–10, 11–15, 16–20
This dataset can be found in verd1985.sav. For more information, see the topic Sample Files in
Appendix A in IBM SPSS Categories 20. The variables of interest are the first six variables, and
they are divided into three sets. Set 1 includes age and marital, set 2 includes pet and news, and
set 3 includes music and live. Pet is scaled as multiple nominal, and age is scaled as ordinal; all
other variables are scaled as single nominal. This analysis requests a random initial configuration.
By default, the initial configuration is numerical. However, when some of the variables are

187
treated as single nominal with no possibility of ordering, it is best to choose a random initial
configuration. This is the case with most of the variables in this study.
Examining the Data
E To obtain a nonlinear canonical correlation analysis for this dataset, from the menus choose:
Figure 11-1
E Select Multiple sets in the Number of Sets of Variables group.
E Click Define.

188
Chapter 11
Figure 11-2
Nonlinear Canonical Correlation Analysis dialog box
E Select Age in years and Marital status as variables for the first set.
E Select age and click Define Range and Scale.
Figure 11-3
E Type 10 as the maximum value for this variable.
E Click Continue.
E In the Nonlinear Canonical Correlation Analysis dialog box, select marital and click Define Range
and Scale.

189
Figure 11-4
E Select Single nominal as the measurement scale.
E Click Continue.
E In the Nonlinear Canonical Correlation Analysis dialog box, click Next to define the next variable
set.
Figure 11-5
E Select Pets owned and Newspaper read most often as variables for the second set.
E Select pet and click Define Range and Scale.

190
Chapter 11
Figure 11-6
E Select Multiple nominal as the measurement scale.
E Click Continue.
E In the Nonlinear Canonical Correlation Analysis dialog box, select news and click Define Range
and Scale.
Figure 11-7
E Click Continue.
E In the Nonlinear Canonical Correlation Analysis dialog box, click Next to define the last variable
set.

191
Figure 11-8
E Select Music preferred and Neighborhood preference as variables for the third set.
E Select music and click Define Range and Scale.
Figure 11-9
E Click Continue.
E In the Nonlinear Canonical Correlation Analysis dialog box, select live and click Define Range
and Scale.

192
Chapter 11
Figure 11-10
E Click Continue.
E In the Nonlinear Canonical Correlation Analysis dialog box, click Options.
Figure 11-11
Options dialog box
E Deselect Centroids and select Weights and component loadings in the Display group.
E Select Category centroids and Transformations in the Plot group.
E Select Use random initial configuration.
E Click Continue.
E In the Nonlinear Canonical Correlation Analysis dialog box, click OK.

193
After a list of the variables with their levels of optimal scaling, categorical canonical correlation
analysis with optimal scaling produces a table showing the frequencies of objects in categories.
This table is especially important if there are missing data, since almost-empty categories are more
likely to dominate the solution. In this example, there are no missing data.
A second preliminary check is to examine the plot of object scores for outliers. Outliers have
such different quantifications from the other objects that they will be at the boundaries of the
plot, thus dominating one or more dimensions.
If you find outliers, you can handle them in one of two ways. You can simply eliminate them
from the data and run the nonlinear canonical correlation analysis again. Alternatively, you can try
recoding the extreme responses of the outlying objects by collapsing (merging) some categories.
As shown in the plot of object scores, there were no outliers for the survey data.
Figure 11-12
Object scores
Accounting for Similarity between Sets
There are several ways to measure the association between sets in a nonlinear canonical correlation
analysis (each way being detailed in a separate table or set of tables).
Summary of Analysis
The fit and loss values tell you how well the nonlinear canonical correlation analysis solution fits
the optimally quantified data with respect to the association between the sets. The summary of
analysis table shows the fit value, loss values, and eigenvalues for the survey example.

194
Chapter 11
Figure 11-13
Summary of analysis
Loss is partitioned across dimensions and sets. For each dimension and set, loss represents
the proportion of variation in the object scores that cannot be accounted for by the weighted
combination of variables in the set. The average loss is labeled Mean. In this example, the
average loss over sets is 0.464. Notice that more loss occurs for the second dimension than for
the first dimension.
The eigenvalue for each dimension equals 1 minus the average loss for the dimension and
indicates how much of the relationship is shown by each dimension. The eigenvalues add up to
the total fit. For Verdegaal’s data, 0.801 / 1.536 = 52% of the actual fit is accounted for by the
first dimension.
The maximum fit value equals the number of dimensions and, if obtained, indicates that the
relationship is perfect. The average loss value over sets and dimensions tells you the difference
between the maximum fit and the actual fit. Fit plus the average loss equals the number of
dimensions. Perfect similarity rarely happens and usually capitalizes on trivial aspects in the data.
Another popular statistic with two sets of variables is the canonical correlation. Since the
canonical correlation is related to the eigenvalue and thus provides no additional information, it is
not included in the nonlinear canonical correlation analysis output. For two sets of variables, the
canonical correlation per dimension is obtained by the following formula:
where d is the dimension number and E is the eigenvalue.
You can generalize the canonical correlation for more than two sets with the following formula:
where d is the dimension number, K is the number of sets, and E is the eigenvalue. For our
example,
and

195
Weights and Component Loadings
Another measure of association is the multiple correlation between linear combinations from each
set and the object scores. If no variables in a set are multiple nominal, you can compute this
measure by multiplying the weight and component loading of each variable within the set, adding
these products, and taking the square root of the sum.
Figure 11-14
Weights
Figure 11-15
Component loadings
These figures give the weights and component loadings for the variables in this example. The
multiple correlation (R) is as follows for the first weighted sum of optimally scaled variables (Age
in years and Marital status) with the first dimension of object scores:
For each dimension, 1 – loss = R2. For example, from the Summary of analysis table, 1 –
0.238 = 0.762, which is 0.873 squared (plus some rounding error). Consequently, small loss
values indicate large multiple correlations between weighted sums of optimally scaled variables
and dimensions. Weights are not unique for multiple nominal variables. For multiple nominal
variables, use 1 – loss per set.

196
Chapter 11
Partitioning Fit and Loss
The loss of each set is partitioned by the nonlinear canonical correlation analysis in several ways.
The fit table presents the multiple fit, single fit, and single loss tables produced by the nonlinear
canonical correlation analysis for the survey example. Note that multiple fit minus single fit
equals single loss.
Figure 11-16
Partitioning fit and loss
Single loss indicates the loss resulting from restricting variables to one set of quantifications (that
is, single nominal, ordinal, or nominal). If single loss is large, it is better to treat the variables
as multiple nominal. In this example, however, single fit and multiple fit are almost equal,
which means that the multiple coordinates are almost on a straight line in the direction given
by the weights.
Multiple fit equals the variance of the multiple category coordinates for each variable. These
measures are analogous to the discrimination measures that are found in homogeneity analysis.
You can examine the multiple fit table to see which variables discriminate best. For example,
look at the multiple fit table for Marital status and Newspaper read most often. The fit values,
summed across the two dimensions, are 1.122 for Marital status and 0.911 for Newspaper read
most often. This information tells us that a person’s marital status provides greater discriminatory
power than the newspaper they subscribe to.
Single fit corresponds to the squared weight for each variable and equals the variance of the
single category coordinates. As a result, the weights equal the standard deviations of the single
category coordinates. By examining how the single fit is broken down across dimensions, we
see that the variable Newspaper read most often discriminates mainly on the first dimension, and
we see that the variable Marital status discriminates almost totally on the second dimension. In
other words, the categories of Newspaper read most often are further apart in the first dimension
than in the second, whereas the pattern is reversed for Marital status. In contrast, Age in years
discriminates in both the first and second dimensions; thus, the spread of the categories is equal
along both dimensions.

197
Component Loadings
The following figure shows the plot of component loadings for the survey data. When there are
no missing data, the component loadings are equivalent to the Pearson correlations between
the quantified variables and the object scores.
The distance from the origin to each variable point approximates the importance of that
variable. The canonical variables are not plotted but can be represented by horizontal and vertical
lines drawn through the origin.
Figure 11-17
Component loadings
The relationships between variables are apparent. There are two directions that do not coincide
with the horizontal and vertical axes. One direction is determined by Age in years, Newspaper read
most often, and Neighborhood preference. The other direction is defined by the variables Marital
status, Music preferred, and Pets owned. The Pets owned variable is a multiple nominal variable,
so there are two points plotted for it. Each quantification is interpreted as a single variable.
The different levels at which each variable can be scaled impose restrictions on the quantifications.
Transformation plots illustrate the relationship between the quantifications and the original
categories resulting from the selected optimal scaling level.
The transformation plot for Neighborhood preference, which was treated as nominal, displays
a U-shaped pattern, in which the middle category receives the lowest quantification, and the
extreme categories receive values that are similar to each other. This pattern indicates a quadratic
relationship between the original variable and the transformed variable. Using an alternative
optimal scaling level is not suggested for Neighborhood preference.

198
Chapter 11
Figure 11-18
Transformation plot for Neighborhood preference (nominal)
The quantifications for Newspaper read most often, in contrast, correspond to an increasing
trend across the three categories that have observed cases. The first category receives the lowest
quantification, the second category receives a higher value, and the third category receives the
highest value. Although the variable is scaled as nominal, the category order is retrieved in the
quantifications.
Figure 11-19
Transformation plot for Newspaper read most often (nominal)

199
Figure 11-20
Transformation plot for Age in years (ordinal)
The transformation plot for Age in years displays an S-shaped curve. The four youngest observed
categories all receive the same negative quantification, whereas the two oldest categories receive
similar positive values. Consequently, collapsing all younger ages into one common category
(that is, below 50) and collapsing the two oldest categories into one category may be attempted.
However, the exact equality of the quantifications for the younger groups indicates that restricting
the order of the quantifications to the order of the original categories may not be desirable.
Because the quantifications for the 26–30, 36–40, and 41–45 groups cannot be lower than the
quantification for the 20–25 group, these values are set equal to the boundary value. Allowing
these values to be smaller than the quantification for the youngest group (that is, treating age as
nominal) may improve the fit. So although age may be considered an ordinal variable, treating it
as such does not appear appropriate in this case. Moreover, treating age as numerical, and thus
maintaining the distances between the categories, would substantially reduce the fit.

200
Chapter 11
Single Category versus Multiple Category Coordinates
For every variable treated as single nominal, ordinal, or numerical, quantifications, single
category coordinates, and multiple category coordinates are determined. These statistics for
Age in years are presented.
Figure 11-21
Coordinates for Age in years
Every category for which no cases were recorded receives a quantification of 0. For Age in years,
this includes the 31–35, 46–50, and 51–55 categories. These categories are not restricted to be
ordered with the other categories and do not affect any computations.
For multiple nominal variables, each category receives a different quantification on each
dimension. For all other transformation types, a category has only one quantification, regardless
of the dimensionality of the solution. Each set of single category coordinates represents the
location of the category on a line in the object space. The coordinates for a given category equal
the quantification multiplied by the variable dimension weights. For example, in the table for Age
in years, the single category coordinates for category 56-60 (-0.142, -0.165) are the quantification
(-0.209) multiplied by the dimension weights (0.680, 0.789).
The multiple category coordinates for variables that are treated as single nominal, ordinal, or
numerical represent the coordinates of the categories in the object space before ordinal or linear
constraints are applied. These values are unconstrained minimizers of the loss. For multiple
nominal variables, these coordinates represent the quantifications of the categories.
The effects of imposing constraints on the relationship between the categories and their
quantifications are revealed by comparing the single category coordinates with the multiple
category coordinates. On the first dimension, the multiple category coordinates for Age in years
decrease to category 2 and remain relatively at the same level until category 9, at which point
a dramatic increase occurs. A similar pattern is evidenced for the second dimension. These
relationships are removed in the single category coordinates, in which the ordinal constraint is
applied. On both dimensions, the coordinates are now nondecreasing. The differing structure of
the two sets of coordinates suggests that a nominal treatment may be more appropriate.

201
Centroids and Projected Centroids
The plot of centroids labeled by variables should be interpreted in the same way as the category
quantifications plot in homogeneity analysis or the multiple category coordinates in nonlinear
principal components analysis. By itself, such a plot shows how well variables separate groups of
objects (the centroids are in the center of gravity of the objects).
Notice that the categories for Age in years are not separated very clearly. The younger age
categories are grouped together at the left of the plot. As suggested previously, ordinal may be too
strict a scaling level to impose on Age in years.
Figure 11-22
Centroids labeled by variables

202
Chapter 11
When you request centroid plots, individual centroid and projected centroid plots for each
variable that is labeled by value labels are also produced. The projected centroids are on a line
in the object space.
Figure 11-23
Centroids and projected centroids for Newspaper read most often
The actual centroids are projected onto the vectors that are defined by the component loadings.
These vectors have been added to the centroid plots to aid in distinguishing the projected centroids
from the actual centroids. The projected centroids fall into one of four quadrants formed by
extending two perpendicular reference lines through the origin. The interpretation of the direction
of single nominal, ordinal, or numerical variables is obtained from the position of the projected
centroids. For example, the variable Newspaper read most often is specified as single nominal.
The projected centroids show that Volkskrant and NRC are contrasted with Telegraaf.

203
Figure 11-24
Centroids and projected centroids for Age in years
The problem with Age in years is evident from the projected centroids. Treating Age in years as
ordinal implies that the order of the age groups must be preserved. To satisfy this restriction, all
age groups below age 45 are projected into the same point. Along the direction defined by Age in
years, Newspaper read most often, and Neighborhood preference, there is no separation of the
younger age groups. Such a finding suggests treating the variable as nominal.

204
Chapter 11
Figure 11-25
Centroids and projected centroids for Neighborhood preference
To understand the relationships among variables, find out what the specific categories (values) are
for clusters of categories in the centroid plots. The relationships among Age in years, Newspaper
read most often, and Neighborhood preference can be described by looking at the upper right and
lower left of the plots. In the upper right, the age groups are the older respondents; they read the
newspaper Telegraaf and prefer living in a village. Looking at the lower left corner of each plot,
you see that the younger to middle-aged respondents read the Volkskrant or NRC and want to live
in the country or in a town. However, separating the younger groups is very difficult.
The same types of interpretations can be made about the other direction (Music preferred,
Marital status, and Pets owned) by focusing on the upper left and the lower right of the centroid
plots. In the upper left corner, we find that single people tend to have dogs and like new wave
music. The married people and other categories for marital have cats; the former group prefers
classical music, and the latter group does not like music.
An Alternative Analysis
The results of the analysis suggest that treating Age in years as ordinal does not appear appropriate.
Although Age in years is measured at an ordinal level, its relationships with other variables are not
monotonic. To investigate the effects of changing the optimal scaling level to single nominal,
you may rerun the analysis.
To Run the Analysis
E Recall the Nonlinear Canonical Correlation Analysis dialog box and navigate to the first set.
E Select age and click Define Range and Scale.

205
E In the Define Range and Scale dialog box, select Single nominal as the scaling range.
E Click Continue.
E In the Nonlinear Canonical Correlation Analysis dialog box, click OK.
The eigenvalues for a two-dimensional solution are 0.806 and 0.757, respectively, with a total fit
of 1.564.
Figure 11-26
Eigenvalues for the two-dimensional solution
The multiple fit and single fit tables show that Age in years is still a highly discriminating
variable, as evidenced by the sum of the multiple fit values. In contrast to the earlier results,
however, examination of the single fit values reveals the discrimination to be almost entirely
along the second dimension.
Figure 11-27
Partitioning fit and loss
Turn to the transformation plot for Age in years. The quantifications for a nominal variable
are unrestricted, so the nondecreasing trend that was displayed when Age in years was treated
ordinally is no longer present. There is a decreasing trend until the age of 40 and an increasing

206
Chapter 11
trend thereafter, corresponding to a U-shaped (quadratic) relationship. The two older categories
still receive similar scores, and subsequent analyses may involve combining these categories.
Figure 11-28
Transformation plot for Age in years (nominal)
The transformation plot for Neighborhood preference is shown here. Treating Age in years as
nominal does not affect the quantifications for Neighborhood preference to any significant degree.
The middle category receives the smallest quantification, with the extreme categories receiving
large positive values.
Figure 11-29
Transformation plot for Neighborhood preference (age nominal)

207
A change is found in the transformation plot for Newspaper read most often. Previously, an
increasing trend was present in the quantifications, possibly suggesting an ordinal treatment
for this variable. However, treating Age in years as nominal removes this trend from the news
quantifications.
Figure 11-30
Transformation plot for Newspaper read most often (age nominal)

208
Chapter 11
This plot is the centroid plot for Age in years. Notice that the categories do not fall in chronological
order along the line joining the projected centroids. The 20–25 group is situated in the middle
rather than at the end. The spread of the categories is much improved over the ordinal counterpart
that was presented previously.
Figure 11-31
Centroids and projected centroids for Age in years (nominal)
Interpretation of the younger age groups is now possible from the centroid plot. The Volkskrant
and NRC categories are also further apart than in the previous analysis, allowing for separate
interpretations of each. The groups between the ages of 26 and 45 read the Volkskrant and prefer
country living. The 20–25 and 56–60 age groups read the NRC; the former group prefers to live
in a town, and the latter group prefers country living. The oldest groups read the Telegraaf and
prefer village living.

209
Interpretation of the other direction (Music preferred, Marital status, and Pets owned) is basically
unchanged from the previous analysis. The only obvious difference is that people with a marital
status of Other have either cats or no pets.
Figure 11-32
Centroids labeled by variables (age nominal)
General Suggestions
After you have examined the initial results, you will probably want to refine your analysis by
changing some of the specifications for the nonlinear canonical correlation analysis. Here are
some tips for structuring your analysis:
 Create as many sets as possible. Put an important variable that you want to predict in a
separate set by itself.
 Put variables that you consider predictors together in a single set. If there are many predictors,
try to partition them into several sets.
 Put each multiple nominal variable in a separate set by itself.
 If variables are highly correlated to each other, and you don’t want this relationship to
dominate the solution, put those variables together in the same set.

210
Chapter 11
See the following texts for more information about nonlinear canonical correlation analysis:
Carroll, J. D. 1968. Generalization of canonical correlation analysis to three or more sets of
variables. In: Proceedings of the 76th Annual Convention of the American Psychological
Association, 3, Washington, D.C.: American Psychological Association, 227–228.
De Leeuw, J. 1984. Canonical analysis of categorical data, 2nd ed. Leiden: DSWO Press.
Horst, P. 1961. Generalized canonical correlations and their applications to experimental data.
Journal of Clinical Psychology, 17, 331–347.
Horst, P. 1961. Relations among m sets of measures. Psychometrika, 26, 129–149.
Kettenring, J. R. 1971. Canonical analysis of several sets of variables. Biometrika, 58, 433–460.
Van der Burg, E. 1988. Nonlinear canonical correlation and some related techniques. Leiden:
DSWO Press.
Van der Burg, E., and J. De Leeuw. 1983. Nonlinear canonical correlation. British Journal of
Mathematical and Statistical Psychology, 36, 54–80.
Van der Burg, E., J. De Leeuw, and R. Verdegaal. 1988. Homogeneity analysis with k sets of
177–197.
Verboon, P., and I. A. Van der Lans. 1994. Robust canonical discriminant analysis. Psychometrika,
59, 485–507.

Chapter
12
Correspondence analysis
A correspondence table is any two-way table whose cells contain some measurement of
correspondence between the rows and the columns. The measure of correspondence can be any
indication of the similarity, affinity, confusion, association, or interaction between the row and
column variables. A very common type of correspondence table is a crosstabulation, where
the cells contain frequency counts.
Such tables can be obtained easily with the Crosstabs procedure. However, a crosstabulation
does not always provide a clear picture of the nature of the relationship between the two variables.
This is particularly true if the variables of interest are nominal (with no inherent order or rank)
and contain numerous categories. Crosstabulation may tell you that the observed cell frequencies
differ significantly from the expected values in a crosstabulation of occupation and breakfast
cereal, but it may be difficult to discern which occupational groups have similar tastes or what
those tastes are.
Correspondence Analysis allows you to examine the relationship between two nominal
variables graphically in a multidimensional space. It computes row and column scores and
produces plots based on the scores. Categories that are similar to each other appear close to
each other in the plots. In this way, it is easy to see which categories of a variable are similar to
each other or which categories of the two variables are related. The Correspondence Analysis
procedure also allows you to fit supplementary points into the space defined by the active points.
If the ordering of the categories according to their scores is undesirable or counterintuitive,
order restrictions can be imposed by constraining the scores for some categories to be equal. For
example, suppose that you expect the variable smoking behavior, with categories none, light,
medium, and heavy, to have scores that correspond to this ordering. However, if the analysis
orders the categories none, light, heavy, and medium, constraining the scores for heavy and
medium to be equal preserves the ordering of the categories in their scores.
The interpretation of correspondence analysis in terms of distances depends on the
normalization method used. The Correspondence Analysis procedure can be used to analyze
either the differences between categories of a variable or the differences between variables. With
the default normalization, it analyzes the differences between the row and column variables.
The correspondence analysis algorithm is capable of many kinds of analyses. Centering
the rows and columns and using chi-square distances corresponds to standard correspondence
analysis. However, using alternative centering options combined with Euclidean distances allows
for an alternative representation of a matrix in a low-dimensional space.
Three examples will be presented. The first employs a relatively small correspondence
table and illustrates the concepts inherent in correspondence analysis. The second example
demonstrates a practical marketing application. The final example uses a table of distances in a
multidimensional scaling approach.

212
Chapter 12
Normalization
Normalization is used to distribute the inertia over the row scores and column scores. Some
aspects of the correspondence analysis solution, such as the singular values, the inertia per
dimension, and the contributions, do not change under the various normalizations. The row and
column scores and their variances are affected. Correspondence analysis has several ways to
spread the inertia. The three most common include spreading the inertia over the row scores only,
spreading the inertia over the column scores only, or spreading the inertia symmetrically over
both the row scores and the column scores.
Row principal. In row principal normalization, the Euclidean distances between the row points
approximate chi-square distances between the rows of the correspondence table. The row scores
are the weighted average of the column scores. The column scores are standardized to have a
weighted sum of squared distances to the centroid of 1. Since this method maximizes the distances
between row categories, you should use row principal normalization if you are primarily interested
in seeing how categories of the row variable differ from each other.
Column principal. On the other hand, you might want to approximate the chi-square distances
between the columns of the correspondence table. In that case, the column scores should be the
weighted average of the row scores. The row scores are standardized to have a weighted sum of
squared distances to the centroid of 1. This method maximizes the distances between column
categories and should be used if you are primarily concerned with how categories of the column
variable differ from each other.
Symmetrical. You can also treat the rows and columns symmetrically. This normalization spreads
inertia equally over the row and column scores. Note that neither the distances between the row
points nor the distances between the column points are approximations of chi-square distances in
this case. Use this method if you are primarily interested in the differences or similarities between
the two variables. Usually, this is the preferred method to make biplots.
Principal. A fourth option is called principal normalization, in which the inertia is spread twice in
the solution—once over the row scores and once over the column scores. You should use this
method if you are interested in the distances between the row points and the distances between
the column points separately but not in how the row and column points are related to each other.
Biplots are not appropriate for this normalization option and are therefore not available if you
have specified the principal normalization method.
Example: Perceptions of Coffee Brands
The previous example involved a small table of hypothetical data. Actual applications often
involve much larger tables. In this example, you will use data pertaining to perceived images of six
iced-coffee brands (Kennedy, Riquier, and Sharp, 1996). This dataset can be found in coffee.sav.
For more information, see the topic Sample Files in Appendix A in IBM SPSS Categories 20.
For each of 23 iced-coffee image attributes, people selected all brands that were described by the
attribute. The six brands are denoted as AA, BB, CC, DD, EE, and FF to preserve confidentiality.

213
Table 12-1
Iced-coffee attributes
Image attribute Label Image attribute Label
good hangover cure cure fattening brand fattening
low fat/calorie brand low fat appeals to men men
brand for children children South Australian brand South Australian
working class brand working traditional/old fashioned
brand
traditional
rich/sweet brand sweet premium quality brand premium
unpopular brand unpopular healthy brand healthy
brand for fat/ugly people ugly high caffeine brand caffeine
very fresh fresh new brand new
brand for yuppies yuppies brand for attractive people attractive
nutritious brand nutritious tough brand tough
brand for women women popular brand popular
minor brand minor
Initially, you will focus on how the attributes are related to each other and how the brands are
related to each other. Using principal normalization spreads the total inertia once over the rows
and once over the columns. Although this prevents biplot interpretation, the distances between the
categories for each variable can be examined.
E The setup of the data requires that the cases be weighted by the variable freq. To do this, from the
menus choose:
Data > Weight Cases...
Figure 12-1
Weight Cases dialog box
E Weight cases by freq.
E Click OK.

214
Chapter 12
E To obtain an initial solution in five dimensions with principal normalization, from the menus
choose:
Analyze > Dimension Reduction > Correspondence Analysis...
Figure 12-2
Correspondence Analysis dialog box
E Select image as the row variable.
E Click Define Range.
Figure 12-3
Define Row Range dialog box
E Type 1 as the minimum value.
E Type 23 as the maximum value.
E Click Update.
E Click Continue.
E Select brand as the column variable.
E Click Define Range in the Correspondence Analysis dialog box.

215
Figure 12-4
Define Column Range dialog box
E Type 1 as the minimum value.
E Type 6 as the maximum value.
E Click Update.
E Click Continue.
E Click Model in the Correspondence Analysis dialog box.

216
Chapter 12
Figure 12-5
Model dialog box
E Select Principal as the normalization method.
E Click Continue.
E Click Plots in the Correspondence Analysis dialog box.

217
Figure 12-6
Plots dialog box
E Select Row points and Column points in the Scatterplots group.
E Click Continue.
E Click OK in the Correspondence Analysis dialog box.
Dimensionality
The inertia per dimension shows the decomposition of the total inertia along each dimension. Two
dimensions account for 83% of the total inertia. Adding a third dimension adds only 8.6% to the
accounted-for inertia. Thus, you elect to use a two-dimensional representation.

218
Chapter 12
Figure 12-7
Inertia per dimension
Contributions
The row points overview shows the contributions of the row points to the inertia of the dimensions
and the contributions of the dimensions to the inertia of the row points. If all points contributed
equally to the inertia, the contributions would be 0.043. Healthy and low fat both contribute a
substantial portion to the inertia of the first dimension. Men and tough contribute the largest
amounts to the inertia of the second dimension. Both ugly and fresh contribute very little to
either dimension.
Figure 12-8
Attribute contributions

219
Two dimensions contribute a large amount to the inertia for most row points. The large
contributions of the first dimension to healthy, new, attractive, low fat, nutritious, and women
indicate that these points are very well represented in one dimension. Consequently, the higher
dimensions contribute little to the inertia of these points, which will lie very near the horizontal
axis. The second dimension contributes most to men, premium, and tough. Both dimensions
contribute very little to the inertia for South Australian and ugly, so these points are poorly
represented.
The column points overview displays the contributions involving the column points. Brands
CC and DD contribute the most to the first dimension, whereas EE and FF explain a large amount
of the inertia for the second dimension. AA and BB contribute very little to either dimension.
Figure 12-9
Brand contributions
In two dimensions, all brands but BB are well represented. CC and DD are represented well in
one dimension. The second dimension contributes the largest amounts for EE and FF. Notice
that AA is represented well in the first dimension but does not have a very high contribution to
that dimension.
Plots
The row points plot shows that fresh and ugly are both very close to the origin, indicating that
they differ little from the average row profile. Three general classifications emerge. Located in
the upper left of the plot, tough, men, and working are all similar to each other. The lower left
contains sweet, fattening, children, and premium. In contrast, healthy, low fat, nutritious, and new
cluster on the right side of the plot.

220
Chapter 12
Figure 12-10
Plot of image attributes (principal normalization)
Notice in the column points plot that all brands are far from the origin, so no brand is similar to
the overall centroid. Brands CC and DD group together at the right, whereas brands BB and FF
cluster in the lower half of the plot. Brands AA and EE are not similar to any other brand.
Figure 12-11
Plot of brands (principal normalization)
Symmetrical Normalization
How are the brands related to the image attributes? Principal normalization cannot address
these relationships. To focus on how the variables are related to each other, use symmetrical
normalization. Rather than spread the inertia twice (as in principal normalization), symmetrical
normalization divides the inertia equally over both the rows and columns. Distances between

221
categories for a single variable cannot be interpreted, but distances between the categories for
different variables are meaningful.
Figure 12-12
Model dialog box
E To produce the following solution with symmetrical normalization, recall the Correspondence
Analysis dialog box and click Model.
E Select Symmetrical as the normalization method.
E Click Continue.
E Click OK in the Correspondence Analysis dialog box.

222
Chapter 12
In the upper left of the resulting biplot, brand EE is the only tough, working brand and appeals to
men. Brand AA is the most popular and also viewed as the most highly caffeinated. The sweet,
fattening brands include BB and FF. Brands CC and DD, while perceived as new and healthy,
are also the most unpopular.
Figure 12-13
Biplot of the brands and the attributes (symmetrical normalization)
For further interpretation, you can draw a line through the origin and the two image attributes
men and yuppies, and project the brands onto this line. The two attributes are opposed to each
other, indicating that the association pattern of brands for men is reversed compared to the pattern
for yuppies. That is, men are most frequently associated with brand EE and least frequently with
brand CC, whereas yuppies are most frequently associated with brand CC and least frequently
with brand EE.
Recommended readings
See the following texts for more information on correspondence analysis:
Fisher, R. A. 1938. Statistical methods for research workers. Edinburgh: Oliver and Boyd.
Fisher, R. A. 1940. The precision of discriminant functions. Annals of Eugenics, 10, 422–429.
Gilula, Z., and S. J. Haberman. 1988. The analysis of multivariate contingency tables by restricted
canonical and restricted association models. Journal of the American Statistical Association, 83,
760–771.

Chapter
13
The purpose of multiple correspondence analysis, also known as homogeneity analysis, is to find
quantifications that are optimal in the sense that the categories are separated from each other as
much as possible. This implies that objects in the same category are plotted close to each other
and objects in different categories are plotted as far apart as possible. The term homogeneity also
refers to the fact that the analysis will be most successful when the variables are homogeneous;
that is, when they partition the objects into clusters with the same or similar categories.
Example: Characteristics of Hardware
To explore how multiple correspondence analysis works, you will use data from Hartigan
(Hartigan, 1975), which can be found in screws.sav. For more information, see the topic Sample
Files in Appendix A in IBM SPSS Categories 20. This dataset contains information on the
characteristics of screws, bolts, nuts, and tacks. The following table shows the variables, along
with their variable labels, and the value labels assigned to the categories of each variable in the
Hartigan hardware dataset.
Table 13-1
Hartigan hardware dataset
thread Thread Yes_Thread, No_Thread
head Head form Flat, Cup, Cone, Round, Cylinder
indhead Indentation of head None, Star, Slit
bottom Bottom shape sharp, flat
length Length in half inches 1/2_in, 1_in, 1_1/2_ in, 2_in, 2_1/2_in
brass Brass Yes_Br, Not_Br
object Object tack, nail1, nail2, nail3, nail4, nail5,
nail6, nail7, nail8, screw1, screw2,
screw3, screw4, screw5, bolt1, bolt2,
bolt3, bolt4, bolt5, bolt6, tack1, tack2,
nailb, screwb
E To obtain a Multiple Correspondence Analysis, from the menus choose:

224
Chapter 13
Figure 13-1
E Make sure All variables are multiple nominal and One set are selected, and click Define.
Figure 13-2
Multiple Correspondence Analysis dialog box
E Select Thread through Length in half-inches as analysis variables.
E Select object as a labeling variable.
E Click Object in the Plots group.

225
Figure 13-3
Object Plots dialog box
E Choose to label objects by Variable.
E Select thread through object as labeling variables.
E Click Continue, and then click Variable in the Plots group of the Multiple Correspondence Analysis
dialog box.

226
Chapter 13
Figure 13-4
Variable Plots dialog box
E Choose to produce a joint category plot for thread through length.
E Click Continue.
E Click OK in the Multiple Correspondence Analysis dialog box.
Model Summary
Homogeneity analysis can compute a solution for several dimensions. The maximum number
of dimensions equals either the number of categories minus the number of variables with no
missing data or the number of observations minus one, whichever is smaller. However, you
should rarely use the maximum number of dimensions. A smaller number of dimensions is
easier to interpret, and after a certain number of dimensions, the amount of additional association
accounted for becomes negligible. A one-, two-, or three-dimensional solution in homogeneity
analysis is very common.

227
Figure 13-5
Model summary
Nearly all of the variance in the data is accounted for by the solution, 62.1% by the first dimension
and 36.8% by the second.
The two dimensions together provide an interpretation in terms of distances. If a variable
discriminates well, the objects will be close to the categories to which they belong. Ideally,
objects in the same category will be close to each other (that is, they should have similar scores),
and categories of different variables will be close if they belong to the same objects (that is, two
objects that have similar scores for one variable should also score close to each other for the other
variables in the solution).
Object Scores
After examining the model summary, you should look at the object scores. You can specify one
or more variables to label the object scores plot. Each labeling variable produces a separate plot
labeled with the values of that variable. We’ll take a look at the plot of object scores labeled by the
variable object. This is just a case-identification variable and was not used in any computations.
The distance from an object to the origin reflects variation from the “average” response pattern.
This average response pattern corresponds to the most frequent category for each variable. Objects
with many characteristics corresponding to the most frequent categories lie near the origin. In
contrast, objects with unique characteristics are located far from the origin.

228
Chapter 13
Figure 13-6
Object scores plot labeled by object
Examining the plot, you see that the first dimension (the horizontal axis) discriminates the screws
and bolts (which have threads) from the nails and tacks (which don’t have threads). This is easily
seen on the plot since screws and bolts are on one end of the horizontal axis and tacks and nails
are on the other. To a lesser extent, the first dimension also separates the bolts (which have flat
bottoms) from all the others (which have sharp bottoms).
The second dimension (the vertical axis) seems to separate SCREW1 and NAIL6 from all other
objects. What SCREW1 and NAIL6 have in common are their values on variable length—they
are the longest objects in the data. Moreover, SCREW1 lies much farther from the origin than
the other objects, suggesting that, taken as a whole, many of the characteristics of this object
are not shared by the other objects.
The object scores plot is particularly useful for spotting outliers. SCREW1 might be considered
an outlier. Later, we’ll consider what happens if you drop this object.
Discrimination Measures
Before examining the rest of the object scores plots, let’s see if the discrimination measures
agree with what we’ve said so far. For each variable, a discrimination measure, which can be
regarded as a squared component loading, is computed for each dimension. This measure is also
the variance of the quantified variable in that dimension. It has a maximum value of 1, which is
achieved if the object scores fall into mutually exclusive groups and all object scores within a
category are identical. (Note: This measure may have a value greater than 1 if there are missing
data.) Large discrimination measures correspond to a large spread among the categories of the
variable and, consequently, indicate a high degree of discrimination between the categories of a
variable along that dimension.

229
The average of the discrimination measures for any dimension equals the percentage of variance
accounted for that dimension. Consequently, the dimensions are ordered according to average
discrimination. The first dimension has the largest average discrimination, the second dimension
has the second largest average discrimination, and so on, for all dimensions in the solution.
Figure 13-7
Plot of discrimination measures
As noted on the object scores plot, the discrimination measures plot shows that the first dimension
is related to variables Thread and Bottom shape. These variables have large discrimination
measures on the first dimension and small discrimination measures on the second dimension.
Thus, for both of these variables, the categories are spread far apart along the first dimension only.
Length in half-inches has a large value on the second dimension but a small value on the first
dimension. As a result, length is closest to the second dimension, agreeing with the observation
from the object scores plot that the second dimension seems to separate the longest objects from
the rest. Indentation of head and Head form have relatively large values on both dimensions,
indicating discrimination in both the first and second dimensions. The variable Brass, located
very close to the origin, does not discriminate at all in the first two dimensions. This makes sense,
since all of the objects can be made of brass or not made of brass.
Category Quantifications
Recall that a discrimination measure is the variance of the quantified variable along a particular
dimension. The discrimination measures plot contains these variances, indicating which variables
discriminate along which dimension. However, the same variance could correspond to all of the
categories being spread moderately far apart or to most of the categories being close together,
with a few categories differing from this group. The discrimination plot cannot differentiate
between these two conditions.

230
Chapter 13
Category quantification plots provide an alternative method of displaying discrimination of
variables that can identify category relationships. In this plot, the coordinates of each category
on each dimension are displayed. Thus, you can determine which categories are similar for
each variable.
Figure 13-8
Category quantifications
Length in half-inches has five categories, three of which group together near the top of the plot.
The remaining two categories are in the lower half of the plot, with the 2_1/2_in category very
far from the group. The large discrimination for length along dimension 2 is a result of this one
category being very different from the other categories of length. Similarly, for Head form, the
category STAR is very far from the other categories and yields a large discrimination measure along
the second dimension. These patterns cannot be illustrated in a plot of discrimination measures.
The spread of the category quantifications for a variable reflects the variance and thus indicates
how well that variable is discriminated in each dimension. Focusing on dimension 1, the
categories for Thread are far apart. However, along dimension 2, the categories for this variable
are very close. Thus, Thread discriminates better in dimension 1 than in dimension 2. In contrast,
the categories for Head form are spread far apart along both dimensions, suggesting that this
variable discriminates well in both dimensions.
In addition to determining the dimensions along which a variable discriminates and how that
variable discriminates, the category quantification plot also compares variable discrimination.
A variable with categories that are far apart discriminates better than a variable with categories
that are close together. For example, along dimension 1, the two categories of Brass are much
closer to each other than the two categories of Thread, indicating that Thread discriminates
better than Brass along this dimension. However, along dimension 2, the distances are very
similar, suggesting that these variables discriminate to the same degree along this dimension. The
discrimination measures plot discussed previously identifies these same relationships by using
variances to reflect the spread of the categories.

231
A More Detailed Look at Object Scores
A greater insight into the data can be gained by examining the object scores plots labeled by
each variable. Ideally, similar objects should form exclusive groups, and these groups should be
far from each other.
Figure 13-9
Object scores labeled with Thread
The plot labeled with Thread shows that the first dimension separates Yes_Thread and No_Thread
perfectly. All of the objects with threads have negative object scores, whereas all of the
nonthreaded objects have positive scores. Although the two categories do not form compact
groups, the perfect differentiation between the categories is generally considered a good result.

232
Chapter 13
Figure 13-10
Object scores labeled with Head form
The plot labeled with Head form shows that this variable discriminates in both dimensions. The
FLAT objects group together in the lower right corner of the plot, whereas the CUP objects group
together in the upper right. CONE objects all lie in the upper left. However, these objects are more
spread out than the other groups and, thus, are not as homogeneous. Finally, CYLINDER objects
cannot be separated from ROUND objects, both of which lie in the lower left corner of the plot.
Figure 13-11
Object scores labeled with Length in half-inches
The plot labeled with Length in half-inches shows that this variable does not discriminate in
the first dimension. Its categories display no grouping when projected onto a horizontal line.
However, Length in half-inches does discriminate in the second dimension. The shorter objects
correspond to positive scores, and the longer objects correspond to large negative scores.

233
Figure 13-12
Object scores labeled with Brass
The plot labeled with Brass shows that this variable has categories that cannot be separated very
well in the first or second dimensions. The object scores are widely spread throughout the space.
The brass objects cannot be differentiated from the nonbrass objects.
Omission of Outliers
In homogeneity analysis, outliers are objects that have too many unique features. As noted earlier,
SCREW1 might be considered an outlier.
To delete this object and run the analysis again, from the menus choose:
Data > Select Cases...

234
Chapter 13
Figure 13-13
Select Cases dialog box
E Select If condition is satisfied.
E Click If.

235
Figure 13-14
If dialog box
E Type object ~= 16 as the condition.
E Click Continue.
E Click OK in the Select Cases dialog box.
E Finally, recall the Multiple Correspondence Analysis dialog box, and click OK.
Figure 13-15
Model summary (outlier removed)
The eigenvalues shift slightly. The first dimension now accounts for a little more of the variance.

236
Chapter 13
Figure 13-16
Discrimination measures
As shown in the discrimination plot, Indentation of head no longer discriminates in the second
dimension, whereas Brass changes from no discrimination in either dimension to discrimination in
the second dimension. Discrimination for the other variables is largely unchanged.
Figure 13-17
Object scores labeled with Brass (outlier removed)
The object scores plot labeled by Brass shows that the four brass objects all appear near the
bottom of the plot (three objects occupy identical locations), indicating high discrimination along
the second dimension. As was the case for Thread in the previous analysis, the objects do not
form compact groups, but the differentiation of objects by categories is perfect.

237
Figure 13-18
Object scores labeled with Indentation of head (outlier removed)
The object scores plot labeled by Indentation of head shows that the first dimension discriminates
perfectly between the non-indented objects and the indented objects, as in the previous analysis.
In contrast to the previous analysis, however, the second dimension cannot now distinguish the
two categories.
Thus, the omission of SCREW1, which is the only object with a star-shaped head, dramatically
affects the interpretation of the second dimension. This dimension now differentiates objects
based on Brass, Head form, and Length in half-inches.
See the following texts for more information on multiple correspondence analysis:
Benzécri, J. P. 1992. Correspondence analysis handbook. New York: Marcel Dekker.
Guttman, L. 1941. The quantification of a class of attributes: A theory and method of scale
construction. In: The Prediction of Personal Adjustment, P. Horst, ed. New York: Social Science
Research Council, 319–348.
Meulman, J. J. 1982. Homogeneity analysis of incomplete data. Leiden: DSWO Press.
Meulman, J. J. 1996. Fitting a distance model to homogeneous subsets of variables: Points of
view analysis of categorical data. Journal of Classification, 13, 249–266.
Meulman, J. J., and W. J. Heiser. 1997. Graphical display of interaction in multiway contingency
tables by use of homogeneity analysis. In: Visual Display of Categorical Data, M. Greenacre, and
J. Blasius, eds. New York: Academic Press, 277–296.
Nishisato, S. 1984. Forced classification: A simple application of a quantification method.

238
Chapter 13
Tenenhaus, M., and F. W. Young. 1985. An analysis and synthesis of multiple correspondence
analysis, optimal scaling, dual scaling, homogeneity analysis, and other methods for quantifying
categorical multivariate data. Psychometrika, 50, 91–119.
Van Rijckevorsel, J. 1987. The application of fuzzy coding and horseshoes in multiple
correspondence analysis. Leiden: DSWO Press.

Chapter
14
Given a set of objects, the goal of multidimensional scaling is to find a representation of the
objects in a low-dimensional space. This solution is found by using the proximities between
the objects. The procedure minimizes the squared deviations between the original, possibly
transformed, object proximities and their Euclidean distances in the low-dimensional space.
The purpose of the low-dimensional space is to uncover relationships between the objects. By
restricting the solution to be a linear combination of independent variables, you may be able to
interpret the dimensions of the solution in terms of these variables. In the following example,
you will see how 15 different kinship terms can be represented in three dimensions and how
that space can be interpreted with respect to the gender, generation, and degree of separation
of each of the terms.
Example: An Examination of Kinship Terms
Rosenberg and Kim (Rosenberg and Kim, 1975) set out to analyze 15 kinship terms (aunt, brother,
cousin, daughter, father, granddaughter, grandfather, grandmother, grandson, mother, nephew,
niece, sister, son, uncle). They asked four groups of college students (two female, two male) to
sort these terms on the basis of similarities. Two groups (one female, one male) were asked to
sort twice, with the second sorting based on a different criteria from the first sort. Thus, a total of
six “sources” were obtained, as outlined in the following table.
Table 14-1
Source structure of kinship data
Source Gender Condition Sample size
1 Female Single sort 85
2 Male Single sort 85
3 Female First sort 80
4 Female Second sort 80
5 Male First sort 80
6 Male Second sort 80
Each source corresponds to a proximity matrix, whose cells are equal to the number of
people in a source minus the number of times that the objects were partitioned together in that
source. This dataset can be found in kinship_dat.sav. For more information, see the topic Sample
Files in Appendix A in IBM SPSS Categories 20.
Choosing the Number of Dimensions
It is up to you to decide how many dimensions the solution should have. The scree plot can
help you make this decision.

240
Chapter 14
E To create a scree plot, from the menus choose:
Analyze > Scale > Multidimensional Scaling (PROXSCAL)...
Figure 14-1
Data Format dialog box
E Select Multiple matrix sources in the Number of Sources group.
E Click Define.
Figure 14-2
Multidimensional Scaling dialog box
E Select Aunt through Uncle as proximities variables.
E Select sourceid as the variable identifying the source.

241
E Click Model.
Figure 14-3
Model dialog box
E Type 10 as the maximum number of dimensions.
E Click Continue.
E Click Restrictions in the Multidimensional Scaling dialog box.

242
Chapter 14
Figure 14-4
E Select Linear combination of independent variables.
E Click File to select the source of the independent variables.
E Select kinship_var.sav.

243
Figure 14-5
E Select gender, gener, and degree as restriction variables.
Note that the variable gender has a user-missing value—9 = missing (for cousin). The procedure
treats this as a valid category. Thus, the default linear transformation is unlikely to be appropriate.
Use a nominal transformation instead.

244
Chapter 14
Figure 14-6
E Select gender.
E Select Nominal from the Independent Variable Transformations drop-down list.
E Click Change.
E Click Continue.
E Click Plots in the Multidimensional Scaling dialog box.

245
Figure 14-7
Plots dialog box
E Select Stress in the Plots group.
E Click Continue.
E Click OK in the Multidimensional Scaling dialog box.

246
Chapter 14
Figure 14-8
Scree plot
The procedure begins with a 10-dimensional solution and works down to a 2-dimensional
solution. The scree plot shows the normalized raw stress of the solution at each dimension. You
can see from the plot that increasing the dimensionality from 2 to 3 and from 3 to 4 offers large
improvements in the stress. After 4, the improvements are rather small. You will choose to
analyze the data by using a 3-dimensional solution, because the results are easier to interpret.
A Three-Dimensional Solution
The independent variables gender, gener (generation), and degree (of separation) were constructed
with the intention of using them to interpret the dimensions of the solution. The independent
variables were constructed as follows:
gender 1 = male, 2 = female, 9 = missing (for cousin)
gener The number of generations from you if the term refers to your kin, with
lower numbers corresponding to older generations. Thus, grandparents
are –2, grandchildren are 2, and siblings are 0.
degree The number of degrees of separation along your family tree. Thus,
your parents are up 1 node, while your children are down 1 node. Your
siblings are up 1 node to your parents and then down 1 node to them, for
2 degrees of separation. Your cousin is 4 degrees away—2 up to your
grandparents and then 2 down through your aunt/uncle to them.
The external variables can be found in kinship_var.sav. Additionally, an initial configuration from
an earlier analysis is supplied in kinship_ini.sav. For more information, see the topic Sample Files
in Appendix A in IBM SPSS Categories 20.

247
Figure 14-9
Model dialog box
E To obtain a three-dimensional solution, recall the Multidimensional Scaling dialog box and click
Model.
E Type 3 as the minimum and maximum number of dimensions.
E Click Continue.
E Click Options in the Multidimensional Scaling dialog box.

248
Chapter 14
Figure 14-10
Options dialog box
E Select Custom as the initial configuration.
E Select kinship_ini.sav as the file to read variables from.
E Select dim01, dim02, and dim03 as variables.
E Click Continue.
E Click Plots in the Multidimensional Scaling dialog box.

249
Figure 14-11
Plots dialog box
E Select Original vs. transformed proximities and Transformed independent variables.
E Click Continue.
E Click Output in the Multidimensional Scaling dialog box.

250
Chapter 14
Figure 14-12
Output dialog box
E Select Input data, Stress decomposition, and Variable and dimension correlations.
E Click Continue.
Stress Measures
The stress and fit measures give an indication of how well the distances in the solution
approximate the original distances.
Figure 14-13
Stress and fit measures

251
Each of the four stress statistics measures the misfit of the data, while the dispersion accounted for
and Tucker’s coefficient of congruence measure the fit. Lower stress measures (to a minimum of
0) and higher fit measures (to a maximum of 1) indicate better solutions.
Figure 14-14
Decomposition of normalized raw stress
The decomposition of stress helps you identify which sources and objects contribute the most to
the overall stress of the solution. In this case, most of the stress among the sources is attributable
to sources 1 and 2, while among the objects, most of the stress is attributable to Brother,
Granddaughter, Grandfather, Grandmother, Grandson, and Sister.
The two sources that are accountable for most of the stress are the two groups that sorted the
terms only once. This information suggests that the students considered multiple factors when
sorting the terms, and those students who were allowed to sort twice focused on a portion of those
factors for the first sort and then considered the remaining factors during the second sort.
The objects that account for most of the stress are those objects with a degree of 2. These
people are relations who are not part of the “nuclear” family (Mother, Father, Daughter, Son)
but are nonetheless closer than other relations. This middle position could easily cause some
differential sorting of these terms.
Final Coordinates of the Common Space
The common space plot gives a visual representation of the relationships between the objects.

252
Chapter 14
Figure 14-15
Common space coordinates
Look at the final coordinates for the objects in dimensions 1 and 3; this is the plot in the lower left
corner of the scatterplot matrix. This plot shows that dimension 1 (on the x axis) is correlated
with the variable gender, and dimension 3 (on the y axis) is correlated with gener. From left to
right, you see that dimension 1 separates the female and male terms, with the genderless term
Cousin in the middle. From the bottom of the plot to the top, increasing values along the axis
correspond to terms that are older.
Now look at the final coordinates for the objects in dimensions 2 and 3; this plot is the plot
on the middle right side of the scatterplot matrix. From this plot, you can see that the second
dimension (along the y axis) corresponds to the variable degree, with larger values along the axis
corresponding to terms that are further from the “nuclear” family.
A Three-Dimensional Solution with Nondefault Transformations
The previous solution was computed by using the default ratio transformation for proximities
and interval transformations for the independent variables gener and degree. The results are
pretty good, but you may be able to do better by using other transformations. For example, the
proximities, gener, and degree all have natural orderings, but they may be better modeled by an
ordinal transformation than a linear transformation.

253
Figure 14-16
Model dialog box
E To rerun the analysis, scaling the proximities, gener, and degree at the ordinal level (keeping ties),
recall the Multidimensional Scaling dialog box and click Model.
E Select Ordinal as the proximity transformation.
E Click Continue.
E Click Restrictions in the Multidimensional Scaling dialog box.

254
Chapter 14
Figure 14-17
E Select gener and degree.
E Select Ordinal (keep ties) from the Independent Variable Transformations drop-down list.
E Click Change.
E Click Continue.
The transformation plots are a good first check to see whether the original transformations were
appropriate. If the plots are approximately linear, the linear assumption is appropriate. If not,
check the stress measures to see whether there is an improvement in fit and check the common
space plot to see whether the interpretation is more useful.
The independent variables each obtain approximately linear transformations, so it may be
appropriate to interpret them as numerical. However, the proximities do not obtain a linear
transformation, so it is possible that the ordinal transformation is more appropriate for the
proximities.

255
Figure 14-18
Transformed proximities
Stress Measures
The stress for the current solution supports the argument for scaling the proximities at the ordinal
level.
Figure 14-19
Stress and fit measures
The normalized raw stress for the previous solution is 0.06234. Scaling the variables by using
nondefault transformations halves the stress to 0.03137.
Final Coordinates of the Common Space
The common space plots offer essentially the same interpretation of the dimensions as the
previous solution.

256
Chapter 14
Figure 14-20
Common space coordinates
Discussion
It is best to treat the proximities as ordinal variables, because there is great improvement in the
stress measures. As a next step, you may want to “untie” the ordinal variables—that is, allow
equivalent values of the original variables to obtain different transformed values. For example, in
the first source, the proximities between Aunt and Son, and Aunt and Grandson, are 85. The “tied”
approach to ordinal variables forces the transformed values of these proximities to be equivalent,
but there is no particular reason for you to assume that they should be. In this case, allowing the
proximities to become untied frees you from an unnecessary restriction.
See the following texts for more information on multidimensional scaling:
Commandeur, J. J. F., and W. J. Heiser. 1993. Mathematical derivations in the proximity scaling
(PROXSCAL) of symmetric data matrices. Leiden: Department of Data Theory, University of
Leiden.
De Leeuw, J., and W. J. Heiser. 1980. Multidimensional scaling with restrictions on the
configuration. In: Multivariate Analysis, Vol. V, P. R. Krishnaiah, ed. Amsterdam: North-Holland,
501–522.
Heiser, W. J. 1981. Unfolding analysis of proximity data. Leiden: Department of Data Theory,
Heiser, W. J., and F. M. T. A. Busing. 2004. Multidimensional scaling and unfolding of symmetric
and asymmetric proximity relations. In: Handbook of Quantitative Methodology for the Social
Sciences, D. Kaplan, ed. Thousand Oaks, Calif.: SagePublications, Inc., 25–48.

257
Kruskal, J. B. 1964. Multidimensional scaling by optimizing goodness of fit to a nonmetric
hypothesis. Psychometrika, 29, 1–28.
Kruskal, J. B. 1964. Nonmetric multidimensional scaling: A numerical method. Psychometrika,
29, 115–129.
Shepard, R. N. 1962. The analysis of proximities: Multidimensional scaling with an unknown
distance function I. Psychometrika, 27, 125–140.
distance function II. Psychometrika, 27, 219–246.

Chapter
15
The Multidimensional Unfolding procedure attempts to find a common quantitative scale that
allows you to visually examine the relationships between two sets of objects.
Example: Breakfast Item Preferences
In a classic study (Green and Rao, 1972), 21 Wharton School MBA students and their spouses
were asked to rank 15 breakfast items in order of preference, from 1 = “most preferred” to 15 =
“least preferred.” This information is collected in breakfast_overall.sav. For more information,
see the topic Sample Files in Appendix A in IBM SPSS Categories 20.
The results of the study provide a typical example of the degeneracy problem inherent in most
multidimensional unfolding algorithms that is solved by penalizing the coefficient of variation
of the transformed proximities (Busing, Groenen, and Heiser, 2005). You will see a degenerate
solution and will see how to solve the problem using Multidimensional Unfolding, allowing you
to determine how individuals discriminate between breakfast items. Syntax for reproducing these
analyses can be found in prefscal_breakfast-overall.sps.
Producing a Degenerate Solution
E To run a Multidimensional Unfolding analysis, from the menus choose:

259
Figure 15-1
E Select Toast pop-up through Corn muffin and butter as proximities variables.
E Click Options.

260
Chapter 15
Figure 15-2
Options dialog box
E Select Spearman as the imputation method for the Classical start.
E In the Penalty Term group, type 1.0 as the value of the Strength parameter and 0.0 as the value of
the Range parameter. This turns off the penalty term.
E Click Continue.
E Click OK in the Multidimensional Unfolding dialog box.
Following is the command syntax generated by these selections:
PREFSCAL
VARIABLES=TP BT EMM JD CT BMM HRB TMd BTJ TMn CB DP GD CC CMB
/INITIAL=CLASSICAL (SPEARMAN)
/TRANSFORMATION=NONE
/PROXIMITIES=DISSIMILARITIES
/CRITERIA=DIMENSIONS(2,2) DIFFSTRESS(.000001) MINSTRESS(.0001)
MAXITER(5000)
/PENALTY=LAMBDA(1.0) OMEGA(0.0)
/PRINT=MEASURES COMMON
/PLOT=COMMON .
 This syntax specifies an analysis on variables tp (Toast pop-up) through cmb (Corn muffin
and butter).

261
 The INITIAL subcommand specifies that the starting values be imputed using Spearman
distances.
 The specified values on the PENALTY subcommand essentially turn off the penalty term, and
as a result, the procedure will minimize Kruskal’s Stress-I. This will result in a degenerate
solution.
 The PLOT subcommand requests plots of the common space.
 All other parameters fall back to their default values.
Measures
Figure 15-3
Measures for degenerate solution
The algorithm converges to a solution after 154 iterations, with a penalized stress (marked final
function value) of 0.0000990. Since the penalty term has been turned off, penalized stress is equal
to Kruskal’s Stress-I (the stress part of the function value is equivalent to Kruskal’s badness-of-fit
measure). Low stress values generally indicate that the solution fits the data well, but there are
several warning signs of a degenerate solution:
 The coefficient of variation for the transformed proximities is very small relative to the
coefficient of variation for the original proximities. This suggests that the transformed
proximities for each row are near-constant, and thus the solution will not provide any
discrimination between objects.

262
Chapter 15
 The sum-of-squares of DeSarbo’s intermixedness indices are a measure of how well the points
of the different sets are intermixed. If they are not intermixed, this is a warning sign that the
solution may be degenerate. The closer to 0, the more intermixed the solution. The reported
value is very large, indicating that the solution is not intermixed.
 Shepard’s rough nondegeneracy index, which is reported as a percentage of different
distances, is equal to 0. This is a clear numerical indication that there are insufficiently
different distances and that the solution is probably degenerate.
Common Space
Figure 15-4
Joint plot of common space for degenerate solution
Visual confirmation that the solution is degenerate is found in the joint plot of the common space
of row and column objects. The row objects (individuals) are situated on the circumference of
a circle centered on the column objects (breakfast items), whose coordinates have collapsed to
a single point.

263
Running a Nondegenerate Analysis
Figure 15-5
Options dialog box
E To produce a nondegenerate solution, click the Dialog Recall tool and select Multidimensional
Unfolding.
E Click Options in the Multidimensional Unfolding dialog box.
E In the Penalty Term group, type 0.5 as the value of the Strength parameter and 1.0 as the value of
the Range parameter. This turns off the penalty term.
E Click Continue.
PREFSCAL
MAXITER(5000)

264
Chapter 15
/PLOT=COMMON .
 The only change is on the PENALTY subcommand. LAMBDA has been set to 0.5, and OMEGA
has been set to 1.0, their default values.
Measures
Figure 15-6
Measures for nondegenerate solution
The problems noted in the measures for the degenerate solution have been corrected here.
 The normalized stress is no longer 0.
 The coefficient of variation for the transformed proximities now has a similar value to the
coefficient of variation for the original proximities.
 DeSarbo’s intermixedness indices are much closer to 0, indicating that the solution is much
better intermixed.
 Shepard’s rough nondegeneracy index, which is reported as a percentage of different
distances, is now nearly 80%. There are sufficiently different distances, and the solution is
probably nondegenerate.

265
Common Space
Figure 15-7
Joint plot of common space for nondegenerate solution
The joint plot of the common space allows for an interpretation of the dimensions. The horizontal
dimension appears to discriminate between soft and hard bread or toast, with softer items as you
move right along the axis. The vertical dimension does not have a clear interpretation, although
it perhaps discriminates based on convenience, with more “formal” items as you move down
along the axis.
This creates several clusters of breakfast items. For example, the donuts, cinnamon buns, and
Danish pastry form a cluster of soft and somewhat informal items. The muffins and cinnamon
toast form a cluster of harder but more formal items. The other toasts and hard rolls form a cluster
of hard and somewhat informal items. The toast pop-up is a hard item that is extremely informal.
The individuals represented by the row objects are clearly split into clusters according to
preference for hard or soft items, with considerable within-cluster variation along the vertical
dimension.
Example: Three-Way Unfolding of Breakfast Item Preferences
In a classic study (Green et al., 1972), 21 Wharton School MBA students and their spouses
were asked to rank 15 breakfast items in order of preference, from 1 = “most preferred” to 15 =
“least preferred.” Their preferences were recorded under six different scenarios, from “Overall
preference” to “Snack, with beverage only.” This information is collected in breakfast.sav. For
more information, see the topic Sample Files in Appendix A in IBM SPSS Categories 20.
The six scenarios can be treated as separate sources. Use PREFSCAL to perform a three-way
unfolding of the rows, columns, and sources. Syntax for reproducing these analyses can be found
in prefscal_breakfast.sps.

266
Chapter 15
Figure 15-8
E Select Toast pop-up through Corn muffin and butter as proximities variables.
E Select Menu scenarios as the source variable.
E Click Model.

267
Figure 15-9
Model dialog box
E Select Weighted Euclidean as the scaling model.
E Click Continue.

268
Chapter 15
Figure 15-10
Options dialog box
E Select Spearman as the imputation method for the Classical start.
E Click Continue.
E Click Plots in the Multidimensional Unfolding dialog box.

269
Figure 15-11
Plots dialog box
E Select Individual spaces in the Plots group.
E Click Continue.
PREFSCAL
/INPUT=SOURCES(srcid )
/CONDITION=ROW
/MODEL=WEIGHTED

270
Chapter 15
MAXITER(5000)
/PLOT=COMMON WEIGHTS INDIVIDUAL ( ALL ) .
 This syntax specifies an analysis on variables tp (Toast pop-up) through cmb (Corn muffin and
butter). The variable srcid is used to identify the sources.
 The INITIAL subcommand specifies that the starting values be imputed using Spearman
distances.
 The MODEL subcommand specifies a weighted Euclidean model, which allows each individual
space to weight the dimensions of the common space differently.
 The PLOT subcommand requests plots of the common space, individual spaces, and individual
space weights.
Measures
Figure 15-12
Measures
The algorithm converges after 481 iterations, with a final penalized stress of 0.8199642. The
variation coefficients and Shepard’s index are sufficiently large, and DeSarbo’s indices are
sufficiently low, to suggest that there are no problems with degeneracy.

271
Common Space
Figure 15-13
Joint plot of common space
The joint plot of the common space shows a final configuration that is very similar to the two-way
analysis on the overall preferences, with the solution flipped over the 45-degree line. Thus,
the vertical dimension now appears to discriminate between soft and hard bread or toast, with
softer items as you move up along the axis. The horizontal dimension now does not have a
clear interpretation, though perhaps it discriminates based on convenience, with more “formal”
items as you move left along the axis.
The individuals represented by the row objects are still clearly split into clusters according
to preference for “hard” or “soft” items, with considerable within-cluster variation along the
horizontal dimension.

272
Chapter 15
Individual Spaces
Figure 15-14
Dimension weights
An individual space is computed for each source. The dimension weights show how the individual
spaces load on the dimensions of the common space. A larger weight indicates a larger distance in
the individual space and thus greater discrimination between the objects on that dimension for
that individual space.
 Specificity is a measure of how different an individual space is from the common space. An
individual space that was identical to the common space would have identical dimension
weights and a specificity of 0, while an individual space that was specific to a particular
dimension would have a single large dimension weight and a specificity of 1. In this case,
the most divergent sources are Breakfast, with juice, bacon and eggs, and beverage, and
Snack, with beverage only.
 Importance is a measure of the relative contribution of each dimension to the solution. In this
case, the dimensions are equally important.

273
Figure 15-15
Dimension weights
The dimension weights chart provides a visualization of the weights table. Breakfast, with juice,
bacon and eggs, and beverage and Snack, with beverage only are the nearest to the dimension
axes, but neither are strongly specific to a particular dimension.
Figure 15-16
Joint plot of individual space “Breakfast, with juice, bacon and eggs, and beverage”
The joint plot of the individual space Breakfast, with juice, bacon and eggs, and beverage
shows the effect of this scenario on the preferences. This source loads more heavily on the first
dimension, so the differentiation between items is mostly due to the first dimension.

274
Chapter 15
Figure 15-17
Joint plot of individual space “Snack, with beverage only”
The joint plot of the individual space Snack, with beverage only shows the effect of this scenario
on the preferences. This source loads more heavily on the second dimension, so the differentiation
between items is mostly due to the second dimension. However, there is still quite a bit of
differentiation along the first dimension because of the fairly low specificity of this source.
Using a Different Initial Configuration
The final configuration can depend on the starting points given to the algorithm. Ideally, the
general structure of the solution should remain the same; otherwise, it can be difficult to ascertain
which is correct. However, details may come into sharper focus as you try different initial
configurations, such as using a correspondence start on the three-way analysis of the breakfast data.
E To produce a solution with a correspondence start, click the Dialog Recall tool and select
Multidimensional Unfolding.

275
Figure 15-18
Options dialog box
E Select Correspondence in the Initial Configuration group.
E Click Continue.
PREFSCAL
/INPUT=SOURCES(srcid )
/INITIAL=CORRESPONDENCE
MAXITER(5000)

276
Chapter 15
/PLOT=COMMON WEIGHTS INDIVIDUAL ( ALL ) .
 The only change is on the INITIAL subcommand. The starting configuration has been set
to CORRESPONDENCE, which uses the results of a correspondence analysis on the reversed
data (similarities instead of dissimilarities), with a symmetric normalization of row and
column scores.
Measures
Figure 15-19
Measures for correspondence initial configuration
The algorithm converges after 385 iterations, with a final penalized stress of 0.8140741. This
statistic, the badness of fit, the goodness of fit, variation coefficients, and Shepard’s index are
all very similar to those for the solution using the classical Spearman start. DeSarbo’s indices
is somewhat different, with a value of 1.7571887 instead of 0.2199287, which suggests that the
solution using the correspondence start is not as well mixed. To see how this affects the solution,
look at the joint plot of the common space.

277
Common Space
Figure 15-20
Joint plot of common space for correspondence initial configuration
The joint plot of the common space shows a final configuration that is similar to the analysis with
the classical Spearman initial configuration; however, the column objects (breakfast items) are
situated around the row objects (individuals) rather than intermixed with them.

278
Chapter 15
Individual Spaces
Figure 15-21
Dimension weights for correspondence initial configuration
Under the correspondence initial configuration, each of the individual spaces has a higher
specificity; that is, each situation under which the participants ranked the breakfast items is more
strongly associated with a specific dimension. The most divergent sources are still Breakfast, with
juice, bacon and eggs, and beverage, and Snack, with beverage only.

279
Figure 15-22
Joint plot of individual space “Breakfast, with juice, bacon and eggs, and beverage” for correspondence
initial configuration
The higher specificity is evident in the joint plot of the individual space Breakfast, with juice,
bacon and eggs, and beverage. The source loads even more heavily on the first dimension than
under the classical Spearman start, so the row and column objects show a little less variation on
the vertical axis and a little more variation on the horizontal axis.

280
Chapter 15
Figure 15-23
Joint plot of individual space “Snack, with beverage only” for correspondence initial configuration
The joint plot of the individual space Snack, with beverage only shows that the row and column
objects lie more closely to a vertical line than under the classical Spearman start.
Example: Examining Behavior-Situation Appropriateness
In a classic example (Price and Bouffard, 1974), 52 students were asked to rate the combinations
of 15 situations and 15 behaviors on a 10-point scale ranging from 0 = “extremely appropriate” to
9 = “extremely inappropriate.” Averaged over individuals, the values are taken as dissimilarities.
This information is collected in behavior.sav. For more information, see the topic Sample Files
in Appendix A in IBM SPSS Categories 20. Use Multidimensional Unfolding to find clusterings
of similar situations and the behaviors with which they are most closely associated. Syntax for
reproducing these analyses can be found in prefscal_behavior.sps.

281
Figure 15-24
E Select Run through Shout as proximities variables.
E Select ROWID as the row variable.
E Click Model.

282
Chapter 15
Figure 15-25
Model dialog box
E Select Linear as the proximity transformation, and choose to Include intercept.
E Choose to apply transformations Across all sources simultaneously.
E Click Continue.

283
Figure 15-26
Options dialog box
E Select Custom in the Initial Configuration group.
E Browse to and choose behavior_ini.sav as the file containing the custom initial configuration. For
more information, see the topic Sample Files in Appendix A in IBM SPSS Categories 20.
E Select dim1 and dim2 as the variables specifying the initial configuration.
E Click Continue.
E Click Plots in the Multidimensional Unfolding dialog box.

284
Chapter 15
Figure 15-27
Plots dialog box
E Select Transformation plots in the Plots group.
E Click Continue.
PREFSCAL
VARIABLES=Run Talk Kiss Write Eat Sleep Mumble Read Fight Belch Argue Jump
Cry Laugh Shout
/INPUT=ROWS(ROWID )
/INITIAL=( 'samplesDirectory/behavior_ini.sav' )
dim1 dim2
/CONDITION=UNCONDITIONAL
/TRANSFORMATION=LINEAR (INTERCEPT)

285
/MODEL=IDENTITY
MAXITER(5000)
/PLOT=COMMON TRANSFORMATIONS .
 This syntax specifies an analysis on variables run through shout. The variable rowid is used
to identify the rows.
 The INITIAL subcommand specifies that the starting values be taken from the file
behavior_ini.sav. The row and column coordinates are stacked, with the column coordinates
following the row coordinates.
 The CONDITION subcommand specifies that all proximities can be compared with each other.
This is true in this analysis, since you should be able to compare the proxmities for running
in a park and running in church and see that one behavior is considered less appropriate
than the other.
 The TRANSFORMATION subcommand specifies a linear transformation of the proximities,
with intercept. This is appropriate if a 1-point difference in proximities is equivalent across
the range of the 10-point scale. That is, if the students have assigned their scores so that
the difference between 0 and 1 is the same as the difference between 5 and 6, then a linear
transformation is appropriate.
 The PLOT subcommand requests plots of the common space and transformation plots.
Measures
Figure 15-28
Measures

286
Chapter 15
The algorithm converges after 169 iterations, with a final penalized stress of 0.6427725. The
variation coefficients and Shepard’s index are sufficiently large, and DeSarbo’s indices are
sufficiently low, to suggest that there are no problems with degeneracy.
Common Space
Figure 15-29
Joint plot of common space
The horizontal dimension appears to be more strongly associated with the column objects
(behaviors) and discriminates between inappropriate behaviors (fighting, belching) and more
appropriate behaviors. The vertical dimension appears to be more strongly associated with the
row objects (situations) and defines different situational-behavior restrictions.
 Toward the bottom of the vertical dimension are situations (church, class) that restrict
behavior to the quieter/introspective types of behaviors (read, write). Thus, these behaviors
are pulled down the vertical axis.
 Toward the top of the vertical dimension are situations (movies, game, date) that restrict
behavior to the social/extroverted types of behaviors (eat, kiss, laugh). Thus, these behaviors
are pulled up the vertical axis.
 At the center of the vertical dimension, situations are separated on the horizontal dimension
based on the general restrictiveness of the situation. Those further from the behaviors
(interview) are the most restricted, while those closer to the behaviors (room, park) are
generally less restricted.

287
Proximity Transformations
Figure 15-30
Transformation plot
The proximities were treated as linear in this analysis, so the plot of the transformed values versus
the original proximities forms a straight line. The fit of this solution is good, but perhaps a better
fit can be achieved with a different transformation of the proximities.
Changing the Proximities Transformation (Ordinal)
E To produce a solution with an ordinal transformation of the proximities, click the Dialog Recall
tool and select Multidimensional Unfolding.

288
Chapter 15
E Click Model in the Multidimensional Unfolding dialog box.
Figure 15-31
Model dialog box
E Select Ordinal as the proximity transformation.
E Click Continue.
PREFSCAL
VARIABLES=Run Talk Kiss Write Eat Sleep Mumble Read Fight Belch Argue Jump
Cry Laugh Shout
/INPUT=ROWS(ROWID )
/INITIAL=( 'samplesDirectory/behavior_ini.sav' )
dim1 dim2
/CONDITION=UNCONDITIONAL
/TRANSFORMATION=ORDINAL (KEEPTIES)
/MODEL=IDENTITY
MAXITER(5000)
/PLOT=COMMON TRANSFORMATIONS .
 The only change is on the TRANSFORMATION subcommand. The transformation has been
set to ORDINAL, which preserves the order of proximities but does not require that the
transformed values be proportional to the original values.

289
Measures
Figure 15-32
Measures for solution with ordinal transformation
The algorithm converges after 268 iterations, with a final penalized stress of 0.6044671. This
statistic and the other measures are slightly better for this solution than the one with a linear
transformation of the proximities.

290
Chapter 15
Common Space
Figure 15-33
Joint plot of common space for solution with ordinal transformation
The interpretation of the common space is the same under both solutions. Perhaps this solution
(with the ordinal transformation) has relatively less variation on the vertical dimension than on the
horizontal dimension than is evident in the solution with the linear transformation.

291
Proximity Transformations
Figure 15-34
Transformation plot for solution with ordinal transformation
Aside from the values with the largest proximities, which bend up from the rest of the values, the
ordinal transformation of proximities is fairly linear. These proximities likely account for most of
the differences between the ordinal and linear solutions; however, there isn’t enough information
here to determine whether this nonlinear trend in the higher values is a true trend or an anomaly.
See the following texts for more information:
Busing, F. M. T. A., P. J. F. Groenen, and W. J. Heiser. 2005. Avoiding degeneracy in
multidimensional unfolding by penalizing on the coefficient of variation. Psychometrika, 70,
71–98.
Green, P. E., and V. Rao. 1972. Applied multidimensional scaling. Hinsdale, Ill.: Dryden Press.
Price, R. H., and D. L. Bouffard. 1974. Behavioral appropriateness and situational constraints as
dimensions of social behavior. Journal of Personality and Social Psychology, 30, 579–586.

Appendix
A
Sample Files
The sample files installed with the product can be found in the Samples subdirectory of the
installation directory. There is a separate folder within the Samples subdirectory for each of
the following languages: English, French, German, Italian, Japanese, Korean, Polish, Russian,
Simplified Chinese, Spanish, and Traditional Chinese.
Not all sample files are available in all languages. If a sample file is not available in a language,
that language folder contains an English version of the sample file.
Descriptions
Following are brief descriptions of the sample files used in various examples throughout the
documentation.
 accidents.sav. This is a hypothetical data file that concerns an insurance company that is
studying age and gender risk factors for automobile accidents in a given region. Each case
corresponds to a cross-classification of age category and gender.
 adl.sav. This is a hypothetical data file that concerns efforts to determine the benefits of a
proposed type of therapy for stroke patients. Physicians randomly assigned female stroke
patients to one of two groups. The first received the standard physical therapy, and the second
received an additional emotional therapy. Three months following the treatments, each
patient’s abilities to perform common activities of daily life were scored as ordinal variables.
 advert.sav. This is a hypothetical data file that concerns a retailer’s efforts to examine the
relationship between money spent on advertising and the resulting sales. To this end, they
have collected past sales figures and the associated advertising costs..
 aflatoxin.sav. This is a hypothetical data file that concerns the testing of corn crops for
aflatoxin, a poison whose concentration varies widely between and within crop yields.
A grain processor has received 16 samples from each of 8 crop yields and measured the
alfatoxin levels in parts per billion (PPB).
 anorectic.sav. While working toward a standardized symptomatology of anorectic/bulimic
behavior, researchers (Van der Ham, Meulman, Van Strien, and Van Engeland, 1997) made a
study of 55 adolescents with known eating disorders. Each patient was seen four times over
four years, for a total of 220 observations. At each observation, the patients were scored for
each of 16 symptoms. Symptom scores are missing for patient 71 at time 2, patient 76 at time
2, and patient 47 at time 3, leaving 217 valid observations.
 bankloan.sav. This is a hypothetical data file that concerns a bank’s efforts to reduce the
rate of loan defaults. The file contains financial and demographic information on 850 past
and prospective customers. The first 700 cases are customers who were previously given
loans. The last 150 cases are prospective customers that the bank needs to classify as good
or bad credit risks.
 bankloan_binning.sav. This is a hypothetical data file containing financial and demographic
information on 5,000 past customers.

293
Sample Files
 behavior.sav. In a classic example (Price and Bouffard, 1974), 52 students were asked to
rate the combinations of 15 situations and 15 behaviors on a 10-point scale ranging from
0=“extremely appropriate” to 9=“extremely inappropriate.” Averaged over individuals, the
values are taken as dissimilarities.
 behavior_ini.sav. This data file contains an initial configuration for a two-dimensional solution
for behavior.sav.
 brakes.sav. This is a hypothetical data file that concerns quality control at a factory that
produces disc brakes for high-performance automobiles. The data file contains diameter
measurements of 16 discs from each of 8 production machines. The target diameter for the
brakes is 322 millimeters.
 breakfast.sav. In a classic study (Green and Rao, 1972), 21 Wharton School MBA students
and their spouses were asked to rank 15 breakfast items in order of preference with 1=“most
preferred” to 15=“least preferred.” Their preferences were recorded under six different
scenarios, from “Overall preference” to “Snack, with beverage only.”
 breakfast-overall.sav. This data file contains the breakfast item preferences for the first
scenario, “Overall preference,” only.
 broadband_1.sav. This is a hypothetical data file containing the number of subscribers, by
region, to a national broadband service. The data file contains monthly subscriber numbers
for 85 regions over a four-year period.
 broadband_2.sav. This data file is identical to broadband_1.sav but contains data for three
additional months.
 car_insurance_claims.sav. A dataset presented and analyzed elsewhere (McCullagh and
Nelder, 1989) concerns damage claims for cars. The average claim amount can be modeled
as having a gamma distribution, using an inverse link function to relate the mean of the
dependent variable to a linear combination of the policyholder age, vehicle type, and vehicle
age. The number of claims filed can be used as a scaling weight.
 car_sales.sav. This data file contains hypothetical sales estimates, list prices, and physical
specifications for various makes and models of vehicles. The list prices and physical
specifications were obtained alternately from edmunds.com and manufacturer sites.
 car_sales_uprepared.sav. This is a modified version of car_sales.sav that does not include any
transformed versions of the fields.
 carpet.sav. In a popular example (Green and Wind, 1973), a company interested in
marketing a new carpet cleaner wants to examine the influence of five factors on consumer
preference—package design, brand name, price, a Good Housekeeping seal, and a
money-back guarantee. There are three factor levels for package design, each one differing
in the location of the applicator brush; three brand names (K2R, Glory, and Bissell); three
price levels; and two levels (either no or yes) for each of the last two factors. Ten consumers
rank 22 profiles defined by these factors. The variable Preference contains the rank of the
average rankings for each profile. Low rankings correspond to high preference. This variable
reflects an overall measure of preference for each profile.
 carpet_prefs.sav. This data file is based on the same example as described for carpet.sav, but it
contains the actual rankings collected from each of the 10 consumers. The consumers were
asked to rank the 22 product profiles from the most to the least preferred. The variables
PREF1 through PREF22 contain the identifiers of the associated profiles, as defined in
carpet_plan.sav.

294
Appendix A
 catalog.sav. This data file contains hypothetical monthly sales figures for three products sold
by a catalog company. Data for five possible predictor variables are also included.
 catalog_seasfac.sav. This data file is the same as catalog.sav except for the addition of a set
of seasonal factors calculated from the Seasonal Decomposition procedure along with the
accompanying date variables.
 cellular.sav. This is a hypothetical data file that concerns a cellular phone company’s efforts
to reduce churn. Churn propensity scores are applied to accounts, ranging from 0 to 100.
Accounts scoring 50 or above may be looking to change providers.
 ceramics.sav. This is a hypothetical data file that concerns a manufacturer’s efforts to
determine whether a new premium alloy has a greater heat resistance than a standard alloy.
Each case represents a separate test of one of the alloys; the heat at which the bearing failed is
recorded.
 cereal.sav. This is a hypothetical data file that concerns a poll of 880 people about their
breakfast preferences, also noting their age, gender, marital status, and whether or not they
have an active lifestyle (based on whether they exercise at least twice a week). Each case
represents a separate respondent.
 clothing_defects.sav. This is a hypothetical data file that concerns the quality control process
at a clothing factory. From each lot produced at the factory, the inspectors take a sample of
clothes and count the number of clothes that are unacceptable.
 coffee.sav. This data file pertains to perceived images of six iced-coffee brands (Kennedy,
Riquier, and Sharp, 1996) . For each of 23 iced-coffee image attributes, people selected all
brands that were described by the attribute. The six brands are denoted AA, BB, CC, DD, EE,
and FF to preserve confidentiality.
 contacts.sav. This is a hypothetical data file that concerns the contact lists for a group of
corporate computer sales representatives. Each contact is categorized by the department of
the company in which they work and their company ranks. Also recorded are the amount of
the last sale made, the time since the last sale, and the size of the contact’s company.
 creditpromo.sav. This is a hypothetical data file that concerns a department store’s efforts to
evaluate the effectiveness of a recent credit card promotion. To this end, 500 cardholders were
randomly selected. Half received an ad promoting a reduced interest rate on purchases made
over the next three months. Half received a standard seasonal ad.
 customer_dbase.sav. This is a hypothetical data file that concerns a company’s efforts to use
the information in its data warehouse to make special offers to customers who are most
likely to reply. A subset of the customer base was selected at random and given the special
offers, and their responses were recorded.
 customer_information.sav. A hypothetical data file containing customer mailing information,
such as name and address.
 customer_subset.sav. A subset of 80 cases from customer_dbase.sav.
 debate.sav. This is a hypothetical data file that concerns paired responses to a survey from
attendees of a political debate before and after the debate. Each case corresponds to a separate
respondent.
 debate_aggregate.sav. This is a hypothetical data file that aggregates the responses in
debate.sav. Each case corresponds to a cross-classification of preference before and after
the debate.

295
Sample Files
 demo.sav. This is a hypothetical data file that concerns a purchased customer database, for
the purpose of mailing monthly offers. Whether or not the customer responded to the offer
is recorded, along with various demographic information.
 demo_cs_1.sav. This is a hypothetical data file that concerns the first step of a company’s
efforts to compile a database of survey information. Each case corresponds to a different city,
and the region, province, district, and city identification are recorded.
 demo_cs_2.sav. This is a hypothetical data file that concerns the second step of a company’s
efforts to compile a database of survey information. Each case corresponds to a different
household unit from cities selected in the first step, and the region, province, district, city,
subdivision, and unit identification are recorded. The sampling information from the first
two stages of the design is also included.
 demo_cs.sav. This is a hypothetical data file that contains survey information collected using a
complex sampling design. Each case corresponds to a different household unit, and various
demographic and sampling information is recorded.
 dmdata.sav. This is a hypothetical data file that contains demographic and purchasing
information for a direct marketing company. dmdata2.sav contains information for a subset of
contacts that received a test mailing, and dmdata3.sav contains information on the remaining
contacts who did not receive the test mailing.
 dietstudy.sav. This hypothetical data file contains the results of a study of the “Stillman diet”
(Rickman, Mitchell, Dingman, and Dalen, 1974). Each case corresponds to a separate
subject and records his or her pre- and post-diet weights in pounds and triglyceride levels
in mg/100 ml.
 dvdplayer.sav. This is a hypothetical data file that concerns the development of a new DVD
player. Using a prototype, the marketing team has collected focus group data. Each case
corresponds to a separate surveyed user and records some demographic information about
them and their responses to questions about the prototype.
 german_credit.sav. This data file is taken from the “German credit” dataset in the Repository of
Machine Learning Databases (Blake and Merz, 1998) at the University of California, Irvine.
 grocery_1month.sav. This hypothetical data file is the grocery_coupons.sav data file with the
weekly purchases “rolled-up” so that each case corresponds to a separate customer. Some of
the variables that changed weekly disappear as a result, and the amount spent recorded is now
the sum of the amounts spent during the four weeks of the study.
 grocery_coupons.sav. This is a hypothetical data file that contains survey data collected by
a grocery store chain interested in the purchasing habits of their customers. Each customer
is followed for four weeks, and each case corresponds to a separate customer-week and
records information about where and how the customer shops, including how much was
spent on groceries during that week.
 guttman.sav. Bell (Bell, 1961) presented a table to illustrate possible social groups. Guttman
(Guttman, 1968) used a portion of this table, in which five variables describing such things
as social interaction, feelings of belonging to a group, physical proximity of members, and
formality of the relationship were crossed with seven theoretical social groups, including
crowds (for example, people at a football game), audiences (for example, people at a theater
or classroom lecture), public (for example, newspaper or television audiences), mobs (like a
crowd but with much more intense interaction), primary groups (intimate), secondary groups

296
Appendix A
(voluntary), and the modern community (loose confederation resulting from close physical
proximity and a need for specialized services).
 health_funding.sav. This is a hypothetical data file that contains data on health care funding
(amount per 100 population), disease rates (rate per 10,000 population), and visits to health
care providers (rate per 10,000 population). Each case represents a different city.
 hivassay.sav. This is a hypothetical data file that concerns the efforts of a pharmaceutical
lab to develop a rapid assay for detecting HIV infection. The results of the assay are eight
deepening shades of red, with deeper shades indicating greater likelihood of infection. A
laboratory trial was conducted on 2,000 blood samples, half of which were infected with
HIV and half of which were clean.
 hourlywagedata.sav. This is a hypothetical data file that concerns the hourly wages of nurses
from office and hospital positions and with varying levels of experience.
 insurance_claims.sav. This is a hypothetical data file that concerns an insurance company
that wants to build a model for flagging suspicious, potentially fraudulent claims. Each
case represents a separate claim.
 insure.sav. This is a hypothetical data file that concerns an insurance company that is studying
the risk factors that indicate whether a client will have to make a claim on a 10-year term
life insurance contract. Each case in the data file represents a pair of contracts, one of which
recorded a claim and the other didn’t, matched on age and gender.
 judges.sav. This is a hypothetical data file that concerns the scores given by trained judges
(plus one enthusiast) to 300 gymnastics performances. Each row represents a separate
performance; the judges viewed the same performances.
 kinship_dat.sav. Rosenberg and Kim (Rosenberg and Kim, 1975) set out to analyze 15 kinship
terms (aunt, brother, cousin, daughter, father, granddaughter, grandfather, grandmother,
grandson, mother, nephew, niece, sister, son, uncle). They asked four groups of college
students (two female, two male) to sort these terms on the basis of similarities. Two groups
(one female, one male) were asked to sort twice, with the second sorting based on a different
criterion from the first sort. Thus, a total of six “sources” were obtained. Each source
corresponds to a proximity matrix, whose cells are equal to the number of people in a
source minus the number of times the objects were partitioned together in that source.
 kinship_ini.sav. This data file contains an initial configuration for a three-dimensional solution
for kinship_dat.sav.
 kinship_var.sav. This data file contains independent variables gender, gener(ation), and degree
(of separation) that can be used to interpret the dimensions of a solution for kinship_dat.sav.
Specifically, they can be used to restrict the space of the solution to a linear combination of
these variables.
 marketvalues.sav. This data file concerns home sales in a new housing development in
Algonquin, Ill., during the years from 1999–2000. These sales are a matter of public record.
 nhis2000_subset.sav. The National Health Interview Survey (NHIS) is a large, population-based
survey of the U.S. civilian population. Interviews are carried out face-to-face in a nationally
representative sample of households. Demographic information and observations about
health behaviors and status are obtained for members of each household. This data
file contains a subset of information from the 2000 survey. National Center for Health
Statistics. National Health Interview Survey, 2000. Public-use data file and documentation.
ftp://ftp.cdc.gov/pub/Health_Statistics/NCHS/Datasets/NHIS/2000/. Accessed 2003.

297
Sample Files
 ozone.sav. The data include 330 observations on six meteorological variables for predicting
ozone concentration from the remaining variables. Previous researchers (Breiman and
Friedman, 1985), (Hastie and Tibshirani, 1990), among others found nonlinearities among
these variables, which hinder standard regression approaches.
 pain_medication.sav. This hypothetical data file contains the results of a clinical trial for
anti-inflammatory medication for treating chronic arthritic pain. Of particular interest is the
time it takes for the drug to take effect and how it compares to an existing medication.
 patient_los.sav. This hypothetical data file contains the treatment records of patients who were
admitted to the hospital for suspected myocardial infarction (MI, or “heart attack”). Each case
corresponds to a separate patient and records many variables related to their hospital stay.
 patlos_sample.sav. This hypothetical data file contains the treatment records of a sample
of patients who received thrombolytics during treatment for myocardial infarction (MI, or
“heart attack”). Each case corresponds to a separate patient and records many variables
related to their hospital stay.
 poll_cs.sav. This is a hypothetical data file that concerns pollsters’ efforts to determine the
level of public support for a bill before the legislature. The cases correspond to registered
voters. Each case records the county, township, and neighborhood in which the voter lives.
 poll_cs_sample.sav. This hypothetical data file contains a sample of the voters listed in
poll_cs.sav. The sample was taken according to the design specified in the poll.csplan plan
file, and this data file records the inclusion probabilities and sample weights. Note, however,
that because the sampling plan makes use of a probability-proportional-to-size (PPS) method,
there is also a file containing the joint selection probabilities (poll_jointprob.sav). The
additional variables corresponding to voter demographics and their opinion on the proposed
bill were collected and added the data file after the sample as taken.
 property_assess.sav. This is a hypothetical data file that concerns a county assessor’s efforts to
keep property value assessments up to date on limited resources. The cases correspond to
properties sold in the county in the past year. Each case in the data file records the township
in which the property lies, the assessor who last visited the property, the time since that
assessment, the valuation made at that time, and the sale value of the property.
 property_assess_cs.sav. This is a hypothetical data file that concerns a state assessor’s efforts
to keep property value assessments up to date on limited resources. The cases correspond
to properties in the state. Each case in the data file records the county, township, and
neighborhood in which the property lies, the time since the last assessment, and the valuation
made at that time.
 property_assess_cs_sample.sav. This hypothetical data file contains a sample of the properties
listed in property_assess_cs.sav. The sample was taken according to the design specified in
the property_assess.csplan plan file, and this data file records the inclusion probabilities
and sample weights. The additional variable Current value was collected and added to the
data file after the sample was taken.
 recidivism.sav. This is a hypothetical data file that concerns a government law enforcement
agency’s efforts to understand recidivism rates in their area of jurisdiction. Each case
corresponds to a previous offender and records their demographic information, some details
of their first crime, and then the time until their second arrest, if it occurred within two years
of the first arrest.

298
Appendix A
 recidivism_cs_sample.sav. This is a hypothetical data file that concerns a government law
enforcement agency’s efforts to understand recidivism rates in their area of jurisdiction. Each
case corresponds to a previous offender, released from their first arrest during the month of
June, 2003, and records their demographic information, some details of their first crime, and
the data of their second arrest, if it occurred by the end of June, 2006. Offenders were selected
from sampled departments according to the sampling plan specified in recidivism_cs.csplan;
because it makes use of a probability-proportional-to-size (PPS) method, there is also a file
containing the joint selection probabilities (recidivism_cs_jointprob.sav).
 rfm_transactions.sav. A hypothetical data file containing purchase transaction data, including
date of purchase, item(s) purchased, and monetary amount of each transaction.
 salesperformance.sav. This is a hypothetical data file that concerns the evaluation of two
new sales training courses. Sixty employees, divided into three groups, all receive standard
training. In addition, group 2 gets technical training; group 3, a hands-on tutorial. Each
employee was tested at the end of the training course and their score recorded. Each case in
the data file represents a separate trainee and records the group to which they were assigned
and the score they received on the exam.
 satisf.sav. This is a hypothetical data file that concerns a satisfaction survey conducted by
a retail company at 4 store locations. 582 customers were surveyed in all, and each case
represents the responses from a single customer.
 screws.sav. This data file contains information on the characteristics of screws, bolts, nuts,
and tacks (Hartigan, 1975).
 shampoo_ph.sav. This is a hypothetical data file that concerns the quality control at a factory
for hair products. At regular time intervals, six separate output batches are measured and their
pH recorded. The target range is 4.5–5.5.
 ships.sav. A dataset presented and analyzed elsewhere (McCullagh et al., 1989) that concerns
damage to cargo ships caused by waves. The incident counts can be modeled as occurring at
a Poisson rate given the ship type, construction period, and service period. The aggregate
months of service for each cell of the table formed by the cross-classification of factors
provides values for the exposure to risk.
 site.sav. This is a hypothetical data file that concerns a company’s efforts to choose new
sites for their expanding business. They have hired two consultants to separately evaluate
the sites, who, in addition to an extended report, summarized each site as a “good,” “fair,”
or “poor” prospect.
 smokers.sav. This data file is abstracted from the 1998 National Household
Survey of Drug Abuse and is a probability sample of American households.
(http://dx.doi.org/10.3886/ICPSR02934) Thus, the first step in an analysis of this data file
should be to weight the data to reflect population trends.
 stocks.sav This hypothetical data file contains stocks prices and volume for one year.
 stroke_clean.sav. This hypothetical data file contains the state of a medical database after it
has been cleaned using procedures in the Data Preparation option.
 stroke_invalid.sav. This hypothetical data file contains the initial state of a medical database
and contains several data entry errors.

299
Sample Files
 stroke_survival. This hypothetical data file concerns survival times for patients exiting a
rehabilitation program post-ischemic stroke face a number of challenges. Post-stroke, the
occurrence of myocardial infarction, ischemic stroke, or hemorrhagic stroke is noted and the
time of the event recorded. The sample is left-truncated because it only includes patients who
survived through the end of the rehabilitation program administered post-stroke.
 stroke_valid.sav. This hypothetical data file contains the state of a medical database after the
values have been checked using the Validate Data procedure. It still contains potentially
anomalous cases.
 survey_sample.sav. This data file contains survey data, including demographic data and
various attitude measures. It is based on a subset of variables from the 1998 NORC General
Social Survey, although some data values have been modified and additional fictitious
variables have been added for demonstration purposes.
 telco.sav. This is a hypothetical data file that concerns a telecommunications company’s
efforts to reduce churn in their customer base. Each case corresponds to a separate customer
and records various demographic and service usage information.
 telco_extra.sav. This data file is similar to the telco.sav data file, but the “tenure” and
log-transformed customer spending variables have been removed and replaced by
standardized log-transformed customer spending variables.
 telco_missing.sav. This data file is a subset of the telco.sav data file, but some of the
demographic data values have been replaced with missing values.
 testmarket.sav. This hypothetical data file concerns a fast food chain’s plans to add a new item
to its menu. There are three possible campaigns for promoting the new product, so the new
item is introduced at locations in several randomly selected markets. A different promotion
is used at each location, and the weekly sales of the new item are recorded for the first four
weeks. Each case corresponds to a separate location-week.
 testmarket_1month.sav. This hypothetical data file is the testmarket.sav data file with the
weekly sales “rolled-up” so that each case corresponds to a separate location. Some of the
variables that changed weekly disappear as a result, and the sales recorded is now the sum of
the sales during the four weeks of the study.
 tree_car.sav. This is a hypothetical data file containing demographic and vehicle purchase
price data.
 tree_credit.sav. This is a hypothetical data file containing demographic and bank loan history
data.
 tree_missing_data.sav This is a hypothetical data file containing demographic and bank loan
history data with a large number of missing values.
 tree_score_car.sav. This is a hypothetical data file containing demographic and vehicle
purchase price data.
 tree_textdata.sav. A simple data file with only two variables intended primarily to show the
default state of variables prior to assignment of measurement level and value labels.
 tv-survey.sav. This is a hypothetical data file that concerns a survey conducted by a TV studio
that is considering whether to extend the run of a successful program. 906 respondents were
asked whether they would watch the program under various conditions. Each row represents a
separate respondent; each column is a separate condition.

300
Appendix A
 ulcer_recurrence.sav. This file contains partial information from a study designed to compare
the efficacy of two therapies for preventing the recurrence of ulcers. It provides a good
example of interval-censored data and has been presented and analyzed elsewhere (Collett,
2003).
 ulcer_recurrence_recoded.sav. This file reorganizes the information in ulcer_recurrence.sav
to allow you model the event probability for each interval of the study rather than simply the
end-of-study event probability. It has been presented and analyzed elsewhere (Collett et
al., 2003).
 verd1985.sav. This data file concerns a survey (Verdegaal, 1985). The responses of 15 subjects
to 8 variables were recorded. The variables of interest are divided into three sets. Set 1
includes age and marital, set 2 includes pet and news, and set 3 includes music and live.
Pet is scaled as multiple nominal and age is scaled as ordinal; all of the other variables are
scaled as single nominal.
 virus.sav. This is a hypothetical data file that concerns the efforts of an Internet service
provider (ISP) to determine the effects of a virus on its networks. They have tracked the
(approximate) percentage of infected e-mail traffic on its networks over time, from the
moment of discovery until the threat was contained.
 wheeze_steubenville.sav. This is a subset from a longitudinal study of the health effects of
air pollution on children (Ware, Dockery, Spiro III, Speizer, and Ferris Jr., 1984). The data
contain repeated binary measures of the wheezing status for children from Steubenville, Ohio,
at ages 7, 8, 9 and 10 years, along with a fixed recording of whether or not the mother was
a smoker during the first year of the study.
 workprog.sav. This is a hypothetical data file that concerns a government works program
that tries to place disadvantaged people into better jobs. A sample of potential program
participants were followed, some of whom were randomly selected for enrollment in the
program, while others were not. Each case represents a separate program participant.
 worldsales.sav This hypothetical data file contains sales revenue by continent and product.

Appendix
B
Notices
This information was developed for products and services offered worldwide.
IBM may not offer the products, services, or features discussed in this document in other countries.
Consult your local IBM representative for information on the products and services currently
available in your area. Any reference to an IBM product, program, or service is not intended to
state or imply that only that IBM product, program, or service may be used. Any functionally
equivalent product, program, or service that does not infringe any IBM intellectual property right
may be used instead. However, it is the user’s responsibility to evaluate and verify the operation
of any non-IBM product, program, or service.
IBM may have patents or pending patent applications covering subject matter described in this
document. The furnishing of this document does not grant you any license to these patents.
You can send license inquiries, in writing, to:
IBM Director of Licensing, IBM Corporation, North Castle Drive, Armonk, NY 10504-1785,
U.S.A.
For license inquiries regarding double-byte character set (DBCS) information, contact the IBM
Intellectual Property Department in your country or send inquiries, in writing, to:
Intellectual Property Licensing, Legal and Intellectual Property Law, IBM Japan Ltd., 1623-14,
Shimotsuruma, Yamato-shi, Kanagawa 242-8502 Japan.
The following paragraph does not apply to the United Kingdom or any other country where such
provisions are inconsistent with local law: INTERNATIONAL BUSINESS MACHINES
PROVIDES THIS PUBLICATION “AS IS” WITHOUT WARRANTY OF ANY KIND,
EITHER EXPRESS OR IMPLIED, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED
WARRANTIES OF NON-INFRINGEMENT, MERCHANTABILITY OR FITNESS FOR A
PARTICULAR PURPOSE. Some states do not allow disclaimer of express or implied warranties
in certain transactions, therefore, this statement may not apply to you.
This information could include technical inaccuracies or typographical errors. Changes are
periodically made to the information herein; these changes will be incorporated in new editions
of the publication. IBM may make improvements and/or changes in the product(s) and/or the
program(s) described in this publication at any time without notice.
Any references in this information to non-IBM Web sites are provided for convenience only and
do not in any manner serve as an endorsement of those Web sites. The materials at those Web sites
are not part of the materials for this IBM product and use of those Web sites is at your own risk.
IBM may use or distribute any of the information you supply in any way it believes appropriate
without incurring any obligation to you.
Licensees of this program who wish to have information about it for the purpose of enabling: (i) the
exchange of information between independently created programs and other programs (including
this one) and (ii) the mutual use of the information which has been exchanged, should contact:
IBM Software Group, Attention: Licensing, 233 S. Wacker Dr., Chicago, IL 60606, USA.

302
Appendix B
Such information may be available, subject to appropriate terms and conditions, including in
some cases, payment of a fee.
The licensed program described in this document and all licensed material available for it are
provided by IBM under terms of the IBM Customer Agreement, IBM International Program
License Agreement or any equivalent agreement between us.
Information concerning non-IBM products was obtained from the suppliers of those products,
their published announcements or other publicly available sources. IBM has not tested those
products and cannot confirm the accuracy of performance, compatibility or any other claims
related to non-IBM products. Questions on the capabilities of non-IBM products should be
addressed to the suppliers of those products.
This information contains examples of data and reports used in daily business operations.
To illustrate them as completely as possible, the examples include the names of individuals,
companies, brands, and products. All of these names are fictitious and any similarity to the names
and addresses used by an actual business enterprise is entirely coincidental.
If you are viewing this information softcopy, the photographs and color illustrations may not
appear.
Trademarks
IBM, the IBM logo, ibm.com, and SPSS are trademarks of IBM Corporation, registered in
many jurisdictions worldwide. A current list of IBM trademarks is available on the Web at
http://www.ibm.com/legal/copytrade.shtml.
Adobe, the Adobe logo, PostScript, and the PostScript logo are either registered trademarks or
trademarks of Adobe Systems Incorporated in the United States, and/or other countries.
Intel, Intel logo, Intel Inside, Intel Inside logo, Intel Centrino, Intel Centrino logo, Celeron, Intel
Xeon, Intel SpeedStep, Itanium, and Pentium are trademarks or registered trademarks of Intel
Corporation or its subsidiaries in the United States and other countries.
Java and all Java-based trademarks and logos are trademarks of Sun Microsystems, Inc. in the
United States, other countries, or both.
Linux is a registered trademark of Linus Torvalds in the United States, other countries, or both.
Microsoft, Windows, Windows NT, and the Windows logo are trademarks of Microsoft
Corporation in the United States, other countries, or both.
UNIX is a registered trademark of The Open Group in the United States and other countries.
This product uses WinWrap Basic, Copyright 1993-2007, Polar Engineering and Consulting,
http://www.winwrap.com.
Other product and service names might be trademarks of IBM or other companies.
Adobe product screenshot(s) reprinted with permission from Adobe Systems Incorporated.

303
Notices
Microsoft product screenshot(s) reprinted with permission from Microsoft Corporation.

Bibliography
Barlow, R. E., D. J. Bartholomew, D. J. Bremner, and H. D. Brunk. 1972. Statistical inference
under order restrictions. New York: John Wiley and Sons.
Bell, E. H. 1961. Social foundations of human behavior: Introduction to the study of sociology.
New York: Harper & Row.
Benzécri, J. P. 1969. Statistical analysis as a tool to make patterns emerge from data. In:
Methodologies of Pattern Recognition, S. Watanabe, ed. New York: Academic Press, 35–74.
Benzécri, J. P. 1992. Correspondence analysis handbook. New York: Marcel Dekker.
Bishop, Y. M., S. E. Feinberg, and P. W. Holland. 1975. Discrete multivariate analysis: Theory
and practice. Cambridge, Mass.: MIT Press.
Blake, C. L., and C. J. Merz. 1998. "UCI Repository of machine learning databases." Available at
http://www.ics.uci.edu/~mlearn/MLRepository.html.
Breiman, L., and J. H. Friedman. 1985. Estimating optimal transformations for multiple
regression and correlation. Journal of the American Statistical Association, 80, 580–598.
Buja, A. 1990. Remarks on functional canonical variates, alternating least squares methods
and ACE. Annals of Statistics, 18, 1032–1069.
Busing, F. M. T. A., P. J. F. Groenen, and W. J. Heiser. 2005. Avoiding degeneracy in
multidimensional unfolding by penalizing on the coefficient of variation. Psychometrika, 70,
71–98.
Carroll, J. D. 1968. Generalization of canonical correlation analysis to three or more sets of
variables. In: Proceedings of the 76th Annual Convention of the American Psychological
Association, 3, Washington, D.C.: American Psychological Association, 227–228.
Collett, D. 2003. Modelling survival data in medical research, 2 ed. Boca Raton: Chapman &
Hall/CRC.
Commandeur, J. J. F., and W. J. Heiser. 1993. Mathematical derivations in the proximity scaling
(PROXSCAL) of symmetric data matrices. Leiden: Department of Data Theory, University of
Leiden.
De Haas, M., J. A. Algera, H. F. J. M. Van Tuijl, and J. J. Meulman. 2000. Macro and micro goal
setting: In search of coherence. Applied Psychology, 49, 579–595.
De Leeuw, J. 1982. Nonlinear principal components analysis. In: COMPSTAT Proceedings in
Computational Statistics, Vienna: Physica Verlag, 77–89.
De Leeuw, J. 1984. Canonical analysis of categorical data, 2nd ed. Leiden: DSWO Press.
De Leeuw, J. 1984. The Gifi system of nonlinear multivariate analysis. In: Data Analysis and
Informatics III, E. Diday, et al., ed., 415–424.
De Leeuw, J., and W. J. Heiser. 1980. Multidimensional scaling with restrictions on the
configuration. In: Multivariate Analysis, Vol. V, P. R. Krishnaiah, ed. Amsterdam: North-Holland,
501–522.
De Leeuw, J., and J. Van Rijckevorsel. 1980. HOMALS and PRINCALS—Some generalizations
of principal components analysis. In: Data Analysis and Informatics, E. Diday,et al., ed.
Amsterdam: North-Holland, 231–242.
De Leeuw, J., F. W. Young, and Y. Takane. 1976. Additive structure in qualitative data: An
alternating least squares method with optimal scaling features. Psychometrika, 41, 471–503.

305
Bibliography
De Leeuw, J. 1990. Multivariate analysis with optimal scaling. In: Progress in Multivariate
Analysis, S. Das Gupta, and J. Sethuraman, eds. Calcutta: Indian Statistical Institute.
Eckart, C., and G. Young. 1936. The approximation of one matrix by another one of lower
rank. Psychometrika, 1, 211–218.
Fisher, R. A. 1938. Statistical methods for research workers. Edinburgh: Oliver and Boyd.
Fisher, R. A. 1940. The precision of discriminant functions. Annals of Eugenics, 10, 422–429.
Gabriel, K. R. 1971. The biplot graphic display of matrices with application to principal
components analysis. Biometrika, 58, 453–467.
Gifi, A. 1985. PRINCALS. Research Report UG-85-02. Leiden: Department of Data Theory,
Gifi, A. 1990. Nonlinear multivariate analysis. Chichester: John Wiley and Sons.
Gilula, Z., and S. J. Haberman. 1988. The analysis of multivariate contingency tables by restricted
canonical and restricted association models. Journal of the American Statistical Association, 83,
760–771.
Gower, J. C., and J. J. Meulman. 1993. The treatment of categorical information in physical
anthropology. International Journal of Anthropology, 8, 43–51.
Green, P. E., and V. Rao. 1972. Applied multidimensional scaling. Hinsdale, Ill.: Dryden Press.
Green, P. E., and Y. Wind. 1973. Multiattribute decisions in marketing: A measurement approach.
Hinsdale, Ill.: Dryden Press.
Guttman, L. 1941. The quantification of a class of attributes: A theory and method of scale
construction. In: The Prediction of Personal Adjustment, P. Horst, ed. New York: Social Science
Research Council, 319–348.
Guttman, L. 1968. A general nonmetric technique for finding the smallest coordinate space for
configurations of points. Psychometrika, 33, 469–506.
Hartigan, J. A. 1975. Clustering algorithms. New York: John Wiley and Sons.
Hastie, T., and R. Tibshirani. 1990. Generalized additive models. London: Chapman and Hall.
Hastie, T., R. Tibshirani, and A. Buja. 1994. Flexible discriminant analysis. Journal of the
American Statistical Association, 89, 1255–1270.
Hayashi, C. 1952. On the prediction of phenomena from qualitative data and the quantification
of qualitative data from the mathematico-statistical point of view. Annals of the Institute of
Statitical Mathematics, 2, 93–96.
Heiser, W. J. 1981. Unfolding analysis of proximity data. Leiden: Department of Data Theory,
Heiser, W. J., and F. M. T. A. Busing. 2004. Multidimensional scaling and unfolding of symmetric
and asymmetric proximity relations. In: Handbook of Quantitative Methodology for the Social
Sciences, D. Kaplan, ed. Thousand Oaks, Calif.: SagePublications, Inc., 25–48.
Heiser, W. J., and J. J. Meulman. 1994. Homogeneity analysis: Exploring the distribution of
variables and their nonlinear relationships. In: Correspondence Analysis in the Social Sciences:
Recent Developments and Applications, M. Greenacre, and J. Blasius, eds. New York: Academic
Press, 179–209.

306
Bibliography
Heiser, W. J., and J. J. Meulman. 1995. Nonlinear methods for the analysis of homogeneity and
heterogeneity. In: Recent Advances in Descriptive Multivariate Analysis, W. J. Krzanowski,
ed. Oxford: Oxford UniversityPress, 51–89.
Horst, P. 1961. Generalized canonical correlations and their applications to experimental data.
Journal of Clinical Psychology, 17, 331–347.
Horst, P. 1961. Relations among m sets of measures. Psychometrika, 26, 129–149.
Israëls, A. 1987. Eigenvalue techniques for qualitative data. Leiden: DSWO Press.
Kennedy, R., C. Riquier, and B. Sharp. 1996. Practical applications of correspondence analysis
to categorical data in market research. Journal of Targeting, Measurement, and Analysis for
Marketing, 5, 56–70.
Kettenring, J. R. 1971. Canonical analysis of several sets of variables. Biometrika, 58, 433–460.
Kruskal, J. B. 1964. Multidimensional scaling by optimizing goodness of fit to a nonmetric
hypothesis. Psychometrika, 29, 1–28.
Kruskal, J. B. 1964. Nonmetric multidimensional scaling: A numerical method. Psychometrika,
29, 115–129.
Kruskal, J. B. 1965. Analysis of factorial experiments by estimating monotone transformations of
the data. Journal of the Royal Statistical Society Series B, 27, 251–263.
Kruskal, J. B. 1978. Factor analysis and principal components analysis: Bilinear methods. In:
International Encyclopedia of Statistics, W. H. Kruskal, and J. M. Tanur, eds. New York: The
Free Press, 307–330.
Kruskal, J. B., and R. N. Shepard. 1974. A nonmetric variety of linear factor analysis.
Krzanowski, W. J., and F. H. C. Marriott. 1994. Multivariate analysis: Part I, distributions,
ordination and inference. London: Edward Arnold.
Lebart, L., A. Morineau, and K. M. Warwick. 1984. Multivariate descriptive statistical analysis.
New York: John Wiley and Sons.
Lingoes, J. C. 1968. The multivariate analysis of qualitative data. Multivariate Behavioral
Research, 3, 61–94.
Max, J. 1960. Quantizing for minimum distortion. Proceedings IEEE (Information Theory), 6,
7–12.
McCullagh, P., and J. A. Nelder. 1989. Generalized Linear Models, 2nd ed. London: Chapman &
Hall.
Meulman, J. J. 1982. Homogeneity analysis of incomplete data. Leiden: DSWO Press.
Meulman, J. J. 1986. A distance approach to nonlinear multivariate analysis. Leiden: DSWO
Press.
Meulman, J. J. 1992. The integration of multidimensional scaling and multivariate analysis with
optimal transformations of the variables. Psychometrika, 57, 539–565.
Meulman, J. J. 1993. Principal coordinates analysis with optimal transformations of the variables:
Minimizing the sum of squares of the smallest eigenvalues. British Journal of Mathematical and
Statistical Psychology, 46, 287–300.
Meulman, J. J. 1996. Fitting a distance model to homogeneous subsets of variables: Points of
view analysis of categorical data. Journal of Classification, 13, 249–266.

307
Bibliography
Meulman, J. J. 2003. Prediction and classification in nonlinear data analysis: Something old,
something new, something borrowed, something blue. Psychometrika, 4, 493–517.
Meulman, J. J., and W. J. Heiser. 1997. Graphical display of interaction in multiway contingency
tables by use of homogeneity analysis. In: Visual Display of Categorical Data, M. Greenacre, and
J. Blasius, eds. New York: Academic Press, 277–296.
Meulman, J. J., and P. Verboon. 1993. Points of view analysis revisited: Fitting multidimensional
structures to optimal distance components with cluster restrictions on the variables.
Meulman, J. J., A. J. Van der Kooij, and A. Babinec. 2000. New features of categorical principal
components analysis for complicated data sets, including data mining. In: Classification,
Automation and New Media, W. Gaul, and G. Ritter, eds. Berlin: Springer-Verlag, 207–217.
Meulman, J. J., A. J. Van der Kooij, and W. J. Heiser. 2004. Principal components analysis
with nonlinear optimal scaling transformations for ordinal and nominal data. In: Handbook of
Quantitative Methodology for the Social Sciences, D. Kaplan, ed. Thousand Oaks, Calif.: Sage
Publications, Inc., 49–70.
Nishisato, S. 1980. Analysis of categorical data: Dual scaling and its applications. Toronto:
University of Toronto Press.
Nishisato, S. 1984. Forced classification: A simple application of a quantification method.
Nishisato, S. 1994. Elements of dual scaling: An introduction to practical data analysis. Hillsdale,
N.J.: Lawrence Erlbaum Associates, Inc.
Pratt, J. W. 1987. Dividing the indivisible: Using simple symmetry to partition variance explained.
In: Proceedings of the Second International Conference in Statistics, T. Pukkila, and S. Puntanen,
eds. Tampere, Finland: Universityof Tampere, 245–260.
Price, R. H., and D. L. Bouffard. 1974. Behavioral appropriateness and situational constraints as
dimensions of social behavior. Journal of Personality and Social Psychology, 30, 579–586.
Ramsay, J. O. 1989. Monotone regression splines in action. Statistical Science, 4, 425–441.
Rao, C. R. 1973. Linear statistical inference and its applications, 2nd ed. New York: John
Wiley and Sons.
Rao, C. R. 1980. Matrix approximations and reduction of dimensionality in multivariate statistical
analysis. In: Multivariate Analysis, Vol. 5, P. R. Krishnaiah, ed. Amsterdam: North-Holland,
3–22.
Rickman, R., N. Mitchell, J. Dingman, and J. E. Dalen. 1974. Changes in serum cholesterol
during the Stillman Diet. Journal of the American Medical Association, 228, 54–58.
Rosenberg, S., and M. P. Kim. 1975. The method of sorting as a data-gathering procedure in
multivariate research. Multivariate Behavioral Research, 10, 489–502.
Roskam, E. E. 1968. Metric analysis of ordinal data in psychology. Voorschoten: VAM.
distance function I. Psychometrika, 27, 125–140.
distance function II. Psychometrika, 27, 219–246.
Shepard, R. N. 1966. Metric structures in ordinal data. Journal of Mathematical Psychology, 3,
287–315.

308
Bibliography
Tenenhaus, M., and F. W. Young. 1985. An analysis and synthesis of multiple correspondence
analysis, optimal scaling, dual scaling, homogeneity analysis, and other methods for quantifying
categorical multivariate data. Psychometrika, 50, 91–119.
Theunissen, N. C. M., J. J. Meulman, A. L. Den Ouden, H. M. Koopman, G. H. Verrips, S.
P. Verloove-Vanhorick, and J. M. Wit. 2003. Changes can be studied when the measurement
instrument is different at different time points. Health Services and Outcomes Research
Methodology, 4, 109–126.
Tucker, L. R. 1960. Intra-individual and inter-individual multidimensionality. In: Psychological
Scaling: Theory & Applications, H. Gulliksen, and S. Messick, eds. NewYork: John Wiley
and Sons, 155–167.
Van der Burg, E. 1988. Nonlinear canonical correlation and some related techniques. Leiden:
DSWO Press.
Van der Burg, E., and J. De Leeuw. 1983. Nonlinear canonical correlation. British Journal of
Mathematical and Statistical Psychology, 36, 54–80.
Van der Burg, E., J. De Leeuw, and R. Verdegaal. 1988. Homogeneity analysis with k sets of
177–197.
Van der Ham, T., J. J. Meulman, D. C. Van Strien, and H. Van Engeland. 1997. Empirically based
subgrouping of eating disorders in adolescents: A longitudinal perspective. British Journal of
Psychiatry, 170, 363–368.
Van der Kooij, A. J., and J. J. Meulman. 1997. MURALS: Multiple regression and optimal
scaling using alternating least squares. In: Softstat ’97, F. Faulbaum, and W. Bandilla, eds.
Stuttgart: Gustav Fisher, 99–106.
Van Rijckevorsel, J. 1987. The application of fuzzy coding and horseshoes in multiple
correspondence analysis. Leiden: DSWO Press.
Verboon, P., and I. A. Van der Lans. 1994. Robust canonical discriminant analysis. Psychometrika,
59, 485–507.
Verdegaal, R. 1985. Meer sets analyse voor kwalitatieve gegevens (in Dutch). Leiden: Department
of Data Theory, University of Leiden.
Vlek, C., and P. J. Stallen. 1981. Judging risks and benefits in the small and in the large.
Organizational Behavior and Human Performance, 28, 235–271.
Wagenaar, W. A. 1988. Paradoxes of gambling behaviour. London: Lawrence Erlbaum
Associates, Inc.
Ware, J. H., D. W. Dockery, A. Spiro III, F. E. Speizer, and B. G. Ferris Jr.. 1984. Passive
smoking, gas cooking, and respiratory health of children living in six cities. American Review of
Respiratory Diseases, 129, 366–374.
Winsberg, S., and J. O. Ramsay. 1980. Monotonic transformations to additivity using splines.
Biometrika, 67, 669–674.
Winsberg, S., and J. O. Ramsay. 1983. Monotone spline transformations for dimension reduction.
Wolter, K. M. 1985. Introduction to variance estimation. Berlin: Springer-Verlag.
Young, F. W. 1981. Quantitative analysis of qualitative data. Psychometrika, 46, 357–387.

309
Bibliography
Young, F. W., J. De Leeuw, and Y. Takane. 1976. Regression with qualitative and quantitative
505–528.
Young, F. W., Y. Takane, and J. De Leeuw. 1978. The principal components of mixed
measurement level multivariate data: An alternating least squares method with optimal scaling
features. Psychometrika, 43, 279–281.
Zeijl, E., Y. te Poel, M. du Bois-Reymond, J. Ravesloot, and J. J. Meulman. 2000. The role of
parents and peers in the leisure activities of young adolescents. Journal of Leisure Research, 32,
281–302.

Index
ANOVA
in Categorical Regression, 21
biplots
in Categorical Principal Components Analysis, 36
in Correspondence Analysis, 52
in Multiple Correspondence Analysis, 63
Categorical Principal Components Analysis, 26, 32, 137,
149
category points, 168
command additional features, 39
component loadings, 146, 149, 165
iteration history, 142
model summary, 142, 148, 164
object scores, 145, 148, 166
optimal scaling level, 28
quantifications, 143, 162
save variables, 35
Categorical Regression, 14, 94
correlations, 106–107
importance, 107
intercorrelations, 105
model fit, 106
plots, 14
regularization, 20
residuals, 111
save, 23
statistics, 14
transformation plots, 109
category coordinates
in Nonlinear Canonical Correlation Analysis, 200
category plots
category points
category quantifications
in Multiple Correspondence Analysis, 61, 229
centroids
in Nonlinear Canonical Correlation Analysis, 44, 201
coefficient of variation
in Multidimensional Unfolding, 261, 264, 270, 276, 285
coefficients
column principal normalization
column scores plots
common space
in Multidimensional Scaling, 251, 255
in Multidimensional Unfolding, 262, 265, 271, 277,
286, 290
common space coordinates
in Multidimensional Scaling, 79
in Multidimensional Unfolding, 90
common space plots
component loadings
in Categorical Principal Components Analysis, 34, 146,
149, 165
component loadings plots
confidence statistics
contributions
correlation matrix
correlations
correlations plots
Correspondence Analysis, 46–49, 51–52, 211–212
column scores plots, 219
contributions, 218
dimensions, 217
normalization, 212
plots, 46
row scores plots, 219
statistics, 46
Cronbach’s alpha
DeSarbo’s intermixedness indices
descriptive statistics
dimension weights
in Multidimensional Unfolding, 272, 278
dimensions
in Correspondence Analysis, 49, 217
discretization
discrimination measures
in Multiple Correspondence Analysis, 61, 228
discrimination measures plots
310

311
Index
distance measures
distances
eigenvalues
in Categorical Principal Components Analysis, 142,
148, 164
elastic net
final common space plots
fit
fit values
generalized Euclidean model
identity model
importance
individual space coordinates
individual space weights
individual space weights plots
individual spaces
individual spaces plots
inertia
initial common space plots
initial configuration
intercorrelations
iteration criteria
iteration history
in Categorical Principal Components Analysis, 34, 142
joint category plots
joint plot of common space
in Multidimensional Unfolding, 262, 265, 271, 277,
286, 290
joint plot of individual spaces
lasso
legal notices, 301
loss values
missing values
model summary
Multidimensional Scaling, 67, 69–73, 239
common space, 251, 255
model, 74
options, 76
output, 79
plots, 67, 78–79
restrictions, 75
statistics, 67
stress measures, 250, 255
transformation plots, 254
Multidimensional Unfolding, 82, 258, 280
common space, 262, 265, 271, 277, 286, 290
degenerate solutions, 258
individual spaces, 272, 278
measures, 261, 264, 270, 276, 285, 289
model, 83
options, 86
output, 90
plots, 82, 88
proximity transformations, 287, 291
restrictions on common space, 85
statistics, 82
three-way unfolding , 265
Multiple Correspondence Analysis, 55, 60, 223
category quantifications, 229
discrimination measures, 228
model summary, 226

312
Index
object scores, 227, 231
outliers, 233
save variables, 63
multiple R
multiple starts plots
Nonlinear Canonical Correlation Analysis, 40, 43, 186
category coordinates, 200
centroids, 201
component loadings, 195, 197
plots, 40
quantifications, 197
statistics, 40
summary of analysis, 193
weights, 195
normalization
in Correspondence Analysis, 49, 212
object points plots
object scores
148, 166
in Multiple Correspondence Analysis, 61, 227, 231
optimal scaling level
outliers
part correlations
partial correlations
penalized stress
penalty term
plots
in Multidimensional Scaling, 78–79
PREFSCAL, 82
principal normalization
projected centroids
projected centroids plots
proximity transformations
quantifications
in Categorical Principal Components Analysis, 143, 162
R2
regression coefficients
relaxed updates
residuals
residuals plots
restrictions
restrictions on common space
ridge regression
row principal normalization
row scores plots
sample files
location, 292
scaling model
scatterplot of fit
Shepard plots
Shepard’s rough nondegeneracy index
space weights plots
standardization
stress measures
in Multidimensional Scaling, 79, 250, 255
stress plots
supplementary objects
symmetrical normalization
three-way unfolding

313
Index
trademarks, 302
transformation plots
in Multidimensional Scaling, 78, 254
in Multidimensional Unfolding, 88, 287, 291
transformed independent variables
transformed proximities
triplots
variable weight
variance accounted for
164
weighted Euclidean model
weights
zero-order correlations

Ibm spss categories

More Related Content

What's hot (15)

Viewers also liked (7)

Similar to Ibm spss categories (20)

Recently uploaded (20)

Ibm spss categories