![xiv preface to the third edition
To a large extent, there are no major differences between the many software
packages that include logistic regression modeling. When a particular approach
is available in a limited number of packages, it will be noted in this text. In
general, analyses in this book have been performed using STATA [Stata Corp.
(2011)]. This easy-to-use package combines excellent graphics and analysis rou-
tines; is fast; is compatible across Macintosh, Windows and UNIX platforms; and
interacts well with Microsoft Word. Other major statistical packages employed
at various points during the preparation of this text include SAS [SAS Institute
Inc. (2009)], OpenBUGS [Lunn et al. (2009)] and R [R Development Core Team
(2010)]. For all intents and purposes the results produced were the same regard-
less of which package we used. Reported numeric results have been rounded from
figures obtained from computer output and thus may differ slightly from those that
would be obtained in a replication of our analyses or from calculations based on
the reported results. When features or capabilities of the programs differed in an
important way, we noted them by the names given rather than by their bibliographic
citation.
We feel that this new edition benefits greatly from the addition of a number of
key topics. These include the following:
1. An expanded presentation of numerous new techniques for model-building,
including methods for determining the scale of continuous covariates and
assessing model performance.
2. An expanded presentation of regression modeling of complex sample survey
data.
3. An expanded development of the use of logistic regression modeling in
matched studies, as well as with multinomial and ordinal scaled responses.
4. A new chapter dealing with models and methods for correlated categorical
response data.
5. A new chapter developing a number of important applications either miss-
ing or expanded from the previous editions. These include propensity score
methods, exact methods for logistic regression, sample size issues, Bayesian
logistic regression, and other link functions for binary outcome regression
models. This chapter concludes with sections dealing with the epidemiologic
concepts of mediation and additive interaction.
As was the case for the second edition, all of the data sets used in the text are
available at a web site at John Wiley & Sons, Inc.
http://wiley.mpstechnologies.com/wiley/BOBContent/searchLPBobContent.do
In addition, the data may also be found, by permission of John Wiley &
Sons Inc., in the archive of statistical data sets maintained at the University of
Massachusetts at http://www.umass.edu/statdata/statdata in the logistic regression
section.
We would like to express our sincere thanks and appreciation to our colleagues,
students, and staff at all of the institutions we have been fortunate to have been
affiliated with since the first edition was conceived more than 25 years ago. This](https://image.slidesharecdn.com/70842-250412101346-80a0998d/85/Applied-Logistic-Regression-3rd-Edition-David-Hosmer-17-320.jpg)
Applied Logistic Regression 3rd Edition David Hosmer
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- 5. Applied Logistic Regression 3rd Edition David Hosmer Digital Instant Download Author(s): David Hosmer, Stanley Lemeshow, Rodney Sturdivant ISBN(s): 9780470582473, 0470582472 Edition: 3rd File Details: PDF, 4.14 MB Year: 2013 Language: english
- 8. Applied Logistic Regression Third Edition DAVID W. HOSMER, JR. Professor of Biostatistics (Emeritus) Division of Biostatistics and Epidemiology Department of Public Health School of Public Health and Health Sciences University of Massachusetts Amherst, Massachusetts STANLEY LEMESHOW Dean, College of Public Health Professor of Biostatistics College of Public Health The Ohio State University Columbus, Ohio RODNEY X. STURDIVANT Colonel, U.S. Army Academy and Associate Professor Department of Mathematical Sciences United States Military Academy West Point, New York
- 9. Copyright © 2013 by John Wiley & Sons, Inc. All rights reserved. Published by John Wiley & Sons, Inc., Hoboken, New Jersey. Published simultaneously in Canada. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400, fax (978) 750-4470, or on the web at www.copyright.com. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley &Sons, Inc., 111 River Street, Hoboken, NJ 07030, (201) 748-6011, fax (201) 748-6008, or online at http://www.wiley.com/go/permission. Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their best efforts in preparing this book, they make no representations or warranties with respect to the accuracy or completeness of the contents of this book and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose. No warranty may be created or extended by sales representatives or written sales materials. The advice and strategies contained herein may not be suitable for your situation. You should consult with a professional where appropriate. Neither the publisher nor author shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages. For general information on our other products and services or for technical support, please contact our Customer Care Department within the United States at (800) 762-2974, outside the United States at (317) 572-3993 or fax (317) 572-4002. Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic formats. For more information about Wiley products, visit our web site at www.wiley.com. Library of Congress Cataloging-in-Publication Data Is Available Hosmer, David W. Applied Logistic Regression / David W. Hosmer, Jr., Stanley Lemeshow, Rodney X. Sturdivant. - 3rd ed. Includes bibliographic references and index. ISBN 978-0-470-58247-3 (cloth) Printed in the United States of America 10 9 8 7 6 5 4 3 2 1
- 10. To our wives, Trina, Elaine, and Mandy, and our sons, daughters, and grandchildren
- 11. Contents Preface to the Third Edition xiii 1 Introduction to the Logistic Regression Model 1 1.1 Introduction, 1 1.2 Fitting the Logistic Regression Model, 8 1.3 Testing for the Significance of the Coefficients, 10 1.4 Confidence Interval Estimation, 15 1.5 Other Estimation Methods, 20 1.6 Data Sets Used in Examples and Exercises, 22 1.6.1 The ICU Study, 22 1.6.2 The Low Birth Weight Study, 24 1.6.3 The Global Longitudinal Study of Osteoporosis in Women, 24 1.6.4 The Adolescent Placement Study, 26 1.6.5 The Burn Injury Study, 27 1.6.6 The Myopia Study, 29 1.6.7 The NHANES Study, 31 1.6.8 The Polypharmacy Study, 31 Exercises, 32 2 The Multiple Logistic Regression Model 35 2.1 Introduction, 35 2.2 The Multiple Logistic Regression Model, 35 2.3 Fitting the Multiple Logistic Regression Model, 37 2.4 Testing for the Significance of the Model, 39 2.5 Confidence Interval Estimation, 42 2.6 Other Estimation Methods, 45 Exercises, 46 vii
- 12. viii contents 3 Interpretation of the Fitted Logistic Regression Model 49 3.1 Introduction, 49 3.2 Dichotomous Independent Variable, 50 3.3 Polychotomous Independent Variable, 56 3.4 Continuous Independent Variable, 62 3.5 Multivariable Models, 64 3.6 Presentation and Interpretation of the Fitted Values, 77 3.7 A Comparison of Logistic Regression and Stratified Analysis for 2 × 2 Tables, 82 Exercises, 87 4 Model-Building Strategies and Methods for Logistic Regression 89 4.1 Introduction, 89 4.2 Purposeful Selection of Covariates, 89 4.2.1 Methods to Examine the Scale of a Continuous Covariate in the Logit, 94 4.2.2 Examples of Purposeful Selection, 107 4.3 Other Methods for Selecting Covariates, 124 4.3.1 Stepwise Selection of Covariates, 125 4.3.2 Best Subsets Logistic Regression, 133 4.3.3 Selecting Covariates and Checking their Scale Using Multivariable Fractional Polynomials, 139 4.4 Numerical Problems, 145 Exercises, 150 5 Assessing the Fit of the Model 153 5.1 Introduction, 153 5.2 Summary Measures of Goodness of Fit, 154 5.2.1 Pearson Chi-Square Statistic, Deviance, and Sum-of-Squares, 155 5.2.2 The Hosmer–Lemeshow Tests, 157 5.2.3 Classification Tables, 169 5.2.4 Area Under the Receiver Operating Characteristic Curve, 173 5.2.5 Other Summary Measures, 182 5.3 Logistic Regression Diagnostics, 186 5.4 Assessment of Fit via External Validation, 202
- 13. contents ix 5.5 Interpretation and Presentation of the Results from a Fitted Logistic Regression Model, 212 Exercises, 223 6 Application of Logistic Regression with Different Sampling Models 227 6.1 Introduction, 227 6.2 Cohort Studies, 227 6.3 Case-Control Studies, 229 6.4 Fitting Logistic Regression Models to Data from Complex Sample Surveys, 233 Exercises, 242 7 Logistic Regression for Matched Case-Control Studies 243 7.1 Introduction, 243 7.2 Methods For Assessment of Fit in a 1–M Matched Study, 248 7.3 An Example Using the Logistic Regression Model in a 1–1 Matched Study, 251 7.4 An Example Using the Logistic Regression Model in a 1–M Matched Study, 260 Exercises, 267 8 Logistic Regression Models for Multinomial and Ordinal Outcomes 269 8.1 The Multinomial Logistic Regression Model, 269 8.1.1 Introduction to the Model and Estimation of Model Parameters, 269 8.1.2 Interpreting and Assessing the Significance of the Estimated Coefficients, 272 8.1.3 Model-Building Strategies for Multinomial Logistic Regression, 278 8.1.4 Assessment of Fit and Diagnostic Statistics for the Multinomial Logistic Regression Model, 283 8.2 Ordinal Logistic Regression Models, 289 8.2.1 Introduction to the Models, Methods for Fitting, and Interpretation of Model Parameters, 289 8.2.2 Model Building Strategies for Ordinal Logistic Regression Models, 305 Exercises, 310
- 14. x contents 9 Logistic Regression Models for the Analysis of Correlated Data 313 9.1 Introduction, 313 9.2 Logistic Regression Models for the Analysis of Correlated Data, 315 9.3 Estimation Methods for Correlated Data Logistic Regression Models, 318 9.4 Interpretation of Coefficients from Logistic Regression Models for the Analysis of Correlated Data, 323 9.4.1 Population Average Model, 324 9.4.2 Cluster-Specific Model, 326 9.4.3 Alternative Estimation Methods for the Cluster-Specific Model, 333 9.4.4 Comparison of Population Average and Cluster-Specific Model, 334 9.5 An Example of Logistic Regression Modeling with Correlated Data, 337 9.5.1 Choice of Model for Correlated Data Analysis, 338 9.5.2 Population Average Model, 339 9.5.3 Cluster-Specific Model, 344 9.5.4 Additional Points to Consider when Fitting Logistic Regression Models to Correlated Data, 351 9.6 Assessment of Model Fit, 354 9.6.1 Assessment of Population Average Model Fit, 354 9.6.2 Assessment of Cluster-Specific Model Fit, 365 9.6.3 Conclusions, 374 Exercises, 375 10 Special Topics 377 10.1 Introduction, 377 10.2 Application of Propensity Score Methods in Logistic Regression Modeling, 377 10.3 Exact Methods for Logistic Regression Models, 387 10.4 Missing Data, 395 10.5 Sample Size Issues when Fitting Logistic Regression Models, 401 10.6 Bayesian Methods for Logistic Regression, 408 10.6.1 The Bayesian Logistic Regression Model, 410 10.6.2 MCMC Simulation, 411
- 15. contents xi 10.6.3 An Example of a Bayesian Analysis and Its Interpretation, 419 10.7 Other Link Functions for Binary Regression Models, 434 10.8 Mediation, 441 10.8.1 Distinguishing Mediators from Confounders, 441 10.8.2 Implications for the Interpretation of an Adjusted Logistic Regression Coefficient, 443 10.8.3 Why Adjust for a Mediator? 444 10.8.4 Using Logistic Regression to Assess Mediation: Assumptions, 445 10.9 More About Statistical Interaction, 448 10.9.1 Additive versus Multiplicative Scale–Risk Difference versus Odds Ratios, 448 10.9.2 Estimating and Testing Additive Interaction, 451 Exercises, 456 References 459 Index 479
- 16. Preface to the Third Edition This third edition of Applied Logistic Regression comes 12 years after the 2000 publication of the second edition. During this interval there has been considerable effort researching statistical aspects of the logistic regression model—particularly when the outcomes are correlated. At the same time, capabilities of computer soft- ware packages to fit models grew impressively to the point where they now provide access to nearly every aspect of model development a researcher might need. As is well-recognized in the statistical community, the inherent danger of this easy-to-use software is that investigators have at their disposal powerful computational tools, about which they may have only limited understanding. It is our hope that this third edition will help bridge the gap between the outstanding theoretical developments and the need to apply these methods to diverse fields of inquiry. As was the case in the first two editions, the primary objective of the third edition is to provide an introduction to the underlying theory of the logistic regression model, with a major focus on the application, using real data sets, of the available methods to explore the relationship between a categorical outcome variable and a set of covariates. The materials in this book have evolved over the past 12 years as a result of our teaching and consulting experiences. We have used this book to teach parts of graduate level survey courses, quarter- or semester-long courses, as well as focused short courses to working professionals. We assume that students have a solid foundation in linear regression methodology and contingency table analysis. The positive feedback we have received from students or professionals taking courses using this book or using it for self-learning or reference, provides us with some assurance that the approach we used in the first two editions worked reasonably well; therefore, we have followed that approach in this new edition. The approach we take is to develop the logistic regression model from a regres- sion analysis point of view. This is accomplished by approaching logistic regression in a manner analogous to what would be considered good statistical practice for linear regression. This differs from the approach used by other authors who have begun their discussion from a contingency table point of view. While the contin- gency table approach may facilitate the interpretation of the results, we believe that it obscures the regression aspects of the analysis. Thus, discussion of the inter- pretation of the model is deferred until the regression approach to the analysis is firmly established. xiii
- 17. xiv preface to the third edition To a large extent, there are no major differences between the many software packages that include logistic regression modeling. When a particular approach is available in a limited number of packages, it will be noted in this text. In general, analyses in this book have been performed using STATA [Stata Corp. (2011)]. This easy-to-use package combines excellent graphics and analysis rou- tines; is fast; is compatible across Macintosh, Windows and UNIX platforms; and interacts well with Microsoft Word. Other major statistical packages employed at various points during the preparation of this text include SAS [SAS Institute Inc. (2009)], OpenBUGS [Lunn et al. (2009)] and R [R Development Core Team (2010)]. For all intents and purposes the results produced were the same regard- less of which package we used. Reported numeric results have been rounded from figures obtained from computer output and thus may differ slightly from those that would be obtained in a replication of our analyses or from calculations based on the reported results. When features or capabilities of the programs differed in an important way, we noted them by the names given rather than by their bibliographic citation. We feel that this new edition benefits greatly from the addition of a number of key topics. These include the following: 1. An expanded presentation of numerous new techniques for model-building, including methods for determining the scale of continuous covariates and assessing model performance. 2. An expanded presentation of regression modeling of complex sample survey data. 3. An expanded development of the use of logistic regression modeling in matched studies, as well as with multinomial and ordinal scaled responses. 4. A new chapter dealing with models and methods for correlated categorical response data. 5. A new chapter developing a number of important applications either miss- ing or expanded from the previous editions. These include propensity score methods, exact methods for logistic regression, sample size issues, Bayesian logistic regression, and other link functions for binary outcome regression models. This chapter concludes with sections dealing with the epidemiologic concepts of mediation and additive interaction. As was the case for the second edition, all of the data sets used in the text are available at a web site at John Wiley & Sons, Inc. http://wiley.mpstechnologies.com/wiley/BOBContent/searchLPBobContent.do In addition, the data may also be found, by permission of John Wiley & Sons Inc., in the archive of statistical data sets maintained at the University of Massachusetts at http://www.umass.edu/statdata/statdata in the logistic regression section. We would like to express our sincere thanks and appreciation to our colleagues, students, and staff at all of the institutions we have been fortunate to have been affiliated with since the first edition was conceived more than 25 years ago. This
- 18. preface to the third edition xv includes not only our primary university affiliations but also the locations where we spent extended sabbatical leaves and special research assignments. For this edition we would like to offer special thanks to Sharon Schwartz and Melanie Wall from Columbia University who took the lead in writing the two final sections of the book dealing with mediation and additive interaction. We benefited greatly from their expertise in applying these methods in epidemiologic settings. We greatly appreci- ate the efforts of Danielle Sullivan, a PhD candidate in biostatistics at Ohio State, for assisting in the preparation of the index for this book. Colleagues in the Division of Biostatistics and the Division of Epidemiology at Ohio State were helpful in their review of selected sections of the book. These include Bo Lu for his insights on propensity score methods and David Murray, Sigrún Alba Jóhannesdóttir, and Morten Schmidt for their thoughts concerning the sections on mediation analysis and additive interaction. Data sets form the basis for the way we present our mate- rials and these are often hard to come by. We are very grateful to Karla Zadnik, Donald O. Mutti, Loraine T. Sinnott, and Lisa A. Jones-Jordan from The Ohio State University College of Optometry as well as to the Collaborative Longitudinal Evaluation of Ethnicity and Refractive Error (CLEERE) Study Group for making the myopia data available to us. We would also like to acknowledge Cynthia A. Fontanella from the College of Social Work at Ohio State for making both the Adolescent Placement and the Polypharmacy data sets available to us. A special thank you to Gary Phillips from the Center for Biostatistics at OSU for helping us identify these valuable data sets (that he was the first one to analyze) as well as for his assistance with some programming issues with Stata. We thank Gordon Fitzgerald of the Center for Outcomes Research (COR) at the University of Mas- sachusetts / Worcester for his help in obtaining the small subset of data used in this text from the Global Longitudinal Study of Osteoporosis in Women (GLOW) Study’s main data set. In addition, we thank him for his many helpful comments on the use of propensity scores in logistic regression modeling. We thank Turner Osler for providing us with the small subset of data obtained from a large data set he abstracted from the National Burn Repository 2007 Report, that we used for the burn injury analyses. In many instances the data sets we used were modified from the original data sets in ways to allow us to illustrate important modeling tech- niques. As such, we issue a general disclaimer here, and do so again throughout the text, that results presented in this text do not apply to the original data. Before we began this revision, numerous individuals reviewed our proposal anonymously and made many helpful suggestions. They confirmed that what we planned to include in this book would be of use to them in their research and teach- ing. We thank these individuals and, for the most part, addressed their comments. Many of these reviewers suggested that we include computer code to run logistic regression in a variety of packages, especially R. We decided not to do this for two reasons: we are not statistical computing specialists and did not want to have to spend time responding to email queries on our code. Also, capabilities of com- puter packages change rapidly and we realized that whatever we decided to include here would likely be out of date before the book was even published. We refer readers interested in code specific to various packages to a web site maintained
- 19. xvi preface to the third edition by Academic Technology Services (ATS) at UCLA where they use a variety of statistical packages to replicate the analyses for the examples in the second edition of this text as well as numerous other statistical texts. The link to this web site is http://www.ats.ucla.edu/stat/. Finally, we would like to thank Steve Quigley, Susanne Steitz-Filler, Sari Fried- man and the production staff at John Wiley & Sons Inc. for their help in bringing this project to completion. David W. Hosmer, Jr. Stanley Lemeshow Rodney X. Sturdivant∗ Stowe, Vermont Columbus, Ohio West Point, New York January 2013 ∗ The views expressed in this book are those of the author and do not reflect the official policy or position of the Department of the Army, Department of Defense, or the U.S. Government.
- 20. C H A P T E R 1 Introduction to the Logistic Regression Model 1.1 INTRODUCTION Regression methods have become an integral component of any data analysis concerned with describing the relationship between a response variable and one or more explanatory variables. Quite often the outcome variable is discrete, tak- ing on two or more possible values. The logistic regression model is the most frequently used regression model for the analysis of these data. Before beginning a thorough study of the logistic regression model it is important to understand that the goal of an analysis using this model is the same as that of any other regression model used in statistics, that is, to find the best fitting and most parsimonious, clinically interpretable model to describe the relationship between an outcome (dependent or response) variable and a set of independent (predictor or explanatory) variables. The independent variables are often called covariates. The most common example of modeling, and one assumed to be familiar to the readers of this text, is the usual linear regression model where the outcome variable is assumed to be continuous. What distinguishes a logistic regression model from the linear regression model is that the outcome variable in logistic regression is binary or dichotomous. This difference between logistic and linear regression is reflected both in the form of the model and its assumptions. Once this difference is accounted for, the methods employed in an analysis using logistic regression follow, more or less, the same general principles used in linear regression. Thus, the techniques used in linear regression analysis motivate our approach to logistic regression. We illustrate both the similarities and differences between logistic regression and linear regression with an example. Applied Logistic Regression, Third Edition. David W. Hosmer, Jr., Stanley Lemeshow, and Rodney X. Sturdivant. © 2013 John Wiley & Sons, Inc. Published 2013 by John Wiley & Sons, Inc. 1
- 21. 2 introduction to the logistic regression model Example 1: Table 1.1 lists the age in years (AGE), and presence or absence of evidence of significant coronary heart disease (CHD) for 100 subjects in a hypo- thetical study of risk factors for heart disease. The table also contains an identifier variable (ID) and an age group variable (AGEGRP). The outcome variable is CHD, which is coded with a value of “0” to indicate that CHD is absent, or “1” to indicate that it is present in the individual. In general, any two values could be used, but we have found it most convenient to use zero and one. We refer to this data set as the CHDAGE data. It is of interest to explore the relationship between AGE and the presence or absence of CHD in this group. Had our outcome variable been continuous rather than binary, we probably would begin by forming a scatterplot of the outcome versus the independent variable. We would use this scatterplot to provide an impres- sion of the nature and strength of any relationship between the outcome and the independent variable. A scatterplot of the data in Table 1.1 is given in Figure 1.1. In this scatterplot, all points fall on one of two parallel lines representing the absence of CHD (y = 0) or the presence of CHD (y = 1). There is some tendency for the individuals with no evidence of CHD to be younger than those with evidence of CHD. While this plot does depict the dichotomous nature of the outcome variable quite clearly, it does not provide a clear picture of the nature of the relationship between CHD and AGE. The main problem with Figure 1.1 is that the variability in CHD at all ages is large. This makes it difficult to see any functional relationship between AGE and CHD. One common method of removing some variation, while still maintaining the structure of the relationship between the outcome and the independent variable, is to create intervals for the independent variable and compute the mean of the outcome variable within each group. We use this strategy by grouping age into the categories (AGEGRP) defined in Table 1.1. Table 1.2 contains, for each age group, the frequency of occurrence of each outcome, as well as the percent with CHD present. By examining this table, a clearer picture of the relationship begins to emerge. It shows that as age increases, the proportion (mean) of individuals with evidence of CHD increases. Figure 1.2 presents a plot of the percent of individuals with CHD versus the midpoint of each age interval. This plot provides considerable insight into the relationship between CHD and AGE in this study, but the functional form for this relationship needs to be described. The plot in this figure is similar to what one might obtain if this same process of grouping and averaging were performed in a linear regression. We note two important differences. The first difference concerns the nature of the relationship between the outcome and independent variables. In any regression problem the key quantity is the mean value of the outcome variable, given the value of the independent variable. This quantity is called the conditional mean and is expressed as “E(Y|x)” where Y denotes the outcome variable and x denotes a specific value of the independent variable. The quantity E(Y|x) is read “the expected value of Y, given the value x”. In linear regression we assume that this mean may be expressed as an equation
- 22. introduction 3 Table 1.1 Age, Age Group, and Coronary Heart Disease (CHD) Status of 100 Subjects ID AGE AGEGRP CHD 1 20 1 0 2 23 1 0 3 24 1 0 4 25 1 0 5 25 1 1 6 26 1 0 7 26 1 0 8 28 1 0 9 28 1 0 10 29 1 0 11 30 2 0 12 30 2 0 13 30 2 0 14 30 2 0 15 30 2 0 16 30 2 1 17 32 2 0 18 32 2 0 19 33 2 0 20 33 2 0 21 34 2 0 22 34 2 0 23 34 2 1 24 34 2 0 25 34 2 0 26 35 3 0 27 35 3 0 28 36 3 0 29 36 3 1 30 36 3 0 31 37 3 0 32 37 3 1 33 37 3 0 34 38 3 0 35 38 3 0 36 39 3 0 37 39 3 1 38 40 4 0 39 40 4 1 40 41 4 0 41 41 4 0 42 42 4 0 43 42 4 0 44 42 4 0 (continued)
- 23. 4 introduction to the logistic regression model Table 1.1 (Continued) ID AGE AGEGRP CHD 45 42 4 1 46 43 4 0 47 43 4 0 48 43 4 1 49 44 4 0 50 44 4 0 51 44 4 1 52 44 4 1 53 45 5 0 54 45 5 1 55 46 5 0 56 46 5 1 57 47 5 0 58 47 5 0 59 47 5 1 60 48 5 0 61 48 5 1 62 48 5 1 63 49 5 0 64 49 5 0 65 49 5 1 66 50 6 0 67 50 6 1 68 51 6 0 69 52 6 0 70 52 6 1 71 53 6 1 72 53 6 1 73 54 6 1 74 55 7 0 75 55 7 1 76 55 7 1 77 56 7 1 78 56 7 1 79 56 7 1 80 57 7 0 81 57 7 0 82 57 7 1 83 57 7 1 84 57 7 1 85 57 7 1 86 58 7 0 87 58 7 1 88 58 7 1 89 59 7 1 90 59 7 1
- 24. introduction 5 Table 1.1 (Continued) ID AGE AGEGRP CHD 91 60 8 0 92 60 8 1 93 61 8 1 94 62 8 1 95 62 8 1 96 63 8 1 97 64 8 0 98 64 8 1 99 65 8 1 100 69 8 1 0 0.2 0.4 0.6 0.8 1 Coronary heart disease 20 30 40 50 60 70 Age (years) Figure 1.1 Scatterplot of presence or absence of coronary heart disease (CHD) by AGE for 100 subjects. linear in x (or some transformation of x or Y), such as E(Y|x) = β0 + β1x. This expression implies that it is possible for E(Y|x) to take on any value as x ranges between −∞ and +∞. The column labeled “Mean” in Table 1.2 provides an estimate of E(Y|x). We assume, for purposes of exposition, that the estimated values plotted in Figure 1.2 are close enough to the true values of E(Y|x) to provide a reasonable assessment of the functional relationship between CHD and AGE. With a dichotomous outcome variable, the conditional mean must be greater than or equal to zero and less than
- 25. 6 introduction to the logistic regression model Table 1.2 Frequency Table of Age Group by CHD Coronary Heart Disease Age Group n Absent Present Mean 20–29 10 9 1 0.100 30–34 15 13 2 0.133 35–39 12 9 3 0.250 40–44 15 10 5 0.333 45–49 13 7 6 0.462 50–54 8 3 5 0.625 55–59 17 4 13 0.765 60–69 10 2 8 0.800 Total 100 57 43 0.430 0 0.2 0.4 0.6 0.8 1 Coronary heart disease (mean) 20 30 40 50 60 70 Age (years) Figure 1.2 Plot of the percentage of subjects with CHD in each AGE group. or equal to one (i.e., 0 ≤ E(Y|x) ≤ 1). This can be seen in Figure 1.2. In addition, the plot shows that this mean approaches zero and one “gradually”. The change in the E(Y|x) per unit change in x becomes progressively smaller as the conditional mean gets closer to zero or one. The curve is said to be S-shaped and resembles a plot of the cumulative distribution of a continuous random variable. Thus, it should not seem surprising that some well-known cumulative distributions have been used to provide a model for E(Y|x) in the case when Y is dichotomous. The model we use is based on the logistic distribution. Many distribution functions have been proposed for use in the analysis of a dichotomous outcome variable. Cox and Snell (1989) discuss some of these. There
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- 27. The centre of these forces is not as might be supposed at the centre of the plane, but at a point between the centre and the front edge called the "centre of pressure." The centre of pressure approaches the front edge as the angle of the plane with the horizontal becomes less. In order to render a better idea of how it is possible for an aeroplane to gain support in the air consider a skater moving swiftly over very thin ice which would not bear his weight, but since he is moving so rapidly that any one portion of the ice does not have time to bend to the breaking point, he is supported. In somewhat the same manner, the planes pass so rapidly on to new and undisturbed bodies of air, and stay over one body for so brief an instant that there is no time to completely overcome the inertia of the air and force it downwards. FIG. 12. The action of the air upon a curved and a flat plane. We have seen that by the effects of the resistance of the air, an aeroplane may be sustained in the atmosphere. We must now see in what manner we can use these effects to the greatest advantage.
- 28. First of all, we have been continually speaking of a "plane" as the supporting surface, which from the definition of the word would lead one to believe that they were flat. If the wings of a bird are examined, it will soon be noticed that they are concave underneath. Since the first attempts at aviation, therefore, machines have been built with planes or wings concave on the underside. The reason for this is very apparent from Fig. 12. The first illustration shows the action of a flat surface moving through the air. The air streams, as represented by the lines do not follow the surface of the plane, but leave a considerable region of dead air. This is the reason that a flat plane is very inefficient and not capable of giving so great a lift as the curved plane in the next figure where the lines follow the outline of the plane. The less disturbance a plane causes in the surrounding air, the closer it is said to approach to "stream line form." A correctly curved plane is considerably more effectual than a flat one, giving at the same time greater "lift" and less "drift." Built-up Planes, that is, planes having a double curve approaching true stream line form, come nearer being the ideal plane than any other from some standpoints, but do not possess any advantages when used on models of less than four feet spread.
- 29. FIG. 13. Section of a built-up plane showing how a rib is made. When made small, they offer greater "drift" or head resistance than a single curved surface plane and cannot because of the delicate structure necessary to make them light, withstand hard knocks. They have the further disadvantage of being from a constructional standpoint very hard to make smooth and rigid. There are innumerable substances which would at first seem to recommend themselves as material for planes, but we may immediately thrust the greater portion aside. By all means avoid tracing cloth or linen, not only because its heavy weight forever precludes it from this use, but because it wrinkles and cockles so as to be absolutely useless when slightly damp or wet. Tissue paper wrinkles easily and is not strong enough. Jap silk is an excellent material for fabric covered planes, being at once light and strong. However, by far the most satisfactory plane of this kind is formed by silk bolting cloth which has been coated with collodion. The collodion is brushed on with a fine camel's hair brush after the fabric is in place and it is thereby rendered both waterproof and air-tight. Fabrics should always be stretched over the planes from end to end and not front to back or vice versa. Make the lap joints or pockets around the end spars as long as possible so that they will not draw "dead air" and impede the forward motion of the machine. Bamboo Paper is one of the best materials for covering the planes of a model aeroplane and is to be highly recommended. It is made in Japan from bamboo fibre and is very strong. It is usually
- 30. stretched tightly over the framework and then given two coats of collodion or, what is much better, bamboo varnish. The framework of the planes may be made of rattan, split bamboo, spruce, or steel piano wire. Piano wire is excellent for small machines since it is springy and light and able to withstand shocks. It is easily bent to any shape and offers considerably less head resistance than rattan because of its small diameter. Rattan can be bent into almost any shape by wetting. Nothing is better for the cross pieces, ribs, etc., of the planes or framework than split bamboo. Bulk for bulk it is heavier but infinitely stronger than other woods. It is easily worked and can be bent into all kinds of shapes. Bamboo must always be bent while hot. The best source of heat is a spirit lamp or a bunsen burner. Always bend toward the hottest side. When bent apply a cold wet rag to cool quickly. If bent more than necessary, it may be straightened by applying heat again and allowing it to straighten itself. In order to make long bends, such as the ends of planes, alighting skids, etc., first wind a strip of wet rag around, the bamboo and allow it to remain on for ten or fifteen minutes. Then remove the rag, heat the bamboo in a flame and bend slowly.
- 31. PLATE III. With a little care, strips several feet long may be easily split from bamboo rods. The best method of accomplishing this is to use a fine saw, but a sharp knife will often be successful.
- 32. FIG. 14. How ribs may be joined to the long members. Planes of any considerable size require ribs to support and hold the fabric in shape. Split bamboo is one of the best materials for this purpose. Two very good methods of joining the ribs to the long members of the planes are illustrated in Fig. 14. In the first, a strip of thin sheet aluminum is bent around the rib and spar and fastened by lashing with silk thread. Care must be taken to file off all sharp edges on the aluminum which might otherwise cut the thread. The second method is the neatest and probably the best, since the rib cannot so easily twist or slip out of place. Wood Planes. In spite of the many advantages of fabric planes they cannot approach wooden planes for efficiency on a small machine. Wood is strong, light and does not change its adjustment. Whitewood and spruce are the best materials for the purpose. Do not endeavor to saw out the wood. Use a carpenter's plane as much as possible in the work. A saw tears the fibres of the wood and will make the finished plane full of tiny splits. The wood, however, may be sawed down to a thickness of 5/32 of an inch and then planed down from that. The finished plane should be about 1/16 of an inch thick. When planing down the wood do not butt one end against a bench stop, because as the wood becomes thin, the pressure exerted by the plane against the wood will cause it to rise in the middle and thereby become thinner at that part. Instead, use a clamp to fasten the wood at one end to the bench and plane away
- 33. from the clamp—Plane down to a smooth surface and avoid the use of sand-paper. FIG. 15. Form for bending the planes. Forming the Curve by steaming and bending the wood is a very poor method. It soon becomes distorted and warped. FIG. 16. A good method of building a wooden plane.
- 34. The best method is illustrated in Fig 16. A piece of wood of the same length as the completed plane and having a cross section like that at A is glued to the forward under edge of a flat plane B. After the glue has hardened, the plane is worked down to the shape shown at D which is very close to the stream line form. The plane is then varnished to prevent it from absorbing moisture and losing its shape. The ends may be covered with thin Jap silk, carefully glued on to prevent splitting. The Wright brothers cover the blades of the propellers on their aeroplanes with silk for the same purpose. Air does not flow smoothly when changing from an interrupted flow to an uninterrupted flow around a square corner and so by rounding the ends of the planes, the disturbance at that point is somewhat eliminated. Planes having rattan or piano wire edges cannot very well be of any other shape than those which are illustrated in Fig. 17. FIG. 17. Various shapes a plane may take. It is a good plan to give wooden planes the shape shown by 3 and 4 in Fig. 17, as the disturbances mentioned above are not so marked.
- 35. FIG. 18. An edgewise view of several planes showing the different ways they may be bent to secure stability. The planes of large man-carrying machines possess the same characteristics, but not to such an alarming extent as in a model. The Voisin aeroplanes overcome the objection by the use of vertical panels set between the planes. The angles at which the planes are set may vary from 1 in 6 to 1 in 20. One in ten might be called the "happy medium." If the planes are given too great an angle, the drift becomes so great that the propeller thrust is severely taxed. The smaller the angle, the less will be the drift and consequently the greater the speed. However, if the surface is curved the angle must not be made too small or not much lift will result.
- 36. FIG. 19. The various ways two planes may be combined to secure stability or form a biplane. The angle of the tail planes should be adjustable. If too great, the machine will slow down and the tail will drop, destroying the equilibrium of the machine and consequently the flight. If the lift of the tail is too great, however, it will cause that part to rise and the machine will dive downwards. Elevators and Tails are usually made of thin wood or fabric stretched over a rattan or wire framework. They are usually rectangular or elliptical in shape. In case they are made of wood one of the best methods of attachment is to fasten the plane to a small stick by means of two or three small rivets. The stick is secured to the framework of the
- 37. machine by two small rubber bands. Then in case the machine strikes head on in alighting, the band will absorb the shock and permit the elevator to move so that it is not damaged by the fall. Vertical Fins. It is a much mooted question whether or not a vertical fin is of any value on a model aeroplane since a good model should be so designed that it will fly in a straight line without the use of a rudder. It has been the author's experience that it is often of decided advantage in correcting the flight of an "erratic machine" or in compensating any little difference that there may result in the drift of the two halves of the planes. FIG. 20. Fins. The fin should be placed well toward the rear of the machine and, whenever possible, stretched both above and below the centre
- 38. line of the machine, so that the pressure due to cross winds will be equal both above and below and there will be no tendency for the machine to twist about its longitudinal axis. When it is not possible to place the fin both above and below the centre line it should be placed above rather than below. Fins may be made out of thin wood, sheet aluminum or fabric stretched over a wire or rattan framework.
- 39. CHAPTER IV. THE FUSELLAGE OR FRAMEWORK. By the term "fusellage" or frames, that part of the aeroplane which serves as the "backbone" and to which all the other members are attached is implied. The fusellage above all must be strong. The second requisite is lightness. The simplest frame for a model aeroplane is a long straight stick. The cross section of the stick may vary and be either round or square. A careful workman, however, can build them of "I" section like a steel girder. Increased lightness and strength is the result. FIG. 21. A simple "motor base" or fusellage.
- 40. A single skein of elastic when wound up tends very strongly to twist the framework of the machine out of true. Since the tail and elevator are usually attached to the ends, the adjustment is thrown out to a marked degree and the flight of the machine is liable to be erratic. We have tried building the fusellage of a network of girders such as the Bleriot and Voisin aeroplanes employ. Nothing could have been prettier than these carefully designed and constructed frames with their little struts and guy wires, but we soon found that for plain ordinary everyday efficiency, the simple stick is the best, provided, of course, that it is of the proper size to resist the twist of the rubber. In some cases it is desirable to retain the framework because of the realistic appearance of the model to the larger machines which it gives. The only practical method then is to employ a plain stick backbone to withstand the torque of the rubber and build a false framework around it. The framework need only be strong enough to support the fabric and resist the shocks of landing. This method of construction is best suited to models of the Bleriot and Antoinette types. The only type of frame consisting of a single member which will resist the torque of powerful rubber bands successfully is a tube. The rubber skein is placed inside the tube which may be of wood, paper or aluminum.
- 41. FIG. 22. Paper Tube Fusellage. Part of the tube is cutaway to show the rubber skein inside. Paper tubes are excellent for small machines, being exceedingly light and very strong. They are formed by wrapping tough, unglazed paper around a rod of the required inside diameter. The paper is well smeared with glue and wrapped tightly. The rod is afterwards removed. Be sure that the glue is thoroughly dry before attempting to use the tube. In larger machines it is preferable to employ some other means of avoiding the nuisance of a single skein rather than to use a tubular frame. There are several ways of accomplishing this, the best one undoubtedly being to balance the torque of one elastic by an equal torque tending to twist in the opposite direction.
- 42. FIG. 23. Two methods of gearing a propeller. In Fig. 23, a second skein of elastic is geared to the first with equal sized gear wheels. The second skein is placed immediately underneath the first and is equal in length and strength. Placing one skein under the other and not side by side as might be the first tendency allows the propeller to be arranged centrally. The lower part of the same figure illustrates a second method. In this, the propeller is attached to a long shaft, the other end of which is fitted with a gear wheel. Two elastic skeins of equal length and strength are attached to a second gear which meshes with the first. The only disadvantage of this form of motor is the long propeller shaft required. The objection, however, is sometimes outweighed by the fact that it is possible to employ a small gear wheel on the shaft meshing with a large one between the bands so that the action of the elastic is multiplied and a greater number of propeller revolutions
- 43. secured where the length of the bands is limited and could not be increased in order to bring about the same result. Skids. It requires only very little experience with model aeroplanes to prove the need of efficient skids on the machine. After the rubber band motor has run down, the propeller offers considerable resistance to the forward travel of the machine so that it does not glide properly and causes it to land on its "nose," often damaging the propeller or front planes. At the least, the framework of the machine is strained by such a shock. FIG. 24. Skids of course weigh something and offer a certain amount of resistance, but the advantages more than outweigh those drawbacks. Skids are usually made of piano wire, split bamboo or rattan. The skids should not be made any larger than is necessary to protect the machine. They do not usually take any special shape but are formed to fit each individual case.
- 44. CHAPTER V. MOTIVE POWER. By far the simplest and most efficient form of power which could be installed in a model aeroplane to drive the propeller is a twisted skein of rubber. Nothing is lighter, or more easily handled and repaired. The word elastic, in physics, is the name given to the tendency which a body exerts, when distorted, to return to its original shape. Rubber possesses more elasticity than any other material known, it being possible to stretch a piece of rubber cord to eight or nine times its original length without fracture. Rubber also possesses the added requisite of lightness and will store up more energy than any form of steel spring. The Simplest Form of Motor is a single skein of elastic stretched between two hooks, one fixed and the other to which the propeller is attached, free to rotate. In some cases it is a decided advantage to divide the motor into one or more parts. One phase of this question has already been considered. The others will be discussed in the following chapter. The type of Elastic which gives at once the longest life and the greatest power is the square rubber, preferably about 3/32 x
- 45. 3/32 inches, and not the flat strip. When examined under the microscope the edges of the square rubber show to be cleaner and sharper and not so ragged as those of the flat strips. To be of any value for use in a model aeroplane, the rubber should be absolutely pure and fresh. There are certain precautions which if observed will add greatly to the power and efficiency of a rubber band motor. Always remove the elastic from the machine when the flights are over for the time being. Rubber spoils very quickly when kept under tension. It also deteriorates if warm, so keep in a cool place. Strong sunlight causes rubber to harden and lose its elasticity, due to the presence of the sulphur used in vulcanizing. If talcum powder or finely powdered soapstone is rubbed on the bands from time to time it will prevent them from sticking together. The strands will then run and slip more easily upon each other, making it possible to store up a greater number of propeller revolutions. In spite of the use of talcum powder, however, when a skein of rubber is twisted very tightly, the strands stick together, causing it to soon break up. This nuisance may be somewhat alleviated if the strands are lubricated with pure redistilled glycerine free from grease, etc. Such a precaution will not always greatly lengthen the life of the rubber, but will increase the number of turns which it is possible to give the skein (and this is a very important advantage in model contests). Due to its sticky nature, however, the glycerine will cause the rubber to gather dust and particles of dirt which, if allowed to grind into the rubber, would soon weaken it. The skein should therefore be washed from time to time in warm soda and water and fresh glycerine
- 46. applied. By all means, avoid all oils or substances of a greasy nature, such as lubricants. They quickly soften and rot the rubber. The Amount of Elastic required for a model will vary considerably for propellers of the same pitch and diameter. There is always a tendency to use too much rather than too little and this fault should be carefully guarded against. In nine cases out of ten it is the cause of the unsatisfactory behavior of a model. The motor should always be "stranded," that is, made up of a skein of bands. It is then possible to secure a larger number of turns than if a single strip were used. Always start a new machine with a small number of strands and gradually add to the number until the proper amount of power is obtained. The distance between the propeller and the fixed hook should always be as great as possible so as to secure the maximum number of turns. Doubling the Number of Elastic Strands increases the power of the motor but cuts down the number of turns which it is possible to give the propeller. That is to say, a certain skein composed of six strands of rubber will take perhaps two hundred and twenty-five turns while a twelve strand skein of he same sized rubber strands strands is only capable of less than half or about one hundred turns before it is wound tight. Doubling the number of strands and at the same time keeping them the same length increases the torque more than three times but diminishes the number of turns from one-half to one-third. Doubling the length of the strands does not materially reduce the torque for the first hundred turns. After two hundred turns have
- 47. been reached, the torque is only about one-half as great as it would be in case the length were not doubled. Doubling the length of the strands doubles the number of turns it is possible to give the skein. It is easy to see from this why it is always advisable to make the motor as long as possible and to compose it of the fewest number of strands if long flights are desirable. By using several separate skeins geared together so as to apply their energy to one screw, it is possible to obtain a greatly increased number of turns. The weight of the gearing is very small and hardly a factor, considering the advantages derived therefrom. Since the skeins revolve in opposite directions the frame of the machine is relieved of the harmful twisting effect so often present in a single skein. The gears should be of steel accurately cut and of no larger diameter than is necessary to separate the rubber skeins the requisite distances so that they will not rub. Holes may be bored in the gears to lighten them. The gears are easily and conveniently cut out of steel pinion wire.
- 48. CHAPTER VI. SCREW PROPELLERS. We might compare a propeller to an ordinary screw or bolt by likening the thread of the screw to the two blades of the propeller. If the screw penetrates wood or metal nut it will advance a certain distance known as the pitch which is always the same, namely, the distance separating two consecutive turns of the threads. The revolving blades of the propeller cut their way through the air in identically the same manner. But since air is a very thin medium as compared to wood or iron the propeller slips a little just like a screw going into an unsteady nut and does not advance the distance it theoretically should considering the angle of the blades. The distance lost in each revolution is called the slip. Thus a screw having a ten-foot pitch in actual operation perhaps only advances the aeroplane eight feet.
- 49. FIG. 25. If a propeller blade had a uniform angle throughout its entire length the portions of the blade near the centre would not have as great a pitch as the extreme tips because the diameter of the circle they travel in one revolution is not as great as that at the tips. For this reason it is usual to give the blades an increasing angle as they approach the centre.
- 50. FIG. 26. Method of laying out a screw propeller, that is, determining the angle of the blades at different points. Fig 26 shows a diagram illustrating the theoretical pitch of a screw, the angle of the blade varying inversely as its radial distance from the centre of the screw. When a propeller revolves it sets in motion a cylinder of air. If the angle of the blades is uniform throughout their length the air in the centre of the cylinder will move much more slowly than that near the outside as shown by the arrow heads in A of Fig. 27. If the blades are given an increasing pitch, the air in all parts of the cylinder will move away from the propeller at the same speed. From a diagram like this it is very easy to calculate the angle of a blade at any point to secure a certain pitch. Suppose that the problem in hand is to design a propeller eight inches in diameter and a pitch of twelve inches. On a sheet of paper draw a vertical line AM twelve inches long to represent the pitch. Draw a long horizontal line AN of indefinite length from the lower end of AM and at right angles
- 51. to it. The diameter of the propeller being eight inches, the tips of the blades must travel in one revolution 8 x 3.1416 (the circumference of an eight inch circle in inches), a distance of 33.1 inches. Lay off on AN the distance AB which is 33.1 inches, draw the line MB. The angle MB forms with AN is the proper angle for the blades at the tips. To find the angle one inch from the tips lay off the distance AC, which is. 8 - 2 x 3.1416 or 24.8 inches. MC gives the right angle. The angle two inches from the tip would be shown by MD where AD is 8 - 4 x 3.1416 or 18.8 inches. Any other points can be located in the same manner. FIG. 27. A propeller of the truly helical type delivers a cylinder of air in which all parts move at the same speed as at A. A propeller having blades of the same angle throughout their length throws the air as in B in which the centre of the cylinder moves more slowly than the outside.
- 52. FIG. 28. Templets for testing and carving a propeller. If desirable, a number of small templets having the proper angle may be cut out of sheet tin and fastened to a board as shown in Fig 28. When making the propeller it can be frequently laid on the templets to see if the proper angle has been secured yet. There are a great many other ways of making propellers for model aeroplanes, the simplest and best of which are described below. Metal Propellers have advantages and disadvantages which may be summed up only to find that as far as efficiency is concerned the advantages outweigh the disadvantages.
- 53. FIG. 29. A simple method of forming a propeller from sheet metal. The simplest method of making a small metal propeller is to cut a piece of sheet aluminum into the shape shown by A in Fig. 29. Fold along the dotted lines so that the result is like B in the same illustration. The shaft may be a small piece of piano wire passed through the hole in the centre and bent around as shown. FIG. 30. A built-up metal propeller made of aluminum. Another method of making a metal propeller which is more suitable for large machines than that just described is illustrated in Fig. 30. The blades are cut out of sheet aluminum to the shape
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