factor-analysis (1).pdf
0 likes626 views
This document provides an introduction to exploratory factor analysis (EFA). It discusses key concepts such as factors, factor loadings, communalities, assumptions of EFA, extraction and rotation methods. An example is provided applying EFA to anthropometric and physical performance data from 21 participants. Three factors were extracted accounting for over 80% of the variance: an anthropometric factor with high loadings for weight, height and leg length; a physical performance factor with high loadings for shuttle run, 50m dash and 12m run/walk; and a third factor with high loading for shoulder width only.
![Variables Component 1 Component 2 Component 3 Communality
Vividness Qu -.198 -.805 .061 69%
Control Qu .173 .751 .306 69%
Preference Qu .353 .577 -.549 76%
Generate Test -.444 .251 .543 55%
Inspect Test -.773 .051 -.051 60%
Maintain .734 -.003 .384 69%
Transform (P&P) Test .759 -.155 .188 64%
Transform (Comp)
Test
-.792 .179 .304 75%
Visual STM Test .792 -.102 .215 69%
Eigenvalues 3.36 1.677 1.018 /
% Variance 37.3% 18.6% 11.3% /
Communality of Variable 1 (Vividness Qu) = (-.198)2 + (-.805)2 + (.061)2 = . 69 or69%
Eigenvalue of Comp 1 = ( [-.198]2 + [.173]2 + [.353]2 + [-.444]2 + [-.773]2 +[.734]2 + [.759]2 + [-
.792]2 + [.792]2 ) =3.36
3.36 / 9 = 37.3%](https://image.slidesharecdn.com/factor-analysis1-230119034900-6bbaebd9/85/factor-analysis-1-pdf-29-320.jpg)
factor-analysis (1).pdf
- 1. A Presentation AN INTRODUCTION TO EXPOLRATORY FACTOR ANALYSIS
- 2. “When you CAN MEASURE what you are speaking about and express it in numbers, you know something about it; but when you CANNOT express it in numbers your knowledge is of a mearge and unsatisfactory kind.” Measurement is necessary. LORD KELVIN, British Scientist
- 3. FIRST NOTABLE MENTION Charles Edward Spearmen was known for his seminal work on testing and measuring of HUMAN INELLIGENCE by using the FACTOR ANALYSIS during World War I. CHARLES EDWARD SPEARMEN (BRITISH PSYCHOLOGIST)
- 4. A factor is a linear combination of variables. It is a construct that is not directly observed but that needs to be inferred from the input variables.
- 5. • Variable reduction technique •Reduces a set of variable in terms of a small number of latent factors(unobservable). •Factor analysis is a correlational method used to find and describe the underlying factors driving data values for a large set of variables.
- 6. SIMPLE PATHDIAGRAM FOR A FACTOR ANALYSIS MODEL •F1 and F2 are two common factors. Y1,Y2,Y3,Y4, and Y5 are observed variables, possibly 5 subtests or measures of other observations such as responses to itemson a survey. •e1,e2,e3,e4, and e5 represent residuals or unique factors, which are assumed to be uncorrelatedwith each other.
- 7. Questionnaireconstruction Test Batteryconstruction
- 8. Testing the Assumptions Interpretation of Factors Construction of correlation Matrix Method of Factor Analysis Determination of Number of Factors Rotation of Factors
- 9. 1. No outliers in the data set. 2. Normality of the data set. 3. Adequate sample size. 4. Multi collinearity and singularity among the variables does not exist. 5. Homoscedasticity does not exist between the variables because factor analysis is a linear function of measuredvariables. 6. Variables should be linear in nature. 7. Data should be metric in nature i.e. on interval and ratio scale. KMO test isused
- 10. Bartlett test of sphericity It test the null hypothesis that all the correlation between the variables is Zero. It also test whether the correlation matrix is a identity matrix or not. If it is an identity matrix then factor analysis becomes in appropriate. Kaiser-Meyer-Olkin (KMO) measure of samplingadequacy This test checks the adequacy of data for running the factor analysis. The value of KMO ranges from 0 to 1. The larger the value of KMO more adequate is the sample for running the factor analysis. Kaiser recommends accepting values greater than 0.5 as acceptable.
- 11. Problem formulation Interpretation of Factors Testing the Assumptions Construction of correlation Matrix Method of Factor Analysis Determination of Number of Factors Rotation of Factors
- 12. •Analyses the pattern of correlations betweenvariables in the correlation matrix •Which variables tend to correlate highlytogether? •If variablesare highly correlated, likely that they represent the same underlyingdimension Factor analysis pinpoints the clusters of high correlations between variables and for each cluster, it will assign a factor
- 13. Q1 Q2 Q3 Q4 Q5 Q6 Q1 1 Q2 .987 1 Q3 .801 .765 1 Q4 -.003 -.088 0 1 Q5 -.051 .044 .213 .968 1 Q6 -.190 -.111 0.102 .789 .864 1 • Q1-3 correlate strongly with each other and hardly at all with 4-6 • Q4-6 correlate strongly with each other and hardly at all with 1-3 • Two factors!
- 14. Testing the Assumptions Problem formulation Constructionof correlation Matrix Method of Factor Analysis Determination of Numberof Factors Rotation of Factors Interpretation of Factors
- 15. Method of FactorAnalysis (A) Principal component analysis •Providesa uniquesolution, so that the original data can be reconstructed from theresults •It looks at the total varianceamong thevariables that is the unique as well as the commonvariance. •In this method, the factorexplaining the maximumvariance is extracted first.
- 16. Uses an estimate of commonvarianceamong theoriginal variables to generate factorsolution. Because of this, the number of factors will always be less than the numberof original variables (B) Common factor analysis Other Method s Includes:- Un weighted least squares, Generalized least squares, Maximum likelihood, Principal axis factoring, Alpha factoring, and Image factoring.
- 17. Variable Specific Variance Error Variance Common Variance Varianceunique to the variable itself Variance due to measurement erroror some random, unknownsource Variance thata variableshares with other variables in a matrix When searching for the factors underlying the relationships between a set of variables, we are interested in detecting and explaining the common variance Total Variance = common variance + specific variance + errorvariance
- 18. Problem formulation Interpretation of Factors Testing the Assumptions Construction of correlation Matrix Method of Factor Analysis Determination of Number of Factors Rotation of Factors
- 19. Determination of Number of Factors EIGEN VALUE •The Eigen value for a given factor measures the variance in allthe variables which is accounted forby that factor. •It is the amount of variance explained by a factor. It is also called as characteristicroot. Kaiser Guttmann Criterion This method states that the number of factors to be extracted should beequal to the numberof factors having an Eigen valueof 1 orgreater than 1.
- 20. The Scree Plot Theexamination of the Screeplot providesavisual of the total variance associated with eachfactor. The steepslope shows the large factors. Thegradual trailing off (scree) shows the restof the factors usually lower than an Eigen valueof 1.
- 21. Take the components abovethe elbow
- 22. Problem formulation Interpretation of Factors Testing the Assumptions Construction of correlation Matrix Method of Factor Analysis Determination of Number of Factors Rotation of Factors
- 23. •Maximizes high item loadings and minimizes low item loadings, thereby producing a more interpretableand simplified solution. orthogonal rotation and Rotation of Factors •Two common rotation techniques obliquerotation. Rotation Orthogonal Oblique Varimax Qudramax Equamax Direct Oblimin Promax
- 26. Problem formulation Interpretation of Factors Testing the Assumptions Construction of correlation Matrix Method of Factor Analysis Determination of Number of Factors Rotation of Factors
- 27. Factor Loading • It can be defined as the correlation coefficient between the variable and the factor. • The squared factor loading of a variable indicates the percentage variability explained by the factor in that variable. A factor loading of 0.7 is considered to be sufficient.
- 28. COMMUNALITY •The communality is the amount of variance each variable in the analysis shares with othervariables. •Squared multiple correlation for the variable as dependent using the factorsas predictorsand isdenoted by h2. •The value of communality may be considered as the indicator of reliability of avariable.
- 29. Variables Component 1 Component 2 Component 3 Communality Vividness Qu -.198 -.805 .061 69% Control Qu .173 .751 .306 69% Preference Qu .353 .577 -.549 76% Generate Test -.444 .251 .543 55% Inspect Test -.773 .051 -.051 60% Maintain .734 -.003 .384 69% Transform (P&P) Test .759 -.155 .188 64% Transform (Comp) Test -.792 .179 .304 75% Visual STM Test .792 -.102 .215 69% Eigenvalues 3.36 1.677 1.018 / % Variance 37.3% 18.6% 11.3% / Communality of Variable 1 (Vividness Qu) = (-.198)2 + (-.805)2 + (.061)2 = . 69 or69% Eigenvalue of Comp 1 = ( [-.198]2 + [.173]2 + [.353]2 + [-.444]2 + [-.773]2 +[.734]2 + [.759]2 + [- .792]2 + [.792]2 ) =3.36 3.36 / 9 = 37.3%
- 30. Field Example
- 36. Click on Descriptives Click on Continue
- 37. Click on Extraction Click on Continue Select Principal components
- 38. Click onRotation Click on Continue Click onRotation Click onContinue Click onOK Select VARIMAXRotation
- 39. Descriptive Statistics Mean Std. Deviation Analysis N Standing Broad Jump 212.3810 15.45793 21 Shuttle Run 10.2514 .51167 21 Fifty Meter Dash 7.6938 .80880 21 Twelve Meter run and walk 2488.9524 222.46696 21 Anerobic capacity 39.9071 12.70207 21 Weight 37.8095 7.67215 21 Height 148.3810 10.18566 21 Leg Length 76.3333 5.18009 21 Calf Girth 28.5238 1.99045 21 Thigh Girth 40.5238 3.51595 21 Shoulder Width 38.1429 4.43041 21
- 40. Correlation Matrix Standin g Broad Jump Shuttle Run Fifty Meter Dash Twelve Meter run and walk Anerob ic capacit y Weigh t Heigh t Leg Lengt h Calf Girth Thigh Girth Shoul der Width Standing 1.000 -.651 -.359 .539 .608 .469 .416 .513 .606 .584 .455 1.000 .277 -.691 -.709 -.087 -.048 -.321 -.495 -.515 -.483 Broad Jump Shuttle Run Fifty Meter Dash 1.000 Twelve Meter run -.492 1.000 and walk Correlation Anerobic capacity -.322 .686 1.000 Weight -.231 -.045 .255 1.000 Height -.358 .010 .142 .947 1.000 Leg Length -.354 .151 .292 .687 .675 1.000 Calf Girth -.400 .366 .602 .577 .522 .739 1.000 Thigh Girth -.186 .269 .589 .632 .543 .646 .773 1.000 Shoulder Width .128 .279 .410 .405 .244 .322 .377 .451 1.000
- 41. KMO and Bartlett's Test Kaiser-Meyer-Olkin Measure of Sampling Adequacy. .687 Approx. Chi-Square 165.579 Bartlett's Test of Sphericity df 55 Sig. .000 Since the value of KMO is more than 0.5 so the sample taken in the study is adequateto run the factoranalysis. Since the value for significance in Bartlett test of sphericity is less than 0.05 so the null hypothesis i.e. all the correlation between the variables is 0 is rejected. So the correlation matrix is not an identity matrix and that is good.
- 42. Total % of Variance Cumulati ve % Total % of Variance Cumulati ve % Total % of Variance Cumulati ve % 1 5.429 49.355 49.355 5.429 49.355 49.355 3.890 35.364 35.364 2 2.157 19.608 68.963 2.157 19.608 68.963 3.692 33.559 68.924 3 1.241 11.285 80.247 1.241 11.285 80.247 1.246 11.324 80.247 4 .595 5.407 85.654 5 .421 3.831 89.485 6 .367 3.336 92.821 7 .243 2.214 95.035 8 .216 1.967 97.001 9 .180 1.637 98.638 10 .137 1.241 99.880 11 .013 .120 100.000 Total Variance Explained Compon ent Initial Eigenvalues Extraction Sums of Squared Loadings Rotation Sums of Squared Loadings Extraction Method: Principal ComponentAnalysis. Weare looking for an Eigen valueabove 1.0 Cumulative percentof varianceexplained.
- 43. These three factorswill beextracted outas they havean eigen value greater than 1.
- 44. Factor loadings of all the variables on each of the two factors have been shown here. Since this is an unrotated factor solution, some of the variables may show their contribution in more than one factor. In order to avoid this situation, the factors are rotated by using the varimax rotation technique. Unrotated Component Matrix Component 1 2 3 Standing Broad Jump .814 -.179 .020 Shuttle Run -.682 .587 -.136 Fifty Meter Dash -.469 .108 .808 Twelve Meter run and walk .549 -.694 -.230 Anerobic capacity .731 -.484 .053 Weight .700 .650 .050 Height .647 .663 -.159 Leg Length .762 .396 -.087 Calf Girth .863 .088 -.051 Thigh Girth .835 .138 .199 Shoulder Width .560 -.082 .660 Extraction Method: Principal Component Analysis. a. 3 components extracted.
- 45. Rotated Component Matrix Component 1 2 3 Standing Broad Jump Shuttle Run Fifty Meter Dash Twelve Meter run and walk Anaerobic capacity Weight Height Leg Length Calf Girth Thigh Girth Shoulder Width .469 -.091 -.292 -.069 .200 .954 .930 .828 .690 .696 .332 .689 -.901 -.356 .868 .855 .010 -.047 .230 .524 .483 .479 -.003 -.090 .820 -.279 .012 .079 -.128 -.074 -.058 .194 .646 Extraction Method: Principal Component Analysis. Rotation Method: Varimax with Kaiser Normalization. a. Rotation converged in 4 iterations. After varimax rotation factors will have non-overlapping variables. If the variable has factor loadings more than 0.7, it indicates that the factor extracts sufficient variance from that variable. Thus, all those variables having loadings more than 0.7 or more on a particularfactor is identified in that factor.
- 46. ANTHROPOMETRIC Weight Height Leg Length Name each factor as per your wish PHYSICAL Shuttle Run Fifty Meter Dash Twelve Meter runand walk
- 47. THANK YOU FOR YOU KIND ATTENTION