Factor analysis ppt
Download as PPTX, PDF•11 likes•5,420 views
This document discusses exploratory factor analysis (EFA). EFA is used to identify underlying factors that explain the pattern of correlations within a set of observed variables. The document outlines the steps of EFA including testing assumptions, constructing a correlation matrix, determining the number of factors, rotating factors, and interpreting the factor loadings. It provides an example of running EFA on a dataset with 11 physical performance and anthropometric variables from 21 participants. The analysis extracts 3 factors that explain over 80% of the total variance.
![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 or 69%
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/factoranalysisppt-170305173421/85/Factor-analysis-ppt-29-320.jpg)
Factor analysis ppt
- 1. • RAKESH KUMAR • MUKESH CHANDRA BISHT(PhD Scholar, LNIPE) A Presentation by AN INTRODUCTION TO EXPLORATORY 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 PATH DIAGRAM 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 items on a survey. • e1,e2,e3,e4, and e5 represent residuals or unique factors, which are assumed to be uncorrelated with each other.
- 7. Questionnaire construction Test Battery construction
- 8. Testing the Assumptions Construction of correlation Matrix Interpretation of Factors Rotation of Factors Determination of Number of Factors Method of Factor Analysis
- 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 measured variables. 6. Variables should be linear in nature. 7. Data should be metric in nature i.e. on interval and ratio scale. KMO test is used
- 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 sampling adequacy 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. Testing the Assumptions Construction of correlation Matrix Problem formulation Interpretation of Factors Rotation of Factors Determination of Number of Factors Method of Factor Analysis
- 12. •Analyses the pattern of correlations between variables in the correlation matrix •Which variables tend to correlate highly together? •If variables are highly correlated, likely that they represent the same underlying dimension 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 Construction of correlation Matrix Problem formulation Interpretation of Factors Rotation of Factors Determination of Number of Factors Method of Factor Analysis
- 15. Method of Factor Analysis (A) Principal component analysis •Provides a unique solution, so that the original data can be reconstructed from the results •It looks at the total variance among the variables that is the unique as well as the common variance. •In this method, the factor explaining the maximum variance is extracted first.
- 16. Uses an estimate of common variance among the original variables to generate factor solution. Because of this, the number of factors will always be less than the number of original variables (B) Common factor analysis Un weighted least squares, Generalized least squares, Maximum likelihood, Principal axis factoring, Alpha factoring, and Image factoring. Other Method s Includes:-
- 17. Variable Specific Variance Error Variance Common Variance Variance unique to the variable itself Variance due to measurement error or some random, unknown source Variance that a variable shares 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 + error variance
- 18. Testing the Assumptions Construction of correlation Matrix Problem formulation Interpretation of Factors Rotation of Factors Determination of Number of Factors Method of Factor Analysis
- 19. Determination of Number of Factors EIGEN VALUE •The Eigen value for a given factor measures the variance in all the variables which is accounted for by that factor. •It is the amount of variance explained by a factor. It is also called as characteristic root. Kaiser Guttmann Criterion This method states that the number of factors to be extracted should be equal to the number of factors having an Eigen value of 1 or greater than 1.
- 20. The Scree Plot The examination of the Scree plot provides a visual of the total variance associated with each factor. The steep slope shows the large factors. The gradual trailing off (scree) shows the rest of the factors usually lower than an Eigen value of 1.
- 21. Take the components above the elbow
- 22. Testing the Assumptions Construction of correlation Matrix Problem formulation Interpretation of Factors Rotation of Factors Determination of Number of Factors Method of Factor Analysis
- 23. • Maximizes high item loadings and minimizes low item loadings, thereby producing a more interpretable and simplified solution. • Two common rotation techniques orthogonal rotation and oblique rotation. Rotation of Factors Rotation Orthogonal Oblique Varimax Qudramax Equamax Direct Oblimin Promax
- 26. Testing the Assumptions Construction of correlation Matrix Problem formulation Interpretation of Factors Rotation of Factors Determination of Number of Factors Method of Factor Analysis
- 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 other variables. •Squared multiple correlation for the variable as dependent using the factors as predictors and is denoted by h2. • The value of communality may be considered as the indicator of reliability of a variable.
- 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 or 69% 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
- 34. Click on this arrow
- 36. Click on Descriptives Click on Continue
- 37. Click on Extraction Click on Continue Select Principal components
- 38. Click on Rotation Click on Continue Click on Rotation Click on Continue Click on OK Select VARIMAX Rotation
- 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 Correlation Standing Broad Jump 1.000 Shuttle Run -.651 1.000 Fifty Meter Dash -.359 .277 1.000 Twelve Meter run and walk .539 -.691 -.492 1.000 Anerobic capacity .608 -.709 -.322 .686 1.000 Weight .469 -.087 -.231 -.045 .255 1.000 Height .416 -.048 -.358 .010 .142 .947 1.000 Leg Length .513 -.321 -.354 .151 .292 .687 .675 1.000 Calf Girth .606 -.495 -.400 .366 .602 .577 .522 .739 1.000 Thigh Girth .584 -.515 -.186 .269 .589 .632 .543 .646 .773 1.000 Shoulder Width .455 -.483 .128 .279 .410 .405 .244 .322 .377 .451 1.000
- 41. KMO and Bartlett's Test Kaiser-Meyer-Olkin Measure of Sampling Adequacy. .687 Bartlett's Test of Sphericity Approx. Chi-Square 165.579 df 55 Sig. .000 Since the value of KMO is more than 0.5 so the sample taken in the study is adequate to run the factor analysis. 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 Variance Explained Compon ent Initial Eigenvalues Extraction Sums of Squared Loadings Rotation Sums of Squared Loadings 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 Extraction Method: Principal Component Analysis. We are looking for an Eigen value above 1.0 Cumulative percent of variance explained.
- 43. These three factors will be extracted out as they have an 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 .469 .689 -.003 Shuttle Run -.091 -.901 -.090 Fifty Meter Dash -.292 -.356 .820 Twelve Meter run and walk -.069 .868 -.279 Anaerobic capacity .200 .855 .012 Weight .954 .010 .079 Height .930 -.047 -.128 Leg Length .828 .230 -.074 Calf Girth .690 .524 -.058 Thigh Girth .696 .483 .194 Shoulder Width .332 .479 .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 particular factor is identified in that factor.
- 46. Shuttle Run Fifty Meter Dash Twelve Meter run and walk ANTHROPOMETRIC Weight Height Leg Length Name each factor as per your wish PHYSICAL
- 47. THANK YOU FOR YOU KIND ATTENTION