Exploratory factor analysis

1. Exploratory factor analysis GHQ-12

2. EGO GHQ-12 EFA 1) Assuming items are continuous Variable: Names are ghq01 ghq02 ghq03 ghq04 ghq05 ghq06 ghq07 ghq08 ghq09 ghq10 ghq11 ghq12 f1 id; Missing are all (-9999) ; usevariables are ghq01 ghq03 ghq05 ghq07 ghq09 ghq11 ghq02 ghq04 ghq06 ghq08 ghq10 ghq12; idvariable = id; Analysis: Type = EFA 1 3 ; 2) Assuming items are categorical Variable: Names are ghq01 ghq02 ghq03 ghq04 ghq05 ghq06 ghq07 ghq08 ghq09 ghq10 ghq11 ghq12 f1 id; Missing are all (-9999) ; usevariables are ghq01 ghq03 ghq05 ghq07 ghq09 ghq11 ghq02 ghq04 ghq06 ghq08 ghq10 ghq12; categorical are ghq01 ghq03 ghq05 ghq07 ghq09 ghq11 ghq02 ghq04 ghq06 ghq08 ghq10 ghq12; idvariable = id; Analysis: Type = EFA 1 3 ;

3. EGO GHQ-12 EFA 1) Assuming items are continuous EIGENVALUES FOR SAMPLE CORRELATION MATRIX 1 2 3 4 • 6.277 1.072 0.803 0.597 5 6 7 8 • 0.565 0.497 0.460 0.445 9 10 11 12 1 0.375 0.365 0.319 0.225 2) Assuming items are categorical EIGENVALUES FOR SAMPLE CORRELATION MATRIX 1 2 3 4 1 7.05 1.107 0.79 0.534 5 6 7 8 1 0.489 0.43 0.365 0.349 9 10 11 12 1 0.289 0.258 0.212 0.128

4. EGO GHQ-12 EFA 1) Assuming items are continuous PROMAX ROTATED LOADINGS 1 2 ________ ________ GHQ01 0.416 0.333 GHQ03 0.727 -0.089 GHQ05 -0.009 0.710 GHQ07 0.3690.348 GHQ09 -0.013 0.871 GHQ11 0.3360.460 GHQ02 -0.089 0.723 GHQ04 0.816 -0.086 GHQ06 0.240 0.569 GHQ08 0.493 0.282 GHQ10 0.229 0.627 GHQ12 0.364 0.457 PROMAX FACTOR CORRELATIONS 1 2 1 1.000 2 0.668 1.000 2) Assuming items are categorical PROMAX ROTATED LOADINGS 1 2 ________ ________ GHQ01 0.529 0.285 GHQ03 0.787 -0.098 GHQ05 0.045 0.718 GHQ07 0.530 0.268 GHQ09 0.069 0.838 GHQ11 0.090 0.780 GHQ02 -0.046 0.732 GHQ04 0.859 -0.059 GHQ06 0.230 0.625 GHQ08 0.527 0.298 GHQ10 0.068 0.842 GHQ12 0.428 0.453 PROMAX FACTOR CORRELATIONS 1 2 1 1.000 2 0.668 1.000

5. Item residual variances

6. Correlation -v- regression coefficient Correlation coefficient: The interdependence between pairs of variables i.e. the extent to which values of the variable change together The strength and direction of the linear relationship A fatter ellipse will result in a greater degree of scatter for a regression line of a given gradient, and a lower correlation

7. Polychoric Correlation - Assumptions • A binary or categorical variable is the observed (or manifest) part of an underlying (or latent) continuous variable • Here we’ll also assume that latent variables are normally distributed • THRESHOLD relates the manifest to the latent variable • Uebersax link: http://ourworld.compuserve.com/homepages/jsuebersax/tetra.htm

8. Thresholds Figure from Uebersax webpage

9. 2 binary variables . tab sumodd_g sumeven_g | sumeven_g sumodd_g | 0 1 | Total -----------+----------------------+---------- 0 | 896 20 | 916 1 | 61 142 | 203 -----------+----------------------+---------- Total | 957 162 | 1,119 This is all we see, however ….

10. … this is what we assume is going on Figure from Uebersax webpage

11. What we are really interested in is the correlation (r) between the continuous latent variables Computer algorithm used to search for a correlation r and thresholds t1 and t2 which best reproduce the cell counts of the 2x2 table

12. Conclusions • EFA can be carried out in Mplus very simply • We have demonstrated that it can be dangerous to ignore the ordinal nature of the data when fitting such a model (a practice followed by many!)