/TITLE 1) Validating Psychometric Assumptions within and between Several Populations Reference: Werts, C.E., Rock, D.A., Linn, R.L., & Joreskog, K.G. (1977). Validating psychometric assumptions within and between several populations. Educational & Psychological Measurement, 37, 863-872. Testing the Basic Model The parameterization in this analysis follows the article, and estimates the unique variance associated with each observed variable as the variance associated with a factor. Thus, there are 2 common factors and 5 unique factors in the solution. /SPECIFICATIONS CASES=900 ; VARIABLES=5 ; METHOD=ML ; MATRIX=COV ; ANALYSIS=COV ; GROUPS=2 ; /LABELS V1=X1 ; V2=X2 ; V3=X3 ; V4=X4 ; V5=X5 ; /MATRIX 67.898 72.391 107.330 58.188 74.013 67.898 40.549 55.347 42.580 63.203 28.976 38.896 30.593 39.261 35.403 /EQUATIONS V1 = 1 F1 + 1 F3 ; V2 = *F1 + 1 F4 ; V3 = *F1 + 1 F5 ; V4 = *F2 + 1 F6 ; V5 = 1 F2 + 1 F7 ; /VARIANCES F1 TO F7 = * ; /COVARIANCES F1,F2 = * ; /PRINT EFFECT=YES ; COVARIANCE=YES ; CORRELATION=YES ; PARAMETER=YES ; DIGITS=3 ; LINESIZE=80 ; FIT=ALL ; /END /TITLE 1) Validating Psychometric Assumptions within and between Several Populations /SPECIFICATIONS CASES=865 ; VARIABLES=5 ; METHOD=ML ; MATRIX=COV ; ANALYSIS=COV ; /LABELS V1=X1 ; V2=X2 ; V3=X4 ; V4=X5 ; V5=X6 ; /MATRIX 63.382 70.984 110.237 41.710 52.747 60.854 30.218 37.489 36.392 32.295 29.139 36.217 35.698 25.698 30.559 /EQUATIONS V1 = 1 F1 + 1 F3 ; V2 = *F1 + 1 F4 ; V3 = *F2 + 1 F5 ; V4 = 1 F2 + 1 F6 ; V5 = *F2 + 1 F7 ; /VARIANCES F1 TO F7 = * ; /COVARIANCES F1,F2 = * ; /PRINT EFFECT=YES ; COVARIANCE=YES ; CORRELATION=YES ; PARAMETER=YES ; DIGITS=3 ; LINESIZE=80 ; FIT=ALL ; /END /TITLE 1) Validating Psychometric Assumptions within and between Several Populations Testing the Basic Model The parameterization in this analysis estimates the unique variance associated with each observed variable in E_. Thus, there are 2 common factors and 5 unique variances in the solution. An index variable is used to specify the units of measurement for each factor. /SPECIFICATIONS CASES=900 ; VARIABLES=5 ; METHOD=ML ; MATRIX=COV ; ANALYSIS=COV ; GROUPS=2 ; /LABELS V1=X1 ; V2=X2 ; V3=X3 ; V4=X4 ; V5=X5 ; /MATRIX 67.898 72.391 107.330 58.188 74.013 67.898 40.549 55.347 42.580 63.203 28.976 38.896 30.593 39.261 35.403 /EQUATIONS V1 = 1 F1 + 1 E1 ; V2 = *F1 + 1 E2 ; V3 = *F1 + 1 E3 ; V4 = *F2 + 1 E4 ; V5 = 1 F2 + 1 E5 ; /VARIANCES F1 TO F2 = * ; E1 TO E5 = * ; /COVARIANCES F1,F2 = * ; /PRINT EFFECT=YES ; COVARIANCE=YES ; CORRELATION=YES ; PARAMETER=YES ; DIGITS=3 ; LINESIZE=80 ; FIT=ALL ; /END /TITLE 1) Validating Psychometric Assumptions within and between Several Populations /SPECIFICATIONS CASES=865 ; VARIABLES=5 ; METHOD=ML ; MATRIX=COV ; ANALYSIS=COV ; /LABELS V1=X1 ; V2=X2 ; V3=X4 ; V4=X5 ; V5=X6 ; /MATRIX 63.382 70.984 110.237 41.710 52.747 60.854 30.218 37.489 36.392 32.295 29.139 36.217 35.698 25.698 30.559 /EQUATIONS V1 = 1 F1 + 1 E1 ; V2 = *F1 + 1 E2 ; V3 = *F2 + 1 E3 ; V4 = 1 F2 + 1 E4 ; V5 = *F2 + 1 E5 ; /VARIANCES F1 TO F2 = * ; E1 TO E5 = * ; /COVARIANCES F1,F2 = * ; /PRINT EFFECT=YES ; COVARIANCE=YES ; CORRELATION=YES ; PARAMETER=YES ; DIGITS=3 ; LINESIZE=80 ; FIT=ALL ; /END /TITLE 1) Validating Psychometric Assumptions within and between Several Populations Testing the Basic Model The parameterization in this analysis estimates the unique variance associated with each observed variable in E_. Thus, there are 2 common factors and 5 unique variances in the solution. The difference between this analysis and the previous analysis is that in the present analysis, the factor variances are set equal to 1, and all specified factor loadings are free to vary. /SPECIFICATIONS CASES=900 ; VARIABLES=5 ; METHOD=ML ; MATRIX=COV ; ANALYSIS=COV ; GROUPS=2 ; /LABELS V1=X1 ; V2=X2 ; V3=X3 ; V4=X4 ; V5=X5 ; /MATRIX 67.898 72.391 107.330 58.188 74.013 67.898 40.549 55.347 42.580 63.203 28.976 38.896 30.593 39.261 35.403 /EQUATIONS V1 = *F1 + 1 E1 ; V2 = *F1 + 1 E2 ; V3 = *F1 + 1 E3 ; V4 = *F2 + 1 E4 ; V5 = *F2 + 1 E5 ; /VARIANCES F1 TO F2 = 1 ; E1 TO E5 = * ; /COVARIANCES F1,F2 = * ; /PRINT EFFECT=YES ; COVARIANCE=YES ; CORRELATION=YES ; PARAMETER=YES ; DIGITS=3 ; LINESIZE=80 ; FIT=ALL ; /END /TITLE 1) Validating Psychometric Assumptions within and between Several Populations /SPECIFICATIONS CASES=865 ; VARIABLES=5 ; METHOD=ML ; MATRIX=COV ; ANALYSIS=COV ; /LABELS V1=X1 ; V2=X2 ; V3=X4 ; V4=X5 ; V5=X6 ; /MATRIX 63.382 70.984 110.237 41.710 52.747 60.854 30.218 37.489 36.392 32.295 29.139 36.217 35.698 25.698 30.559 /EQUATIONS V1 = *F1 + 1 E1 ; V2 = *F1 + 1 E2 ; V3 = *F2 + 1 E3 ; V4 = *F2 + 1 E4 ; V5 = *F2 + 1 E5 ; /VARIANCES F1 TO F2 = 1 ; E1 TO E5 = * ; /COVARIANCES F1,F2 = * ; /PRINT EFFECT=YES ; COVARIANCE=YES ; CORRELATION=YES ; PARAMETER=YES ; DIGITS=3 ; LINESIZE=80 ; FIT=ALL ; /END /TITLE 1) Validating Psychometric Assumptions within and between Several Populations Testing the Units of Measurement between Tests The parameterization in this analysis estimates the unique variance associated with each observed variable in Theta Delta (TD). Thus, there are 2 common factors and 5 unique variances in the solution. Factor loadings are fixed to certain values to correspond to the relative lengths of the tests involved. /SPECIFICATIONS CASES=900 ; VARIABLES=5 ; METHOD=ML ; MATRIX=COV ; ANALYSIS=COV ; GROUPS=2 ; /LABELS V1=X1 ; V2=X2 ; V3=X3 ; V4=X4 ; V5=X5 ; /MATRIX 67.898 72.391 107.330 58.188 74.013 67.898 40.549 55.347 42.580 63.203 28.976 38.896 30.593 39.261 35.403 /EQUATIONS V1 = 1 F1 + 1 E1 ; V2 = 1.25 F1 + 1 E2 ; V3 = *F1 + 1 E3 ; V4 = 1.4 F2 + 1 E4 ; V5 = 1 F2 + 1 E5 ; /VARIANCES F1 TO F2 = * ; E1 TO E5 = * ; /COVARIANCES F1,F2 = * ; /PRINT EFFECT=YES ; COVARIANCE=YES ; CORRELATION=YES ; PARAMETER=YES ; DIGITS=3 ; LINESIZE=80 ; FIT=ALL ; /END /TITLE 1) Validating Psychometric Assumptions within and between Several Populations /SPECIFICATIONS CASES=865 ; VARIABLES=5 ; METHOD=ML ; MATRIX=COV ; ANALYSIS=COV ; /LABELS V1=X1 ; V2=X2 ; V3=X4 ; V4=X5 ; V5=X6 ; /MATRIX 63.382 70.984 110.237 41.710 52.747 60.854 30.218 37.489 36.392 32.295 29.139 36.217 35.698 25.698 30.559 /EQUATIONS V1 = 1 F1 + 1 E1 ; V2 = 1.25 F1 + 1 E2 ; V3 = 1.4 F2 + 1 E3 ; V4 = 1 F2 + 1 E4 ; V5 = *F2 + 1 E5 ; /VARIANCES F1 TO F2 = * ; E1 TO E5 = * ; /COVARIANCES F1,F2 = * ; /PRINT EFFECT=YES ; COVARIANCE=YES ; CORRELATION=YES ; PARAMETER=YES ; DIGITS=3 ; LINESIZE=80 ; FIT=ALL ; /END /TITLE 1) Validating Psychometric Assumptions within and between Several Populations Testing Equality of Parameters between Populations The parameterization in this analysis estimates the unique variance associated with each observed variable in E_. Thus, there are 2 common factors and 5 unique variances in the solution. Factor variances and covariances are now set equal across populations. /SPECIFICATIONS CASES=900 ; VARIABLES=5 ; METHOD=ML ; MATRIX=COV ; ANALYSIS=COV ; GROUPS=2 ; /LABELS V1=X1 ; V2=X2 ; V3=X3 ; V4=X4 ; V5=X5 ; /MATRIX 67.898 72.391 107.330 58.188 74.013 67.898 40.549 55.347 42.580 63.203 28.976 38.896 30.593 39.261 35.403 /EQUATIONS V1 = 1 F1 + 1 E1 ; V2 = 1.25 F1 + 1 E2 ; V3 = *F1 + 1 E3 ; V4 = 1.4 F2 + 1 E4 ; V5 = 1 F2 + 1 E5 ; /VARIANCES F1 TO F2 = * ; E1 TO E5 = * ; /COVARIANCES F1,F2 = * ; /PRINT EFFECT=YES ; COVARIANCE=YES ; CORRELATION=YES ; PARAMETER=YES ; DIGITS=3 ; LINESIZE=80 ; FIT=ALL ; /END /TITLE 1) Validating Psychometric Assumptions within and between Several Populations /SPECIFICATIONS CASES=865 ; VARIABLES=5 ; METHOD=ML ; MATRIX=COV ; ANALYSIS=COV ; /LABELS V1=X1 ; V2=X2 ; V3=X4 ; V4=X5 ; V5=X6 ; /MATRIX 63.382 70.984 110.237 41.710 52.747 60.854 30.218 37.489 36.392 32.295 29.139 36.217 35.698 25.698 30.559 /EQUATIONS V1 = 1 F1 + 1 E1 ; V2 = 1.25 F1 + 1 E2 ; V3 = 1.4 F2 + 1 E3 ; V4 = 1 F2 + 1 E4 ; V5 = *F2 + 1 E5 ; /VARIANCES F1 TO F2 = * ; E1 TO E5 = * ; /COVARIANCES F1,F2 = * ; /CONSTRAINTS (1,F1,F1) = (2,F1,F1) ; (1,F2,F2) = (2,F2,F2) ; (1,F1,F2) = (2,F1,F2) ; /PRINT EFFECT=YES ; COVARIANCE=YES ; CORRELATION=YES ; PARAMETER=YES ; DIGITS=3 ; LINESIZE=80 ; FIT=ALL ; /END /TITLE 1) Validating Psychometric Assumptions within and between Several Populations Testing Equality of Parameters between Populations The parameterization in this analysis estimates the unique variance associated with each observed variable in E_. Thus, there are 2 common factors and 5 unique variances in the solution. Error variances are now set equal across populations. NOTE how this is done, given the fact that the tests used in the two groups differ. /SPECIFICATIONS CASES=900 ; VARIABLES=5 ; METHOD=ML ; MATRIX=COV ; ANALYSIS=COV ; GROUPS=2 ; /LABELS V1=X1 ; V2=X2 ; V3=X3 ; V4=X4 ; V5=X5 ; /MATRIX 67.898 72.391 107.330 58.188 74.013 67.898 40.549 55.347 42.580 63.203 28.976 38.896 30.593 39.261 35.403 /EQUATIONS V1 = 1 F1 + 1 E1 ; V2 = 1.25 F1 + 1 E2 ; V3 = *F1 + 1 E3 ; V4 = 1.4 F2 + 1 E4 ; V5 = 1 F2 + 1 E5 ; /VARIANCES F1 TO F2 = * ; E1 TO E5 = * ; /COVARIANCES F1,F2 = * ; /PRINT EFFECT=YES ; COVARIANCE=YES ; CORRELATION=YES ; PARAMETER=YES ; DIGITS=3 ; LINESIZE=80 ; FIT=ALL ; /END /TITLE 1) Validating Psychometric Assumptions within and between Several Populations /SPECIFICATIONS CASES=865 ; VARIABLES=5 ; METHOD=ML ; MATRIX=COV ; ANALYSIS=COV ; /LABELS V1=X1 ; V2=X2 ; V3=X4 ; V4=X5 ; V5=X6 ; /MATRIX 63.382 70.984 110.237 41.710 52.747 60.854 30.218 37.489 36.392 32.295 29.139 36.217 35.698 25.698 30.559 /EQUATIONS V1 = 1 F1 + 1 E1 ; V2 = 1.25 F1 + 1 E2 ; V3 = 1.4 F2 + 1 E3 ; V4 = 1 F2 + 1 E4 ; V5 = *F2 + 1 E5 ; /VARIANCES F1 TO F2 = * ; E1 TO E5 = * ; /COVARIANCES F1,F2 = * ; /CONSTRAINTS (1,F1,F1) = (2,F1,F1) ; (1,F2,F2) = (2,F2,F2) ; (1,F1,F2) = (2,F1,F2) ; (1,E1,E1) = (2,E1,E1) ; (1,E2,E2) = (2,E2,E2) ; (1,E4,E4) = (2,E3,E3) ; (1,E5,E5) = (2,E4,E4) ; /PRINT EFFECT=YES ; COVARIANCE=YES ; CORRELATION=YES ; PARAMETER=YES ; DIGITS=3 ; LINESIZE=80 ; FIT=ALL ; /END /TITLE 1) Validating Psychometric Assumptions within and between Several Populations Testing Published Standard Error Estimates The parameterization in this analysis estimates the unique variance associated with each observed variable in E_. Thus, there are 2 common factors and 5 unique variances in the solution. The factor loadings are fixed as before, but the factor variances are free to vary across groups. The unique variances for X1, X2, X4, and X5 are set equal to the square of the published standard errors of measurement -- in the first group only. /SPECIFICATIONS CASES=900 ; VARIABLES=5 ; METHOD=ML ; MATRIX=COV ; ANALYSIS=COV ; GROUPS=2 ; /LABELS V1=X1 ; V2=X2 ; V3=X3 ; V4=X4 ; V5=X5 ; /MATRIX 67.898 72.391 107.330 58.188 74.013 67.898 40.549 55.347 42.580 63.203 28.976 38.896 30.593 39.261 35.403 /EQUATIONS V1 = 1 F1 + 1 E1 ; V2 = 1.25 F1 + 1 E2 ; V3 = *F1 + 1 E3 ; V4 = 1.4 F2 + 1 E4 ; V5 = 1 F2 + 1 E5 ; /VARIANCES F1 TO F2 = * ; E1 = 9.922 ; E2 = 12.960 ; E3 = * ; E4 = 8.644 ; E5 = 5.905 ; /COVARIANCES F1,F2 = * ; /PRINT EFFECT=YES ; COVARIANCE=YES ; CORRELATION=YES ; PARAMETER=YES ; DIGITS=3 ; LINESIZE=80 ; FIT=ALL ; /END /TITLE 1) Validating Psychometric Assumptions within and between Several Populations /SPECIFICATIONS CASES=865 ; VARIABLES=5 ; METHOD=ML ; MATRIX=COV ; ANALYSIS=COV ; /LABELS V1=X1 ; V2=X2 ; V3=X4 ; V4=X5 ; V5=X6 ; /MATRIX 63.382 70.984 110.237 41.710 52.747 60.854 30.218 37.489 36.392 32.295 29.139 36.217 35.698 25.698 30.559 /EQUATIONS V1 = 1 F1 + 1 E1 ; V2 = 1.25 F1 + 1 E2 ; V3 = 1.4 F2 + 1 E3 ; V4 = 1 F2 + 1 E4 ; V5 = *F2 + 1 E5 ; /VARIANCES F1 TO F2 = * ; E1 TO E5 = * ; /COVARIANCES F1,F2 = * ; /CONSTRAINTS /PRINT EFFECT=YES ; COVARIANCE=YES ; CORRELATION=YES ; PARAMETER=YES ; DIGITS=3 ; LINESIZE=80 ; FIT=ALL ; /END /TITLE 1) Validating Psychometric Assumptions within and between Several Populations Testing Error Variances The parameterization in this analysis follows the article, and estimates the unique variance associated with each observed variable as the variance associated with a factor. Thus, there are 2 common factors and 5 unique factors in the solution. In this case, the factor loading for X2 is fixed at 1.118 in the first group only. /SPECIFICATIONS CASES=900 ; VARIABLES=5 ; METHOD=ML ; MATRIX=COV ; ANALYSIS=COV ; GROUPS=2 ; /LABELS V1=X1 ; V2=X2 ; V3=X3 ; V4=X4 ; V5=X5 ; /MATRIX 67.898 72.391 107.330 58.188 74.013 67.898 40.549 55.347 42.580 63.203 28.976 38.896 30.593 39.261 35.403 /EQUATIONS V1 = 1 F1 + 1 F3 ; V2 = 1.25 F1 + 1.118 F4 ; V3 = *F1 + 1 F5 ; V4 = 1.4 F2 + 1 F6 ; V5 = 1 F2 + 1 F7 ; /VARIANCES F1 TO F7 = * ; /COVARIANCES F1,F2 = * ; /CONSTRAINTS (F3,F3) = (F4,F4) ; /PRINT EFFECT=YES ; COVARIANCE=YES ; CORRELATION=YES ; PARAMETER=YES ; DIGITS=3 ; LINESIZE=80 ; FIT=ALL ; /END /TITLE 1) Validating Psychometric Assumptions within and between Several Populations /SPECIFICATIONS CASES=865 ; VARIABLES=5 ; METHOD=ML ; MATRIX=COV ; ANALYSIS=COV ; /LABELS V1=X1 ; V2=X2 ; V3=X4 ; V4=X5 ; V5=X6 ; /MATRIX 63.382 70.984 110.237 41.710 52.747 60.854 30.218 37.489 36.392 32.295 29.139 36.217 35.698 25.698 30.559 /EQUATIONS V1 = 1 F1 + 1 F3 ; V2 = 1.25 F1 + 1 F4 ; V3 = 1.4 F2 + 1 F5 ; V4 = 1 F2 + 1 F6 ; V5 = *F2 + 1 F7 ; /VARIANCES F1 TO F7 = * ; /COVARIANCES F1,F2 = * ; /PRINT EFFECT=YES ; COVARIANCE=YES ; CORRELATION=YES ; PARAMETER=YES ; DIGITS=3 ; LINESIZE=80 ; FIT=ALL ; /END /TITLE 1) Validating Psychometric Assumptions within and between Several Populations Testing Error Variances The parameterization in this analysis estimates the unique variance associated with each observed variable in E_. Thus, there are 2 common factors and 5 unique variances in the solution. Factor loadings are fixed to certain values to correspond to the relative lengths of the tests involved. Then, the /CONSTRIANTS command is used to test for the equality of the error variances in X1 and X2, as above. /SPECIFICATIONS CASES=900 ; VARIABLES=5 ; METHOD=ML ; MATRIX=COV ; ANALYSIS=COV ; GROUPS=2 ; /LABELS V1=X1 ; V2=X2 ; V3=X3 ; V4=X4 ; V5=X5 ; /MATRIX 67.898 72.391 107.330 58.188 74.013 67.898 40.549 55.347 42.580 63.203 28.976 38.896 30.593 39.261 35.403 /EQUATIONS V1 = 1 F1 + 1 E1 ; V2 = 1.25 F1 + 1 E2 ; V3 = *F1 + 1 E3 ; V4 = 1.4 F2 + 1 E4 ; V5 = 1 F2 + 1 E5 ; /VARIANCES F1 TO F2 = * ; E1 = 9.922 ; E2 = 12.960 ; E3 = * ; E4 = 8.644 ; E5 = 5.905 ; /COVARIANCES F1,F2 = * ; /CONSTRAINTS (E1,E1)*1.25 = (E2,E2) ; /PRINT EFFECT=YES ; COVARIANCE=YES ; CORRELATION=YES ; PARAMETER=YES ; DIGITS=3 ; LINESIZE=80 ; FIT=ALL ; /END /TITLE 1) Validating Psychometric Assumptions within and between Several Populations /SPECIFICATIONS CASES=865 ; VARIABLES=5 ; METHOD=ML ; MATRIX=COV ; ANALYSIS=COV ; /LABELS V1=X1 ; V2=X2 ; V3=X4 ; V4=X5 ; V5=X6 ; /MATRIX 63.382 70.984 110.237 41.710 52.747 60.854 30.218 37.489 36.392 32.295 29.139 36.217 35.698 25.698 30.559 /EQUATIONS V1 = 1 F1 + 1 E1 ; V2 = 1.25 F1 + 1 E2 ; V3 = 1.4 F2 + 1 E3 ; V4 = 1 F2 + 1 E4 ; V5 = *F2 + 1 E5 ; /VARIANCES F1 TO F2 = * ; E1 TO E5 = * ; /COVARIANCES F1,F2 = * ; /CONSTRAINTS /PRINT EFFECT=YES ; COVARIANCE=YES ; CORRELATION=YES ; PARAMETER=YES ; DIGITS=3 ; LINESIZE=80 ; FIT=ALL ; /END