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Multi-sample Equality of two covariance matrices

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Testing equality of a factor correlation Data on mathematical and reading skills, at two points of time

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Multiple Group Data Head Start Data Lisrel’s manual Ex94.ls8 EQS manul10.eqs

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Sample moments

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Purpose of the analysis

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Estimated results

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EQS code /TITLE MULTIPLE GROUPS AND STRUCTURED MEANS -- GROUP 1 HEAD START DATA -- LISREL 7 MANUAL, P. 254 HEAD START GROUP EXAMPLE IN EQS MANUAL P.186 /SPECIFICATIONS CASES=148; VARIABLES=6; ANALYSIS=MOMENT; MATRIX=CORRELATION; METHOD=ML; GROUPS=2; /EQUATIONS V1 = 3.9*V999 + F1 + E1; V2 = 3.3*V *F1 + E2; V3 = 2.6*V *F1 + E3; V4 = 6.4*V *F1 + E4; V5 = 20.3*V999 + F2 + E5; V6 = 10.1*V *F2 + E6; F2 = *V *F1 + D2; F1 = -0.4*V999 + D1; /VARIANCES E1 TO E6 = 1.5*; D1 TO D2 = 0.3*; /COVARIANCES E2,E1=*; /PRINT EFFECT=YES; /MATRIX /STANDARD DEVIATIONS /MEANS /END /TITLE MULTIPLE GROUPS AND STRUCTURED MEANS -- GROUP 2 CONTROL GROUP /SPECIFICATIONS CASES=155; VARIABLES=6; ANALYSIS=MOMENT; MATRIX=CORRELATION; METHOD=ML; /EQUATIONS V1 = 3.9*V999 + F1 + E1; V2 = 3.3*V *F1 + E2; V3 = 2.6*V *F1 + E3; V4 = 6.4*V *F1 + E4; V5 = 20.3*V999 + F2 + E5; V6 = 10.1*V *F2 + E6; F2 = 2.10*F1 + D2; F1 = 0V999 + D1; /VARIANCES E1 TO E6 = 1.5*; D1 TO D2 = 0.3*; /COVARIANCES E2,E1=*; /PRINT EFFECT=YES; /MATRIX /STANDARD DEVIATIONS /MEANS /CONSTRAINTS (1,V1,V999)=(2,V1,V999); (1,V2,V999)=(2,V2,V999); (1,V3,V999)=(2,V3,V999); (1,V4,V999)=(2,V4,V999); (1,V5,V999)=(2,V5,V999); (1,V6,V999)=(2,V6,V999); (1,V2,F1)=(2,V2,F1); (1,V3,F1)=(2,V3,F1); (1,V4,F1)=(2,V4,F1); (1,V6,F2)=(2,V6,F2); (1,F2,F1)=(2,F2,F1); /LMTEST /END

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/PRINT EFFECT=YES; /MATRIX /STANDARD DEVIATIONS /MEANS /END /TITLE MULTIPLE GROUPS AND STRUCTURED MEANS -- GROUP 2 CONTROL GROUP /SPECIFICATIONS CASES=155; VARIABLES=6; ANALYSIS=MOMENT; MATRIX=CORRELATION; METHOD=ML; /EQUATIONS V1 = 3.9*V999 + F1 + E1; V2 = 3.3*V *F1 + E2; V3 = 2.6*V *F1 + E3; V4 = 6.4*V *F1 + E4; V5 = 20.3*V999 + F2 + E5; V6 = 10.1*V *F2 + E6; F2 = 2.10*F1 + D2; F1 = 0V999 + D1; /VARIANCES E1 TO E6 = 1.5*; D1 TO D2 = 0.3*; /COVARIANCES E2,E1=*; /PRINT EFFECT=YES; /MATRIX /STANDARD DEVIATIONS /MEANS /CONSTRAINTS (1,V1,V999)=(2,V1,V999); (1,V2,V999)=(2,V2,V999); (1,V3,V999)=(2,V3,V999); (1,V4,V999)=(2,V4,V999); (1,V5,V999)=(2,V5,V999); (1,V6,V999)=(2,V6,V999); (1,V2,F1)=(2,V2,F1); (1,V3,F1)=(2,V3,F1); (1,V4,F1)=(2,V4,F1); (1,V6,F2)=(2,V6,F2); (1,F2,F1)=(2,F2,F1); /LMTEST /END

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/VARIANCES E1 TO E6 = 1.5*; D1 TO D2 = 0.3*; /COVARIANCES E2,E1=*; /PRINT EFFECT=YES; /MATRIX /STANDARD DEVIATIONS /MEANS /CONSTRAINTS (1,V1,V999)=(2,V1,V999); (1,V2,V999)=(2,V2,V999); (1,V3,V999)=(2,V3,V999); (1,V4,V999)=(2,V4,V999); (1,V5,V999)=(2,V5,V999); (1,V6,V999)=(2,V6,V999); (1,V2,F1)=(2,V2,F1); (1,V3,F1)=(2,V3,F1); (1,V4,F1)=(2,V4,F1); (1,V6,F2)=(2,V6,F2); (1,F2,F1)=(2,F2,F1); /LMTEST /END

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Multiple group model for liberal- conservative attitudes at three time points Judd and Milburn (1980) used a latent variable analysis to examine attitudes in a nation-wide sample of individuals who were surveyed on three occasions, in 1972, 1974 and (Dunn et al. P. 140)

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Part of the data involved recording attitudes on three topics: busing- a policy designed to achieve school integration; criminals - the protection for the legal rights of those accused of crimes; jobs- whether government should guarantee jobs and standard of living. The sample consisted of 143 individuals each with four years of college education, and 203 individuals who had no college education.

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college education n = BCJBCJBCJ B1 C.43 1 J B C J B C J SD B Busing C Criminals J Jobs

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No-College education n = BCJBCJBCJ B1 C.24 1 J B C J B C J SD

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V1V6V2V3V4V5V7V8V9 T1T3 T2 * * * * * Path diagram for effects across time

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EQS code for multiple sample /TITLE liberalism-conservatism exmple factor loadings and latent variable regression coefficients constrained to be equal across groups group 1 - four years of college education /SPECIFICATIONS GROUPS = 2; CAS=143; VAR=9; MATRIX = CORR; ANALYSIS = COV; /EQUATIONS V1 = 1*F1 + E1; ! F1 is liberalism in 1972 V2 = 1*F1 + E2; V3 = 1*F1 + E3; V4 = F2 + E4; !F2 is liberalism in 1974 !scale of F2 set to that of V4 !scale can not be set in /VARIANCE ! since F2 appears later as a depend. var V5 = 1*F2 + E5; V6 = 1*F2 + E6; V7 = F3 + E7; !F3 is liberalism in 1976, again scale.. V8 = 1*F3 + E8; V9 = 1*F3 + E9; F2 = 1*F1 + D1; !Regression of 1974 kuberakusn ib 1972 F3 = 1*F2 + D2; !Regression of 1976 liberalism on 1974 /VARIANCES F1 = 1; E1 TO E9 = 1*; D1 TO D2 =.2*; /COVARIANCES E1,E4 =.5*; E1,E7 =.5*; E2,E5 =.5*; E2,E8 =.5*; E3,E6 =.5*; E3,E9 =.5*; E4,E7 =.5*; E5,E8 =.5*; E6,E9 =.5*; /STANDARD DEVIATIONS /MATRIX /END /TITLE Group 2 - no college education /SPECIFICATIONS CAS=203; VAR=9; MATRIX = CORR; ANALYSIS = COV; /EQUATIONS V1 = 1*F1 + E1; ! F1 is liberalism in 1972 V2 = 1*F1 + E2; V3 = 1*F1 + E3; V4 = F2 + E4; !F2 is liberalism in 1974 !scale of F2 set to that of V4 !scale can not be set in /VARIANCE ! since F2 appears later as a depend. var V5 = 1*F2 + E5; V6 = 1*F2 + E6; V7 = F3 + E7; !F3 is liberalism in 1976, again scale.. V8 = 1*F3 + E8; V9 = 1*F3 + E9; F2 = 1*F1 + D1; !Regression of 1974 kuberakusn ib 1972 F3 = 1*F2 + D2; !Regression of 1976 liberalism on 1974 /VARIANCES F1 = 1; E1 TO E9 = 1*; D1 TO D2 =.2*; /COVARIANCES E1,E4 =.5*; E1,E7 =.5*; E2,E5 =.5*; E2,E8 =.5*; E3,E6 =.5*; E3,E9 =.5*; E4,E7 =.5*; E5,E8 =.5*; E6,E9 =.5*; /CONSTRAINTS (1,V1,F1) = (2,V1,F1); ! constraint factor model (1,V2,F1) = (2,V2,F1); (1,V3,F1) = (2,V3,F1); (1,V5,F2) = (2,V5,F2); (1,V6,F2) = (2,V6,F2); (1,V8,F3) = (2,V8,F3); (1,V9,F3) = (2,V9,F3); (1,F2,F1) = (2,F2,F1); ! constrain regression coefficient (1,F3,F2) = (2,F3,F2); /MATRIX /STANDARD DEVIATIONS /END

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/TITLE Group 2 - no college education /SPECIFICATIONS CAS=203; VAR=9; MATRIX = CORR; ANALYSIS = COV; /EQUATIONS V1 = 1*F1 + E1; ! F1 is liberalism in 1972 V2 = 1*F1 + E2; V3 = 1*F1 + E3; V4 = F2 + E4; !F2 is liberalism in 1974 !scale of F2 set to that of V4 !scale can not be set in /VARIANCE ! since F2 appears later as a depend. var V5 = 1*F2 + E5; V6 = 1*F2 + E6; V7 = F3 + E7; !F3 is liberalism in 1976, again scale.. V8 = 1*F3 + E8; V9 = 1*F3 + E9; F2 = 1*F1 + D1; !Regression of 1974 kuberakusn ib 1972 F3 = 1*F2 + D2; !Regression of 1976 liberalism on 1974 /VARIANCES F1 = 1; E1 TO E9 = 1*; D1 TO D2 =.2*; /COVARIANCES E1,E4 =.5*; E1,E7 =.5*;.....

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E2,E5 =.5*; E2,E8 =.5*; E3,E6 =.5*; E3,E9 =.5*; E4,E7 =.5*; E5,E8 =.5*; E6,E9 =.5*; /CONSTRAINTS (1,V1,F1) = (2,V1,F1); ! constraint factor model (1,V2,F1) = (2,V2,F1); (1,V3,F1) = (2,V3,F1); (1,V5,F2) = (2,V5,F2); (1,V6,F2) = (2,V6,F2); (1,V8,F3) = (2,V8,F3); (1,V9,F3) = (2,V9,F3); (1,F2,F1) = (2,F2,F1); ! constrain regression coefficient (1,F3,F2) = (2,F3,F2); /MATRIX /STANDARD DEVIATIONS /END

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Estimated Time effects F2 =F2 =.932*F D F3 =F3 = 1.003*F D CHI-SQUARE = BASED ON 41 DEGREES OF FREEDOM PROBABILITY VALUE FOR THE CHI-SQUARE STATISTIC IS

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V1 =V1 = 1.114*F E V2 =V2 =.839*F E V3 =V3 = 1.005*F E V4 =V4 = F E4 V5 =V5 =.773*F E V6 =V6 =.907*F E V7 =V7 = F E7 V8 =V8 =.552*F E V9 =V9 =.836*F E

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Multiple group Equality of Factors

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EQS code /TITLE 2 GROUP EXAMPLE FROM WERTS ET AL GROUP 1 (EXAMPLE IN EQS MANUAL P. 158) 1 FACTOR MODEL WITH UNEQUAL FACTOR CORRELATIONS /SPECIFICATIONS CASES = 865; VARIABLES = 4; GROUPS = 2; /EQUATIONS V1=5*F1+E1; V2=5*F1+E2; V3=5*F2+E3; V4=5*F2+E4; /VARIANCES F1 TO F2 = 1; E1 TO E4 = 50*; /COVARIANCES F2,F1=.5*; /MATRIX /END /TITLE 2 GROUP EXAMPLE FROM WERTS ET AL GROUP 2 /SPECIFICATIONS CASES = 900; VARIABLES = 4; /EQUATIONS V1=5*F1+E1; V2=5*F1+E2; V3=5*F2+E3; V4=5*F2+E4; /VARIANCES F1 TO F4 = 1; E1 TO E4 = 50*; /COVARIANCES F2,F1=.5*; /MATRIX /CONSTRAINTS (1,V1,F1)=(2,V1,F1); (1,V2,F1)=(2,V2,F1); (1,V3,F2)=(2,V3,F2); (1,V4,F2)=(2,V4,F2); (1,F2,F1)=(2,F2,F1); /LMTEST /END

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