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Multi-sample Equality of two covariance matrices.

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Presentation on theme: "Multi-sample Equality of two covariance matrices."— Presentation transcript:

1 Multi-sample Equality of two covariance matrices

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

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

6 Sample moments

7 Purpose of the analysis

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9 Estimated results

10 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

11 /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

12 /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

13 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)

14 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.

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

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

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

18 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

19 /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*;.....

20 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

21 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

22 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

23 Multiple group Equality of Factors

24 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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