# Multi-sample Equality of two covariance matrices.

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Multi-sample Equality of two covariance matrices

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

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

Sample moments

Purpose of the analysis

Estimated results

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*V999 + 0.85*F1 + E2; V3 = 2.6*V999 + 1.21*F1 + E3; V4 = 6.4*V999 + 2.16*F1 + E4; V5 = 20.3*V999 + F2 + E5; V6 = 10.1*V999 + 0.85*F2 + E6; F2 = *V999 + 2.10*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 1.000 0.441 1.000 0.220 0.203 1.000 0.304 0.182 0.377 1.000 0.274 0.265 0.208 0.084 1.000 0.270 0.122 0.251 0.198 0.664 1.000 /STANDARD DEVIATIONS 1.332 1.281 1.075 2.648 3.764 2.677 /MEANS 3.520 3.081 2.088 5.358 19.672 9.562 /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*V999 + 0.85*F1 + E2; V3 = 2.6*V999 + 1.21*F1 + E3; V4 = 6.4*V999 + 2.16*F1 + E4; V5 = 20.3*V999 + F2 + E5; V6 = 10.1*V999 + 0.85*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 1.000 0.484 1.000 0.224 0.342 1.000 0.268 0.215 0.387 1.000 0.230 0.215 0.196 0.115 1.000 0.265 0.297 0.234 0.162 0.635 1.000 /STANDARD DEVIATIONS 1.360 1.195 1.193 3.239 3.900 2.719 /MEANS 3.839 3.290 2.600 6.435 20.415 10.070 /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

/PRINT EFFECT=YES; /MATRIX 1.000 0.441 1.000 0.220 0.203 1.000 0.304 0.182 0.377 1.000 0.274 0.265 0.208 0.084 1.000 0.270 0.122 0.251 0.198 0.664 1.000 /STANDARD DEVIATIONS 1.332 1.281 1.075 2.648 3.764 2.677 /MEANS 3.520 3.081 2.088 5.358 19.672 9.562 /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*V999 + 0.85*F1 + E2; V3 = 2.6*V999 + 1.21*F1 + E3; V4 = 6.4*V999 + 2.16*F1 + E4; V5 = 20.3*V999 + F2 + E5; V6 = 10.1*V999 + 0.85*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 1.000 0.484 1.000 0.224 0.342 1.000 0.268 0.215 0.387 1.000 0.230 0.215 0.196 0.115 1.000 0.265 0.297 0.234 0.162 0.635 1.000 /STANDARD DEVIATIONS 1.360 1.195 1.193 3.239 3.900 2.719 /MEANS 3.839 3.290 2.600 6.435 20.415 10.070 /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

/VARIANCES E1 TO E6 = 1.5*; D1 TO D2 = 0.3*; /COVARIANCES E2,E1=*; /PRINT EFFECT=YES; /MATRIX 1.000 0.484 1.000 0.224 0.342 1.000 0.268 0.215 0.387 1.000 0.230 0.215 0.196 0.115 1.000 0.265 0.297 0.234 0.162 0.635 1.000 /STANDARD DEVIATIONS 1.360 1.195 1.193 3.239 3.900 2.719 /MEANS 3.839 3.290 2.600 6.435 20.415 10.070 /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

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 1976. (Dunn et al. P. 140)

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.

college education n = 143 1972 1974 1976 BCJBCJBCJ B1 C.43 1 J.47.291 B.79.43.481 C.39.54.38.451 J.50.28.56.56.351 B.71.37.49.78.44.591 C.27.53.18.35.60.20.341 J.47.29.49.48.32.61.53.281 SD2.031.841.671.761.681.481.741.831.54 B Busing C Criminals J Jobs

No-College education n = 203 1972 1974 1976 BCJBCJBCJ B1 C.24 1 J.39.251 B.44.22.221 C.20.53.16.251 J.31.21.62.30.211 B.54.21.22.58.28.211 C.14.40.13.13.44.23.171 J.30.25.48.33.16.41.28.14 1 SD1.252.111.901.311.971.821.3421.79

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

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 2.031.841.671.761.681.481.741.831.54 /MATRIX 1.43 1.47.291.79.43.481.39.54.38.451.50.28.56.56.351.71.37.49.78.44.591.27.53.18.35.60.20.341.47.29.49.48.32.61.53.281 /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 1.24 1.39.251.44.22.221.20.53.16.251.31.21.62.30.211.54.21.22.58.28.211.14.40.13.13.44.23.171.30.25.48.33.16.41.28.14 1 /STANDARD DEVIATIONS 1.252.111.901.311.971.821.3421.79 /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 1.24 1.39.251.44.22.221.20.53.16.251.31.21.62.30.211.54.21.22.58.28.211.14.40.13.13.44.23.171.30.25.48.33.16.41.28.14 1 /STANDARD DEVIATIONS 1.252.111.901.311.971.821.3421.79 /END

Estimated Time effects F2 =F2 =.932*F1 + 1.000 D1.102 9.106 F3 =F3 = 1.003*F2 + 1.000 D2.085 11.800 CHI-SQUARE = 47.577 BASED ON 41 DEGREES OF FREEDOM PROBABILITY VALUE FOR THE CHI-SQUARE STATISTIC IS 0.22257

V1 =V1 = 1.114*F1 + 1.000 E1.113 9.897 V2 =V2 =.839*F1 + 1.000 E2.120 6.999 V3 =V3 = 1.005*F1 + 1.000 E3.115 8.730 V4 =V4 = 1.000 F2 + 1.000 E4 V5 =V5 =.773*F2 + 1.000 E5.131 5.922 V6 =V6 =.907*F2 + 1.000 E6.147 6.174 V7 =V7 = 1.000 F3 + 1.000 E7 V8 =V8 =.552*F3 + 1.000 E8.123 4.496 V9 =V9 =.836*F3 + 1.000 E9.142 5.890

Multiple group Equality of Factors

EQS code /TITLE 2 GROUP EXAMPLE FROM WERTS ET AL 1976 - 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 63.382 70.984 110.237 41.710 52.747 60.584 30.218 37.489 36.392 32.295 /END /TITLE 2 GROUP EXAMPLE FROM WERTS ET AL 1976 - 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 67.898 72.301 107.330 40.549 55.347 63.203 28.976 38.896 39.261 35.403 /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