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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*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

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/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

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/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

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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 1976. (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 = 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

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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

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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 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

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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 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

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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

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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

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Multiple group Equality of Factors

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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

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