Multi-sample

Equality of two covariance matrices

/TITLETesting the equality of two population covar. Matrices , group 1/SPECIFICATIONS CASES=373; VARIABLES=4; ANALYSIS=COVAR;MATRIX=COVAR; METHOD=ML; GROUPS=2;/EQUATIONS V1 = E1; V2 = E2; V3 = E3; V4 = E4; /VARIANCES E1 TO E4 = *;/COVARIANCES E1 TO E4 = *;/MATRIX281184 182216 171 283198 153 208 246/END/TITLE group 2/SPECIFICATIONS CASES=249; VARIABLES=4; ANALYSIS=COVAR;MATRIX=COVAR; METHOD=ML;/EQUATIONS V1 = E1; V2 = E2; V3 = E3; V4 = E4; /VARIANCES E1 TO E4 = *;/COVARIANCES E1 TO E4 = *;/CONSTRAINTS

Testing equality of a factor correlation

Data on mathematical and reading skills, at two points of time

/TITLEtesting equality of a correlation among factos , group 1/SPECIFICATIONS CASES=373; VARIABLES=4; ANALYSIS=COVAR;MATRIX=COVAR; METHOD=ML; GROUPS=2;/EQUATIONS V1 = *F1+ E1; V2 = *F1 + E2; V3 = *F2 + E3; V4 = *F2 + E4; /VARIANCES E1 TO E4 = *; F1 TO F2 = 1;/COVARIANCES F2,F1 = *;/MATRIX281184 182216 171 283198 153 208 246/END/TITLEtesting equality of a correlation among factors, group 2/SPECIFICATIONS CASES=249; VARIABLES=4; ANALYSIS=COVAR;MATRIX=COVAR; METHOD=ML;/EQUATIONS V1 = *F1+ E1; V2 = *F1 + E2; V3 = *F2 + E3; V4 = *F2 + E4; /VARIANCES E1 TO E4 = *; F1 TO F2 = 1;/COVARIANCES F2,F1 = *;/CONSTRAINTS! (1,F2,F1) = (2,F2,F1) ;

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 1976B C J B C J B C J

B 1C .43 1J .47 .29 1B .79 .43 .48 1C .39 .54 .38 .45 1J .50 .28 .56 .56 .35 1B .71 .37 .49 .78 .44 .59 1C .27 .53 .18 .35 .60 .20 .34 1J .47 .29 .49 .48 .32 .61 .53 .28 1SD 2.03 1.84 1.67 1.76 1.68 1.48 1.74 1.83 1.54

B BusingC CriminalsJ Jobs

No-College education n = 203

1972 1974 1976

B C J B C J B C JB 1C .24 1J .39 .25 1B .44 .22 .22 1C .20 .53 .16 .25 1J .31 .21 .62 .30 .21 1B .54 .21 .22 .58 .28 .21 1C .14 .40 .13 .13 .44 .23 .17 1J .30 .25 .48 .33 .16 .41 .28 .14 1SD 1.25 2.11 1.90 1.31 1.97 1.82 1.34 2 1.79

V1 V6V2 V3 V4 V5 V7 V8 V9

T1 T3T2*

*

**

*

Path diagram for effects across time

EQS code for multiple sample /TITLE liberalism-conservatism exmplefactor loadings and latent variable regression coefficients constrained to be equal across groupsgroup 1 - four years of college education /SPECIFICATIONS GROUPS = 2; CAS=143; VAR=9; MATRIX = CORR; ANALYSIS = COV; /EQUATIONSV1 = 1*F1 + E1; ! F1 is liberalism in 1972V2 = 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. varV5 = 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 1972F3 = 1*F2 + D2; !Regression of 1976 liberalism on 1974

/VARIANCESF1 = 1;E1 TO E9 = 1*;D1 TO D2 = .2*;/COVARIANCESE1,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 DEVIATIONS2.03 1.84 1.67 1.76 1.68 1.48 1.74 1.83 1.54 /MATRIX 1 .43 1 .47 .29 1 .79 .43 .48 1 .39 .54 .38 .45 1 .50 .28 .56 .56 .35 1 .71 .37 .49 .78 .44 .59 1 .27 .53 .18 .35 .60 .20 .34 1 .47 .29 .49 .48 .32 .61 .53 .28 1 /END/TITLEGroup 2 - no college education/SPECIFICATIONS CAS=203; VAR=9; MATRIX = CORR; ANALYSIS = COV; /EQUATIONSV1 = 1*F1 + E1; ! F1 is liberalism in 1972V2 = 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. varV5 = 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 1972F3 = 1*F2 + D2; !Regression of 1976 liberalism on 1974/VARIANCESF1 = 1;E1 TO E9 = 1*;D1 TO D2 = .2*;/COVARIANCESE1,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 .25 1 .44 .22 .22 1 .20 .53 .16 .25 1 .31 .21 .62 .30 .21 1 .54 .21 .22 .58 .28 .21 1 .14 .40 .13 .13 .44 .23 .17 1 .30 .25 .48 .33 .16 .41 .28 .14 1/STANDARD DEVIATIONS1.25 2.11 1.90 1.31 1.97 1.82 1.34 2 1.79/END

/TITLEGroup 2 - no college education/SPECIFICATIONS CAS=203; VAR=9; MATRIX = CORR; ANALYSIS = COV; /EQUATIONSV1 = 1*F1 + E1; ! F1 is liberalism in 1972V2 = 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. varV5 = 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 1972F3 = 1*F2 + D2; !Regression of 1976 liberalism on 1974/VARIANCESF1 = 1;E1 TO E9 = 1*;D1 TO D2 = .2*;/COVARIANCESE1,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 .25 1 .44 .22 .22 1 .20 .53 .16 .25 1 .31 .21 .62 .30 .21 1 .54 .21 .22 .58 .28 .21 1 .14 .40 .13 .13 .44 .23 .17 1 .30 .25 .48 .33 .16 .41 .28 .14 1/STANDARD DEVIATIONS1.25 2.11 1.90 1.31 1.97 1.82 1.34 2 1.79/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

/TITLE2 GROUP EXAMPLE FROM WERTS ET AL 1976 - GROUP 1 (EXAMPLE IN EQS MANUAL P. 158)1 FACTOR MODEL WITH UNEQUAL FACTOR CORRELATIONS/SPECIFICATIONSCASES = 865; VARIABLES = 4; GROUPS = 2;/EQUATIONSV1=5*F1+E1;V2=5*F1+E2;V3=5*F2+E3;V4=5*F2+E4;/VARIANCESF1 TO F2 = 1;E1 TO E4 = 50*;/COVARIANCESF2,F1=.5*;/MATRIX63.38270.984 110.23741.710 52.747 60.58430.218 37.489 36.392 32.295/END/TITLE2 GROUP EXAMPLE FROM WERTS ET AL 1976 - GROUP 2/SPECIFICATIONSCASES = 900; VARIABLES = 4;/EQUATIONSV1=5*F1+E1;V2=5*F1+E2;V3=5*F2+E3;V4=5*F2+E4;/VARIANCESF1 TO F4 = 1;E1 TO E4 = 50*;/COVARIANCESF2,F1=.5*;/MATRIX67.89872.301 107.33040.549 55.347 63.20328.976 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