Mx PRACTICAL Measurement Invariance Test for measurement Invariance

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Mx PRACTICAL: Measurement Invariance Test for measurement Invariance & Differences in latent means

Mx PRACTICAL: Measurement Invariance Test for measurement Invariance & Differences in latent means

Phenotypic common factor model One Group: Full Model L p P T 0 F

Phenotypic common factor model One Group: Full Model L p P T 0 F 1 L 21 L 31 L 41 M 1 M 2 M 3 M 4 1 1 res res T 11 T 22 T 33 T 44 M B C Means Covariance M+ C*B; L*P*L'+T;

Phenotypic common factor model One Group: Full Model Open: MIphenotypic_1 G. mx Standardized estimates:

Phenotypic common factor model One Group: Full Model Open: MIphenotypic_1 G. mx Standardized estimates: Unstandardized estimates: 1 171. 59 F F 1. 00 . 34 1 1 1 res res 245. 03 . 67 . 41 . 24 1 1 1 res res res . 90 61. 28 58. 97 10593 92 78 165. 63 FIT STATISTICS: . 58 . 76 . 57 . 43 . 46 10593 92 78 . 54 -2 LL = 9287. 688 , DF = 1160 (np=16) vs. sat: Δχ2(2) = 0. 619 , p = 0. 734

Phenotypic common factor model Multigroup: Full Model (M 1) p p 0 F 1

Phenotypic common factor model Multigroup: Full Model (M 1) p p 0 F 1 L 21 L 31 L 41 M 1 M 2 M 3 M 4 1 1 res res T 11 T 22 T 33 T 44 Reference Group: males D=0 F 1 res 1 1 1 res res T 11 T 22 T 33 T 44 Comparison Group: Females

Mx matrices: Full model Reference Group: males L Comparison Group: Females P T L

Mx matrices: Full model Reference Group: males L Comparison Group: Females P T L P T D M M B B = C Means Covariance M+ C*B; L*P*L'+T; C Means (M+(L*D)')+ (C*B); Covariance L*P*L'+T;

Phenotypic common factor model Multigroup: Full Model (M 1) Open: MIphenotypic. mx Unstandardized estimates

Phenotypic common factor model Multigroup: Full Model (M 1) Open: MIphenotypic. mx Unstandardized estimates Males: Unstandardized estimates Females: 1 171. 59 F F 1. 00 . 29 1 1 1 res res 237. 03 . 62 1. 00 . 39 1 1 1 res res res . 69 63. 76 52. 04 10894 94 83 173. 54 FIT STATISTICS: 247. 72 . 76 1. 04 100 92 90 74 59. 25 62. 43 148. 92 -2 LL = 9260. 058 , DF = 1148 (np=28) vs. sat: Δχ2(4) = 0. 854 , p = 0. 931

Phenotypic common factor model Metric Invariance (M 2) p p 1 1 L 21

Phenotypic common factor model Metric Invariance (M 2) p p 1 1 L 21 L 31 L 41 M 1 M 2 M 3 M 4 1 1 1 res res T 11 T 22 T 33 T 44 Reference Group: males 1 = L 21 L 31 L 41 M 1 M 2 M 3 M 4 1 1 res D=0 F 1 res T 11 1 res T 22 res T 33 T 44 Comparison Group: Females

Mx matrices: Metric Invariance Reference Group: males L Comparison Group: Females P T =L

Mx matrices: Metric Invariance Reference Group: males L Comparison Group: Females P T =L P T D M M B B = C Means Covariance M+C*B; L*P*L'+T; C Means (M+(L*D)')+(C*B); Covariance L*P*L'+T;

Phenotypic common factor model Strong Factorial Invariance (M 3) p p L 11 L

Phenotypic common factor model Strong Factorial Invariance (M 3) p p L 11 L 21 L 31 L 41 M 1 M 2 M 3 M 4 1 1 res res T 11 T 22 T 33 T 44 Reference Group: males D=d D=0 F L 11 L 21 L 31 L 41 M 1 M 2 M 3 M 4 = 1 1 res res T 11 T 22 T 33 T 44 Comparison Group: Females

Strong Factorial Invariance (M 3): Identify the latent means Reference Group Latent mean as

Strong Factorial Invariance (M 3): Identify the latent means Reference Group Latent mean as difference between groups 8 8 Equations : : 5 Unknown 10 Unknownparameters

Mx matrices: Strong Factorial invariance Reference Group: males L Comparison Group: Females P T

Mx matrices: Strong Factorial invariance Reference Group: males L Comparison Group: Females P T =L P T D = M B B = C Means Covariance M M+C*B; L*P*L'+T; C Means (M+(L*D)')+(C*B); Covariance L*P*L'+T;

Phenotypic common factor model Strict Factorial Invariance (M 4) p p 1 res T

Phenotypic common factor model Strict Factorial Invariance (M 4) p p 1 res T 11 1 res T 22 T 33 1 1 1 res T 44 Reference Group: males D=d L 11 L 21 L 31 L 41 M 1 M 2 M 3 M 4 1 F 0 F res = 1 res T 11 1 res T 22 res T 33 T 44 Comparison Group: Females

Mx matrices: Strict Factorial invariance Reference Group: males L Comparison Group: Females P T

Mx matrices: Strict Factorial invariance Reference Group: males L Comparison Group: Females P T =L P T D = M M B B = C Means M+C*B; Covariance L*P*L'+T; C Means (M+(L*D)')+ (C*B); Covariance L*P*L'+T;

Mx Option Multiple: Testing MI ! METRIC INVARIANCE: Constrain factor loadings accross males and

Mx Option Multiple: Testing MI ! METRIC INVARIANCE: Constrain factor loadings accross males and females eq L 1 1 1 L 2 1 1 eq L 1 2 1 L 2 2 1 eq L 1 3 1 L 2 3 1 eq L 1 4 1 L 2 4 1 End ! STRONG FACTORIAL INVARIANCE: Constrain the intercepts to be equal for males and females eq m 1 1 1 m 2 1 1 eq m 1 1 2 m 2 1 2 eq m 1 1 3 m 2 1 3 eq m 1 1 4 m 2 1 4 end

Mx Option Multiple: Testing MI ! Estimate the latent FREE D 2 1 ST

Mx Option Multiple: Testing MI ! Estimate the latent FREE D 2 1 ST 0. 5 D 2 END the mean difference accross groups in factor 1 1 1 ! STRICT FACTORIAL INVARIANCE: Constrain residuals accross males and females EQ T 1 1 1 T 2 1 1 EQ T 1 2 2 T 2 2 2 EQ T 1 3 3 T 2 3 3 EQ T 1 4 4 T 2 4 4 END

Practical: MIphenotypic. mx -2 LL DF CTM 1. Full Model(M 1) SAT 2. Metric

Practical: MIphenotypic. mx -2 LL DF CTM 1. Full Model(M 1) SAT 2. Metric Invar. (M 2) 1 SAT 3. Strong Inv. (M 3) 2 SAT 4. Strict Inv. (M 4) 3 SAT ∆X 2 ∆ DF 1. Can we assume Metric invariance, Strong factorial Invariance or Strict Factorial Invariance? p

Phenotypic common factor model -2 LL DF 1. Full Model 9260. 05 2. Metric

Phenotypic common factor model -2 LL DF 1. Full Model 9260. 05 2. Metric Invar. L=L 3. Strong Inv. M=M 5. Strict Inv. T=T CTM ∆X 2 ∆ DF p AIC 1148 SAT 0. 85 4 . 931 -7. 146 9262. 77 1151 1 SAT 2. 72 3. 56 3 7 . 436. 828 -10. 43 9271. 01 1154 2 SAT 8. 24 11. 80 3 10 . 041. 298 -8. 19 1158 3 SAT 0. 71 4 14 . 950. 565 -15. 48 9271. 72

Common pathway model Full Model: Path Diagram for Opposite sex pairs MF 1 1

Common pathway model Full Model: Path Diagram for Opposite sex pairs MF 1 1 1 Ec Cc Z Y X 1 0 1 Ac 1/. 5/W Ac 1 F 1 1 1 F M 3 M 4 F M 11 F 21 F 31 F 41 M 2 1 Cc X 1 Ec 1 Y Z F S=1 F D=0 F 11 F 21 F 31 F 41 M 1 M 2 M 3 M 4 1 V V Es U Cs T As As V V U Es T C TC TC T C s s As As 1 1/. 5/W Reference Group: males Comparison Group: Females

Mx matrices: Full model Reference Group: males F T V X Y U Z

Mx matrices: Full model Reference Group: males F T V X Y U Z Comparison Group: Females F X Y U T V S Z

Mx matrices: Full model Reference Group: males Comparison Group: Females Covariance (DZM) (I@F) &

Mx matrices: Full model Reference Group: males Comparison Group: Females Covariance (DZM) ([email protected]) & (A+C+E| [email protected]+C _ [email protected]+C| A+C+E ) + (G+J+K| [email protected]+J_ [email protected]+J| G+J+K ) ; Covariance (MZF) ([email protected]) & (O&(A+C+E| A+C _ A+C | A+C+E )) + (G+J+K | G+J_ G+J | G+J+K ) ; B B = N N M M D Means (M | M)+ (N*B) ; (M+(F*D)' | M+(F*D)')+ (N*B) ;

Practical: MIcommonpathway. mx/mxo -2 LL DF CTM 1. Full Model 2. Metric Invar. L=L

Practical: MIcommonpathway. mx/mxo -2 LL DF CTM 1. Full Model 2. Metric Invar. L=L 1 3. Strong Inv. M=M 2 4. Free D 3 5. Strict Inv. TUV=TUV 4 ∆X 2 ∆ DF p

Practical: MIcommonpathway. mx/mxo 1. Can we assume Metric invariance, Strong factorial Invariance or Strict

Practical: MIcommonpathway. mx/mxo 1. Can we assume Metric invariance, Strong factorial Invariance or Strict Factorial Invariance? 2. 2. Is there a difference in the means between the reference and the comparison group? 3. 3. If there is a difference, is it due to the latent factor? How large is it? How do you interpret it? 4. 4. Looking at the output of the last fitted model in the output, summary group 11 for females: 5. - Proportion of Variance of BD explained by the common A factor? 6. - Proportion of Covariance BD-IP explained by the common A factor? 7. - Proportion of Covariance IP-MX explained by the common A factor?

Common pathway model Example: Results -2 LL DF 1. Full Model 18254. 32 2297

Common pathway model Example: Results -2 LL DF 1. Full Model 18254. 32 2297 2. Metric Invar. L=L 1825. 70 2301 1 1. 38 4 0. 84 -161. 24 3. Strong Inv. M=M 18302. 93 2305 2 47. 13 4 0. 000 -122. 12 18277. 09 2304 3 25. 84 1 . 000 -145. 85 5. Strict Inv. 18280. 82 TUV=TUV 2312 4 3. 72 8 . 881 -158. 128 4. Free D CTM ∆X 2 ∆ DF p AIC -154. 62

Practical: MIcommonpathway. mx/mxo 1. Can we assume Metric invariance, Strong factorial Invariance or Strict

Practical: MIcommonpathway. mx/mxo 1. Can we assume Metric invariance, Strong factorial Invariance or Strict Factorial Invariance? 2. 2. Is there a difference in the means between the reference and the comparison group? Yes 3. 3. If there is a difference, is it due to the latent factor? How large is it? How do you interpret it? 4. -0. 557 5. 4. Looking at the output of the last fitted model in the output, summary group 11 for females: 6. - Proportion of Variance of BD explained by the common A factor? 0. 437 7. - Proportion of Covariance BD-IP explained by the common A factor? 0. 947 8. - Proportion of Covariance IP-MX explained by the common A factor? 0. 947

Independent Pathway model Full Model 1 1 E C 1 0 C 0 A

Independent Pathway model Full Model 1 1 E C 1 0 C 0 A A A W M 1 W 1 W 2 3 W 4 X 1 X 2 X 3 X 4 M Y 1 Y 3 MM 43 2 Y 4 V 1 V 2 res Z 1 res Z 2 1 C E I=1 S=1 1 0 1 1 A 1 E 1/. 5/L G=0 X 4 X 1 X X 3 2 Y 1 Y 2 T=1 J=0 E C M W 1 W W W M 12 2 3 4 M 3 Y 4 M 4 V 3 V 4 V 1 V 2 V 3 V 4 res res res Z 3 Z 4 Z 1 K=0 Z 2 Z 3 Z 4

Mx matrices: Full model Reference Group: males X Z Y W Comparison Group: Females

Mx matrices: Full model Reference Group: males X Z Y W Comparison Group: Females X Y W Z S I T

Mx matrices: Full model Reference Group: males Covariance (A+C+E+D | H@A+C _ H@A+C |

Mx matrices: Full model Reference Group: males Covariance (A+C+E+D | [email protected]+C _ [email protected]+C | A+C+E+D) Comparison Group: Females Covariance (S*S')@(A|A_ A|A) + (I*I')@(C|C_ C|C) + (T*T')@(E|Q_ Q|E) +(D|Q_ Q|D); ; B B = F F M M G Means (M | M)+ (F*B) ; J K Means (M+(X*G)'+(Y*J)'+(W*K)' | M+(X*G)'+(Y*J)'+(W*K)')+ (F*B) ;

Independent Pathway model Metric Invariance 1 1 E C 1 0 C 0 1

Independent Pathway model Metric Invariance 1 1 E C 1 0 C 0 1 0 A V 2 res Z 1 res Z 2 A C S=1 I=1 A = W M 1 W 1 W 2 3 W 4 X 1 X 2 X 3 X 4 M Y 1 Y 3 MM 43 2 Y 4 V 1 1 1 A 1 E 1/. 5/L=0. 5 1/. 5/L G=0 X 4 X 1 X X 3 2 Y 1 Y 2 1 E T=1 J=0 C E M W 1 W W W M 12 2 3 4 M 3 Y 4 M 4 V 3 V 4 V 1 V 2 V 3 V 4 res res res Z 3 Z 4 Z 1 K=0 Z 2 Z 3 Z 4

Mx Option Multiple: Testing MI ! Model 1: Quantitative Heterogeneity FIX L 7 1

Mx Option Multiple: Testing MI ! Model 1: Quantitative Heterogeneity FIX L 7 1 1 te the OS genetic correlation to. 5 VALUE. 5 L 7 1 1 end ! Model 2: Homogeneity EQ X 1 1 1 X 2 1 1 EQ X 1 2 1 X 2 2 1 EQ X 1 3 1 X 2 3 1 EQ X 1 4 1 X 2 4 1 ! EQ Y 1 1 1 Y 2 1 1 ! EQ Y 1 2 1 Y 2 2 1 ! EQ Y 1 3 1 Y 2 3 1 ! EQ Y 1 4 1 Y 2 4 1 EQ W 1 1 1 W 2 1 1 EQ W 1 2 1 W 2 2 1 EQ W 1 3 1 W 2 3 1 EQ W 1 4 1 W 2 4 1 End

Independent Pathway model Strong Factorial Invariance 1 1 E C 1 0 C 0

Independent Pathway model Strong Factorial Invariance 1 1 E C 1 0 C 0 1 0 A V 2 res Z 1 res Z 2 A C S=1 I=1 A = W M 1 W 1 W 2 3 W 4 X 1 X 2 X 3 X 4 M Y 1 Y 3 MM 43 2 Y 4 V 1 1 1 A 1 E 1/. 5 G=g G=0 X 4 X 1 X X 3 2 Y 1 Y 2 1 E T=1 J=0 J=j C E M W 1 W W W M 12 2 3 4 M 3 Y 4 M 4 V 3 V 4 V 1 V 2 V 3 V 4 res res res Z 3 Z 4 Z 1 K=0 K=k Z 2 Z 3 Z 4

Mx Option Multiple: Testing MI !!STRONG FACTORIAL INVARIANCE!! !At this point, choose the best

Mx Option Multiple: Testing MI !!STRONG FACTORIAL INVARIANCE!! !At this point, choose the best fitting model: homogeneity or scalar heterogeneity (1 or 2 scalars) and go on with modeling the means ! Model EQ M 1 End 4: Homogeneity, equality of intercepts 1 1 M 2 1 1 1 2 M 2 1 3 1 4 M 2 1 4 ! Model 5: G&E Heterogeneity Free G 2 1 1 K 2 1 1 st 0. 5 G 2 1 1 K 2 1 1 End

Independent Pathway model Strict Factorial Invariance 1 1 E C 1 0 C 0

Independent Pathway model Strict Factorial Invariance 1 1 E C 1 0 C 0 1 0 A V 2 res Z 1 res Z 2 A C S=1 I=1 A W M 1 W 1 W 2 3 W 4 X 1 X 2 X 3 X 4 M Y 1 Y 3 MM 43 2 Y 4 V 1 1 1 A 1 E 1/. 5 G=g X 4 X 1 X X 3 2 Y 1 Y 2 1 E T=1 J=j C E M W 1 W W W M 12 2 3 4 M 3 Y 4 M 4 V 3 V 4 V 1 V 2 V 3 V 4 res res res Z 3 Z 4 = Z 1 K=k Z 2 Z 3 Z 4

Mx Option Multiple: Testing MI !!STRICT FACTORIAL INVARIANCE!! !Model 6: Equality of residual variances

Mx Option Multiple: Testing MI !!STRICT FACTORIAL INVARIANCE!! !Model 6: Equality of residual variances !Choose the model for the means that explains the data best: Homogeneity, genetic and environmental heterogeneity !Final test of strict factorial invariance by constraining the residual variances to be equal EQ Z 1 1 1 Z 2 1 1 EQ Z 1 2 2 Z 2 2 2 EQ Z 1 3 3 Z 2 3 3 EQ Z 1 4 4 Z 2 4 4 End

Practical: MIindepenpathway. mx/mxo -2 LL DF CTM 1. Full Model 2. Metric Invar. L=0.

Practical: MIindepenpathway. mx/mxo -2 LL DF CTM 1. Full Model 2. Metric Invar. L=0. 5 1 3. Metric Invar. XW=XW 2 4. Strong Inv. M=M 3 5. Free G & K 4 6. Strict Inv. T=T 5 ∆X 2 ∆ DF p

Practical: MIindepenpathway. mx/mxo 1. Can we assume Metric invariance, Strong factorial Invariance or Strict

Practical: MIindepenpathway. mx/mxo 1. Can we assume Metric invariance, Strong factorial Invariance or Strict Factorial Invariance? 2. 2. Is there a difference in the means between the reference and the comparison group? 3. 3. If there is a difference, is it due to the A and or E latent factors? How large is it? How do you interpret it? 4. 4. Looking at the output of the last fitted model in the output, summary group 10 for females: 5. - Proportion of Variance of BD explained by the common A factor? 6. - Proportion of Covariance BD-IP explained by the common A factor? 7. - Proportion of Covariance IP-MX explained by the common A factor?

Practical: MIindepenpathway. mx/mxo -2 LL DF CTM ∆X 2 ∆ DF p AIC 1.

Practical: MIindepenpathway. mx/mxo -2 LL DF CTM ∆X 2 ∆ DF p AIC 1. Full Model 18332. 54 2298 2. Metric Invar. L=0. 5 18332. 54 2299 1 . 000 1 1. 000 -80. 418 3. Metric Invar. XW=XW 18338. 79 2307 2 6. 250 8 . 619 -90. 158 4. Strong Inv. M=M 18385. 76 2311 3 46. 97 4 . 000 -51. 188 5. Free G & K 18339. 39 2309 4 46. 37 2 . 000 -93. 558 6. Strict Inv. T=T 18340. 96 2313 5 1. 57 4 . 814 -99. 988 -78. 408

Practical: MIindepenpathway. mx/mxo 1. Can we assume Metric invariance, Strong factorial Invariance or Strict

Practical: MIindepenpathway. mx/mxo 1. Can we assume Metric invariance, Strong factorial Invariance or Strict Factorial Invariance? 2. 2. Is there a difference in the means between the reference and the comparison group? yes 3. 3. If there is a difference, is it due to the A and or E latent factors? How large is it? How do you interpret it? G=-0. 828 K=0. 786 4. 4. Looking at the output of the last fitted model in the output, summary group 10 for females: 5. - Proportion of Variance of BD explained by the common A factor? 0. 421 6. - Proportion of Covariance BD-IP explained by the common A factor? 0. 858 7. - Proportion of Covariance IP-MX explained by the common A factor? 0. 723

Last notes Including additional variables would allow to estimate additional latent common factors i.

Last notes Including additional variables would allow to estimate additional latent common factors i. e. C and E. One needs NV=NF+1. NF: Number of common factors that contribute to the means. But, including additional variables will not allow to estimate absolute latent means for the two groups, instead of differences, due to the equality of factor loadings. Never forget about the Common Causation Assumption: Jensen(1973): the sources of between group variation and the sources of within group variation are identical. If the Model of Heterogeneity for the means does not improve the fit with respect to the means’ homogeneity model, it means that the assumption can not be held and, thus the sources of between group differences are different from the sources of within group differences.

Useful References • Byrne, B. M. , Shavelson, R. J. , & Muthen, B.

Useful References • Byrne, B. M. , Shavelson, R. J. , & Muthen, B. (1989). Testing for the equivalence of factor covariance and mean structures: the issue of partial measurement invariance. Psychological Bulletin, 105, 456 -466. • Dolan, C. V. , Molenaar, P. C. M. , & Boomsma, D. I. (1991). Simultaneous genetic analysis of longitudinal means and covariance structure in the simplex model using twin data. Behavior Genetics, 21, 49 -65. • Dolan, C. V. , Molenaar, P. C. M. , & Boomsma, D. I. (1992). Decomposition of Multivariate phenotypic means in multigroup genetic covariance structure analysis. Behavior Genetics, 22, 319 -335. • Dolan, C. V. , Moolenar, P. C. M. , & Boomsma, D. I. (1994). Simultaneous genetic analysis of means and covariance structure: Pearson-Lawley selection rules. Behavior Genetics, 24, 17 -24. • Dolan, C. V. & Molenaar, P. C. M. (1994). Testing specific hypothesis cocerning latent group differences in multi-group covariance structure analysis sith structured means. Multivariate Behavioral Research, 29, 203 -222. • Rowe, D. & Cleveland, H. (1996). Academic achievement in blacks and whites: are the developmental processes similar? Intelligence, 23, 205 -228. • Wicherts, J. M. & Dolan, C. V. (2004). A cautionary note on the use of information fit indexes in covariance structure modeling with means. Structural Equation Modeling, 11, 4550.