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General Structural Equations

Explore different forms of constraints, including equality and inequality constraints, and learn how to apply them in running separate models in multiple groups. Understand parameter constraints and programming options like LISREL for variance fixing and constraint imposition. Discover techniques for handling more complex constraints and parameters across different groups, such as gender, occupation, or nationality. Enhance your understanding through practical examples and regression equivalences. Enhance your SEM analysis by effectively managing constraints in your models.

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General Structural Equations

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  1. General Structural Equations Week 2 #5 Different forms of constraints Introduction for models estimated in multiple groups

  2. Multiple Group Models(Hayduk: “Stacked” models) • Constraints on parameters • Running separate models in different groups • Applying equality constraints across groups

  3. Parameter constraints • Technically, any “fixed” parameter is constrained. • Trivially, b1=0 is a constraint • Another constraint: b1=1 (e.g., reference indicator) • or b1=-1 • “Fixing” the variance of an error term (usually because only 1 single indicator available) var(e1) = 7.0

  4. Inequality constraints • Can approximate an inequality constraint “manually” (check value, if –ve, “fix” it to some small +ve value) • Or, can re-express model so error variance is now the square of a coefficient (see yesterday’s class) • Inequality constrain may only be necessary “early” in the iteration process 0 Iteration Number Parameter value

  5. Inequality constraints Programming: (e.g. LISREL)… there will still be an epsilon error… must fix the variance of this error to 0. Variance of Ksi-1 = what in earlier model had been variance of epsilon-1

  6. Inequality constraints (other y-var’s) The above model can be reformulated as: Note var(Ksi-1) = 1.0

  7. Inequality constraints VAR(Y1) = lambda-12 VAR(Eta-1) + lambda-22 (1.0) What used to be VAR(Ksi) = error variance for Y1 – is now contained in the expression lambda22. Note, however, that no matter what the value of lambda-2 is, the entire expression will be positive. In other words, it is impossible for the error variance to drop below 0. Note var(Ksi-1) = 1.0

  8. Inequality constraints In AMOS, instead of a 1 in the path from the error term to the manifest variable, use a parameter name, but fix the variance of the error to 1.0.

  9. Equality constraints in single group models • This equality constraint in LISREL: • EQ LY 2 1 LY 3 1 • The constraint would only make sense if var(y2) = var(y3) • To impose the constraint that LY 1 1 = LY 2 1, we would fix LY 2 1 to 1.0 • (EQ LY 1 1 LY 2 1 would do this too)

  10. Equality constraints in the context of dummy variables X1 = Protestant X2 = Catholic X3 = Jewish X4 = Ref. All others (Atheist, Muslim, etc.) Tests of Prot vs. Catholic: b1=b2 (LISREL: EQ GA 1 1 GA 1 2 Test of Cath. vs. Jewish: b2=b3 (LISREL: EQ GA 1 2 GA 1 3 (Prot + Cath) vs. Jewish: Model 1: EQ GA 1 1 GA 1 2 Model 2: Above constraint, ADD: EQ GA 1 2 GA 1 3

  11. Equality constraints in the context of dummy variables X1 = Protestant X2 = Catholic X3 = Jewish X4 = Ref. All others (Atheist, Muslim, etc.) (Prot + Cath) vs. Jewish: Model 1: EQ GA 1 1 GA 1 2 Model 2: Above constraint, ADD: EQ GA 1 2 GA 1 3 Alternative, use LISREL “constraint” facility: CO GA 1 3 = GA(1,1)*0.5 + GA(1,2)*0.5 2b3 = b1 + b2 == can’t do this with AMOS

  12. More complex constraints when the software doesn’t seem to want to allow them: b1 = 2*b2 LISREL CO LY(2,1)=2*LY(3,1) AMOS only allows equality constraints

  13. More complex constraints when the software doesn’t seem to want to allow them: b1 = 2*b2 Re-express as New model: X3 = 2*b2LV1 + e3 Fix variance to 1.0

  14. AN EVEN MORE IMPORTANT (VIZ., MORE FREQUENTLY USED) FORM OF CONSTRAINT INVOLVES CONSTRAINING PARAMETERS ACROSS GROUPS Group 1 Group 2 Constraint: b1group1 = b1group2

  15. AN EVEN MORE IMPORTANT (VIZ., MORE FREQUENTLY USED) FORM OF CONSTRAINT INVOLVES CONSTRAINING PARAMETERS ACROSS GROUPS • What constitutes a group? • Males, females (esp. in psychological research) • Managers, workers (in management studies) • Country (in any form of cross-national / cross-cultural research) • City (in studies involving replications in a small number of cities, where cities are internally homogeneous but quite different from each other)

  16. AN EVEN MORE IMPORTANT (VIZ., MORE FREQUENTLY USED) FORM OF CONSTRAINT INVOLVES CONSTRAINING PARAMETERS ACROSS GROUPS • What constitutes a group? • Males, females (esp. in psychological research) • Managers, workers (in management studies) • Country (in any form of cross-national / cross-cultural research) • City (in studies involving replications in a small number of cities, where cities are internally homogeneous but quite different from each other) • Firms (e.g., in business studies, a 10-firm study, with different firms from different sectors of the economy) • Immigrant group

  17. AN EVEN MORE IMPORTANT (VIZ., MORE FREQUENTLY USED) FORM OF CONSTRAINT INVOLVES CONSTRAINING PARAMETERS ACROSS GROUPS • Regression equivalences: • X1: Male=1 Female=0 • X2: continuous variables of the sort used in typical SEM models (e.g., edcation) • Y = b0 + b1 X1 + b2 Educ • we can handle this in the SEM frame by using a dummy variable for X1 • Y = b0 + b1 X1 + b2 Educ + b3 (X1*Educ) • we could handle this if Educ is single-indicator (manually construction interaction term) • better way to deal with this: a multiple-group model

  18. A simple multiple-group example: Key question: b1(males) = b1(females)? males Notation: H0: b1[1] = b1[2] females

  19. Equivalences: Regression: X1=male/female X2 = Education Y = b0 + b1 X1 + b2 X2 + e SEM: Group 1 Group 2 Eta1 = gamma1 Ksi1 + zeta Eta1 = gamma1 Ksi + zeta Constraint: gamma1[1] = gamma1[2] Gamma1 in group 1 = Gamma1 in group 2 LISREL: EQ GA 1 1 1 GA 2 1 1

  20. Equivalences: • Regression: X1=male/female Male=1 Female=0 • X2 = Education • Y = b0 + b1 X1 + b2 X2 + b3 X1*X2 + e • SEM: Group 1 {male} Group 2 {female} • Eta1 = gamma1 Ksi1 + zeta Eta1 = gamma1 Ksi + zeta • What is b3 above is the difference between • gamma1[1] and gamma1[2] in SEM multiple-group • model. • [what is b2 in regression model is gamma1[2] (gamma1 in reference • group] • There is no equivalent to b1 in SEM framework • we could run a “pooled” model with a gender dummy variable though

  21. Multiple Group Models Group 2 (female) Group 1 (male) Equivalence of measurement coefficients H0: Λ[1] = Λ[2] lambda 1 [1] = lambda 1 [2] df=2 lambda 2 [1] = lambda 2 [2]

  22. Multiple Group Models • Other equivalence tests possible: • Equivalence of variances of latent variables • H0: PSI-1[1] = PSI-1[2] • This test will depend upon which ref. indicator used • Equivalence of error variances * • H0: Theta-eps[1] = Theta-eps[2] {entire matrix} • df=3 *and covariances if there are correlated errors

  23. Multiple Group Models • Measurement model equivalence does not imply same mean levels • Measurement model for Group 1 can be identical to Group 2, yet the two groups can differ radically in terms of level. • Example: Group 1Group 2 Load mean Load mean • Always trust gov’t .80 2.3 .78 3.9 • Govern. Corrupt -.75 3.8 -.80 2.3 • Politicians don’t care (where 1=agree strongly through 10=disagree strongly)

  24. Multiple Group Models • It is possible to have multiple group models with both common and unique items • Example: • Y1 Both countries: We should always trust our elected leaders • Y2 Both countries: If my government told me to go to war, I’d go • Y3 Both countries: We need more respect for government & authority Y4 (US): George Bush commands my respect because he is our President Y4 (Canada) Paul Martin commands my respect because he is our Prime Minister

  25. Multiple Group Models • It is possible to have multiple group models with both common and unique items • Example: • Y1 Both countries: We should always trust our elected leaders • Y2 Both countries: If my government told me to go to war, I’d go • Y3 Both countries: We need more respect for government & authority • Y4 (US): George Bush commands my respect because he is our President • Y4 (Canada) Paul Martin commands my respect because he is our Prime Minister We might expect (if measurement equivalence holds): lambda1[1] = lambda1[2] lambda2[1] = lambda2[2] BUT lambda3[1] ≠ lambda3[2]

  26. Multiple Group Models • Should be careful with the use of reference indicators (and/or sensitive to the fact that apparently non-equivalent models might appear to be so simply because of a single (reference) indicator • Example: • Group 1 Group 2 • Lambda-1 1.0* 1.0* • Lambda-2 .50 1.0 • Lambda-3 .75 1.5 • Lambda-4 1.0 2.0 • These two groups appear to have measurement models that are very different, but….

  27. Multiple Group Models • Group 1 Group 2 • Lambda-1 1.0* 1.0* • Lambda-2 .50 1.0 • Lambda-3 .75 1.5 • Lambda-4 1.0 2.0 • These two groups appear to have measurement models that are very different, but…. • If we change the reference indicator to Y2, we find: Gr 1 Gr 2 Lambda1 2.0 1.0 Lambda2 1.0* 1.0* Lambda3 1.5 1.5 Lambda4 2.0 2.0

  28. Multiple Group Models Modification Indices and what they mean in multiple-group models Assuming LY[1] = LY[2] (entire matrix) Example: MODIFICATION INDICES: Group 1 Group 2 Eta 1 Eta 1 Y1 --- Y1 --- Y2 .382 Y2 .382 Y3 1.24 Y3 1.24 Y4 45.23 Y4 45.23

  29. Multiple Group Models Modification Indices and what they mean in multiple-group models Assuming LY[1] = LY[2] (entire matrix) Example: MODIFICATION INDICES: Group 1 Group 2 Eta 1 Eta 1 Y1 --- Y1 --- Y2 .382 Y2 .382 Y3 1.24 Y3 1.24 Y4 45.23 Y4 45.23 Improvement in chi-square if equality constraint released

  30. Multiple Group Models : Modification Indices MODIFICATION Group 1 Group 2 INDICES eta1 eta2 eta1 eta2 Y1 --- 2.42 --- 3.89 Y2 1.42 3.44 1.42 1.01 Y3 0.43 2.11 0.43 40.89 Y4 0.11 --- 0.98 --- Y5 2.32 1.49 1.22 1.49 Y6 1.01 29.23 3.21 29.23 Tests equality constraint lambda5[1]=lambda5[2]

  31. Multiple Group Models : Modification Indices MODIFICATION Group 1 Group 2 INDICES eta1 eta2 eta1 eta2 Y1 --- 2.42 --- 3.89 Y2 1.42 3.44 1.42 1.01 Y3 0.43 2.11 0.43 40.89 Y4 0.11 --- 0.98 --- Y5 2.32 1.49 1.22 1.49 Y6 1.01 29.23 3.21 29.23 Tests equality constraint lambda5[1]=lambda5[2] Wald test (MI) for adding parameter LY(3,3) to the model in group 2 only

  32. MULTIPLE GROUP MODELS: parameter significance tests • When a parameter is constrained to equality across 2 (or more) groups, “pooled” significance test (more power) • Possible to have a coefficient non-signif. In each of 2 groups yet significant when equality constraint imposed

  33. MULTIPLE GROUP MODELS: Modification Indices (again) Model: LY[1]=LY[2]=LY[3] Group 1 MOD INDICES Lambda 1 3.01 Lambda 2 1.52 Lambda 3 3.22 Group 2 MOD INDICES Lambda 1 4.22 Lambda 2 3.99 Lambda 3 5.22 Group 3 MOD INDICES Lambda 1 89.22 Lambda 2 6.11 Lambda 3 1.22 Free LY(2,1) in group 3 but LY(2,1) in group 1 = LY(2,1) in group 2

  34. When do we have measurement equivalence • STRONG equivalence: • all matrices identical, all groups • (might possibly exclude variance of LV’s from this … i.e., the PHI or PSI matrices) • WEAKER equivalence (usually accepted) • Lambda matices identical, all groups • Theta matrices could be different (and probably are), either having the same form or not • WEAKER YET: • Lambda matrices have the same form, some identical coefficients

  35. Measurement coefficients, construct equation coefficients in multiple group models • We usually need the measurement equation coefficients to be equivalent in order to proceed with comparison of construct equation coefficients

  36. Measurement coefficients, construct equation coefficients in multiple group models • We usually need the measurement equation coefficients to be equivalent in order to proceed with comparison of construct equation coefficients • For this reason, tests for measurement equivalence are usually not as rigorous as the “substantive” tests for construct equation coefficient equivalence (though instances of poor fit should be noted in any report of results)

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