General Structural Equation (LISREL) Models Week 3 #1 Multiple Group Models An extended multiple-group model: Religiosity & Sexual Morality in 2 countries (LISREL example) Computer programming for multiple-group models: a) LISREL b) AMOS See Week3Examples

Constraint: b1group1 = b1group2 MOST IMPORTANT FORM OF CONSTRAINT INVOLVES CONSTRAINING PARAMETERS ACROSS GROUPS Group 1 Group 2 Constraint: b1group1 = b1group2

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] From last Friday

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 From last Friday

Measurement model equivalence does not imply same mean levels 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 1 Group 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) From last Friday

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] From last Friday

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…. From last Friday

Multiple Group Models Gr 1 Gr 2 Lambda1 2.0 1.0 Lambda2 1.0* 1.0* 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 From last Friday

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 From last Friday

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

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]

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

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 Possible to have a coefficient that is not significant in each of two groups (e.g., +ve coefficient, NS, in one group, -ve, NS, in another) yet the difference between the groups is statistically significant

Tests in 3+ groups 2-group model can be extended to m groups (in theory, infinite number as long as minimum sample size requirements met in each group; in practice, some software packages have limits) Models: Group1=Group2=Group3 Group1=Group2≠Group3 Group1 ≠ Group2≠Group3 Must be careful about interpretation of Modification Indices In a model with [1]=[2]=[3] ≠ 0, then a MI will provide an indication of how much the model improve if the parameter constraint is removed in the mth group only (e.g., MI in group 2 would test against a model in which the group 1 & group 3 {same} parameter are constrained to equality but the group 2 is allowed to differ)

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

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

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

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)

LISREL PROGRAMMING FOR MULTIPLE GROUP MODELS Basics: “Stack” the groups (one program after the next) In DA statement of first group, specify total number of groups: e.g., DA NG=2 NI=23 NO=1246 NI specification (# of input var’s) applies only to group 1 NO specification (# of observations) applies only to group 1

LISREL PROGRAMMING FOR MULTIPLE GROUP MODELS DA NG=2 … Specify group 1 model as usual Title for Group 2 DA statement group 2 NI= NO= CM FI= [location of group 2 cov mtx] SE 1 3 4 6 8 / LABELS [optional] MO NY= NX= NK= NE= + special options for matrix specification OU (as usual)

LISREL PROGRAMMING: MULTIPLE GROUPS MO specification: LY=PS same pattern as prev. group - for example, if group 1 specifies 2 LV’s with first three indicators on LV1, next 6 on LV2, this same specification will be copied to group 2 LY= IN invariant - same pattern and all free coefficients in this matrix constrained to equality with corresponding coefficients in previous group

LISREL PROGRAMMING: MULTIPLE GROUPS Adding equality constraints to a matrix that is otherwise allowed to differ from the same matrix in a previous group: Group 1 (e.g.) LY=FU,FI VA 1.0 LY 2 1 FR LY 2 1 LY 3 1 LY 4 1 Group 2 LY=PS EQ LY 1 2 1 LY 2 2 1 EQ LY 1 3 1 LY 3 1

LISREL PROGRAMMING: MULTIPLE GROUPS Releasing equality constraints on a single parameter when matrix is otherwise specified as Invariant: Group 2: LY=IN FR LY 4 1  LY remains invariant except for parameter LY(4,1)

Examples: Example (separate handouts) (religion & sexual morality in 2 countries) Multiple group example #1: USA DA NO=1150 NI=19 MA=CM NG=2 CM FI=G:\CLASSES\ICPSR2005\WEEK3EXAMPLES\USA.COV LA A006 F028 F066 F063 F118 F119 F120 F121 GENDER AGE EDUC TOWNSIZE MARR1 MARR2 MARR3 OCC1 OCC2 OCC3 OCC4 SE 1 2 3 4 5 6 7 8/ MO NY=8 NE=2 LY=FU,FI PS=SY,FR TE=SY VA 1.0 LY 2 1 LY 5 2 FR LY 1 1 LY 3 1 LY 4 1 FR LY 6 2 LY 7 2 LY 8 2 FR TE 4 3 OU ND=4 SC MI Basic program See handout for variable list

Program & output: TwoGroup1b.ls8, *.out Multiple group example #1: USA Number of Input Variables 19 Number of Y - Variables 8 Number of X - Variables 0 Number of ETA - Variables 2 Number of KSI - Variables 0 Number of Observations 1150 Number of Groups 2 Group #2: Canada DA NO=1763 NI=19 MA=CM CM FI=G:\CLASSES\ICPSR2005\WEEK3EXAMPLES\CDN.COV LA A006 F028 F066 F063 F118 F119 F120 F121 GENDER AGE EDUC TOWNSIZE MARR1 MARR2 MARR3 OCC1 OCC2 OCC3 OCC4 SE 1 2 3 4 5 6 7 8/ MO LY=IN PS=PS TE=PS OU ND=4 SC MI Number of Observations 1763 Program & output: TwoGroup1b.ls8, *.out

Multiple group example #1: USA Parameter Specifications LAMBDA-Y EQUALS LAMBDA-Y IN THE FOLLOWING GROUP PSI ETA 1 ETA 2 -------- -------- ETA 1 7 ETA 2 8 9 THETA-EPS A006 F028 F066 F063 F118 F119 -------- -------- -------- -------- -------- -------- A006 10 F028 0 11 F066 0 0 12 F063 0 0 13 14 F118 0 0 0 0 15 F119 0 0 0 0 0 16 F120 0 0 0 0 0 0 F121 0 0 0 0 0 0 F120 F121 F120 17 F121 0 18

Covariance Matrix of ETA ETA 1 ETA 2 -------- -------- ETA 1 2.4328 ETA 2 1.9415 4.5847 PSI (0.1554) 15.6531 (0.1464) (0.3118) 13.2596 14.7036 Group 1 Covariance Matrix of ETA ETA 1 ETA 2 -------- -------- ETA 1 3.5419 ETA 2 2.3997 5.2741 PSI (0.1924) 18.4093 (0.1529) (0.3175) 15.6932 16.6128 Group 2

Global Goodness of Fit Statistics Group Goodness of Fit Statistics (USA) Contribution to Chi-Square = 120.0772 Percentage Contribution to Chi-Square = 51.3912 Root Mean Square Residual (RMR) = 0.2698 Standardized RMR = 0.04414 Goodness of Fit Index (GFI) = 0.9756 Global Goodness of Fit Statistics Degrees of Freedom = 42 Minimum Fit Function Chi-Square = 233.6530 (P = 0.0) Normal Theory Weighted Least Squares Chi-Square = 229.0169 (P = 0.0) Estimated Non-centrality Parameter (NCP) = 187.0169 90 Percent Confidence Interval for NCP = (143.2744 ; 238.2786) Normed Fit Index (NFI) = 0.9840 Non-Normed Fit Index (NNFI) = 0.9824 Parsimony Normed Fit Index (PNFI) = 0.7380 Comparative Fit Index (CFI) = 0.9868 Incremental Fit Index (IFI) = 0.9868 Relative Fit Index (RFI) = 0.9787 Group Goodness of Fit Statistics (CANADA) Contribution to Chi-Square = 113.5759 Percentage Contribution to Chi-Square = 48.6088 Root Mean Square Residual (RMR) = 0.2222 Standardized RMR = 0.03069 Goodness of Fit Index (GFI) = 0.9841

Multiple group example #1: USA Modification Indices and Expected Change Modification Indices for LAMBDA-Y ETA 1 ETA 2 -------- -------- A006 5.5766 6.6058 F028 16.2024 26.0322 F066 0.0376 5.6663 F063 1.4466 1.0440 F118 0.5472 4.7638 F119 0.0715 0.0993 F120 2.9330 0.0982 F121 8.8507 2.3763 Group #2: Canada A006 3.7307 0.3247 F028 16.2027 0.0282 F066 0.0262 3.8083 F063 1.0022 1.3223 F118 18.8935 4.7637 F119 1.4474 0.0831 F120 21.8972 0.0808 F121 5.7255 1.9721

F121 - - 0.7996 LISREL Estimates (Maximum Likelihood) LAMBDA-Y ETA 1 ETA 2 -------- -------- A006 0.4590 - - (0.0119) 38.4181 F028 1.0000 - - F066 0.8854 - - (0.0257) 34.4187 F063 -1.1710 - - (0.0332) -35.2593 F118 - - 1.0000 F119 - - 0.7174 (0.0262) 27.3386 F120 - - 1.0684 (0.0326) 32.8141 F121 - - 0.7996 (0.0267) 29.9686

Expected Change for LAMBDA-Y (USA) ETA 1 ETA 2 -------- -------- A006 -0.0334 -0.0296 F028 0.2040 0.1633 F066 0.0041 -0.0492 F063 0.0339 -0.0272 F118 0.0458 0.1229 F119 0.0135 -0.0089 F120 0.0974 -0.0110 F121 -0.1477 -0.0442 Expected Change for LAMBDA-Y (Canada) A006 0.0150 0.0057 F028 -0.2040 0.0045 F066 -0.0024 -0.0398 F063 -0.0173 -0.0296 F118 -0.1956 -0.1229 F119 0.0444 0.0054 F120 0.1835 0.0059 F121 -0.0842 0.0253

Group #2: Canada Within Group Completely Standardized Solution LAMBDA-Y ETA 1 ETA 2 -------- -------- A006 0.8727 - - F028 0.7325 - - F066 0.7209 - - F063 -0.7439 - - F118 - - 0.6738 F119 - - 0.6078 F120 - - 0.8202 F121 - - 0.6876 Multiple group example #1: USA A006 0.8201 - - F028 0.6690 - - F066 0.7403 - - F063 -0.7512 - - F118 - - 0.6711 F119 - - 0.6047 F120 - - 0.7647 F121 - - 0.6674

More variance in Canada Multiple group example #1: USA Common Metric Completely Standardized Solution LAMBDA-Y ETA 1 ETA 2 -------- -------- A006 0.8554 - - F028 0.7109 - - F066 0.7267 - - F063 -0.7461 - - F118 - - 0.6728 F119 - - 0.6067 F120 - - 0.7988 F121 - - 0.6800 Covariance Matrix of ETA ETA 1 0.7837 ETA 2 0.4927 0.9166 Group #2: Canada Common Metric Standardized Solution A006 0.8086 - - F028 1.7618 - - F066 1.5599 - - F063 -2.0631 - - F118 - - 2.2365 F119 - - 1.6044 F120 - - 2.3894 F121 - - 1.7883 More variance in Canada Connection stronger in Canada Covariance Matrix of ETA ETA 1 ETA 2 -------- -------- ETA 1 1.1410 ETA 2 0.6090 1.0544

Testing measurement equivalence Group #2: Canada DA NO=1763 NI=19 MA=CM CM FI=G:\CLASSES\ICPSR2005\WEEK3EXAMPLES\CDN.COV LA A006 F028 F066 F063 F118 F119 F120 F121 GENDER AGE EDUC TOWNSIZE MARR1 MARR2 MARR3 OCC1 OCC2 OCC3 OCC4 SE 1 2 3 4 5 6 7 8 / MO LY=PS PS=PS TE=PS OU ND=4 SC MI Global Goodness of Fit Statistics Degrees of Freedom = 36 Minimum Fit Function Chi-Square = 209.7783 (P = 0.0) Normal Theory Weighted Least Squares Chi-Square = 207.3408 (P = 0.0) Estimated Non-centrality Parameter (NCP) = 171.3408 90 Percent Confidence Interval for NCP = (129.7727 ; 220.4219) Normed Fit Index (NFI) = 0.9856 Non-Normed Fit Index (NNFI) = 0.9814 Parsimony Normed Fit Index (PNFI) = 0.6336 Comparative Fit Index (CFI) = 0.9881 Incremental Fit Index (IFI) = 0.9881 Relative Fit Index (RFI) = 0.9777

Some Model/Test Results: Matrices Chi-square df IFI TwoGroup1b LY=IN 233.653 42 .987 TwoGroup1c LY=PS 209.778 36 .988 TwoGroup1d LY=IN PS=IN 272.740 45 .984 TwoGroup1e LY=IN PS=IN but 267.472 44 .985 PS 2,1 free From TwoGroup1d, in Group #2 (Canada): Modification Indices for PSI ETA 1 ETA 2 -------- -------- ETA 1 31.1069 ETA 2 4.8672 4.0391 Expected Change for PSI ETA 1 0.3466 ETA 2 -0.1158 0.2188

Models with exogenous variables in construct equations: Multiple group example #1: USA DA NO=1150 NI=19 MA=CM NG=2 CM FI=G:\CLASSES\ICPSR2005\WEEK3EXAMPLES\USA.COV LA A006 F028 F066 F063 F118 F119 F120 F121 GENDER AGE EDUC TOWNSIZE MARR1 MARR2 MARR3 OCC1 OCC2 OCC3 OCC4 SE 1 2 3 4 5 6 7 8 9 10 11 12/ MO NY=8 NE=2 LY=FU,FI PS=SY,FR TE=SY NX=4 NK=4 LX=ID C PH=SY,FR TD=ZE GA=FU,FR LE RELIG MORAL VA 1.0 LY 2 1 LY 5 2 FR LY 1 1 LY 3 1 LY 4 1 FR LY 6 2 LY 7 2 LY 8 2 FR TE 4 3 OU ND=4 SC MI Group #2: Canada DA NO=1763 NI=19 MA=CM CM FI=G:\CLASSES\ICPSR2005\WEEK3EXAMPLES\CDN.COV LA A006 F028 F066 F063 F118 F119 F120 F121 GENDER AGE EDUC TOWNSIZE MARR1 MARR2 MARR3 OCC1 OCC2 OCC3 OCC4 SE 1 2 3 4 5 6 7 8 9 10 11 12/ MO LY=IN PS=PS TE=PS LX=IN PH=PS TD=IN GA=PS LE RELIG MORAL OU ND=4 SC MI

LAMBDA-Y EQUALS LAMBDA-Y IN THE FOLLOWING GROUP GAMMA GENDER AGE EDUC TOWNSIZE -------- -------- -------- -------- RELIG 7 8 9 10 MORAL 11 12 13 14 PHI GENDER 15 AGE 16 17 EDUC 18 19 20 TOWNSIZE 21 22 23 24 PSI RELIG MORAL -------- -------- RELIG 25 MORAL 26 27

Multiple group example #1: USA LISREL Estimates (Maximum Likelihood) LAMBDA-Y EQUALS LAMBDA-Y IN THE FOLLOWING GROUP GAMMA GENDER AGE EDUC TOWNSIZE -------- -------- -------- -------- RELIG 0.6772 -0.0169 0.0818 0.0182 (0.1001) (0.0031) (0.0351) (0.0293) 6.7629 -5.5270 2.3342 0.6194 MORAL 0.0671 -0.0143 0.3045 -0.0218 (0.1451) (0.0045) (0.0515) (0.0428) 0.4623 -3.1968 5.9077 -0.5080 Group #2: Canada Number of Iterations = 8 RELIG 0.9517 -0.0292 0.1184 0.0636 (0.0926) (0.0028) (0.0397) (0.0173) 10.2755 -10.4921 2.9799 3.6641 MORAL -0.0967 -0.0222 0.4775 0.0934 (0.1182) (0.0036) (0.0526) (0.0226) -0.8176 -6.1962 9.0759 4.1312

Models/Tests Model Description Chi-square df CFI Group2A LY=IN GA=PS 645.0492 90 .968 Group2B LY=IN GA=IN 675.2567 98 .966 From Model Group2B (Group #2, Canada): Modification Indices for GAMMA GENDER AGE EDUC TOWNSIZE -------- -------- -------- -------- RELIG 7.8388 7.6672 -0.0001 0.0965 MORAL 3.1242 0.0477 41.6948 1.3675 Modification Indices for GAMMA (Group #1, USA) RELIG 7.7220 7.4528 0.0002 0.2770 MORAL 3.0729 0.0516 5.5255 4.2116

Model Description Chi-square df CFI Group3A GA=IN 747.942 154 .971 Group3B GA=PS 712.264 138 .972 Group3C GA=IN 739.907 146 .971 Free occ. in group 2 Group3D GA=PS 720.507 146 .972 Occ FI in group 2 Group3E GA=PS 731.286 154 .972 Occ Fi both groups Group3F GA=PS 715.659 142 .972 Occ coefficients = For 1st Eta variable Tests: Group3A vs. Group 3B Equality of all GA coefficients Group 3A vs Group 3B Equality of occupation GA coefficients Group 3B vs. Group 3D Is occupation stat. significant in group 2 (Canada)? Group 3B vs. Group 3E Is occupation stat. sign. in both groups? Group 3D vs. Group 3E Is occupation stat. sign. in group 1? Group 3B vs. Group 3F Equality of occupation effect for First eta variable.

GAMMA USA GENDER AGE EDUC TOWNSIZE OCC1 OCC2 -------- -------- -------- -------- -------- -------- RELIG 0.6576 -0.0165 0.1054 0.0189 0.3237 0.1832 (0.1023) (0.0031) (0.0382) (0.0293) (0.3118) (0.3063) 6.4303 -5.3480 2.7565 0.6438 1.0382 0.5982 MORAL 0.1103 -0.0147 0.2874 -0.0247 -0.0462 0.1877 (0.1484) (0.0045) (0.0561) (0.0429) (0.4558) (0.4479) 0.7435 -3.2723 5.1225 -0.5766 -0.1013 0.4192 GAMMA CANADA RELIG 0.9358 -0.0287 0.1329 0.0640 -0.2922 -0.2209 (0.0944) (0.0029) (0.0465) (0.0174) (0.3589) (0.3319) 9.9177 -10.0111 2.8594 3.6850 -0.8143 -0.6655 MORAL -0.0903 -0.0239 0.4201 0.0927 0.2387 0.2938 (0.1206) (0.0037) (0.0610) (0.0226) (0.4668) (0.4317) -0.7488 -6.4577 6.8844 4.0967 0.5112 0.6805 GAMMA OCC3 OCC4 -------- -------- RELIG 0.4312 0.3988 (0.3019) (0.3000) 1.4285 1.3290 MORAL -0.0230 0.2741 (0.4412) (0.4386) GAMMA OCC3 OCC4 -------- -------- RELIG -0.1334 -0.2529 (0.3125) (0.3123) -0.4269 -0.8097 MORAL -0.0750 0.0260 (0.4065) (0.4063) -0.1846 0.0640

(insert AMOS demo here)