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General Structural Equation (LISREL) Models

General Structural Equation (LISREL) Models . Week 3 # 3 MODELS FOR MEANS AND INTERCEPTS. Refer to slides from previous class (Week 3 #2) if not covered in full on Tuesday. Models with Means and Intercepts. Review of material from last class

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General Structural Equation (LISREL) Models

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  1. General Structural Equation(LISREL) Models Week 3 # 3 MODELS FOR MEANS AND INTERCEPTS

  2. Refer to slides from previous class (Week 3 #2) if not covered in full on Tuesday.

  3. Models with Means and Intercepts • Review of material from last class • (detail of coverage to depend on progress from Tuesday’s class) • Consider a measurement model: Equations: V1 = 1.0 L1 + E1 V2 = b1L1 + E2 V3 = b2L1 + E3 V4 = b3L1 + E4

  4. Models with Means and Intercepts • The covariance matrix upon which this model is based:

  5. Models with Means and Intercepts • Simple replacements in this matrix: • 1. For any element, covariance replaced by moment: 2. And an “augmented moment matrix” is created by letting the first (or the last) element of the data matrix (the “X” in X’X) be a vector of 1’s

  6. Models for Means and Intercepts • Augmented moment matrix: Each of the above divided by N-1

  7. Means and intercepts in SEM Models Working from this matrix instead of working from S, we can add intercepts back into equations (reproduce M instead of S).

  8. Models for Means and Intercepts • MEASUREMENT EQUATIONS NOW BECOME: • V1 = a1 + 1.0L1 + E1 • V2 = a2 + b1 L1 + E2 • V3 = a3 + b2 L1 + E3 • V4 = a4 + b3 L1 + E4 • And there is a final equation for the mean of the latent variable: • L1 = a5

  9. Means and intercepts in SEM Models Conventional Model: X1 = 1.0 LV1 + e1 X2 = b2 LV1 + e2 X3 = b3 LV1 + e3 Extended to include intercepts: X1 = a1 + 1.0 LV1 + e1 X2 = a2 + b2 LV1 + e2 X3 = a3 + b3 LV1 + e3 [LV1 = a4] EQS calls this “V999”. Other programs do not explicitly model “1” as if it were a variable

  10. Means and intercepts in SEM Models Three new pieces of information: Means of X1, X2, X3 Equations: X1 = a1 + 1.0 L1 + e1 X2 = a2 + b2 L1 + e2 X3 = a3 + b3 L1 + e3 Other parameters: Var(e1) Var(e2) Var(e3) Var(L1) Mean(L1) One of the following parameters needs to be fixed: a1,a2,a3, mean(L1)

  11. Models for Means and Intercepts • From the augmented moment matrix, 4 new pieces of information • 5 new (possible) parameters: • a1 through a5 •  cannot identify equation intercepts (under-identified) •  but we can identify differences between intercepts.

  12. Means and intercepts in SEM Models Equations: X1 = a1 + 1.0 L1 + e1 X2 = a2 + b2 L1 + e2 X3 = a3 + b3 L1 + e3 Conventions: a1 = 0 Then Mean(L1) = Mean(X1) and a2 is difference between means X1,X2 (not usually of interest) a3 is difference between means X1, X3 (not usually of interest)

  13. Means and intercepts in SEM Models Equations: X1 = a1 + 1.0 L1 + e1 X2 = a2 + b2 L1 + e2 X3 = a3 + b3 L1 + e3 Conventions: Mean(L1) = 0 Then a1=mean of X1 a2 = mean of X2 a3 = mean of X3 Not particularly useful: means of LV’s by definition =0

  14. Means and intercepts in SEM Models In longitudinal case, more interesting possibilities: Equations: X1 = a1 + 1.0 L1 + e1 X2 = a2 + b1 L1 + e2 X3 = a3 + b2 L1 + e3 X4 = a4 + 1.0 L2 + e4 X5 = a5 + b3 L2 + e5 X6 = a6 + b4 L2 + e6 Constrain measurement models: b1=b3 b2=b4 Constrain intercepts: a1 = a4 a2 = a5 a3 = a6 Fix Mean(L1) to 0 Can now estimate parameter for Mean (L2)

  15. Means and intercepts in SEM Models Equations: X1 = a1 + 1.0 L1 + e1 X2 = a2 + b1 L1 + e2 X3 = a3 + b2 L1 + e3 Y1 = a4 + 1.0 L2 + e4 Y2 = a5 + b3 L2 + e5 Y3 = a6 + b4 L2 + e6 Constrain measurement models: b1=b3 b2=b4 Constrain intercepts: a1 = a4 a2 = a5 a3 = a6 Fix Mean(L1) to 0 Can now estimate parameter for Mean (L2) Example: X1 X2 X3 X4 X5 X6 Means: 2 3 2.5 3 4 3.5 Y1 = a4 + 1.0 L2 + e4 (E(L2)=a7 Estimate: a7=1.0 Y1 = 2 + 1.0*1 + 0 (expected value of L2=1.0) Y2 = 3 + b3*1 + 0 (expected value of L2 = 1.0) New parameter:a7

  16. Means and intercepts in SEM Models Equations: X1 = a1 + 1.0 L1 + e1 X2 = a2 + b1 L1 + e2 X3 = a3 + b2 L1 + e3 Y1 = a4 + 1.0 L2 + e4 Y2 = a5 + b3 L2 + e5 Y3 = a6 + b4 L2 + e6 There can be a construct equation intercept parameter in causal models L2 = a7 + b1 L1 + D2 If mean(L1) fixed to 0 E(L2) = a7 + b1*0 = a7 As before, a7 represents the expected difference between the mean of L1 and the mean of L2

  17. Means and intercepts in SEM Models L2 = a7 + b1 L1 + D2 If mean(L1) fixed to 0 E(L2) = a7 + b1*0 = a7 In practice, if L1 and L2 represent time 1 and time 2 measures of the same thing, we would expect correlated errors:

  18. Means and intercepts in SEM Models Same principle can be applied to multiple group models: Group 1 a1[1] = a1[2] X1 = a1 + 1.0 L1 + e1 X2 = a2 + b2 L1 + e2 X3 = a3 + b3 L1 + e3 a2[1]=a2[2] a3[1]=a3[2] Mean(L1)=0 Group 2 X1 = a1 + 1.0 L1 + e1 X2 = a2 + b2 L1 + e2 X3 = a3 + b3 L1 + e3 We usually constrain measurement coefficients: b2[1]=b2[2] & b3[1]=b3[2] Mean(L1) = a4

  19. Models for Means and Intercepts • Applications: • #1: A two-group model Group 1 V1 = a1 + 1.0L1 + E1 V2 = a2 + b1 L1 + E2 V3 = a3 + b2 L1 + E3 V4 = a4 + b3 L1 + E4 Group 2 V1 = a1 + 1.0 L1 + E1 V2 = a2 + b1 L1 + E2 V3 = a3 + b2 L1 + E3 V4 = a4 + b3 L1 + E4 Group 1 Group 2 Mean(L1) =a5 Mean(L1) =a5

  20. Models for Means and Intercepts • Group 1 Group 2 Group 1 V1 = a1 + 1.0L1 + E1 V2 = a2 + b1 L1 + E2 V3 = a3 + b2 L1 + E3 V4 = a4 + b3 L1 + E4 Group 2 V1 = a1 + 1.0 L1 + E1 V2 = a2 + b1 L1 + E2 V3 = a3 + b2 L1 + E3 V4 = a4 + b3 L1 + E4 Mean(L1) =a5 Mean(L1) =a5 Constraints: 1. Measurement model 2. intercepts: a1[1] = a1[2] ; a2[1] = a2[2] etc. 3. a5[1] = 0 THIS MEANS THAT a5[2] represents between-group mean differences.

  21. A practical example: • Differences in religiosity, World Values Study 1990 In PRELIS, generate mean vectors as well as covariances

  22. Looking at item means • Means: • U.S.: • Means • v9 v147 v175 v176 • -------- -------- -------- -------- • 1.700 3.854 1.401 8.126 • Canada: • Means high = less religious except for V176 • v9 v147 v175 v176 • -------- -------- -------- -------- • 2.193 4.811 1.750 7.005

  23. Factor Means: • We cannot establish a factor mean for each group, but we CAN get a coefficient representing the difference between the factor means • (factor mean in each group can be established trivially as equal to the mean of one of the indicators – not particularly helpful though).

  24. LISREL TERMINOLOGY • Equations: • X1 = τx1 + λ11ξ1 + δ1 • X2 = τx2 + λ21ξ1 + δ2 • X3 = τx3 + λ31ξ1 + δ3 • X4 = τx4 + λ41ξ1 + δ4 • New vector: Tau-X (TX) • Normally, λ11 = 1.0 (reference indicator) • Variances, covariances, means: • VAR(δ1), VAR(δ2), VAR(δ3), VAR(δ4), MEAN(ξ1) • New vector: Kappa (vector of means of ξ’s)

  25. LISREL TERMINOLOGY • Constraints: • Group 1 Group 2 • TX(1) = TX(1) • TX(2) = TX(2) • TX(3) = TX(3) • TX(4) = TX(4) • Kappa1 = 0 Kappa1 = free*

  26. LISREL TERMINOLOGY • Constraints: • Group 1 Group 2 • TX(1) = TX(1) • TX(2) = TX(2) • TX(3) = TX(3) • TX(4) = TX(4) • Kappa1 = 0 Kappa1 = free* • Tau-X : vector of manifest variable intercepts • Kappa: vector of latent (exogenous) variable means • PROGRAMMING: • Group 1: TX=FR KA=FI • Group 2: TX=IN KA=FR

  27. LISREL TERMINOLOGY • Equivalent for Y-variables: • Tau-Y: intercepts for manifest variable eq’s • Alpha: intercepts for construct equations • Eta1 = alpha1 + gamma ksi + zeta • Important Note: • When gammas are constrained to equality across groups, alphas represent a between-group differences in means controlling for differences in Ksi.

  28. Factor Mean differences • Variances: • PHI USA Canada • USA KSI 1 KSI 1 • -------- • 2.751 3.268 • (0.187) (0.162) • TAU-X TAU-X is constrained to equality (both groups) • v9 v147 v175 v176 • -------- -------- -------- -------- • 1.715 3.828 1.428 8.197 • (0.023) (0.058) (0.016) (0.065) • 74.484 65.924 86.556 127.025 • KAPPA Kappa is zero in group 1 • KSI 1 Lambda-X V9 .458 • -------- V147 1.00 • 1.005 V175 .276 • (0.072) V176 -1.289 • 13.927

  29. Models for Means and Intercepts • Testing assumptions • we have assumed that the pattern of differences between corresponding measurement equation intercepts can be expressed by a single coefficient V1 = a1 + 1.0 L1 + e1 V2 = a2 + b2 L1 + e2 V3 = a3 + b3 L1 + e3 V4 = a4 + b4 L1 + e4 L1=a5 We constrain a1,a2,a3,a4 to equality across groups and estimate a5 to represent between-group differences

  30. Models for Means and Intercepts • What if the pattern is: • Group 1 Group 2 • v1 3.2 4.2 • v2 2.2 3.2 • v3 1.9 2.8 • v4 2.0 1.5 • a5 will be positive, but the fact that the group1-group2 difference on V4 is not consistent will lead to poorer fit Could estimate model with a1[1]=a1[2], a2[1]=a2[2], a3[1]=a3[2] BUT a4[1]a4[2]

  31. LISREL TERMINOLOGY • LISREL: • Equations: • X1 = τx1 + λ11ξ1 + δ1 • X2 = τx2 + λ21ξ1 + δ2 • X3 = τx3 + λ31ξ1 + δ3 • X4 = τx4 + λ41ξ1 + δ4 • Normally, τx1 = 1.0 (reference indicator) • Variances, covariances, means: • VAR(δ1), VAR(δ2), VAR(δ3), VAR(δ4), MEAN(ξ1)

  32. Models for Means and Intercepts • Modification Indices for TAU-X • v9 v147 v175 v176 • -------- -------- -------- -------- • 4.941 0.613 13.006 16.218 • We could estimate a model with TX 4 free (not essential; would be more important if chi-square really large) • Expected Change for TAU-X • v9 v147 v175 v176 • -------- -------- -------- -------- • 0.045 -0.036 0.063 0.236 • TAU-X (repeated from previous slide): • v9 v147 v175 v176 • -------- -------- -------- -------- • 1.715 3.828 1.428 8.197 • (0.023) (0.058) (0.016) (0.065) • 74.484 65.924 86.556 127.025

  33. LISREL PROGRAMMING CODE FOR PREVIOUS EXAMPLE: • 2 group model for relig 1:USA • DA NG=2 NI=23 NO=1456 • CM FI=g:\Means&Intercepts\usa.cov • ME FI=g:\Means&Intercepts\usa.mn • LABELS • v9 v147 v151 v175 v176 v304 v305 v307 v308 v309 v310 v355 v356 sex • occup1 occup2 occup3 occup4 occup5 occup6 occup7 occup8 occup9 • SE • 1 2 4 5 / • MO NX=4 NK=1 LX=FU,FI PH=SY,FR TD=SY C • TX=FR KA=FI • VA 1.0 LX 2 1 • FR LX 1 1 LX 3 1 LX 4 1 • OU ME=ML SE TV MI SC ND=3 • Group 2: Canada • DA NI=23 NO=1474 • CM FI=g:\Means&Intercepts\cdn.cov • ME FI=g:\Means&Intercepts\cdn.mn • LABELS • v9 v147 v151 v175 v176 v304 v305 v307 v308 v309 v310 v355 v356 sex • occup1 occup2 occup3 occup4 occup5 occup6 occup7 occup8 occup9 • SE • 1 2 4 5 / • MO LX=IN PH=PS TD=PS KA=FR TX=IN • OU ME=ML SE TV MI SC ND=3 Do not include MA=CM New

  34. Doing it in AMOS: Add this

  35. AMOS: For each exogenous variable, the Object Properties box will now have Mean and Variance

  36. AMOS: For each endogenous variable, the Object Properties box will now have an Intercept For all indicators, type in a parameter name here. For all indicators, click “all groups” to impose equality constraint.

  37. AMOS Constraints: Group 1 Group 2 b1 = b1 b2 = b2 b3 = b3 a1 = a1 a2 = a2 a3 = a3 a4 = a4 a5=0 a5 free (parameter for mean differences)

  38. AMOS • Group: Canada • Means • Estimate S.E. C.R. P Label • RELIG 1.005 0.072 13.931 0.000 a5 • Group: United States • Intercepts • Estimate S.E. C.R. P Label • V9 1.715 0.023 74.512 0.000 a1 • V147 3.828 0.058 65.947 0.000 a2 • V175 1.428 0.016 86.585 0.000 a3 • V176 8.197 0.065 127.068 0.000 a4 REFER TO: Model2.amw for more extended example

  39. Moving to Y, Eta and adding a 3rd country • 2 group model for relig 1:USA • DA NG=3 NI=23 NO=1456 • CM FI=H:\Means&Intercepts\usa.cov • ME FI=H:\Means&Intercepts\usa.mn • LABELS • v9 v147 v151 v175 v176 v304 v305 v307 v308 v309 v310 v355 v356 sex • occup1 occup2 occup3 occup4 occup5 occup6 occup7 occup8 occup9 • SE • 1 2 4 5 / • MO NY=4 NE=1 LY=FU,FI PS=SY,FR TE=SY C • TY=FR AL=FI • VA 1.0 LY 2 1 • FR LY 1 1 LY 3 1 LY 4 1 • OU ME=ML SE TV MI SC ND=3 • Group 2: Canada • DA NI=23 NO=1474 • CM FI=H:\Means&Intercepts\cdn.cov • ME FI=H:\Means&Intercepts\cdn.mn • LABELS • v9 v147 v151 v175 v176 v304 v305 v307 v308 v309 v310 v355 v356 sex • occup1 occup2 occup3 occup4 occup5 occup6 occup7 occup8 occup9 • SE • 1 2 4 5 / • MO LY=IN PS=PS TE=PS AL=FR TY=IN • OU ME=ML SE TV MI SC ND=3 • Group 3: Netherlands • DA NI=23 NO=909 • CM FI=H:\Means&Intercepts\neth.cov • ME FI=H:\Means&Intercepts\neth.mn • LABELS • v9 v147 v151 v175 v176 v304 v305 v307 v308 v309 v310 v355 v356 sex • occup1 occup2 occup3 occup4 occup5 occup6 occup7 occup8 occup9 • SE • 1 2 4 5 / • MO LY=IN PS=PS TE=PS AL=FR TY=IN • OU ME=ML SE TV MI SC ND=3

  40. Mean comparisons USA=0 • ALPHA Canada • ETA 1 • -------- • 0.896 • (0.064) • 14.077 • ALPHA Netherlands • ETA 1 • -------- • 2.069 • (0.087) • 23.889 Chi-square = 280.733, df=18 With AL(1)=AL(1)=AL(1) 3 groups (i.e., AL=0 in all three groups) Chi-square = 888. 794 df=20

  41. Mean comparisons USA=0 In USA Modification Indices for TAU-Y v9 v147 v175 v176 -------- -------- -------- -------- 0.855 6.130 13.121 2.882 In Canada: Modification Indices for TAU-Y v9 v147 v175 v176 -------- -------- -------- -------- 20.003 18.756 3.873 69.008 In the Netherlands: Modification Indices for TAU-Y v9 v147 v175 v176 -------- -------- -------- -------- 19.044 60.570 4.629 62.667

  42. Models for Means and Intercepts: Interpreting Mean differences with exogenous variables GROUP 1 GROUP 2 Equations: L3 = a1 + b1 L1 + b2 L 2 + D3 In group 1, we will hold a1 fixed to 0. In group 2, a1 will be free. IF b1 group 1 = b1 group 2 AND b2 group 1 = b2 group 2 THEN a1 is the between-group difference in L3, controlling for the effects of L1 and L2

  43. Lisrel model for mean comparisons with controls • 2 group model for relig 1:USA • DA NG=3 NI=23 NO=1456 • CM FI=H:\Means&Intercepts\usa.cov • ME FI=H:\Means&Intercepts\usa.mn • LABELS • v9 v147 v151 v175 v176 v304 v305 v307 v308 v309 v310 v355 v356 sex • occup1 occup2 occup3 occup4 occup5 occup6 occup7 occup8 occup9 • SE • 1 2 4 5 12 13 14 / • MO NY=4 NX=3 NK=3 FIXEDX NE=1 LY=FU,FI PS=SY,FR TE=SY C • TY=FR AL=FI GA=FU,FR KA=FI TX=FR • VA 1.0 LY 2 1 • FR LY 1 1 LY 3 1 LY 4 1 • OU ME=ML SE TV MI SC ND=3 • Group 2: Canada • DA NI=23 NO=1474 • CM FI=H:\Means&Intercepts\cdn.cov • ME FI=H:\Means&Intercepts\cdn.mn • LABELS • v9 v147 v151 v175 v176 v304 v305 v307 v308 v309 v310 v355 v356 sex • occup1 occup2 occup3 occup4 occup5 occup6 occup7 occup8 occup9 • SE • 1 2 4 5 12 13 14 / • MO LY=IN PS=PS TE=PS AL=FR TY=IN FIXEDX GA=IN KA=FR TX=IN • OU ME=ML SE TV MI SC ND=3 • Group 3: Netherlands • DA NI=23 NO=909 • CM FI=H:\Means&Intercepts\neth.cov • ME FI=H:\Means&Intercepts\neth.mn • LABELS • v9 v147 v151 v175 v176 v304 v305 v307 v308 v309 v310 v355 v356 sex • occup1 occup2 occup3 occup4 occup5 occup6 occup7 occup8 occup9 • SE • 1 2 4 5 12 13 14 / • MO LY=IN PS=PS TE=PS AL=FR TY=IN FIXEDX GA=IN KA=FR TX=IN • OU ME=ML SE TV MI SC ND=3

  44. Lisrel model for mean comparisons with controls GROUP #1 SPECIFICATION: MO NY=4 NX=3 NK=3 FIXEDX NE=1 LY=FU,FI PS=SY,FR TE=SY C TY=FR AL=FI GA=FU,FR KA=FI TX=FR • Group 3: Netherlands • DA NI=23 NO=909 • CM FI=H:\Means&Intercepts\neth.cov • ME FI=H:\Means&Intercepts\neth.mn • LABELS • v9 v147 v151 v175 v176 v304 v305 v307 v308 v309 v310 v355 v356 sex • occup1 occup2 occup3 occup4 occup5 occup6 occup7 occup8 occup9 • SE • 1 2 4 5 12 13 14 / • MO LY=IN PS=PS TE=PS AL=FR TY=IN FIXEDX GA=IN KA=FR TX=IN • OU ME=ML SE TV MI SC ND=3 TX parameters constrained to = Exogenous variable mean =0 in group 1 Exogenous variable mean reflects difference from group 1

  45. Lisrel model for mean comparisons with controls GROUP #1 SPECIFICATION: MO NY=4 NX=3 NK=3 FIXEDX NE=1 LY=FU,FI PS=SY,FR TE=SY C TY=FR AL=FI GA=FU,FR KA=FI TX=FR • Group 3: Netherlands • DA NI=23 NO=909 • CM FI=H:\Means&Intercepts\neth.cov • ME FI=H:\Means&Intercepts\neth.mn • LABELS • v9 v147 v151 v175 v176 v304 v305 v307 v308 v309 v310 v355 v356 sex • occup1 occup2 occup3 occup4 occup5 occup6 occup7 occup8 occup9 • SE • 1 2 4 5 12 13 14 / • MO LY=IN PS=PS TE=PS AL=FR TY=IN FIXEDX GA=IN KA=FR TX=IN • OU ME=ML SE TV MI SC ND=3 Alpha zero in group 1 Group m coefficient represents differences from group 1 GA matrix fixed to invariance (ksi- variables have same effect in each group)

  46. Mean differences, with controls • ALPHA CANADA • ETA 1 • -------- • 0.854 • (0.062) • 13.822 • KAPPA • v355 v356 sex • -3.465 -0.391 0.008 • (0.621) (0.085) (0.018) • -5.579 -4.592 0.410 • ALPHA • ETA 1 • -------- • 2.080 • (0.086) • 24.324 • KAPPA • v355 v356 sex • -------- -------- -------- • -4.017 -0.908 -0.059 • (0.703) (0.111) (0.021) • -5.711 -8.179 -2.812

  47. Models for Means and Intercepts GROUP 1 GROUP 2 L3 = a1[1] + b1[1]L1 + b2[1]L2 + D3 L3 = a1[2] + b1[2]L1 + b2[2]L2 + D3 Models/constraints: {1} a1[1]=0 (always) {2} b1[1] = b1[2] and b2[1]=b2[2] (normally; parallel slopes) a1[2]=0 vs. a1[2]  0 under {2}: mean diff’s controlling for L1,L2 a1[2]=0 vs. a1[2] 0 under b1=b2=0: mean diff’s without controls

  48. Models for Means and Intercepts • If slopes of all exogenous variables (L1 and L2 in this example) are parallel, a1 is the mean difference controlling for exog. var’s b1 b1 a1

  49. Models for Means and Intercepts • What if slopes are not parallel? L3 L1 A1 only represents between-group difference when L1=0 Between-group difference contingent upon value of L1

  50. Models for Means and Intercepts • #2 A longitudinal model Fix measurement model intercepts to equality (We would also normally fix measurement model b coefficients to equality) b5 Equations: LVTime2 = a6 + b5*LVTime1 + D2 LVTime1 = a5 Fix a5=0; a6 represents change in level over time

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