1. 1 General Structural Equation (LISREL) Models Week 3 #2 A.Multiple Group Models with > 2 groups B.Relationship to ANOVA, ANCOVA models C.Introduction to mean & intercept models

2. 2 LISREL PROGRAMMING: MULTIPLE GROUPS Special considerations for 3 or more groups Group 1:[specification for matrix] Group 2: LY=IN Group 3: LY=PS You would think this means, LY[1]=LY[2] but LY[3] ≠ LY[1] = LY[2] But this is not the case: LY[1] = LY[2] = LY[3] (PS in group 3 “copies” the group 2 specification, which is IN!)

3. 3 LISREL PROGRAMMING: MULTIPLE GROUPS Special considerations for 3 or more groups Group 1:[specification for matrix] Group 2: LY=IN Group 3: LY=PS You would think this means, LY[1]=LY[2] but LY[3] ≠ LY[1] = LY[2] But this is not the case: LY[1] = LY[2] = LY[3] (PS in group 3 “copies” the group 2 specification, which is IN!) Possibilities: Re-organize input so Group 3 is now group 1 Group 1: [specification for matrix] Group 2: LY=PS Group 3: LY=IN

4. 4 LISREL PROGRAMMING: MULTIPLE GROUPS Possibilities: Re-organize input so Group 3 is now group 1 Group 1: [specification for matrix] Group 2: LY=PS Group 3: LY=IN OR: If Group1=Group2≠Group 3 Group 2 LY=IN Group 3 LY=IN or LY=PS (will do same thing) THEN use a FR statement for all parameters in matrix

5. 5 LISREL PROGRAMMING: MULTIPLE GROUPS If Group1=Group2≠Group 3 Group 2 LY=PS Group 3 LY=IN or LY=PS (will do same thing) THEN use a FR statement for all parameters in matrix Eg: LY=IN FR LY 2 1 LY 3 1 LY 4 1

6. 6 A three group example: Religion & Sexual Morality Data USA Canada Britain See file: /Week3Examples/ThreeGroupLISREL

7. 7 Quick notes on more complex multiple-group models Any number of groups can be modeled, subject to software limitations: –EQS: version 5 max. of 10 (version 6???) –AMOS: no apparent max. –LISREL: had a “Fortran file maximum” restriction of 17 but could be worked around if covariance matrix pasted into program itself: CM * (INSERT MATRIX)

8. 8 Quick notes on more complex multiple-group models Four group models could be 4 categories of one variable OR 2 x 2 design Could consider the equivalent of a 3-way interaction Eg: Sex (male/female) Country (Canada/US) Example: effect of education on attitudes, each of 4 groups Interested in Gender*Educ*Country interaction

9. 9 Quick notes on more complex multiple-group models Eg: Sex (male/female) Country (Canada/US) Example: effect of education on attitudes, each of 4 groups Interested in Gender*Educ*Country interaction Coefficients: Gamma1[1] US male Gamma1[2] US female Gamma1[3] Cdn male Gamma1[4] Cdn female

10. 10 Notes on more complex multiple- group models Coefficients: Gamma1[1] US male Gamma1[2] US female Gamma1[3] Cdn male Gamma1[4] Cdn female TEST for male/female differences in effect of education: Model 1 all gammas free Model 2 gamma1[1]=gamma1[2] gamma1[3]=gamma1[4] Other tests possible (e.g., all gammas fixed, then allow ga1[1]≠ga1[2] and ga1[3]≠ga1[4]

11. 11 Notes on more complex multiple- group models Coefficients: Gamma1[1] US maleGamma1[3] Cdn male Gamma1[2] US female Gamma1[4] Cdn female TEST for male/female differences in effect of education: Model 1 all gammas free Model 2 gamma1[1]=gamma1[2] gamma1[3]=gamma1[4] Other tests possible (e.g., all gammas fixed, then allow ga1[1]≠ga1[2] and ga1[3]≠ga1[4] SIMILAR TEST FOR effect of Country Three way interaction !!! ga1[1] – ga1[2] = ga1[3]-ga1[4] allows males, females to be different but extent of difference must be the same in each country Vs. a model where these constraints are freed. LISREL CO statement could be used to program this (more difficult in AMOS) CO GA 1 1 1 = ( GA 3 1 1 – GA 4 1 1 ) + GA 2 1 1 [re-expression of: ga 1 1 1 – ga 2 1 1 = ga 3 11 – ga 4 1 1

12. 12 Means and intercepts in SEM models If we work with X d and y d in a regression model instead of X and y, then the intercept drops out.

13. 13 Means and intercepts in SEM Models

14. 14 Means and intercepts in SEM Models

15. 15 Means and intercepts in SEM Models

16. 16 Means and intercepts in SEM Models This is the variance-covariance matrix of the X’s

17. 17 Means and intercepts in SEM Models By contrast, the X’X matrix is: divide by N, “Moment Matrix”

18. 18 Means and intercepts in SEM Models But the X matrix in a regular regression model has a vector of 1s:

19. 19 Means and intercepts in SEM Models  Augmented Moment Matrix This matrix has k more pieces of information

20. 20 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).

21. 21 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

22. 22 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)

23. 23 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)

24. 24 Means and intercepts in SEM Models 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 Equations: X1 = a1 + 1.0 L1 + e1 X2 = a2 + b2 L1 + e2 X3 = a3 + b3 L1 + e3

25. 25 Means and intercepts in SEM Models Construct equation now: L2 = a1 + b1 L1 + D1 (also: new parameter: mean of L1)

26. 26 Means and intercepts in SEM Models In longitudinal case, more interesting possibilities: 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) 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

27. 27 Means and intercepts in SEM Models 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) 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 Example: X1 X2 X3 X4 X5 X6 Means: 2 3 2.5 3 4 3.5 X4 = a4 + 1.0 L2 + e4 (E(L2)=a7 Estimate: a7=1.0 X4 = 2 + 1.0*1 + 0 (expected value of L2=1.0) X5 = 3 + b3*1 + 0 (expected value of L2 = 1.0) New parameter:a7

28. 28 Means and intercepts in SEM Models 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 There can be a construct equation intercept parameter in causal models L2 = a7 + b5 L1 + D2 If mean(L1) fixed to 0 E(L2) = a7 + b5*0 = a7 As before, a7 represents the expected difference between the mean of L1 and the mean of L2

29. 29 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:

30. 30 Means and intercepts in SEM Models Same principle can be applied to multiple group models: Group 1 Group 2 X1 = a1 + 1.0 L1 + e1 X2 = a2 + b2 L1 + e2 X3 = a3 + b3 L1 + e3 X1 = a1 + 1.0 L1 + e1 X2 = a2 + b2 L1 + e2 X3 = a3 + b3 L1 + e3 a1[1] = a1[2] a2[1]=a2[2] a3[1]=a3[2] Mean(L1)=0 Mean(L1) = a4 We usually constrain measurement coefficients: b2[1]=b2[2] & b3[1]=b3[2]

31. 31 LAST SLIDE