1. 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

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

3. 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

4. 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

5. 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
Govern. Corrupt
Politicians don’t
care
(where 1=agree strongly through 10=disagree strongly)
From last Friday

6. 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

7. 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 * 1.0*
Lambda
Lambda
Lambda
These two groups appear to have measurement models that are very different, but….
From last Friday

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

9. 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
Y Y1 ---
Y Y
Y Y
Y Y
From last Friday

10. 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
Y Y1 ---
Y Y
Y Y
Y Y
Improvement in chi-square if equality constraint released

11. Multiple Group Models : Modification Indices
MODIFICATION Group Group 2
INDICES eta1 eta2 eta1 eta2
Y Y Y Y Y Y
Tests equality constraint
lambda5[1]=lambda5[2]

12. Multiple Group Models : Modification Indices
MODIFICATION Group Group 2
INDICES eta1 eta2 eta1 eta2
Y Y Y Y Y Y
Tests equality constraint
lambda5[1]=lambda5[2]
Wald test (MI) for adding parameter LY(3,3) to the model in group 2 only

13. 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

14. 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)

15. MULTIPLE GROUP MODELS: Modification Indices (again)
Model: LY[1]=LY[2]=LY[3]
Group 1 MOD INDICES
Lambda
Lambda
Lambda
Group 2 MOD INDICES
Lambda
Lambda
Lambda
Group 3 MOD INDICES
Lambda
Lambda
Lambda
Free LY(2,1) in group 3 but
LY(2,1) in group 1 = LY(2,1) in group 2

16. 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

17. 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

18. 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)

19. 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

20. 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 /
LABELS [optional]
MO NY= NX= NK= NE= + special options for matrix specification
OU (as usual)

21. 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

22. 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 LY 2 2 1
EQ LY LY 3 1

23. 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)

24. 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 /
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

25. 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
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 /
MO LY=IN PS=PS TE=PS
OU ND=4 SC MI
Number of Observations 1763
Program & output:
TwoGroup1b.ls8, *.out

26. Multiple group example #1: USA
Parameter Specifications
LAMBDA-Y EQUALS LAMBDA-Y IN THE FOLLOWING GROUP
PSI
ETA ETA 2
ETA
ETA
THETA-EPS
A F F F F F119 A F F F F F F F
F F121 F F

27. Covariance Matrix of ETA
ETA ETA 2
ETA
ETA
PSI
(0.1554)
(0.1464) (0.3118)
Group 1
Covariance Matrix of ETA
ETA ETA 2
ETA
ETA
PSI
(0.1924)
(0.1529) (0.3175)
Group 2

28. Global Goodness of Fit Statistics
Group Goodness of Fit Statistics (USA)
Contribution to Chi-Square =
Percentage Contribution to Chi-Square =
Root Mean Square Residual (RMR) =
Standardized RMR =
Goodness of Fit Index (GFI) =
Global Goodness of Fit Statistics
Degrees of Freedom = 42
Minimum Fit Function Chi-Square = (P = 0.0)
Normal Theory Weighted Least Squares Chi-Square = (P = 0.0)
Estimated Non-centrality Parameter (NCP) =
90 Percent Confidence Interval for NCP = ( ; )
Normed Fit Index (NFI) =
Non-Normed Fit Index (NNFI) =
Parsimony Normed Fit Index (PNFI) =
Comparative Fit Index (CFI) =
Incremental Fit Index (IFI) =
Relative Fit Index (RFI) =
Group Goodness of Fit Statistics (CANADA)
Contribution to Chi-Square =
Percentage Contribution to Chi-Square =
Root Mean Square Residual (RMR) =
Standardized RMR =
Goodness of Fit Index (GFI) =

29. Multiple group example #1: USA
Modification Indices and Expected Change
Modification Indices for LAMBDA-Y
ETA ETA 2 A F F F F F F F
Group #2: Canada A F F F F F F F

30. F121 - - 0.7996 LISREL Estimates (Maximum Likelihood) LAMBDA-Y
ETA ETA 2 A
(0.0119) F F
(0.0257) F
(0.0332) F F
(0.0262) F
(0.0326) F
(0.0267)

31. Expected Change for LAMBDA-Y (USA)
ETA ETA 2 A F F F F F F F
Expected Change for LAMBDA-Y (Canada) A F F F F F F F

32. Group #2: Canada
Within Group Completely Standardized Solution
LAMBDA-Y
ETA ETA 2 A F F F F F F F
Multiple group example #1: USA A F F F F F F F

33. More variance in Canada
Multiple group example #1: USA
Common Metric Completely Standardized Solution
LAMBDA-Y
ETA ETA 2 A F F F F F F F
Covariance Matrix of ETA
ETA
ETA
Group #2: Canada
Common Metric Standardized Solution A F F F F F F F
More variance in Canada
Connection stronger
in Canada
Covariance Matrix of ETA
ETA ETA 2
ETA
ETA

34. 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 /
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 = (P = 0.0)
Normal Theory Weighted Least Squares Chi-Square = (P = 0.0)
Estimated Non-centrality Parameter (NCP) =
90 Percent Confidence Interval for NCP = ( ; )
Normed Fit Index (NFI) =
Non-Normed Fit Index (NNFI) =
Parsimony Normed Fit Index (PNFI) =
Comparative Fit Index (CFI) =
Incremental Fit Index (IFI) =
Relative Fit Index (RFI) =

35. Some Model/Test Results: Matrices Chi-square df IFI
TwoGroup1b LY=IN
TwoGroup1c LY=PS
TwoGroup1d LY=IN PS=IN
TwoGroup1e LY=IN PS=IN but
PS 2,1 free
From TwoGroup1d, in Group #2 (Canada):
Modification Indices for PSI
ETA ETA 2
ETA
ETA
Expected Change for PSI
ETA
ETA

36. 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 /
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 /
MO LY=IN PS=PS TE=PS LX=IN PH=PS TD=IN GA=PS LE
RELIG MORAL
OU ND=4 SC MI

37. LAMBDA-Y EQUALS LAMBDA-Y IN THE FOLLOWING GROUP
GAMMA
GENDER AGE EDUC TOWNSIZE
RELIG
MORAL
PHI
GENDER
AGE
EDUC
TOWNSIZE
PSI
RELIG MORAL
RELIG
MORAL

38. Multiple group example #1: USA
LISREL Estimates (Maximum Likelihood)
LAMBDA-Y EQUALS LAMBDA-Y IN THE FOLLOWING GROUP
GAMMA
GENDER AGE EDUC TOWNSIZE
RELIG
(0.1001) (0.0031) (0.0351) (0.0293)
MORAL
(0.1451) (0.0045) (0.0515) (0.0428)
Group #2: Canada
Number of Iterations = 8
RELIG
(0.0926) (0.0028) (0.0397) (0.0173)
MORAL
(0.1182) (0.0036) (0.0526) (0.0226)

39. Models/Tests
Model Description Chi-square df CFI
Group2A LY=IN GA=PS
Group2B LY=IN GA=IN
From Model Group2B (Group #2, Canada):
Modification Indices for GAMMA
GENDER AGE EDUC TOWNSIZE
RELIG
MORAL
Modification Indices for GAMMA (Group #1, USA)
RELIG
MORAL

40. Model Description Chi-square df CFI
Group3A GA=IN
Group3B GA=PS
Group3C GA=IN
Free occ. in group 2
Group3D GA=PS
Occ FI in group 2
Group3E GA=PS
Occ Fi both groups
Group3F GA=PS
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.

41. GAMMA USA
GENDER AGE EDUC TOWNSIZE OCC OCC2
RELIG
(0.1023) (0.0031) (0.0382) (0.0293) (0.3118) (0.3063)
MORAL
(0.1484) (0.0045) (0.0561) (0.0429) (0.4558) (0.4479)
GAMMA CANADA
RELIG
(0.0944) (0.0029) (0.0465) (0.0174) (0.3589) (0.3319)
MORAL
(0.1206) (0.0037) (0.0610) (0.0226) (0.4668) (0.4317)
GAMMA
OCC OCC4
RELIG
(0.3019) (0.3000)
MORAL
(0.4412) (0.4386)
GAMMA
OCC OCC4
RELIG
(0.3125) (0.3123)
MORAL
(0.4065) (0.4063)

42. (insert AMOS demo here)