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

(Hayduk: “Stacked” models)
1.
2.
3.
2





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
3




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
Parameter value
Iteration Number
4

Inequality constraints lambda-1
ETA-1
Y1
1
Eta2
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
Eta-1
Lambda-1
E1
1
0
Y1
1
Ksi-1
5

Inequality constraints
(other y-var’s)
Eta-1
Lambda-1
E1
1
0
Y1
1
Ksi-1
The above model can be reformulated as:
E1
1
0
Y1
Eta-1
Lambda-1
Note var(Ksi-1) = 1.0
lambda-2
Ksi-1
1
6

Inequality constraints
VAR(Y1) = lambda-1 2 VAR(Eta-1) + lambda-2 2 (1.0)
What used to be VAR(Ksi) = error variance for Y1 – is now contained in the expression lambda2 2 .
Note, however, that no matter is, the entire expression will be what the value of lambda-2 positive . In other words, it is impossible for the error variance to drop below 0.
Eta-1
Lambda-1
E1
1
0
Y1
Note var(Ksi-1) = 1.0
lambda-2
Ksi-1
1
7

Inequality constraints
1 b1
1
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.
8

Eta-1
1 b1 b1 y1
1 y2
1 y3
1 e1 e2 e3
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)
9

X1
X2
X3 b2 b1 b3
Eta-1
1 y1
1 y2
1 y3
1
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
10

X1
X2
X3 b2 b1 b3
Eta-1
1 y1
1 y2
1 y3
1
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
11

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

More complex constraints when the software doesn’t seem to want to allow them:
1 1 1
1 b1 b2
Re-express as b1 = 2*b2
1
1
1 b1 2 var=0
1
Var=1.0
b2
1
X3
LV1
New model:
X3 = 2*b2LV1 + e3
Fix variance to 1.0
13

AN EVEN MORE IMPORTANT (VIZ., MORE FREQUENTLY USED)
FORM OF CONSTRAINT INVOLVES CONSTRAINING PARAMETERS
ACROSS GROUPS
1
1
1 1 b1
1
1
1 1
Group 1
1
1
1 1 b1
Constraint: b1 group1
= b1 group2
1
1
1 1
Group 2
14

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 / crosscultural research)
• City (in studies involving replications in a small number of cities, where cities are internally homogeneous but quite different from each other)
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)
• Firms (e.g., in business studies, a 10-firm study, with different firms from different sectors of the economy)
• Immigrant group
16

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
17

Key question:
1 b1 b1(males) = b1(females)?
males
Notation:
H0: b1
[1]
= b1
[2]
1 b1 females
18

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

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 20

Multiple Group Models
Eta1[1]
1 y1
1 ly1[1] y2 ly2[1] y3
1
1
Eta1[2]
1 y1 ly1[2] y2 ly2[2] y3
1
1
1
Group 1 (male)
Group 2 (female)
Equivalence of measurement coefficients
H
0
: Λ[1] = Λ[2] lambda 1 [1] = lambda 1 [2] lambda 2 [1] = lambda 2 [2] df=2
21

Multiple Group Models
Eta1[1]
1 y1
1 ly1[1] y2 ly2[1] y3
1
1 e1 e2 e3
Eta1[2]
1 y1 ly1[2] y2 ly2[2] y3
1
1
1 e1 e2 e3
Other equivalence tests possible:
1. Equivalence of variances of latent variables
H0: PSI-1[1] = PSI-1[2]
• This test will depend upon which ref. indicator used
2. Equivalence of error variances *
H0: Theta-eps[1] = Theta-eps[2] {entire matrix} df=3 *and covariances if there are correlated errors
22

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)
23

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
Eta1
1 lambda-1 lambda-2 lambda-3 y1 y2 y3 y4
1
1
1
1
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
We might expect (if measurement equivalence holds): lambda1[1] = lambda1[2] lambda2[1] = lambda2[2]
BUT lambda3[1] ≠ lambda3[2]
Eta1
1 lambda-1 lambda-2 lambda-3 y1 y2 y3 y4
1
1
1
1
25

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:
Lambda-1
Lambda-2
Lambda-3
Lambda-4
Group 1 Group 2
1.0*
.50
.75
1.0
1.0*
1.0
1.5
2.0
• These two groups appear to have measurement models that are very different, but….
Eta1 lambda-1 lambda-2 lambda-3 lambda-4 y1 y2
1
1
1 y3 y4
1
26

Multiple Group Models
Lambda-1
Lambda-2
Group 1 Group 2
1.0*
.50
1.0*
1.0
Lambda-3
Lambda-4
.75
1.0
1.5
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:
Lambda1
Lambda2
Lambda3
Lambda4
Gr 1
2.0
1.0*
1.5
2.0
Gr 2
1.0
1.0*
1.5
2.0
Eta1 lambda-1 lambda-2 lambda-3 lambda-4 y1 y2
1
1
1 y3 y4
1
27

Multiple Group Models
Modification Indices and what they mean in multiplegroup models
Assuming LY[1] = LY[2]
(entire matrix)
Y1
Y2
Y3
Y4
Example:
MODIFICATION INDICES:
Group 1
Eta 1
---
.382
1.24
45.23
Y1
Y2
Y3
Y4
Eta1
Group 2
Eta 1
---
.382
1.24
45.23
1 lambda-1 lambda-2 lambda-3 y1
1 y2
1 y3
1 y4
1
28

Multiple Group Models
Modification Indices and what they mean in multiplegroup models
Assuming LY[1] = LY[2]
(entire matrix)
Y1
Y2
Y3
Y4
Example:
MODIFICATION INDICES:
Group 1
Eta 1
---
.382
1.24
45.23
Y1
Y2
Y3
Y4
Eta1
Group 2
Eta 1
---
.382
1.24
45.23
1 lambda-1 lambda-2 lambda-3 y1
1 y2
1 y3
1 y4
1
Improvement in chisquare if equality constraint released
29

Multiple Group Models : Modification Indices eta1
1 lambda-2 lambda-3 y1
1 y2
1
1 y3 eta2
1 lambda-4 lambda-5 y4
1
1 y5 y6
1
MODIFICATION Group 1
INDICES
Y1
Y2
Y3 eta1
---
1.42
0.43
Group 2 eta2 eta1
2.42
3.44
2.11
---
1.42
0.43 eta2
3.89
1.01
40.89
Y4
Y5
Y6
0.11
2.32
1.01
---
1.49
0.98 ---
1.22
1.49
29.23
3.21 29.23
Tests equality constraint lambda5[1]=lambda5[2]
30

Multiple Group Models : Modification Indices eta1
1 lambda-2 lambda-3 y1
1 y2
1
1 y3 eta2
1 lambda-4 lambda-5 y4
1
1 y5 y6
1
MODIFICATION Group 1
INDICES
Y1
Y2
Y3 eta1
---
1.42
0.43
Group 2 eta2 eta1
2.42
3.44
2.11
---
1.42
0.43 eta2
3.89
1.01
40.89
Y4
Y5
Y6
0.11
2.32
1.01
---
1.49
0.98 ---
1.22
1.49
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
31

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
32

MULTIPLE GROUP MODELS: Modification Indices (again)
Model: LY[1]=LY[2]=LY[3]
Group 1 MOD INDICES
Lambda 1
Lambda 2
Lambda 3
3.01
1.52
3.22
Group 2 MOD INDICES
Lambda 1 4.22
Lambda 2 3.99
Lambda 3 5.22
Group 3 MOD INDICES
Lambda 1
Lambda 2
Lambda 3
89.22
6.11
1.22
Eta1
1 lambda-1 lambda-2 lambda-3 y1
1 y2
1 y3
1 y4
1
Free LY(2,1) in group 3 but
LY(2,1) in group 1 = LY(2,1) in group 2
33

 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
34

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
35

Measurement coefficients, construct equation coefficients in multiple group models
1 1 1
1 1 1


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)
1 gamma1[1]
1 1 1
1
1
1 1 1
1
1 gamma1[2]
1
36