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General Structural EquationsGeneral Structural Equations
Week 2 #5Week 2 #5
Different forms of constraintsDifferent forms of constraintsIntroduction for models estimated in multiple groupsIntroduction for models estimated in multiple groups
2
Multiple Group Models(Hayduk: “Stacked” models)
1. Constraints on parameters
2. Running separate models in different groups
3. Applying equality constraints across groups
3
Parameter constraints
Technically, any “fixed” parameter is constrained.
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
4
Inequality constraints 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
Parameter valueIteration Number
0
5
Inequality constraints
Y1 Eta21ETA-1 lambda-1
Y1
0
E11
Eta-1 Ksi-1
Lambda-1 1
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
6
Inequality constraints
Y1
0
E11
Eta-1 Ksi-1
Lambda-1 1
The above model can be reformulated as:
Y1
0
E11
Eta-1
1
Ksi-1
Lambda-1 lambda-2
Note var(Ksi-1) = 1.0
(other y-var’s)
7
Inequality constraints
Y1
0
E11
Eta-1
1
Ksi-1
Lambda-1 lambda-2
Note var(Ksi-1) = 1.0
VAR(Y1) = lambda-12 VAR(Eta-1) + lambda-22 (1.0)
What used to be VAR(Ksi) = error variance for Y1 – is now contained in the expression lambda22.
Note, however, that no matter what the value of lambda-2 is, the entire expression will be positive. In other words, it is impossible for the error variance to drop below 0.
8
Inequality constraints
1
1
b1
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.
9
Equality constraints in single group models
Eta-1
y1 e111
y2 e2b1 1
y3 e3b1
1
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)
10
Equality constraints in the context of dummy variables
Eta-1
y111
y21
y31
X1
X2
X3
b1
b2
b3
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
11
Equality constraints in the context of dummy variables
Eta-1
y111
y21
y31
X1
X2
X3
b1
b2
b3
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
12
More complex constraints when the software doesn’t seem to want to allow
them:
1
1
b1
1
b2
1b1 = 2*b2
LISREL
CO LY(2,1)=2*LY(3,1)
AMOS only allows equality constraints
13
More complex constraints when the software doesn’t seem to want to allow
them:
1
1
b1
1
b2
1 b1 = 2*b2
LV1
1
1
b1
1
X3Var=1.02
b2
var=01 1
Fix variance to 1.0
New model:
X3 = 2*b2LV1 + e3
Re-express as
14
AN EVEN MORE IMPORTANT (VIZ., MORE FREQUENTLY USED) FORM OF CONSTRAINT INVOLVES CONSTRAINING PARAMETERS ACROSS GROUPS
1
1 1 1
1
1 1 1
b1
1
1 1 1
1
1 1 1
b1
Group 1
Group 2
Constraint: b1group1 = b1group2
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)
16
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
17
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
18
A simple multiple-group example:
1
b1
1
b1
males
females
Key question:
b1(males) = b1(females)?
Notation:
H0: b1[1] = b1[2]
19
Equivalences:
Regression: X1=male/female
X2 = Education
Y = b0 + b1 X1 + b2 X2 + e
SEM: Group 1 Group 2
Eta1 = gamma1 Ksi1 + zeta Eta1 = gamma1 Ksi + zeta
Constraint: gamma1[1] = gamma1[2]
Gamma1 in group 1 = Gamma1 in group 2
LISREL: EQ GA 1 1 1 GA 2 1 1
20
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
21
Multiple Group Models
Group 1 (male)Group 2 (female)
Equivalence of measurement coefficients
H0: Λ[1] = Λ[2]
lambda 1 [1] = lambda 1 [2] df=2
lambda 2 [1] = lambda 2 [2]
Eta1[1]
y111
y2ly1[1] 1
y3
ly2[1]1
Eta1[2]
y1
y2
y3
11
ly1[2] 1
ly2[2]1
22
Multiple Group Models
Eta1[1]
y1 e111
y2 e2ly1[1] 1
y3 e3
ly2[1]1
Eta1[2]
y1 e1
y2 e2
y3 e3
11
ly1[2] 1
ly2[2]1
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
23
Multiple Group Models
Measurement model equivalence does not imply same mean levelsMeasurement 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)
24
Multiple Group Models
Eta1
y11
1
y2lambda-1 1
y3lambda-2 1
y4
lambda-3
1
• 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 & authorityY4 (US): George Bush commands my respect because he is our President
Y4 (Canada) Paul Martin commands my respect because he is our Prime Minister
25
Multiple Group Models
Eta1
y11
1
y2lambda-1 1
y3lambda-2 1
y4
lambda-3
1
• 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]
26
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….
Eta1
y1lambda-1
1
y2lambda-2 1
y3lambda-3 1
y4
lambda-4
1
27
Multiple Group ModelsGroup 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:
Eta1
y1lambda-1
1
y2lambda-2 1
y3lambda-3 1
y4
lambda-4
1
Gr 1 Gr 2
Lambda1 2.0 1.0
Lambda2 1.0* 1.0*
Lambda3 1.5 1.5
Lambda4 2.0 2.0
28
Multiple Group Models
Modification Indices and what they mean in multiple-group models
Assuming LY[1] = LY[2]
(entire matrix)
Eta1
y11
1
y2lambda-1 1
y3lambda-2 1
y4
lambda-3
1
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
29
Multiple Group Models
Modification Indices and what they mean in multiple-group models
Assuming LY[1] = LY[2]
(entire matrix)
Eta1
y11
1
y2lambda-1 1
y3lambda-2 1
y4
lambda-3
1
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
30
Multiple Group Models : Modification Indices
eta1
y111
y2lambda-2 1
y3lambda-3 1
eta2
y4
y5
y6
11
lambda-4 1
lambda-5 1
MODIFICATION Group 1 Group 2
INDICES eta1 eta2 eta1 eta2
Y1 --- 2.42 --- 3.89Y2 1.42 3.44 1.42 1.01Y3 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]
31
Multiple Group Models : Modification Indices
eta1
y111
y2lambda-2 1
y3lambda-3 1
eta2
y4
y5
y6
11
lambda-4 1
lambda-5 1
MODIFICATION Group 1 Group 2
INDICES eta1 eta2 eta1 eta2
Y1 --- 2.42 --- 3.89Y2 1.42 3.44 1.42 1.01Y3 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
32
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
33
MULTIPLE GROUP MODELS: Modification Indices (again)
Eta1
y11
1
y2lambda-1 1
y3lambda-2 1
y4
lambda-3
1
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
Model: LY[1]=LY[2]=LY[3]
Free LY(2,1) in group 3 but
LY(2,1) in group 1 = LY(2,1) in group 2
34
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
35
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
36
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)
1
1 1 1
1
1 1 1
gamma1[1]
1
1
1 1 1
1
1 1 1
gamma1[2]
1