General Structural Equations

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 parameters2. Running separate models in

different groups3. 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


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


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


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 = ProtestantX2 = CatholicX3 = JewishX4 = Ref. All others (Atheist, Muslim, etc.)

Tests of Prot vs. Catholic: b1=b2 (LISREL: EQ GA 1 1 GA 1 2Test 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 2Model 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 = ProtestantX2 = CatholicX3 = JewishX4 = Ref. All others (Atheist, Muslim, etc.)

(Prot + Cath) vs. Jewish:Model 1: EQ GA 1 1 GA 1 2Model 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


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


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


Regression equivalences:X1: Male=1 Female=0X2: 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 = EducationY = b0 + b1 X1 + b2 X2 + e

SEM: Group 1 Group 2Eta1 = gamma1 Ksi1 + zeta Eta1 = gamma1 Ksi +

zetaConstraint: 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 = EducationY = 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 betweengamma1[1] and gamma1[2] in SEM multiple-

groupmodel.

[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 coefficientsH0: Λ[1] = Λ[2]

lambda 1 [1] = lambda 1 [2] df=2lambda 2 [1] = lambda 2 [2]

Eta1[1]

y111

y2ly1[1] 1

y3ly2[1]

1

Eta1[2]

y1

y2

y3

11

ly1[2] 1

ly2[2]1

22


Eta1[1]

y1 e111

y2 e2ly1[1] 1

y3 e3ly2[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


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)

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Eta1

y11

1

y2lambda-1 1

y3lambda-2 1

y4

lambda-31

• 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 PresidentY4 (Canada) Paul Martin commands my respect because he is our Prime Minister

25


Eta1

y11

1

y2lambda-1 1

y3lambda-2 1

y4

lambda-31

• 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]BUTlambda3[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 2Lambda-1 1.0* 1.0*Lambda-2 .50 1.0Lambda-3 .75 1.5Lambda-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-41

27

Multiple Group ModelsGroup 1 Group 2

Lambda-1 1.0* 1.0*Lambda-2 .50 1.0Lambda-3 .75 1.5Lambda-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-41

Gr 1 Gr 2Lambda1 2.0 1.0Lambda2 1.0* 1.0*Lambda3 1.5 1.5Lambda4 2.0 2.0

28


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-31

Example:MODIFICATION INDICES:Group 1 Group 2

Eta 1 Eta 1Y1 --- Y1 ---Y2 .382 Y2 .382Y3 1.24 Y3 1.24Y4 45.23 Y4 45.23

29


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-31

Example:MODIFICATION INDICES:Group 1 Group 2

Eta 1 Eta 1Y1 --- Y1 ---Y2 .382 Y2 .382Y3 1.24 Y3 1.24Y4 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 2INDICES eta1 eta2 eta1

eta2Y1 --- 2.42 --- 3.89Y2 1.42 3.44 1.42 1.01Y3 0.43 2.11 0.43 40.89Y4 0.11 --- 0.98 ---Y5 2.32 1.49 1.22 1.49Y6 1.01 29.23 3.21 29.23

Tests equality constraintlambda5[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 2INDICES eta1 eta2 eta1

eta2Y1 --- 2.42 --- 3.89Y2 1.42 3.44 1.42 1.01Y3 0.43 2.11 0.43 40.89Y4 0.11 --- 0.98 ---Y5 2.32 1.49 1.22 1.49Y6 1.01 29.23 3.21 29.23

Tests equality constraintlambda5[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-31

Group 1 MOD INDICESLambda 1 3.01Lambda 2 1.52Lambda 3 3.22Group 2 MOD INDICESLambda 1 4.22Lambda 2 3.99Lambda 3 5.22Group 3 MOD INDICESLambda 1 89.22Lambda 2 6.11Lambda 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

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General Structural Equations