1 General Structural Equation (LISREL) Models Week 3 #2 A.Multiple Group Models with > 2 groups...

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General Structural General Structural EquationEquation

(LISREL) Models(LISREL) ModelsWeek 3 #2Week 3 #2

A.A. Multiple Group Models with > 2 Multiple Group Models with > 2 groupsgroups

B.B. Relationship to ANOVA, ANCOVA Relationship to ANOVA, ANCOVA modelsmodels

C.C. Introduction to mean & intercept Introduction to mean & intercept modelsmodels

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LISREL PROGRAMMING: MULTIPLE GROUPS

Special considerations for 3 or more groupsGroup 1: [specification for matrix]Group 2: LY=INGroup 3: LY=PSYou 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!)

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Special considerations for 3 or more groupsGroup 1: [specification for matrix]Group 2: LY=INGroup 3: LY=PSYou 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 1Group 1: [specification for matrix]Group 2: LY=PSGroup 3: LY=IN

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

matrix

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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

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A three group example:

Religion & Sexual Morality Data

USA

Canada

Britain

See file: /Week3Examples/ThreeGroupLISREL

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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)

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• 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

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• 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

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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]

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Notes on more complex multiple-group models

Coefficients:Gamma1[1] US male Gamma1[3] Cdn maleGamma1[2] US female Gamma1[4] Cdn female

TEST for male/female differences in effect of education:Model 1 all gammas freeModel 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 CountryThree 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

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Means and intercepts in SEM models

If we work with Xd and yd in a regression model instead of X and y, then the intercept drops out.

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Means and intercepts in SEM Models

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This is the variance-covariance matrix of the X’s

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By contrast, the X’X matrix is:

divide by N,

“Moment Matrix”

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But the X matrix in a regular regression model has a vector of 1s:

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Augmented Moment Matrix

This matrix has k more pieces of information

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Working from this matrix instead of working from S, we can add intercepts back into equations (reproduce M instead of S).

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LV1X1 e11

X2 e2b2

X3 e3

b3Conventional Model:

X1 = 1.0 LV1 + e1

X2 = b2 LV1 + e2

X3 = b3 LV1 + e3

LV1X1 e11

X2 e2b2

X3 e3

b3

1a4 a1

a2 a3

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

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LV1X1 e11

X2 e2b2

X3 e3

b3

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)

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LV1X1 e11

X2 e2b2

X3 e3

b3

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)

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LV1X1 e11

X2 e2b2

X3 e3

b3

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 + e1X2 = a2 + b2 L1 + e2X3 = a3 + b3 L1 + e3

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L1 L2

D11

b1

Construct equation now:

L2 = a1 + b1 L1 + D1

(also: new parameter: mean of L1)

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L1

X1

e1

1

X2

e2

b1X3

e3

b2

L2

Y1

e4

Y2

e5

Y3

e6

1 b3 b4

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

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L1

X1

e1

1

X2

e2

b1X3

e3

b2

L2

Y1

e4

Y2

e5

Y3

e6

1 b3 b4

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

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L1

X1

e1

1

X2

e2

b1X3

e3

b2

L2

Y1

e4

Y2

e5

Y3

e6

1 b3 b4

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

L1

X1

e1

1

X2

e2

b1X3

e3

b2

L2

Y1

e4

Y2

e5

Y3

e6

1 b3 b4

b5

D2

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

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L1

X1

e1

1

X2

e2

b1X3

e3

b2

L2

Y1

e4

Y2

e5

Y3

e6

1 b3 b4

b5

D2

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:

L1

X1

e1

1

X2

e2

b1X3

e3

b2

L2

Y1

e4

Y2

e5

Y3

e6

1 b3 b4

b5

D2

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Same principle can be applied to multiple group models:

L1X1 e11

X2 e2b2

X3 e3

b3

L1X1 e11

X2 e2b2

X3 e3

b3

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) = a4We usually constrain measurement coefficients:

b2[1]=b2[2] & b3[1]=b3[2]

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