General Structural Equation (LISREL) Models

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General Structural Equation(LISREL) Models

• Week 3 2
• Multiple Group Models with gt 2 groups
• Relationship to ANOVA, ANCOVA models
• Introduction to mean intercept models

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LISREL PROGRAMMING MULTIPLE GROUPS

• Special considerations for 3 or more groups
• Group 1 specification for matrix
• Group 2 LYIN
• Group 3 LYPS
• You would think this means, LY1LY2 but LY3
  ? LY1 LY2
• But this is not the case
• LY1 LY2 LY3
• (PS in group 3 copies the group 2
  specification, which is IN!)

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LISREL PROGRAMMING MULTIPLE GROUPS

• Special considerations for 3 or more groups
• Group 1 specification for matrix
• Group 2 LYIN
• Group 3 LYPS
• You would think this means, LY1LY2 but LY3
  ? LY1 LY2
• But this is not the case
• LY1 LY2 LY3
• (PS in group 3 copies the group 2
  specification, which is IN!)
• Possibilities
• Re-organize input so Group 3 is now group 1
• Group 1 specification for matrix
• Group 2 LYPS
• Group 3 LYIN

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LISREL PROGRAMMING MULTIPLE GROUPS

• Possibilities
• Re-organize input so Group 3 is now group 1
• Group 1 specification for matrix
• Group 2 LYPS
• Group 3 LYIN
• OR
• If Group1Group2?Group 3
• Group 2 LYIN
• Group 3 LYIN or LYPS (will do same thing)
• THEN use a FR statement for all parameters in
  matrix

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LISREL PROGRAMMING MULTIPLE GROUPS

• If Group1Group2?Group 3
• Group 2 LYPS
• Group 3 LYIN or LYPS (will do same thing)
• THEN use a FR statement for all parameters in
  matrix
• Eg LYIN
• 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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Quick notes on more complex multiple-group models

• 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 GenderEducCountry interaction

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Quick notes on more complex multiple-group models

• Eg Sex (male/female) Country (Canada/US)
• Example effect of education on attitudes,
• each of 4 groups
• Interested in GenderEducCountry interaction
• Coefficients
• Gamma11 US male
• Gamma12 US female
• Gamma13 Cdn male
• Gamma14 Cdn female

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Notes on more complex multiple-group models

• Coefficients
• Gamma11 US male
• Gamma12 US female
• Gamma13 Cdn male
• Gamma14 Cdn female
• TEST for male/female differences in effect of
  education
• Model 1 all gammas free
• Model 2 gamma11gamma12
• gamma13gamma14
• Other tests possible (e.g., all gammas fixed,
  then allow ga11?ga12 and ga13?ga14

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Notes on more complex multiple-group models

• Coefficients
• Gamma11 US male Gamma13 Cdn male
• Gamma12 US female Gamma14 Cdn female
• TEST for male/female differences in effect of
  education
• Model 1 all gammas free
• Model 2 gamma11gamma12
• gamma13gamma14
• Other tests possible (e.g., all gammas fixed,
  then allow ga11?ga12 and ga13?ga14
• SIMILAR TEST FOR effect of Country
• Three way interaction !!!
• ga11 ga12 ga13-ga14 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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Means and intercepts in SEM Models
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Means and intercepts in SEM Models
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Means and intercepts in SEM Models
This is the variance-covariance matrix of the Xs
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Means and intercepts in SEM Models
By contrast, the XX matrix is
divide by N, Moment Matrix
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Means and intercepts in SEM Models
But the X matrix in a regular regression model
has a vector of 1s
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Means and intercepts in SEM Models
? Augmented Moment Matrix
This matrix has k more pieces of information
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Means and intercepts in SEM Models
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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Means and intercepts in SEM Models
Conventional Model X1 1.0 LV1 e1 X2 b2 LV1
e2 X3 b3 LV1 e3
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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Means and intercepts in SEM Models
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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Means and intercepts in SEM Models
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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Means and intercepts in SEM Models
Equations X1 a1 1.0 L1 e1 X2 a2 b2
L1 e2 X3 a3 b3 L1 e3
Conventions Mean(L1) 0 Then a1mean of
X1 a2 mean of X2 a3 mean of X3 Not
particularly useful means of LVs by
definition 0
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Means and intercepts in SEM Models
Construct equation now L2 a1 b1 L1
D1 (also new parameter mean of L1)
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Means and intercepts in SEM Models
In longitudinal case, more interesting
possibilities
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
Constrain measurement models b1b3 b2b4 Constr
ain intercepts a1 a4 a2 a5 a3 a6 Fix
Mean(L1) to 0 Can now estimate parameter for Mean
(L2)
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Means and intercepts in SEM Models
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
Constrain measurement models b1b3 b2b4 Constr
ain intercepts a1 a4 a2 a5 a3 a6 Fix
Mean(L1) to 0 Can now estimate parameter for Mean
(L2)
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 a71.0 X4 2
1.01 0 (expected value of L21.0) X5 3
b31 0 (expected value of L2 1.0)
New parametera7
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Means and intercepts in SEM Models
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
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 b50 a7
As before, a7 represents the expected difference
between the mean of L1 and the mean of L2
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Means and intercepts in SEM Models
L2 a7 b1 L1 D2
If mean(L1) fixed to 0 E(L2) a7 b10 a7
In practice, if L1 and L2 represent time 1 and
time 2 measures of the same thing, we would
expect correlated errors
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Means and intercepts in SEM Models
Same principle can be applied to multiple group
models
Group 1
a11 a12
X1 a1 1.0 L1 e1 X2 a2 b2 L1 e2 X3
a3 b3 L1 e3
a21a22
a31a32
Mean(L1)0
Group 2
X1 a1 1.0 L1 e1 X2 a2 b2 L1 e2 X3
a3 b3 L1 e3
We usually constrain measurement
coefficients b21b22 b31b32
Mean(L1) a4
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