General Structural Equation (LISREL) Models

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General Structural Equation (LISREL)Models

• ICPSR D. Baer
• Week 1 2

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Todays class

• Translating Diagrams to Equations and vice-versa
• Implications of measurement error (more)
• Testable vs. non-testable hypotheses
• Covariance algebra for structural equation models
  (the scalar version)
• Introduction to the AMOS computer program

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Todays class

• Readings and exercises
• Read chapter 2 of class text
• A short AMOS exercise will be provided at the end
  of the class (this is not a formal assignment
  not to be handed in)
• A short covariance algebra exercise (again, not a
  formal assignment)

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Path Models

• Familiar to some, though prior to the widespread
  diffusion of SEM models in the social sciences,
  path models were seldom seen after the 1970s
• Early path models based on OLS estimation other
  estimation methods possible
• More sophisticated path models involved
  simultaneous equations

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Path Models

• A Regular path model
• Variables in rectangles (V1, V2, V3)
• Error terms in circles (alternative convention
  leave error terms unenclosed)
• This model can be estimated with 2 OLS equations
• V2 b1 V1 e2
• V3 b2 V1 b3 V2 e3

No intercepts in equations variables are
mean-centered.
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Path Models
With 3 variables, V1, V2 and V3, other models are
possible
Model on bottom is non-recursive, makes different
assumptions about the causal relationships among
variables.
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Path Models
With 3 variables, V1, V2 and V3, other models are
possible
Empirical adjudication NOT possible
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Path Models
With 3 variables, V1, V2 and V3, other models are
possible
In previous slide, models not testable. Here,
models are nested and thus can be tested
against each other Model 1 H0 b30 Model 2
b3? 0 Test between 2 models involves test of 1
parameter (it is a df1 test)
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Implications of Measurement error Three quick
examples

• A two-variable example

LV1, LV2 imperfectly measured Assume reliability
of .80 How does b1 compare with the coefficient
we would have obtained if we had regressed X2 on
X1? If true b1 .80 Observed X2 b1X1 e
b1 .512 If true b1 .5 b1 .32
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Implications of Measurement error A
three-variable example
Different degrees of measurement error for LV1,
LV2, LV3 LV3 .70 LV2 .90 LV1 .60
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Implications of Measurement error A
three-variable example
Parameter True Observed
Cov(LV1,LV2) -.40 -.22
b1 .86 .31
B2 .54 .27
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Implications of Measurement error A second
three-variable example
Parameter True Observed
Cov(LV1,LV2) -.60 -.32
b1 .50 .16
B2 .00 -.13
Same reliabilities/measurement error variances as
previous examples LV1 .6 LV2 .90 LV3 .70
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Implications of Measurement error

• Previous examples all assumed random measurement
  error
• Error can be systematic too.
• We will discuss hypothetical and actual examples
  of systematic measurement error later

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Translating Diagrams to Equationsand Equations
to DiagramsTwo quick exercises
Write out equations List all model
parameters (enlarged version on next slide)
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Write out equations for this model
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Write out equations for this model
Measurement equations X1 1.0 LV1 e1 (or X1
LV1 e1) X2 b1 LV1 e2 X3 b2 LV2 e3 X4
1.0 LV2 e4 X5 1.0 LV3 e5 X6 b3 LV3
e6 Structural equations among latent
variables (construct equations) LV3 b4 LV1
b5 LV2 d1
Notes It is common to have one indicator with
a fixed measurement equation parameter value of
1.0. The error term d1 could have been called
e7 instead. Usually, we distinguish between
measurement equation errors and construct eq.
errors, but not necessary to do so.
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Model parameters
b1, b2, b3, b4, b5 Error term variances Var
(e1) var (e2) var (e3) var (e4) Var (e5) var
(e6) var (d1) Exogenous latent variable
variances Var (LV1) var (LV2) Exogenous latent
variable covariance Cov (LV1, LV2) Error term
covariances (NONE IN THIS MODEL)
Total number of model parameters 15
(error term is a type of exogenous variable)
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Another example
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Equations X4 1.0 LV1 e4 X5 b1 LV1 e5 X6
b2 LV1 e6 X7 b3 LV1 e7 X8 b4 LV2
e8 X9 b5 LV2 e9 X10 1.0 LV2 e10 LV1 b6
X1 b7 X2 D1 LV2 b8 X2 b9 X3 D2
Parameters in addition to 9 coefficients Variance
s of e4,e5,e6,e7,e8,e9,e10, d1, d2 Variances
of X1,X2,X3 Covariances X1,X2 X1,X3
X2,X3 D1,D2 e5, e6
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Taking equations ? diagrams

• X1 1.0 LV1 e1
• X2 b2 LV1 e2
• X3 b3 LV1 b4 LV2 e3
• X4 1.0 LV2 e4
• X5 b5 LV2 e5
• X6 b6 LV2 e6
• X7 1.0 LV3 e7
• LV1 b7 LV2 b8 LV3 d1
• LV2 b9 LV1 d2

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Taking equations ? diagrams
Equations alone dont specify whether there are
any exogenous variable covariances (probably
needed cov(D1,D2)
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Different notation systems

• AMOS
• Indic1 b1 LVar1 1 e1
• EQS
• V1 F1 E1
• LISREL
• X1 ?1 ?1 d1

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Covariance algebra for structural equation models

• WHY?
• For any model, we use as input the observed
  covariance matrix, S
• But we also use the model parameters to calculate
  a reproduced covariance matrix, S.
• This matrix is important for a number of reasons
  (for one thing, the best estimates of model
  parameters are those that in some way minimize S-
  S).

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Covariance algebra for structural equation models

• For any model, we can take the model parameters,
  and calculate the elements of S as a function of
  these model parameters using the rules of
  covariance algebra.

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Covariance algebra for structural equation
models Four simple rules

• Null rule covariance between a variable and a
  constant is 0
• COV(X,a) 0 X variable a constant
• Variance definition covariance of a variable
  with itself is its variance
• VAR(X) COV(X,X)

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Covariance algebra for structural equation
models Four simple rules

• Constant Rule
• COV(X,aY) a COV(X,Y)
• We can factor constants out of covariances
• The covariance between 2 variables is multiplied
  by a constant when either of the two variables is
  multiplied by that constant

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Covariance algebra for structural equation
models Four simple rules

• Sum Rule
• COV(X,YZ) COV(X,Y) COV(X,Z)
• The covariance of a variable with the sum of two
  variables is the sum of the covariances between
  that variable and each of these two variables.

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Covariance algebra for structural equation
models Four simple rules

• COV(X,a) 0 null rule
• VAR(X) COV(X,X) variance def.
• COV(X,aY) a COV(x,y) constant rule
• COV(X,YZ) COV(X,Y) COV(X,Z) sum rule

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Applications
X2 b1 X1 e2 X3 b2 X2 b3 X1 e3 We can
use covariance algebra to solve for the
reproduced covariance for any of the 6
variances-covariances among manifest variables.
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Application
X2 b1 X1 e2 X3 b2 X2 b3 X1 e3
Example cov(x2,X1) X1 is already exogenous,
can be left. X2 is endogenous, so we must
replace COV(X2,X1) COV (b1 X1 e2, X1)
COV (B1 X1,X1) COV(e2,X1) sum rule
b1 COV (X1,X1) COV(e2,X1) b1 VAR
(X1) 0 variance definition
model specification
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Application
X2 b1 X1 e2 X3 b2 X2 b3 X1 e3
We can solve for each of the 6 variances and
covariances VAR(X3) COV(X3,X3) COV (b2 X2
b3 X1 e3, b2 X2 b3 X1 e3) Note that X2 is
not exogenous, so we still have work to do! COV
(b2 b1 X1 e2 b3 X1 e 3, b2 b1 X1 e2
b3 X1 e 3) COV (b2b1 X1 b2e2 b3 X1
e3, b2b1 X1 b2e2 b3 X1 e3) b12b22
COV(X1,X1) b22b1 COV(e2,X1) b3b2b1
(COV(X1,X1) b22b1 COV(e2,X1) b22 COV (e2,e2)
b2b3 COV(e2,X1) b2 COV(e2,e3) b3b2b1
COV(X1,X1) b3b2 COV(X1,e2) b32 COV(X1,X1)
b3(COV(X1,e3) b2b1 COV(e3,X1) b2 COV(e3,e2)
b3 COV(e3,X1) COV(e3,e3)
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Application
X2 b1 X1 e2 X3 b2 X2 b3 X1 e3
We can solve for each of the 6 variances and
covariances VAR(X3) b12b22 COV(X1,X1) b22b1
COV(e2,X1) b3b2b1 (COV(X1,X1) b22b1
COV(e2,X1) b22 COV (e2,e2) b2b3 COV(e2,X1)
b2 COV(e2,e3) b3b2b1 COV(X1,X1) b3b2
COV(X1,e2) b32 COV(X1,X1) b3(COV(X1,e3)
b2b1 COV(e3,X1) b2 COV(e3,e2) b3 COV(e3,X1)
COV(e3,e3) Red reduces to zero Result VAR(X3
) b12b22VAR(X1) b3b2b1 VAR(x1) b22var(e2)
b3b2b1 VAR(X1) b32var(X1) VAR(e3)
(b12b22 2b3b2b1 b32) VAR(X1) b22var(e2)
var(e3)
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Application
Equations X1 1.0 LV1 e1 X2 b2 LV1 e2 X3
b3 LV1 e3
COV (X1,X3) COV(LV1 e1, b3 LV1 e3) b3
COV(LV1,LV1) b3 COV(e1,LV1) COV(LV1,e3)
COV(e1,e3) b3 VAR(LV1) Note cov(e1,e3) may be
non-zero in some models look for curved arrow
connecting error terms
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Application
Equations X1 1.0 LV1 e1 X2 b2 LV1 e2 X3
b3 LV1 e3
VAR(X3) COV(X3,X3) COV(b3 LV1 e3, b3 LV1
e3) b32 COV(L1,L1) b3 COV(LV1,e3) b3
COV(LV1,e3) COV(e3,e3) b32 VAR(L1) VAR(e3)
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Homework (to be discussed in class
tomorrow)not a formalassignment
Express the following covariances as a function
of model parameters COV(X1,X2) COV(X3,X4) VAR(x6
)
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Computer programmingThe AMOS program

• We will start with an SPSS file, which AMOS will
  read
• AMOS calculates covariance matrices internally
• The SPSS file was set up with program
  ReligSexMoral1.sps located in a folder called
  Setups.
• Missing cases have been deleted listwise (see
  program to see how this is done)

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Amos example
1990 World Values Study Models Religiosity and
Sexual Morality Example The Variables Religiosity
Indicators V9 Importance of Religion 1Very
important 2quite imp.3not very 4not at
all V147 Church attendance 1more than 1/week
21/week31/month4ChristmasEaster 5other
holy days 6once/year 7less 8never V175
Views 1there is a personal God 2some sort of
spirit/life force 3dont know 4 dont think
there is any sort of God V176 Importance of God
in your life 1Not at all through 10very Sexual
Morality Can each be always justified (10),
never justified (1) or something in
between. V304 Married men/women having an
affair. V305 Sex under legal age of consent V307
Homosexuality V308 Prostitution V309
Abortion V310 Divorce
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AMOS example

• File USA1M
• Do NOT use file USA1 there are missing cases in
  this file
• USA1M has had cases deleted listwise (no variable
  has missing cases)
• We will discuss how to deal with files with
  missing cases later.
• Location \baer\Week1Examples\ReligSexMoral

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LAST Slide

• AMOS DEMONSTRATION