This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike License. Your use of this
material constitutes acceptance of that license and the conditions of use of materials on this site.
Copyright 2007, The Johns Hopkins University and Qian-Li Xue. All rights reserved. Use of these materials
permitted only in accordance with license rights granted. Materials provided “AS IS”; no representations or
warranties provided. User assumes all responsibility for use, and all liability related thereto, and must independently
review all materials for accuracy and efficacy. May contain materials owned by others. User is responsible for
obtaining permissions for use from third parties as needed.
Commonly Applied Structural
Models with Latent Variables
Statistics for Psychosocial Research II:
Structural Models
Qian-Li Xue
Outline
1. Three useful types of SEMs
a) Multiple Indicators Multiple Causes (MIMIC) model
b) Multitrait-Multimethod SEM
c) Causal vs. Effect Indicators
2. Group comparison models
1a) Multiple Indicators Multiple
Causes (MIMIC) Model
The MIMIC Structural Equation Model
(Jöreskog & Goldberger, 1975)
Structural
StructuralPart
Part
Regressors
Regressors
Measurement
MeasurementPart
Part
Latent
LatentConstruct
Construct
XX1
1
Response
ResponseItems
Items
YY1
1
ηη
XXq q
Adapted from Muthén (1988)
YYp
p
A MIMIC Structural Probit Model
(Muthén, 1988, 1989)
ƒ Extension of standard Item Response Theory
(IRT) modeling of dichotomous items to include
covariates that
ƒ Simultaneously addresses four issues:
ƒ Estimation of IRT measurement parameters (internal
validity)
ƒ Assessment of associations between covariates and
the latent trait (external validity)
ƒ Detection of item bias (differential item performance)
ƒ Relaxation of the assumptions of unidimensionality
and conditional independence
Example: Sadness in Older Adults
(Gallo, J.J., P. V. Rabins, and J. C. Anthony. 1999. Psychological Medicine 29:341-350)
ƒ Background: Earlier studies reported that major
depression is less prevalent among the elderly and
incidence rates also decline with age
ƒ Study Hypothesis: older adults are less likely than
younger adults to report sadness
ƒ Methods:
ƒ A population-based 13-year follow-up study of community-living
adults in Baltimore (part of ECA)
ƒ A subset of 1651 adults who participated at both initial interview
in 1981 and follow-up interview in 1994 and were younger than
65 years of age in 1981
ƒ Cross-sectional analysis at follow-up using the MIMIC model
Example: Sadness in Older Adults
(Gallo, J.J., P. V. Rabins, and J. C. Anthony. 1999. Psychological Medicine 29:341-350)
age
sadness
appetite criterion
sex
sleep criterion
minority
fatigue criterion
education
cognition
depression
psychomotor criterion
anhedonia criterion
worthlessness criterion
married
trouble concentrating
working
thoughts of death or suicide
Example: Sadness in Older Adults
(Gallo, J.J., P. V. Rabins, and J. C. Anthony. 1999. Psychological Medicine 29:341-350)
Model 1
Model 2
Model 3
Direct effect of age ≥ 65 on
sadness
-0.335*
(-0.643, -0.027)
-0.298
(-0.602, 0.006)
-0.317*
(-0.623, -0.011)
Difference in the mean level of
depression comparing age ≥ 65
to < 65
-0.361*
(-0.512, -0.210)
-0.581*
(-0.746, -0.416)
-0.639*
(-0.808, -0.470)
Model 1: adjusting for age only
Model 2: adjusting for sex, race, education, cognitive impairment, marital status, and
current working status
Model 3: Model 2 – cognitive impairment
* p-value<0.05
1b) Multitrait-Multimethod SEM
Multitrait-Multimethod SEM
(Campbell & Fiske, 1959)
ƒ Goal: separate out true variance from variance
due to measurement methods
ƒ Method: study a common set of traits by multiple
methods
ƒ Inference on
ƒ Convergent validity – “the tendency for different
measurement operations to converge on the same
underlying trait”
ƒ Discriminant validity – “ the ability to discriminate
among different traits
Example: Personality Traits
P
e
e
r
1
2
3
4
Peer
1 2 3 4
(.rr)
.aa (.rr)
.aa .aa (.rr)
.aa .aa .aa (.rr)
Teacher
1 2 3 4
Self
1 2 3
4
aa – within-method cross-trait corr.
cc – validity diagonals
bb – cross-method cross-trait corr.
rr = reliability coefficient
T
e
a
c
1
2
3
4
.cc .bb .bb .bb
.bb .cc .bb .bb
.bb .bb .cc .bb
.bb .bb .bb .cc
(.rr)
.aa (.rr)
.aa .aa (.rr)
.aa .aa .aa (.rr)
S
e
l
f
1
2
3
4
.cc .bb .bb .bb
.bb .cc .bb .bb
.bb .bb .cc .bb
.bb .bb .bb .cc
.cc .bb .bb .bb
.bb .cc .bb .bb
.bb .bb .cc .bb
.bb .bb .bb .cc
1 – Extraversion
2 – Anxiety
3 – impulsivity
(.rr)
4 – Academic
.aa (.rr) Achievement
.aa .aa (.rr)
.aa .aa .aa (.rr)
Example: Personality Traits
T1
scale 1,
peer
T3
T2
scale 2,
peer
scale 3,
peer
Peer
bias
scale 4,
peer
scale 1,
teacher
scale 2,
teacher
scale 3,
teacher.
Teacher
bias
scale 4,
teacher.
T4
scale 1,
self
scale 2,
self
scale 3,
self
Self
bias
scale 4,
self
Example: Personality Traits
h
T1
T3
T2
T4
Within-method cross-trait Corr:
e.g. r(1p,3p)=cf+ahg
b
a
Within-trait cross-method Corr:
g
e.g. r(1p,1q) =ab+ced
s1,
peer
s2,
peer
s3,
peer
s1,
s1,
s2,
sc3, s4,
teach. teach teach. teach. self
s4,
peer
s2,
self
s3,
self
s4,
self
Cross-method cross-trait Corr.
e.g. r(3p,1q)=bhg+fed
c
f
Peer
bias
d
e
Teacher
bias
Self
bias
Example: Model Identification
h
T1
T3
T2
T4
# equations = 12*13/2=78
# unknowns = 45
b
a
d.f.=78-45=33 for a χ2 goodnessof-fit test
g
s1,
peer
c
s2,
peer
s3,
peer
s1,
s1,
s2,
sc3, s4,
teach. teach teach. teach. self
s4,
peer
f
Peer
bias
s2,
self
s3,
self
d
e
Teacher
bias
Self
bias
s4,
self
Example: Inter-Personal Violence Scale
Methods
Peer
Teacher
Self
Extraversion
0.98
0.62
0.42
Anxiety
0.77
0.91
0.35
Impulsivity
0.78
0.64
0.42
Motivation
0.72
0.89
0.66
1. Trait Factors
Traits
2. Method Factors
E
A
I
M
Peer ratings
0.15
-0.25
0.32
-0.68
Teacher ratings
0.74
-0.19
0.49
0.13
Self ratings
0.34
0.17
0.89
-0.22
(Bentler & Lee, 1979)
Example: Inter-Personal Violence Scale
Methods
E
A
I
M
1.0
-0.35
0.52
-0.24
1.0
-0.26
0.74
1.0
-0.48
3. Trait Factor
Intercorrelations
Extraversion
Anxiety
Impulsivity
Motivation
1.0
Traits
2. Method Factors
Peer ratings
Teacher ratings
Self ratings
P
T
S
1.0
0.08
0.04
1.0
0.32
1.0
(Bentler & Lee, 1979)
1c) Causal vs. Effect Indicators
Cause vs. Effect Indicators
1
1
1
1
1
1
1
A) Effect Indicator
1
B) Causal Indicator
Cause vs. Effect Indicators
ƒ Effect indicators
ƒ manifestations of a the latent variable (LV)
ƒ The LV explains the intercorrelations among the effect indicators
ƒ Observed correlations among the effect indicators are
appropriate basis for estimating their association with the LV
ƒ “Cause” indicators
ƒ Causes of the LV or
ƒ Relate to the LV via unmeasured common causes
ƒ No reason why the LV should explain the correlations among the
cause indicators
ƒ No reason why the cause indicators should be correlated
ƒ Observed correlations among the cause indicators are irrelevant
to their association with the LV
Example: Cause vs. Effect Indicators
1
Life
Satisfaction
1
Life
Satisfaction
Health
Joint Dis.
1
Heart Dis.
1
Health
Digestive
Dis.
Joint Dis.
×
1
Heart Dis. Digestive
Dis.
1
Loehlin, pp. 219-220
Example: Cause vs. Effect Indicators
Life
Satisfaction
1
1
Financial
Res.
Financial
Res.
Current
Earnings
1
Life
Satisfaction
Retirement
Income
Current
Earnings
Retirement
Income
1
Loehlin, pp. 219-220
2. Group Comparison Models
Multiple-Group Comparisons: Example
ƒ Suppose you had a four-indicator measure of
self-esteem (exogenous) and you hypothesized
that it had a different relationship with
depression (endogenous) for high SES and low
SES adults
ƒ How would you test this hypothesis with regular
regression analysis?
1) You would have to a priori add the self-esteem
indicators to make a scale.
2) You would have to use product terms, which
compound problems of reliability.
™
™
a) we know that the V(t1)=ρ1V(x1) and V(t2)=ρ2V(x2)
b) so, then, the true variance of t1 times t2 (if they are not
correlated) = ρ1ρ2V(x1)V(x2)
Hierarchy of Multiple-Group Comparisons
ξ1
ξ1
Group 1
1 λ21
λ311
1
Different Forms
ξ2
λ311
1 λ212
X1
X2
X3
X1
X2
X3
δ1
δ1
δ1
δ1
δ1
δ1
ξ1
ξ1
Group 1
1 λ21
1
λ311
Same Forms
Group 2
1 λ212
Group 2
λ312
X1
X2
X3
X1
X2
X3
δ1
δ1
δ1
δ1
δ1
δ1
(Bollen, p. 357)
Steps in Multiple-Group Comparisons
1. Model specification for each group
2. Determine hierarchy of invariance in
models
ƒ
ƒ
Similarity in model form
Similarity in parameter values
3. Model estimation
4. Multiple-group comparisons
Multiple-Group Comparisons: Model Fitting
ƒ Object of analysis: Sg, i.e. group g’s covariance
structure, g=1, …, G
ƒ Assumption: Σg= Σ(θg)
ƒ Fit function:
⎛ Ng ⎞
⎟⎟ Fg ( S g , Σ g (θ g ))
F = Σ⎜⎜
⎝ N ⎠
i.e. a weighted sum of the fit function for each
group of size Ng.
ƒ The Fg for ML, ULS, and GLS are the same as
before
Multiple-Group Comparisons
ƒ To assess global goodness-of-fit based on
the fact: (N-1)F ~ χ2 with d.f.=G*[n(n+1)/2]t, where n is the number of observed
variables, and t is the number of free
parameters in ALL groups
ƒ To compare nested models using the
difference in χ2 with d.f. =difference in d.f.
for the two models
ƒ Other fit statistics apply as well
Example: Ratings of a political debate
Does gender modify the relationship between personableness and the
democratic candidate’s success in winning a debate?
Step 1. Model specification
δ1
1
p1
1
personable
δ2
1
ζ11
1
q1
1
q2
δ5
1
s2
δ6
1
quality
δ4
1
1
p2
success
δ3
s1
s3
1
δ7
Women
δ1
δ2
1
1
p1
lp1f
personable
lp2f
1
ζ1
ponsfem
ls2f
success
δ3
1
q1
δ6
1
lq1f
ls3f
1
s3
lq2f
δ4
1
s2
quality
1
δ5
ls1f
1
p2
1
s1
δ7
q2
Men
δ1
δ2
1
1
p1
lp1m
personable
1
lp2m
ζ1
ponsmale
1
p2
success
δ3
1
q1
δ4
ls1m
ls2m
1
s2
δ5
δ6
1
lq1m
ls3m
quality
1
1
s1
lq2m
q2
Group 2: men
Number of parameters to estimate:
s3
1
δ7
Example: Ratings of a political debate
Step 2. Hierarchy of model invariance
ƒ Model #1: Measurement model, in which CFAs are constrained to
be equal across groups (males and females)
ƒ lp1f=lp1m; lp2f=lp2m;
ƒ lq1f=lq1m; lq2f=lq2m;
ƒ ls1f=ls1m; ls2f=ls2m; ls3f=ls3m
ƒ Model #2: Same as Model #1, except that the parameter of interest
is constrained to be equal across groups (i.e. ponsmale=ponsfem)
ƒ Model #3: Same as Model #1, except that the variances of residual
errors of latent variable indicators are constrained to be equal
across groups
ƒ Θδ(m)=Θδ(f)
Example: Ratings of a political debate
MPLUS syntax for Model #2:
TITLE: Ratings of a political debate
DATA: FILE is c:/teaching/140.658.2007/twogrp.dat;
TYPE IS COVARIANCE;
NOBSERVATIONS ARE 154 125;
NGROUPS=2;
VARIABLE: NAMES ARE S1 S2 S3 P1 P2 Q1 Q2;
USEVARIABLES ARE S1-Q2;
MODEL:
fs BY S1* S2 S3;
fp BY P1* P2;
fq BY Q1* Q2;
fs on fp (1);
fs on fq;
fp@1 fq@1 fs@1;
OUTPUT:
TECH1
Example: Ratings of a political debate
MPLUS syntax
ƒ Model #3:
MODEL:
fs BY S1* S2 S3;
fp BY P1* P2;
fq BY Q1* Q2;
fs on fp fq;
S1 (1)
S2 (2)
S3 (3)
P1 (4)
P2 (5)
Q1 (6)
Q2 (7);
fp@1 fq@1 fs@1;
OUTPUT:
TECH1
Example: Ratings of a political debate
Results of two-group (males vs. females) comparisons
DF
χ2
Model I
29
Model II
TLI
RMSEA
SRMR BIC
Loglikelihood
16.9 (0.96)
1.01
0
0.02
8882.3
-4365.1
30
42.0 (0.07)
0.99
0.05
0.15
8901.7
-4377.7
Model III 36
24.1 (0.93)
1.01
0.00
0.02
8850.1
-4368.7
(p value)
Model I: CFAs are constrained to be equal across groups
Model II: Model I plus the association between “peronableness” and success
is constrained to be equal across groups
Model III: Model I plus the variances of residual errors of LV indicators are
constrained to be equal across groups