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Advanced Structural Equations
Models I
Statistics for Psychosocial Research II:
Structural Models
Qian-Li Xue
Test of causal
hypotheses?
No
Yes
Ordinary
Regression
SEM (Origin: Path Models)
Yes
Continuous
endogenous var. and
Continuous LV?
Yes
Classic SEM
Yes
Categorical indicators
and Categorical LV?
Latent Class Reg.
Longitudinal Data?
No
Multilevel Data ?
Adv. SEM II:
Multilevel Models
No
Latent Trait
Latent Profile
No
Yes
Adv. SEM I: latent
growth curves)
No
Classic SEM
Outline
1. Estimating means of observed and latent
variables
2. Modeling repeated measures of outcome over
time
The Simplex-Growth Over Time
3. Non-Recursive Models
4. Modeling repeated measures of outcome and
covariate over time
Cross-Lag Panel Analysis
Latent Growth Curve Models (Next Lecture)
1. Estimating Means of Observed
and Latent Variables
Estimating Means of Observed and
Latent Variables
So far, we have largely ignored intercept
terms in our analyses
What has happened to the alpha
coefficient?
Estimating Means of Observed and
Latent Variables
Up to now, information on means and intercepts
has not been of interest
It is possible to estimate levels of association without
information on these parameters
If of interest, these parameters can be
estimated using a “mean model.”
In addition to covariances, these models also require
information on mean of variables
These parameters are of key interest in group
comparisons and growth curve models
Estimating Means of Observed and
Latent Variables
Does the mean score on the latent variable ξ (e.g.
depression) differ between men and women?
Man
Women
1
e
d
ξ11
0.6 0.8
X11
.64
4.0
a
b
c
0.7
X12
.36
5.0
X13
.51
6.0
a
b
ξ12
c
0.6
0.8
X21
Y22
.64
4.3
.36
5.4
0.7
X23
.51
6.35
Resid. Var.
Means
(Loehlin p.139)
Estimating Means of Observed and
Latent Variables
Man
Women
1
e
d
ξ11
0.6 0.8
X11
.64
4.0
a
b
c
0.7
X12
.36
5.0
X13
.51
6.0
a
b
d=0 (reference)
a=4.0,b=5.0,c=6.0
(baseline values,
same across groups)
e – difference
between the means of
the latent variable
e*0.6+a=4.3 ⇒ e=0.5
ξ12
c
0.6
0.8
X21
Y22
.64
4.3
.36
5.4
0.7
X23
.51
6.35
Resid. Var.
Means
(Loehlin p.139)
Example: Stress, Resources, and
Depression (Holahan & Moos, 1991)
“How do the high-stressor and the low-stressor
groups compare on the two latent variables:
depression (D) and resources (R)
r
1
g
f
D
High-Stressor
R
i
h
l
a
b
DM
DF
m
n
j
c
d
k
SC
o
e
EG
FS
p
q
Example: Stress, Resources, and
Depression (Holahan & Moos, 1991)
High-stressor group: above diagonal (underlined)
Low-stressor group: below diagonal
DM
DF
SC
EG
FS
SD
M
Depressed Mood
1
.84
-.36
-.45
-.51
5.97
8.82
Depressive Features
.71
1
-.32
-.41
-.50
7.98
13.87
Self-confidence
-.35
-.16
1
.26
.47
3.97
15.24
Easygoingness
-.35
-.21
.11
1
.34
2.27
7.92
Family support
-.38
-.26
.30
.28
1
4.91
19.03
Standard Deviation
4.84
6.33
3.84
2.14
4.43
N
128
Mean
6.15
9.96
15.14
8.80
20.43
126
Example: Stress, Resources, and
Depression (Holahan & Moos, 1991)
MPLUS code
Low-Stressor
r
1
g
f
D
R
i
h
l
a
b
DM
DF
m
n
j
c
d
k
SC
o
e
EG
FS
p
q
TITLE: Stress, resources, and depression (Loehlin, p.142)
DATA: FILE is c:/teaching/140.658.2007/depression.dat;
TYPE IS CORRELATION MEANS STDEVIATIONS;
NOBSERVATIONS ARE 126 128;
NGROUPS=2;
VARIABLE: NAMES ARE DM DF SC EG FS;
USEVARIABLES ARE DM-FS;
MODEL:
D BY DM* DF;
R BY SC* EG FS;
DM (1);
Equate the measurement models
DF (2);
SC (3);
across the groups
EG (4);
FS (5);
MODEL g1:
[D@0 R@0];
Set reference group (i.e. lowD@1 R@1;
stressor)
OUTPUT:
TECH1;
Example: Stress, Resources, and
Depression (Holahan & Moos, 1991)
Measurement Model
Latent Variables
LowStressor
HighStressor
Path
Coeff.
Residual
Var.
Baseline
means
Depression: Mean
f
[0]*
a
4.42
m 2.91
h 6.09
Resources: Mean
g
[0]
b
5.22
n 16.04
i 10.27
Depression: SD
[1]
c
1.56
o 11.76
j 15.59
Resources: SD
[1]
d
1.01
p 3.61
k 8.61
e
2.67
q 12.25
l 20.40
correlation
r
-0.72
Depression: Mean
f
0.63
Resources: Mean
g
-0.50
Depression: SD
1.30
Resources: SD
1.29
correlation
r
-0.78
* Numbers in [ ] are prefixed in order to make the model identified
Same as above
Example: Stress, Resources, and
Depression (Holahan & Moos, 1991)
TESTS OF MODEL FIT
Chi-Square Test of Model Fit
Value
Degrees of Freedom
P-Value
CFI/TLI
CFI
TLI
27.245
19
0.0991
0.979
0.978
RMSEA (Root Mean Square Error Of Approximation)
Estimate
0.058
90 Percent C.I.
0.000
SRMR (Standardized Root Mean Square Residual)
Value
0.055
The model fits reasonably well to the data!
0.104
2. Modeling Repeated Measures of
Outcome Over Time
The Simplex-Growth Over Time
Modeling growth over (e.g. height)
Measurements taken repeatedly over time
In general, measurements made closer together
in time would be more highly correlated (called
“simplex” by Guttman, 1954)
Smaller
E.g.
Correlation
1
2
3
4
1
1
0.73
0.72
0.68
1
0.79
0.76
1
0.84
2
3
4
1
The Simplex-Growth Over Time
Example: Scores on standardized tests of academic achievement at
grades 1-7 (Bracht & Hopkins, 1972)
Test score (Y) is a measure of the latent academic achievement (η)
Achievement at grade t is a function of achievement at t-1 via β, and
other factors ζ
ζ2
η1
1
β21
ζ3
η2
1
β32
ζ4
η3
β43
ζ5
η4
1
β54
ζ6
η5
1
1
β65
ζ7
η6
β76
η7
1
1
Y1
Y2
Y3
Y4
Y5
Y6
Y7
ε1
ε2
ε3
ε4
ε5
ε6
ε7
Loehlin p.125
The Simplex-Growth Over Time
ζ2
η1
1
β21
ζ3
η2
1
β32
ζ4
η3
β43
ζ5
η4
1
β54
ζ6
η5
1
1
β65
ζ7
η6
1
β76
η7
1
Y1
Y2
Y3
Y4
Y5
Y6
Y7
ε1
ε2
ε3
ε4
ε5
ε6
ε7
Yi = ηi + ε i
ηi = β iηi −1 + ς i
εi are uncorrelated, εi⊥ηi, and ζi⊥ηi-1
The Simplex-Growth Over Time
Var(η1), Var(ζ7), Var(ε1), Var(ε2), β21 are unidentified
To achieve identification, set Var(ε1)=Var(ε2) AND
Var(ε6)=Var(ε7), reasonable if Ys are on the same scale
# free parameters = 3p-3, where p=# of Ys
For testing a simplex model, p>3 !!!
ζ2
η1
1
β21
ζ3
η2
1
β32
ζ4
η3
β43
ζ5
η4
1
β54
ζ6
η5
1
1
β65
ζ7
η6
1
β76
η7
1
Y1
Y2
Y3
Y4
Y5
Y6
Y7
ε1
ε2
ε3
ε4
ε5
ε6
ε7
3. Non-Recursive Models
Non-Recursive Models
So far, there has been little discussion of models
with feedback loops
Non-recursive models deal with reciprocal
causal relationships
Can not be analyzed by ordinary regression
analysis due to correlated errors
Non-recursive models may not be identified
even if the T-rule is met
Non-Recursive Models
Time 1
Time 2
A
A
A
B
B
B
Reciprocal
Lagged
What do you mean by “reciprocal causation”?
Alternative: Lagged model
Assumption: the principal of “finite causal lag”
Roles of the variables in the bidirectional relationship change
over time (e.g. A is a cause at Time 1, but effect at Time 2)
The reciprocal causation model becomes the only choice
if only cross-sectional data are available
Non-Recursive Models: Model Identification
Recall: recursive path models without measurement
error are always identified
Not true for non-recursive models
Definition: Instrumental variable – a predictor is an
instrument for an endogenous variable if it has a direct
path to other endogenous variables but not the
endogenous variable of interest
X1
Y1
X3 is an instrument for Y1
X2
X3
Y2
Maruyama, 1998; p.106
Non-Recursive Models: Model Identification
Order condition (necessary but not sufficient) – For any
system of N endogenous variables, a particular equation
is identified only if at least N-1 variables are left out of
that equation
Rank condition (necessary AND sufficient) – is met for a
particular equation if there is at least one non-zero
determinant of rank N-1 from the coefficients of the
variables omitted from that equation
X1
Y1
X2
X3
Y2
Maruyama, 1998; p.106
4.
Modeling Repeated Measures of
Outcome and Covariate Over Time
Cross-Lagged Panel Analysis:
Terminology
Time 1
Time 2
eX1
X1
eX2
Synchronous correlations:
Corr(X1,Y1) and Corr(X2,Y2)
Autocorrelations (i.e. stability):
Corr(X1,X2) and Corr(Y1,Y2)
X2
Cross-lagged: Corr(X1,Y2) and
Corr(Y1,X2)
Y1
eY1
Y2
eY2
Residual correlations (due to
measure-specific variance):
Corr(ex1,eX2) and Corr(eY1,eY2)
Here Corr. denotes total correlation!
Cross-Lagged Panel Analysis:
Identification
Time 1
Time 2
eX1
X1
eX2
X2
Is this model identified?
# equations = 4*5/2=10
# unknowns = 11
Not identified!
What is the problem?
The repeated assessment of the
same measure leads to two
sources of common variance
Y1
Y2
eY1
eY2
construct variance
Measure-specific variance
Model would be identified if
delete residual correlations or
Build multiple-indicator models
Cross-Lagged Panel Analysis:
Key Issues (Maruyama, pp.112-120)
Time 1
Time 2
1. Stability of a variable
For example, if Y is perfectly stable,
Y2 is perfectly determined by Y1
If data is only available at Time 2,
then Y1 is not available
Any variable correlated with Y or
caused by Y could be included as
predictors, leading to a misspecified
model!
Low stability over time may result
from poor reliability (if so, we’re in
trouble!) or
Real change in the measure
eY2
Y1
Y2
X2
eX2
Cross-Lagged Panel Analysis:
Key Issues
Time 1
Time 2
eY2
eX1
X1
Y1
eY1
2.
Temporal Lags
How long is the causal lag?
It the sampling interval > causal
lag ⇒ attenuated effect
If the sampling interval < causal
lag ⇒ no effect or
underestimated effect
What if the causal lag from X1 to
Y2 is different from Y1 to X2?
Solution: three-wave data with
different intervals
X2
Y2
eY2
Cross-Lagged Panel Analysis:
Key Issues
3. Growth Across Time
When to use covariance vs. correlation
data in SEM
Covariance allows for “growth” by focusing
on raw scores
Correlation focuses on standardized
relationships
If no change in variability of any of the
variables over time, the results are identical
Using covariance is highly recommended!
Cross-Lagged Panel Analysis:
Key Issues
3. Stability of Causal Process
Causal dynamics between variables remain
stable across time intervals of the same length
If not true, the relationships would differ
depending on the particular interval sampled
On the other hand, modeling unstable
processes may be warranted when studying
Developmental processes
Time-varying interventions
Cross-Lagged Panel Analysis with Latent
Variables: Example
0.39
0.52
0.53
Nervous
or upset
Nervous
or upset
Often get
scared
0.63
Often get
scared
0.72
0.73
Grade 7
Anxiety
0.54
0.51
Grade 8
Anxiety
0.69
Nervous
or upset
0.48
0.63
Grade 9
Anxiety
0.73
0.64
…
0.73
Often get
scared
0.53
(Ma & Xu, Journal of Adolescence 27 (2): 165-179 APR 2004 )
Cross-Lagged Panel Analysis with Latent
Variables: Example
0.77
0.31
Basic skills
0.88
0.46
Algebra
0.56
0.64
Geometry
0.68
Grade 7
Achieve
Literacy
0.80
0.98
0.89
Basic skills
0.79
0.88
0.77
0.90
0.85
Algebra
…
0.92
Grade 8
Achieve
Geometry
0.72
Literacy
0.81
(Ma & Xu, Journal of Adolescence 27 (2): 165-179 APR 2004 )
Anxiety Grade
0.39
0.39
77
0.55
0.55
88
0.57
0.57
99
10
10
-0.05
-0.05
-0.01
-0.20
-0.12
-0.14
77
88
0.98
0.98
99
0.91
0.91
0.59
0.59
0.57
0.57
11
11
-0.02
-0.02
-0.15
10
10
0.95
0.95
12
12
-0.11
11
11
0.97
0.97
12
12
0.97
0.97
Achievement Grade
Example of cross-lagged panel analysis with latent variables. Structural equation model estimating
the causal relationship between mathematics anxiety & mathematics achievement across Grades
7–12. Large ovals represent latent factors & unidirectional arrows represent casual links. All
parameter estimates for unidirectional paths are standardized. Pink boxes indicated P < 0.001).
Adapted from Ma & Xu, Journal of Adolescence 2004;27:165-179
