Transcript ICPSR General Structural Equation Models

ICPSR General Structural
Equation Models
Week 4 #4
(last class)
•Interactions in latent variable models
•An introduction to MPLUS software
• An introduction to latent class models
• Models for (conceptually!) categorical dependent variables
1
Article discussion:
“Reexamination and Extension of Kleine,
Llein and Kerman’s Social Identity Model
of Mundane Consumption: the Mediating
Role of the Appraisal Process”
J. Of Consumer Research, 28, 2002, 659660.
2
Article discussion:
“Reexamination and Extension of Kleine, Llein and Kerman’s Social Identity Model of
Mundane Consumption: the Mediating Role of the Appraisal Process”
J. Of Consumer Research, 28, 2002, 659-660.
Data pooled, 2 groups: tennis players; aerobics group
Tested H0: S[1] = S[2] (p>.50)
Tennis players, 68% response, listwise N=213 vs. N of 318.
Aerobics, 73% response, listwise N= 329 vs. N of 359
3
“Reexamination and Extension of Kleine, Llein and Kerman’s Social Identity Model of Mundane
Consumption: the Mediating Role of the Appraisal Process”
J. Of Consumer Research, 28, 2002, 659-660.
Measurement model fit to data: “fit the aerobics data well, … residuals
normally distributed”
Common method variance… to test, allowed covariances among
residuals of identically worded questions… “mimimal effect (change
in r <.01) on interfactor correlations.
Original model: identity importance DV.
2nd model reverses direction entirely: 3 commitment variables as DVs
“significant reduction in model fit” (table 1, model 2 Xsq=15605,
df=373 vs. 1541.6 df=370 for “a priori structural model” and 1422.1
with post-hoc modifications to this). Question not answered:
which additional restrictions were in the “reversed” model?
4
A
b
E
D
C
The chi-square value reflects, among other things, the restrictions
in this model, eg. AE coefficient = 0.
5
E
b
A
D
C
In this model, another set of restrictions is imposed (e.g., EA
direct path =0).
If the true model involves reciprocal causation, neither model
is specified correctly
“Tests” – chi-square comparisons” – are not formal (not nested)
Moreover, they reflect the “other” restrictions in the model and not
an AE vs. E—A test.
6
True model
7
INTERACTIONS IN LATENT
VARIABLE STRUCTURAL
EQUATION MODELS
Y = b0 + b1 X1 + b2 X2 + b3 (X1*X2) + e
If X is categorical: multiple group modeling
If X is continuous: more complicated
• Categorical: can also model as dummy
variables.
8
Interactions
Easiest case: X1 is 0/1
X2 ix 0/1
Options: 1. Manually construct X3=X1*X2 outside
SEM software, estimate model with X1,X2,X3
exogenous. Test for interaction: fix regression coefficient
for X3 to 0.
2. Create two groups: X1=0 and X1=1. In each group,
X2 as exogenous variable. Test for interaction would be
H0: gamma[1] = gamma[2].
Extensions for X1, X2 >2 categories straightfoward (more
groups/dummy variables)
9
Interactions
Option 3: Model as a 4-group problem.
X1
1
0
X2 1
gr1 gr2
0
gr3 gr4
AL[1]=0 al[2], al[3],al[4] parameters to be
estimated.
Main effects model (no interaction) would allow for
al[2]≠al[3] ≠al[4] but pattern of differences would
be constrained such that…..
10
Model as a 4-group problem. Interactions
X1
1
0
X2
1
gr1
gr2
0
gr3 gr4
AL[1]=0 al[2], al[3],al[4] parameters to be estimated.
Main effects model (no interaction) would allow for al[2]≠al[3] ≠al[4] but
pattern of differences would be constrained such that…..
The group1 vs. group 2 difference = group 3 vs. group 4 difference
(or group 1 vs. 3 difference = group 2 vs. group 4).
Programming in LISREL would be:
Al[1] – Al[2] = al[3]- al[4]
0 – al[2] = al[3] – al[4]
Al[2] = al[4]-al[3] LISREL: CO al 2 1 = al 4 1 – al 3 1
Test for interaction: run another model removing this constraint (all AL
completely free except group 1)
… more examples provided in text
11
Interactions
Interactions involving continuous variables.
Case 1: One continuous (single or multiple indicator) and one categorical variable
EASY: categorical variable becomes basis for grouping.
Group 1 Eta = gamma[1] Ksi + zeta
Group 2 Eta = gamma[2] Ksi + zeta
Test for interaction: H0: gamma[1] = gamma[2]
Case 2: Two continuous single indicator variables
Also somewhat straightforward:
Create single-indicator X3 = X2*X1
Case 3: Two continuous multiple indicator latent variables
This is not so easy! Substantial literature on this question
See course outline for extended list. (Schumacker and Mracoulides, eds., Interaction
and Nonlinear Effects in Structural Equation Modeling).
Case 3A, not talked about much: X1 single indicator Ksi1 (X2, X3,X4)
Create: X1X2 , X1X3, X1,X4
12
Latent variable interactions
Major approaches:
• Kenny-Judd
• Simplified variants of Kenny-Judd,
modifications, etc. (Joreskog & Yang, 1996; Ping)
• Two-stage least squares (get instrumental
variables)
• Use SEM to estimate 2 factor model, save
latent variable “scores” (analogous to
factor scores), then use these
13
Latent variable interactions
• Use SEM to estimate 2 factor model, save latent
variable “scores” (analogous to factor scores),
then use these
In LISREL:
Mo nx=6 nk=2 lx=fu,fi ph-sy,fr td=sy
Va 1.0 lx 1 1 lx 4 2
Fr lx 2 1 lx 3 1 lx 5 2 lx 6 2
PS=Newfile.psf
OU
14
Latent variable interactions
•
Use SEM to estimate 2 factor model, save latent variable “scores”
(analogous to factor scores), then use these
In LISREL:
Mo nx=6 nk=2 lx=fu,fi ph-sy,fr td=sy
Va 1.0 lx 1 1 lx 4 2
Fr lx 2 1 lx 3 1 lx 5 2 lx 6 2
PS=Newfile.psf
OU
LISREL documentation suggests that a simple regression can be
estimated in PRELIS:
Sy=newfile.psf
ne inter=ksi1*ksi2
rg y on ksi1 ksi2 ksi1ksi2
ou
15
Latent variable interactions
LISREL documentation suggests that a simple regression can be
estimated in PRELIS:
Sy=newfile.psf
ne inter=ksi1*ksi2
rg y on ksi1 ksi2 ksi1 ksi2
ou
…. But it should also be possible to a) construct “inter” (=ksi1*ksi2) and read
the 3 new “single indicator” variables back into LISREL for use with other
variables (including those which form the basis of multiple-indicator
endogenous variables.
If all else fails, construct a LISREL model for Ksi1, Ksi2, and put FS (factor
score regressions) on the OU line:
OU ME=ML FS MI ND=4
.. And use factor score regressions to compute estimated factor scores in any
stat package (incl. PRELIS)
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Example:
INTERACTION MODEL WITH INTERACTION TERM CREATED EXTERNALLY
SINGLE INDICATORS FOR EXOGENOUS LVS INVOLVED IN INTERACTION
DA NO=1111 NI=10 MA=CM
CM FI=G:\ICPSR\INTERACTIONS\INT5b.COV FU FO
(10F10.7)
LABELS
lv1 lv2 interact
sex race v217 v216 v125 v127 v130
se
8 9 10 1 2 3 4 5 6 7/
mo ny=3 ne=1 LY=FU,FI PS=SY,FR TE=SY c
nx=7 nk=7 fixedx ga=fu,fr
va 1.0 ly 1 1
fr ly 2 1 ly 3 1
ou me=ml se tv mi sc
17
LISREL Estimates (Maximum Likelihood)
Example:
LAMBDA-Y
v125
ETA 1
-------1.00
v127
1.34
(0.24)
5.59
v130
Dep var = inequality att’s (high
score  “more individual effort”)
0.65
(0.11)
5.74
GAMMA
ETA 1
lv1
--------0.04
(0.06)
-0.65
lv2
--------0.21
(0.08)
-2.57
interact
-------0.85
(0.45)
1.89
sex
-------0.22
(0.11)
2.10
race
--------0.30
(0.13)
-2.27
v217
-------0.05
(0.03)
1.75
GAMMA
ETA 1
v216
-------0.09
(0.03)
2.92
Lv1=relig. Lv2=econ. status
18
Kenny-Judd model
Typically, literature (e.g., Kenny-Judd, 1984; Hayduk, 1987) starts with 2indicator example (2 LV’s each with 2 indicators).
Ksi1
Ksi2
Ksi1*Ksi2 (interaction term)
Indicators:
Ksi1:
x1
x2
Ksi2:
x3
x4
Possible product terms:
x1*x3
x1*x4
x2*X3 X2*x4
Kenny-Judd model use 4 product terms but Joreskog and Yang show that the
model can be constructed with 1 product term.
19
Kenny-Judd model
Typically, literature (e.g., Kenny-Judd, 1984; Hayduk, 1987) starts with 2-indicator example (2 LV’s
each with 2 indicators).
Ksi1
Ksi2
Ksi1*Ksi2 (interaction term)
Indicators: Ksi1:
x1
x2
Ksi2:
x3
x4
Possible product terms:
x1*x3
x1*x4
x2*X3
X2*x4
Kenny-Judd model use 4 product terms but Joreskog and Yang show that the
model can be constructed with 1 product term.
Kenny-Judd do not include constant intercept terms (alpha, tau).. But even if
dependent variable, Ksi1, Ksi2 and zeta have zero means, alpha will still be
nonzero. - means of observed variables functions of other parameters in
the model and therefore intercept terms have to be included.
- Nonnormality even if x’s are normal (ADF estimation often recommended if
sample size acceptable)
20
Kenny-Judd model
21
Kenny-Judd model
alpha=1 term
22
Kenny-Judd model, mod.
INTERACTION MODEL KENNY JUDD MODIFICATION (JORESKOG AND YANG)
ONE INTERACTION INDICATOR 3 INDICATORS PER L.V.
DA NO=1111 NI=22
CM FI=G:\ICPSR2000\INTERACTIONS\INT5c.COV FU FO
(22F20.11)
ME FI=G:\ICPSR2000\INTERACTIONS\INT5C.MN FO
(22F20.11)
LABELS
v181 v9 v190 v221 v226 v227
relinc1 relinc2 relinc3 relinc4 relinc5
relinc6 relinc7 relinc8 reling9
sex race v217 v216 v125 v127 v130
se
20 21 22 1 2 3 4 5 6 9 16 17 18 19/
mo ny=3 ne=1 NX=11 NK=7 LY=FU,FI PS=SY,FR C
TE=SY TX=FR KA=FI C
LX=FU,FI GA=FU,FR PH=SY,FR TD=SY AL=FI TY=FR
va 1.0 ly 1 1
fr ly 2 1 ly 3 1
FI PH 3 1 PH 3 2
FR KA 3
VA 1.0 LX 1 1 LX 4 2 LX 7 3 LX 8 4 LX 9 5 LX 10 6 LX 11 7
FR TD 1 1 TD 2 2 TD 3 3 TD 4 4 TD 5 5 TD 6 6 TD 7 7
FR LX 2 1 LX 3 1 LX 5 2 LX 6 2 LX 7 1 LX 7 2
CO LX(7,1)=TX(1)
CO LX(7,2)=TX(4)
CO KA(3) = PH(2,1)
FI PH 3 1 PH 3 2
CO PH(3,3) = PH(1,1)*PH(2,2) + PH(2,1)**2
CO TX(6) = TX(1)*TX(4)
FI TD(8,8) TD(9,9) TD(10,10) TD(11,11)
CO TD(7,7) = TX(1)**2*TD(3,3) + TX(4)**2*TD(1,1) + PH(1,1)*TX(4) + C
PH(2,2)*TX(1) + TD(1,1)*TD(4,4)
OU ME=ML SE TV ND=3 AD=off
23
Kenny-Judd model, modified Joreskog/Yang
Parameter Specifications
LAMBDA-Y
ETA 1
-------0
1
2
v125
v127
v130
LAMBDA-X
v181
v9
v190
v221
v226
v227
relinc3
sex
race
v217
v216
KSI 1
-------0
3
4
0
0
0
Constr'd
0
0
0
0
KSI 2
-------0
0
0
0
5
6
Constr'd
0
0
0
0
KSI 3
-------0
0
0
0
0
0
0
0
0
0
0
KSI 4
-------0
0
0
0
0
0
0
0
0
0
0
KSI 5
-------0
0
0
0
0
0
0
0
0
0
0
KSI 6
-------0
0
0
0
0
0
0
0
0
0
0
24
Kenny-Judd model, modified Joreskog/Yang
GAMMA
ETA 1
KSI 1
--------0.023
(0.009)
-2.557
KSI 2
--------0.003
(0.015)
-0.198
KSI 3
--------0.008
(0.004)
-1.984
KSI 4
-------0.209
(0.098)
2.130
KSI 5
--------0.324
(0.125)
-2.593
KSI 6
-------0.051
(0.024)
2.094
GAMMA
ETA 1
KSI 7
-------0.080
(0.029)
2.735
25
Latent class models
Basic parameters:
1. Latent class probabilities
2. Conditional probabilities (given one is in
latent class A, what are the probabilities
that one will be in cat i of indicator j? …
prob’s sum to 1.0).
Parameter constraints are possible (in some
cases, needed for identification).
26
A latent class model
•
Software: MLLSA
NUMBER OF LATENT CLASSES REQUESTED:
5
START VALUES ENTERED FOR LATENT CLASS PROBABILITIES:
.630000
.110000
.160000
.020000
.080000
START VALUES ENTERED FOR CONDITIONAL PROBABILITIES:
.000000
1.000000
1.000000
.000000
.000000
1.000000
.060000
.000000
1.000000
.000000
.000000 1.000000
.250000 .750000
.000000 .000000
.500000 .250000
.800000 .200000
.000000 .000000
.540000 .300000
.900000 .100000
.000000 .000000
.000000 .600000
.450000 .550000
.000000 .000000
.000000 .000000
.250000 1.000000
.000000
.000000 .020000
.100000 1.000000
.000000
.000000 .600000
.400000 1.000000
.000000 .000000
.000000 1.000000
.300000 .350000
.000000 .000000
.000000
.370000
.000000
.400000
.000000
.310000
.000000
.400000 1.000000
.000000 .000000
.000000
.350000
.000000
27
A latent class model
•
Software: MLLSA
***** ITERATION STEPS *****
DEVIATION =
.00306576
DEVIATION =
.00078193
DEVIATION =
.00041910
DEVIATION =
.00024801
DEVIATION =
.00015106
DEVIATION =
.00009318
DEVIATION =
.00005788
DEVIATION =
.00004791
ITERATION =
ITERATION =
ITERATION =
ITERATION =
ITERATION =
ITERATION =
ITERATION =
ITERATION =
10
20
30
40
50
60
70
74
-------------------------------------------------------------------------------
-------------------------------------------------------------------------------
FINAL LIKELIHOOD RATIO CHI-SQUARE =
155.032400
FINAL PEARSON CHI-SQUARE =
157.236800
INDEX OF DISSIMILARITY =
.034417
------------------------------------------------------------------------------FINAL LATENT CLASS PROBABILITIES:
.627384
.110530
.160754
.018552
.082779
28
Latent class model
1.
FINAL CONDITIONAL PROBABILITIES:
2.
LATENT CLASS =
1
2
3
4 . . .
3.
PLAN
ENTIRE
.0000
.3546
.0000
.2394
.0000
4.
PLAN
PART
.0000
.6454
.0000
.7606
.0000
5.
PLAN
NOT
1.0000
.0000
1.0000
.0000
1.0000
6.
SUPTIME
NOT
1.0000
.0000
.0000
1.0000
.0000
7.
SUPTIME
1/4
.0000
.4019
.3166
.0000
.8308
8.
SUPTIME
1/4-1/2
.0000
.3333
.2867
.0000
.1692
9.
SUPTIME
1/2+
.0000
.2648
.3966
.0000
.0000
10.
NSUPER
0
1.0000
.0213
.0975
1.0000
.0000
11.
NSUPER
1-4
.0000
.3688
.3032
.0000
.9990
12.
NSUPER
5-19
.0000
.4019
.4514
.0000
.0010
13.
NSUPER
20+
.0000
.2080
.1479
.0000
.0000
14.
TIMEPLAN
NOT
1.0000
.0000
1.0000
.0000
1.0000
15.
TIMEPLAN
UP TO 1/
.0000
.5745
.0000
.5915
.0000
16.
TIMEPLAN
1/4+
.0000
.4255
.0000
.4085
.0000
FINAL LATENT CLASS PROBABILITIES:
.627384
.110530
.160754
.018552
.082779
29
Latent class model
ASSIGNMENT OF RESPONDENTS TO LATENT CLASS:
CELL
OBSERVED
PROBABILITY
1
.00
2
.00
3
2401.00
4
.00
5
.00
6
42.00
7
.00
8
.00
9
9.00
10
.00
11
.00
EXPECTED
.00
.00
2401.00
.00
.00
19.00
.00
.00
17.20
.00
.00
ASSIGN TO CLASS
1
1
1
1
1
3
1
1
3
1
1
MODAL
.0000
.0000
1.0000
.0000
.0000
1.0000
.0000
.0000
1.0000
.0000
.0000
30
MPlus software
See director /Week4Examples/MPlus
TITLE: categorical #1
DATA:
FILE IS
H:\ICPSR2003\Week4Examples\MPlus\Categor.dat
VARIABLE:
NAMES ARE REGION V166-V175 EDUC AGE SEX;
USEV = V166-V175;
CATEGORICAL = V166-V175;
ANALYSIS:
TYPE = EFA 1 3;
ESTIMATOR WLSMV;
Exploratory factor analysis with binary variables
31
V166
V167
V168
V169
V170
V171
V172
V173
V174
V175
VARIMAX ROTATED LOADINGS
1
2
________
________
0.853
0.127
0.488
0.693
0.655
0.408
0.533
0.019
0.370
0.041
0.626
0.192
0.531
0.071
0.693
0.336
0.002
0.836
-0.739
-0.019
3
________
0.427
0.397
0.406
0.753
0.993
0.662
0.598
0.473
-0.080
-0.330
V166
V167
V168
V169
V170
V171
V172
V173
V174
V175
PROMAX ROTATED LOADINGS
1
2
________
________
0.996
-0.064
0.360
0.623
0.659
0.281
0.358
-0.089
-0.024
-0.019
0.512
0.068
0.432
-0.039
0.691
0.198
-0.116
0.880
-0.904
0.153
3
________
-0.015
0.196
0.093
0.646
1.082
0.459
0.439
0.158
-0.111
0.067
1
2
3
PROMAX FACTOR CORRELATIONS
1
2
________
________
1.000
0.370
1.000
0.746
0.197
MPlus software
3
________
1.000
Exploratory factor analysis with binary variables
32
MPlus reads raw data
write outfile = 'h:\icpsr2003\Week4Examples\Mplus\catmiss.dat' /region
v166 v167 v168 v169 v170 v171 v172 v173 v174 v175 v356 v355
v353 (14F3.0).
- Must use WRITE command in SPSS (or PUT command in SAS) to
write raw data to file.
- Initially, listwise delete, though MPlus will handle missing data
33
Latent class model using MPlus
TITLE: latent class model #1
DATA:
FILE IS H:\ICPSR2003\Week4Examples\MPlus\Categor.dat
VARIABLE:
NAMES ARE REGION V166-V175 EDUC AGE SEX;
USEV = V166-V169;
CLASSES = C(2);
CATEGORICAL = V166-V169;
ANALYSIS:
TYPE = MIXTURE;
MITERATIONS=100;
MODEL:
%OVERALL%
[v166$1*-1 V167$1*1 V168$1*1 V169$1*1];
%c#2%
[V166$1*-2 V167$1*0 v168$1*0 v169$1*0];
OUTPUT:
TECH8;
34
Latent class model using MPlus
Chi-Square Test of Model Fit for the Latent Class Indicator
Model Part
Pearson Chi-Square
Value
Degrees of Freedom
P-Value
72.161
6
0.0000
Likelihood Ratio Chi-Square
Value
Degrees of Freedom
P-Value
77.561
6
0.0000
35
Latent class model using MPlus
FINAL CLASS COUNTS AND PROPORTIONS OF TOTAL SAMPLE SIZE
BASED ON ESTIMATED POSTERIOR PROBABILITIES
Class 1
Class 2
540.13069
1833.86931
0.22752
0.77248
CLASSIFICATION OF INDIVIDUALS BASED ON THEIR MOST LIKELY CLASS
MEMBERSHIP
Class Counts and Proportions
Class 1
Class 2
557
1817
0.23463
0.76537
36
Latent class model using MPlus
LATENT CLASS INDICATOR MODEL PART
Class 1
Thresholds
V166$1
V167$1
V168$1
V169$1
-0.640
1.317
0.141
2.244
0.115
0.142
0.121
0.200
-5.561
9.297
1.162
11.232
-6.577
-2.152
-5.320
-0.999
1.239
0.106
0.610
0.063
-5.307
-20.388
-8.718
-15.803
Class 2
Thresholds
V166$1
V167$1
V168$1
V169$1
LATENT CLASS REGRESSION MODEL PART
Means
C#1
-1.222
0.073
-16.713
37
LATENT CLASS INDICATOR MODEL PART IN PROBABILITY SCALE
Latent class model using MPlus
Class 1
V166
Category 1
Category 2
V167
Category 1
Category 2
V168
Category 1
Category 2
V169
Category 1
Category 2
0.345
0.655
0.026
0.026
13.266
25.164
0.789
0.211
0.024
0.024
33.406
8.952
0.535
0.465
0.030
0.030
17.735
15.403
0.904
0.096
0.017
0.017
52.210
5.536
Class 2
V166
Category 1
Category 2
V167
Category 1
Category 2
V168
Category 1
Category 2
V169
Category 1
Category 2
V166=God
V167=Life after death
V168=A soul
0.001
0.999
0.002
0.002
0.808
580.389
0.104
0.896
0.010
0.010
10.576
90.960
0.005
0.995
0.003
0.003
1.647
336.475
0.269
0.731
0.012
0.012
21.651
58.781
V169 = The devil
38
LATENT CLASS INDICATOR MODEL PART IN PROBABILITY SCALE
Class 1
V166
Category 1
Category 2
V167
Category 1
Category 2
V168
Category 1
Category 2
V169
Category 1
Category 2
Latent class model using MPlus
0.179
0.821
0.031
0.031
5.843
26.794
0.650
0.350
0.050
0.050
12.922
6.973
0.335
0.665
0.051
0.051
6.520
12.948
0.825
0.175
0.030
0.030
27.419
5.831
0.000
1.000
0.000
0.000
0.000
0.000
V166=God
0.082
0.918
0.010
0.010
7.818
87.908
V167=Life after death
0.000
1.000
0.000
0.000
0.000
0.000
V168=A soul
0.238
0.762
0.015
0.015
15.732
50.278
V169 = The devil
0.791
0.209
0.147
0.147
5.367
1.417
1.000
0.000
0.000
0.000
0.000
0.000
0.979
0.021
0.110
0.110
8.918
0.193
1.000
0.000
0.000
3 class model
Class 2
V166
Category 1
Category 2
V167
Category 1
Category 2
V168
Category 1
Category 2
V169
Category 1
Category 2
Class 3
V166
Category 1
Category 2
V167
Category 1
Category 2
V168
Category 1
Category 2
V169
Category 1
FINAL CLASS COUNTS AND PROPORTIONS OF TOTAL
SAMPLE SIZE
BASED ON ESTIMATED POSTERIOR PROBABILITIES
Class 1
Class 2
Class 3
565.88686
1697.27605
110.83709
0.23837
0.71494
0.04669
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40