ICPSR General Structural Equation Models

1
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

2
Article discussion

• Reexamination and Extension of Kleine, Llein and
  Kermans Social Identity Model of Mundane
  Consumption the Mediating Role of the Appraisal
  Process
• J. Of Consumer Research, 28, 2002, 659-660.

3
Article discussion

• Reexamination and Extension of Kleine, Llein and
  Kermans 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 S1 S2 (pgt.50)
• Tennis players, 68 response, listwise N213 vs.
  N of 318.
• Aerobics, 73 response, listwise N 329 vs. N of
  359

4

• Reexamination and Extension of Kleine, Llein and
  Kermans 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 lt.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
  Xsq15605, df373 vs. 1541.6 df370 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?

5
A
b
E
D
C
The chi-square value reflects, among other
things, the restrictions in this model, eg. A?E
coefficient 0.
6
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. EA
test.
7
True model
8
INTERACTIONS IN LATENT VARIABLE STRUCTURAL
EQUATION MODELS

• Y b0 b1 X1 b2 X2 b3 (X1X2) e
• If X is categorical multiple group modeling
• If X is continuous more complicated
• Categorical can also model as dummy variables.

9
Interactions

• Easiest case X1 is 0/1
• X2 ix 0/1
• Options 1. Manually construct X3X1X2 outside
  SEM software, estimate model with X1,X2,X3
  exogenous. Test for interaction fix regression
  coefficient for X3 to 0.
• 2. Create two groups X10 and X11. In each
  group, X2 as exogenous variable. Test for
  interaction would be H0 gamma1 gamma2.
• Extensions for X1, X2 gt2 categories
  straightfoward (more groups/dummy variables)

10
Interactions

• Option 3 Model as a 4-group problem.
• X1
• 1 0
• X2 1 gr1 gr2
• 0 gr3 gr4
• AL10 al2, al3,al4 parameters to be
  estimated.
• Main effects model (no interaction) would allow
  for al2?al3 ?al4 but pattern of differences
  would be constrained such that..

11
Interactions

• Model as a 4-group problem.
• X1
• 1 0
• X2 1 gr1 gr2
• 0 gr3 gr4
• AL10 al2, al3,al4 parameters to be
  estimated.
• Main effects model (no interaction) would allow
  for al2?al3 ?al4 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
• Al1 Al2 al3- al4
• 0 al2 al3 al4
• Al2 al4-al3 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
12
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 gamma1 Ksi zeta
• Group 2 Eta gamma2 Ksi zeta
• Test for interaction H0 gamma1 gamma2
• Case 2 Two continuous single indicator
  variables
• Also somewhat straightforward
• Create single-indicator X3 X2X1
• 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

13
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

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 nx6 nk2 lxfu,fi ph-sy,fr tdsy
• Va 1.0 lx 1 1 lx 4 2
• Fr lx 2 1 lx 3 1 lx 5 2 lx 6 2
• PSNewfile.psf
• OU

15
Latent variable interactions

• Use SEM to estimate 2 factor model, save latent
  variable scores (analogous to factor scores),
  then use these
• In LISREL
• Mo nx6 nk2 lxfu,fi ph-sy,fr tdsy
• Va 1.0 lx 1 1 lx 4 2
• Fr lx 2 1 lx 3 1 lx 5 2 lx 6 2
• PSNewfile.psf
• OU
• LISREL documentation suggests that a simple
  regression can be estimated in PRELIS
• Synewfile.psf
• ne interksi1ksi2
• rg y on ksi1 ksi2 ksi1ksi2
• ou

16
Latent variable interactions

• LISREL documentation suggests that a simple
  regression can be estimated in PRELIS
• Synewfile.psf
• ne interksi1ksi2
• rg y on ksi1 ksi2 ksi1 ksi2
• ou
• . But it should also be possible to a) construct
  inter (ksi1ksi2) 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 MEML FS MI ND4
• .. And use factor score regressions to compute
  estimated factor scores in any stat package
  (incl. PRELIS)

17
Example

• INTERACTION MODEL WITH INTERACTION TERM CREATED
  EXTERNALLY
• SINGLE INDICATORS FOR EXOGENOUS LVS INVOLVED IN
  INTERACTION
• DA NO1111 NI10 MACM
• CM FIG\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 ny3 ne1 LYFU,FI PSSY,FR TESY c
• nx7 nk7 fixedx gafu,fr
• va 1.0 ly 1 1
• fr ly 2 1 ly 3 1
• ou meml se tv mi sc

18
Example

• LISREL Estimates (Maximum Likelihood)
  
• LAMBDA-Y
• ETA 1
• --------
• v125 1.00
• v127 1.34
• (0.24)
• 5.59
• v130 0.65
• (0.11)
• 5.74
• GAMMA

Dep var inequality atts (high score ? more
individual effort)
Lv1relig. Lv2econ. status
19
Kenny-Judd model

• Typically, literature (e.g., Kenny-Judd, 1984
  Hayduk, 1987) starts with 2-indicator example (2
  LVs each with 2 indicators).
• Ksi1
• Ksi2 Ksi1Ksi2 (interaction term)
• Indicators Ksi1 x1
• x2
• Ksi2 x3
• x4
• Possible product terms
• x1x3 x1x4
• x2X3 X2x4
• Kenny-Judd model use 4 product terms but Joreskog
  and Yang show that the model can be constructed
  with 1 product term.

20
Kenny-Judd model

• Typically, literature (e.g., Kenny-Judd, 1984
  Hayduk, 1987) starts with 2-indicator example (2
  LVs each with 2 indicators).
• Ksi1
• Ksi2 Ksi1Ksi2 (interaction term)
• Indicators Ksi1 x1
• x2
• Ksi2 x3
• x4
• Possible product terms
• x1x3 x1x4
• x2X3 X2x4
• 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 xs are normal (ADF
  estimation often recommended if sample size
  acceptable)

21
Kenny-Judd model
22
Kenny-Judd model
alpha1 term
23
Kenny-Judd model, mod.
INTERACTION MODEL KENNY JUDD MODIFICATION
(JORESKOG AND YANG) ONE INTERACTION INDICATOR
3 INDICATORS PER L.V. DA NO1111 NI22 CM
FIG\ICPSR2000\INTERACTIONS\INT5c.COV FU FO
(22F20.11) ME FIG\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 ny3 ne1 NX11 NK7 LYFU,FI
PSSY,FR C TESY TXFR KAFI C LXFU,FI
GAFU,FR PHSY,FR TDSY ALFI TYFR 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)2TD(3,3)
TX(4)2TD(1,1) PH(1,1)TX(4) C
PH(2,2)TX(1) TD(1,1)TD(4,4) OU MEML SE TV
ND3 ADoff
24
Kenny-Judd model, modified Joreskog/Yang
Parameter Specifications LAMBDA-Y
ETA 1 --------
v125 0 v127 1 v130
2 LAMBDA-X
KSI 1 KSI 2 KSI 3 KSI 4 KSI 5
KSI 6 -------- --------
-------- -------- -------- --------
v181 0 0 0 0
0 0 v9 3
0 0 0 0 0
v190 4 0 0
0 0 0 v221 0
0 0 0 0
0 v226 0 5 0
0 0 0 v227 0
6 0 0 0
0 relinc3 Constr'd Constr'd 0
0 0 0 sex
0 0 0 0 0
0 race 0 0 0
0 0 0 v217
0 0 0 0 0
0 v216 0 0
0 0 0 0
25
Kenny-Judd model, modified Joreskog/Yang
GAMMA KSI 1 KSI 2
KSI 3 KSI 4 KSI 5 KSI 6
-------- -------- -------- --------
-------- -------- ETA 1 -0.023
-0.003 -0.008 0.209 -0.324
0.051 (0.009) (0.015) (0.004)
(0.098) (0.125) (0.024)
-2.557 -0.198 -1.984 2.130
-2.593 2.094 GAMMA
KSI 7 -------- ETA 1
0.080 (0.029)
2.735
26
Latent class models

• Basic parameters
• Latent class probabilities
• Conditional probabilities (given one is in latent
  class A, what are the probabilities that one will
  be in cat i of indicator j? probs sum to 1.0).
• Parameter constraints are possible (in some
  cases, needed for identification).

27
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 .000000 1.000000 .450000 .550000
  .000000 .000000 .000000
• 1.000000 .250000 .750000 .000000 .000000
  .000000 1.000000
• 1.000000 .000000 .000000 .000000 .000000
  .300000 .350000 .350000
• .000000 .500000 .250000 .250000 1.000000
  .000000 .000000 .000000
• .000000 .800000 .200000 .000000
• 1.000000 .000000 .000000 .000000 .020000
  .370000 .400000 .310000
• .060000 .540000 .300000 .100000 1.000000
  .000000 .000000 .000000
• .000000 .900000 .100000 .000000
• 1.000000 .000000 .000000 .000000 .600000
  .400000 1.000000 .000000

28
A latent class model

• Software MLLSA
• ITERATION STEPS
• DEVIATION .00306576 ITERATION 10
• DEVIATION .00078193 ITERATION 20
• DEVIATION .00041910 ITERATION 30
• DEVIATION .00024801 ITERATION 40
• DEVIATION .00015106 ITERATION 50
• DEVIATION .00009318 ITERATION 60
• DEVIATION .00005788 ITERATION 70
• DEVIATION .00004791 ITERATION 74
• -------------------------------------------------
  ------------------------------
• -------------------------------------------------
  ------------------------------
• FINAL LIKELIHOOD RATIO CHI-SQUARE
  155.032400

29
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

• FINAL LATENT CLASS PROBABILITIES
• .627384 .110530 .160754 .018552 .082779

30
Latent class model

• ASSIGNMENT OF RESPONDENTS TO LATENT CLASS
• CELL OBSERVED EXPECTED ASSIGN TO CLASS
  MODAL PROBABILITY
• 1 .00 .00 1
  .0000
• 2 .00 .00 1
  .0000
• 3 2401.00 2401.00 1
  1.0000
• 4 .00 .00 1
  .0000
• 5 .00 .00 1
  .0000
• 6 42.00 19.00 3
  1.0000
• 7 .00 .00 1
  .0000
• 8 .00 .00 1
  .0000
• 9 9.00 17.20 3
  1.0000
• 10 .00 .00 1
  .0000
• 11 .00 .00 1
  .0000

31
MPlus software

• See director /Week4Examples/MPlus

TITLE categorical 1 DATA FILE IS
H\ICPSR2003\Week4Examples\MPlus\Categor.dat VARIA
BLE 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
32
MPlus software
VARIMAX ROTATED LOADINGS
1 2 3
________ ________ ________ V166
0.853 0.127 0.427 V167
0.488 0.693 0.397 V168
0.655 0.408 0.406 V169
0.533 0.019 0.753 V170
0.370 0.041 0.993 V171
0.626 0.192 0.662 V172
0.531 0.071 0.598 V173
0.693 0.336 0.473 V174
0.002 0.836 -0.080 V175
-0.739 -0.019 -0.330
PROMAX ROTATED LOADINGS 1
2 3 ________
________ ________ V166 0.996
-0.064 -0.015 V167 0.360
0.623 0.196 V168 0.659
0.281 0.093 V169 0.358
-0.089 0.646 V170 -0.024
-0.019 1.082 V171 0.512
0.068 0.459 V172 0.432
-0.039 0.439 V173 0.691
0.198 0.158 V174 -0.116
0.880 -0.111 V175 -0.904
0.153 0.067 PROMAX FACTOR
CORRELATIONS 1 2
3 ________ ________
________ 1 1.000 2
0.370 1.000 3 0.746
0.197 1.000
Exploratory factor analysis with binary variables
33
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

34
Latent class model using MPlus

• TITLE latent class model 1
• DATA
• FILE IS H\ICPSR2003\Week4Examples\MPlus\Categ
  or.dat
• VARIABLE
• NAMES ARE REGION V166-V175 EDUC AGE SEX
• USEV V166-V169
• CLASSES C(2)
• CATEGORICAL V166-V169
• ANALYSIS
• TYPE MIXTURE
• MITERATIONS100
• MODEL
• OVERALL
• v1661-1 V16711 V16811 V16911
• c2
• V1661-2 V16710 v16810 v16910
• OUTPUT
• TECH8

35
Latent class model using MPlus
Chi-Square Test of Model Fit for the Latent Class
Indicator Model Part Pearson
Chi-Square Value
72.161 Degrees of Freedom
6 P-Value
0.0000 Likelihood Ratio
Chi-Square Value
77.561 Degrees of Freedom
6 P-Value
0.0000
36
Latent class model using MPlus
FINAL CLASS COUNTS AND PROPORTIONS OF TOTAL
SAMPLE SIZE BASED ON ESTIMATED POSTERIOR
PROBABILITIES Class 1 540.13069
0.22752 Class 2 1833.86931
0.77248 CLASSIFICATION OF INDIVIDUALS BASED ON
THEIR MOST LIKELY CLASS MEMBERSHIP Class Counts
and Proportions Class 1 557
0.23463 Class 2 1817
0.76537
37
Latent class model using MPlus
LATENT CLASS INDICATOR MODEL PART Class 1
Thresholds V1661 -0.640 0.115
-5.561 V1671 1.317 0.142
9.297 V1681 0.141 0.121
1.162 V1691 2.244 0.200
11.232 Class 2 Thresholds V1661
-6.577 1.239 -5.307 V1671
-2.152 0.106 -20.388 V1681
-5.320 0.610 -8.718 V1691
-0.999 0.063 -15.803 LATENT CLASS
REGRESSION MODEL PART Means C1
-1.222 0.073 -16.713
38
Latent class model using MPlus
LATENT CLASS INDICATOR MODEL PART IN PROBABILITY
SCALE Class 1 V166 Category 1
0.345 0.026 13.266 Category 2
0.655 0.026 25.164 V167 Category 1
0.789 0.024 33.406 Category 2
0.211 0.024 8.952 V168 Category
1 0.535 0.030 17.735 Category
2 0.465 0.030 15.403 V169
Category 1 0.904 0.017 52.210
Category 2 0.096 0.017 5.536
Class 2 V166 Category 1 0.001
0.002 0.808 Category 2 0.999
0.002 580.389 V167 Category 1
0.104 0.010 10.576 Category 2
0.896 0.010 90.960 V168 Category 1
0.005 0.003 1.647 Category 2
0.995 0.003 336.475 V169 Category
1 0.269 0.012 21.651 Category
2 0.731 0.012 58.781
V166God V167Life after death V168A soul V169
The devil
39
Latent class model using MPlus
LATENT CLASS INDICATOR MODEL PART IN PROBABILITY
SCALE Class 1 V166 Category 1
0.179 0.031 5.843 Category 2
0.821 0.031 26.794 V167 Category 1
0.650 0.050 12.922 Category 2
0.350 0.050 6.973 V168 Category
1 0.335 0.051 6.520 Category
2 0.665 0.051 12.948 V169
Category 1 0.825 0.030 27.419
Category 2 0.175 0.030 5.831
Class 2 V166 Category 1 0.000
0.000 0.000 Category 2 1.000
0.000 0.000 V167 Category 1
0.082 0.010 7.818 Category 2
0.918 0.010 87.908 V168 Category 1
0.000 0.000 0.000 Category 2
1.000 0.000 0.000 V169 Category
1 0.238 0.015 15.732 Category
2 0.762 0.015 50.278 Class 3
V166 Category 1 0.791 0.147
5.367 Category 2 0.209 0.147
1.417 V167 Category 1 1.000 0.000
0.000 Category 2 0.000 0.000
0.000 V168 Category 1 0.979
0.110 8.918 Category 2 0.021
0.110 0.193 V169 Category 1
1.000 0.000 0.000 Category 2
0.000 0.000 0.000
3 class model
V166God V167Life after death V168A soul V169
The devil
FINAL CLASS COUNTS AND PROPORTIONS OF TOTAL
SAMPLE SIZE BASED ON ESTIMATED POSTERIOR
PROBABILITIES Class 1 565.88686
0.23837 Class 2 1697.27605
0.71494 Class 3 110.83709
0.04669
40
Last slide

• Email baer_at_uvic.ca
• Put ICPSR in subject heading