1 icpsr general structural equation models week 4 #4 (last class) interactions in latent variable...

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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

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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.

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Article discussion:

“Reexamination and Extension of Kleine, Llein and Kerman’s Social Identity Model of

Mundane Consumption: the Mediating Role of the Appraisal Process”


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

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“Reexamination and Extension of Kleine, Llein and Kerman’s Social Identity Model of Mundane Consumption: the Mediating Role of the Appraisal Process”


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?

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A b

C

ED

The chi-square value reflects, among other things, the restrictions in this model, eg. AE coefficient = 0.

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E b

C

AD

In this model, another set of restrictions is imposed (e.g., EA 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 AE vs. E—A test.

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True model

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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.

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Interactions

Easiest case: X1 is 0/1X2 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)

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Interactions

Option 3: Model as a 4-group problem.X11 0

X2 1 gr1 gr20 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…..

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InteractionsModel as a 4-group problem.X11 0

X2 1 gr1 gr20 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

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Interactions

Interactions involving continuous variables.

Case 1: One continuous (single or multiple indicator) and one categorical variableEASY: categorical variable becomes basis for grouping.

Group 1 Eta = gamma[1] Ksi + zetaGroup 2 Eta = gamma[2] Ksi + zetaTest for interaction: H0: gamma[1] = gamma[2]

Case 2: Two continuous single indicator variablesAlso somewhat straightforward:

Create single-indicator X3 = X2*X1

Case 3: Two continuous multiple indicator latent variablesThis 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

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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

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• 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

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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

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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

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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

lv1 lv2 interact sex race v217 -------- -------- -------- -------- -------- -------- ETA 1 -0.04 -0.21 0.85 0.22 -0.30 0.05 (0.06) (0.08) (0.45) (0.11) (0.13) (0.03) -0.65 -2.57 1.89 2.10 -2.27 1.75

GAMMA

v216 -------- ETA 1 0.09 (0.03) 2.92

Dep var = inequality att’s (high score “more individual effort”)

Lv1=relig. Lv2=econ. status

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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.

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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)

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Kenny-Judd model

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Kenny-Judd model

alpha=1 term

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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

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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

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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

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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).

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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 .000000 .000000 .600000 .400000 1.000000 .000000 .000000

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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 FINAL PEARSON CHI-SQUARE = 157.236800 INDEX OF DISSIMILARITY = .034417

-------------------------------------------------------------------------------

FINAL LATENT CLASS PROBABILITIES:

.627384 .110530 .160754 .018552 .082779

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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 .0000FINAL LATENT CLASS PROBABILITIES:

.627384 .110530 .160754 .018552 .082779

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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

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MPlus software

See director /Week4Examples/MPlus

TITLE: categorical #1DATA: FILE IS H:\ICPSR2003\Week4Examples\MPlus\Categor.datVARIABLE: 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

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MPlus software

Exploratory factor analysis with binary variables

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

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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

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Latent class model using MPlus

TITLE: latent class model #1DATA: FILE IS H:\ICPSR2003\Week4Examples\MPlus\Categor.datVARIABLE: 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;

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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

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FINAL CLASS COUNTS AND PROPORTIONS OF TOTAL SAMPLE SIZEBASED 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

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LATENT CLASS INDICATOR MODEL PART

Class 1

Thresholds V166$1 -0.640 0.115 -5.561 V167$1 1.317 0.142 9.297 V168$1 0.141 0.121 1.162 V169$1 2.244 0.200 11.232

Class 2

Thresholds V166$1 -6.577 1.239 -5.307 V167$1 -2.152 0.106 -20.388 V168$1 -5.320 0.610 -8.718 V169$1 -0.999 0.063 -15.803

LATENT CLASS REGRESSION MODEL PART

Means C#1 -1.222 0.073 -16.713

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Latent class model using MPlusLATENT 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=God

V167=Life after death

V168=A soul

V169 = The devil

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V166=God

V167=Life after death

V168=A soul

V169 = The devil

LATENT CLASS INDICATOR MODEL PART IN PROBABILITY SCALE Class 1


Class 2


Class 3


3 class model

FINAL CLASS COUNTS AND PROPORTIONS OF TOTAL SAMPLE SIZEBASED ON ESTIMATED POSTERIOR PROBABILITIES

Class 1 565.88686 0.23837 Class 2 1697.27605 0.71494 Class 3 110.83709 0.04669

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