Muthen LCA

8/18/2019 Muthen LCA

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Statistical Analysiswith Latent Variables

using Mplus Version 3

Bengt Muthen, UCLA

[email protected]


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Overview

• Overview of Mplus modeling capabilitiesn Emphasis on growth models

• Future seminars in this series:n Latent class analysis (Karen Nylund)

n Discrete-time survival analysis (Katherine Masyn)n Modeling with a preponderance of zeros (Frauke Kreuter)

• The Mplus web site

• Examples of Mplus analysesn

Real datan Monte Carlo simulations

• Recorded


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Program Background

• Inefficient dissemination of statistical methods:n Many good methods contributions from biostatistics,

psychometrics, etc are underutilized in practice

• Fragmented presentation of methods:

n Technical descriptions in many different journals

n Many different pieces of limited software

• Mplus: Integration of methods in one framework

n Easy to use: Simple language, graphicsn Powerful: General modeling capabilities


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General Latent Variable Modeling Framework


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General Latent Variable

Modeling Framework

• Muthén, B. (2002). Beyond SEM: General latentvariable modeling. Behaviormetrika, 29, 81-117

• Asparouhov & Muthen (2004). Maximum-likelihoodestimation in general latent variable modeling

• Muthen & Muthen (1998-2004). Mplus Version 3

• Mplus team: Linda Muthen, Bengt Muthen, Tihomir Asparouhov, Thuy Nguyen, Michelle Conn

(see www.statmodel.com)


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Continuous Latent Variables

• Factor analysis, structural equation modeling

- Constructs measured with multiple indicators

• Growth modeling

- Growth factors, random effects: random intercepts andrandom slopes representing individual differences of

development over time (unobserved heterogeneity)

• Survival analysis

- Frailties

• Missing data modeling


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femalemothedhomeresexpectlunchexpelarrest

droptht7hisp

black math7math10

hsdrop

femalemothedhomeresexpectlunchexpel

arrestdroptht7hisp

black math7

hsdrop

math10

Path Analysis with a Categorical Outcome

and Missing Data on a Mediator

Logistic Regression Path Analysis


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Continuous Latent Variables:

Two Examples

• Muthen (1992). Latent variable modeling in

epidemiology. Alcohol Health & Research World,

16, 286-292

- Blood pressure predicting coronary heart disease

• Nurses’ Health Study (Rosner, Willet & Spiegelman,

1989). Nutritional study of 89,538 women.

- Dietary fat intake questionnaire for everyone

- Dietary diary for 173 women for 4 1-week periods at 3-

month intervals


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Measurement Error in a Covariate

Blood Pressure (millimeters of mercury)

P r o p

o r t

i o n

W i t h C o r o n a r y

H e a r t

D i s e a s e

0.0

20 40 60 80 100 120

0.2

0.4

0.6

0.8

1.0

0

Without measurement error (latent variable)

With measurement error (observed variable)


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Measurement Error in a Covariate

y1

f

y2

y3


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y7 y8 y11 y12

f1

f2

y1

f4

y2

y5

y6

y4

y3

y10y9

f3

Structural Equation Model


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y7 y8 y11 y12

f1

f2

y1

f4

y2

y5

y6

y4

y3

y10y9

f3

Structural Equation Model with

Interaction between Latent Variables


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Growth Modeling

• Exploring the data using Mplus graphics

• Math achievement in grades 7 – 10 of the

Longitudinal Study of American Youth (LSAY)


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Growth Modeling with Time-Varying Covariates

i

s

stvc

mothed

homeres

female

mthcrs7 mthcrs8 mthcrs9 mthcrs10

math7 math8 math9 math10


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A Generalized Growth Model

i

s

stvc

mothed

homeres

female



f


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i

s

stvc

mothed

homeres

female



f


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i

s

stvc

mothed

homeres

female



f

dropout


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i s


mthcrs7

Growth Modeling with a

Latent Variable Interaction


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f1 f2

x0

i

u11-u14

x1 x2 x3 x4

f3 f4

u21-u24 u31-u34 u41-u44

s

Growth in Factors Measured

by Multiple Categorical Indicators


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Modeling with a Preponderance of Zeros

0 1 2 3 4 y 0 1 2 3 4 y

• Outcomes: non-normal continuous – count – categorical

• Censored-normal modeling

• Two-part (semicontinuous modeling): Duan et al. (1983),

Olsen & Shafer (2001)• Onset (survival) followed by growth: Albert & Shih (2003)

• Mixture models, e.g. zero-inflated (mixture) Poisson

(Roeder et al., 1999), censored-inflated, mover-stayer

latent transition models, growth mixture models


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Two-Part (Semicontinuous) Growth Modeling

u1 u2 u3 u4

iu

iy sy

y1 y2 y3 y4

x c

su

0 1 2 3 4 y


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Onset (Survival) Followed by Growth

u1 u2 u3 u4

f

iy sy

y1 y2 y3 y4

Event History

Growth

x c


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Categorical Latent Variables

• Mixture regression

• Latent class analysis

• Latent transition analysis• Missing data modeling


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Randomized Preventive Interventions and

Complier-Average Causal Effect Estimation

(CACE)

• Angrist, Imbens & Rubin (1996)

• Yau & Little (1998, 2001)

• Jo (2002)• Dunn et al. (2003)

• Compliance status observed for those invited for

treatment

• Compliance status unobserved for controls


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y

c

txx

CACE Mixture Modeling


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Latent Class Analysis

Item Profiles

I t e m u

1

I t e m u

2

I t e m u

3

I t e m u

4

Class 1

Class 2

Class 3

u1

c

u2 u3 u4

x


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Hidden Markov Modeling

c1

u1 u2 u3 u4

c2 c3 c4


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u21 u22 u23 u24u11 u12 u13 u14

c1 c2

c

Latent Transition Analysis

Transition Probabilities

0.6 0.4

0.3 0.7

c21 2

1 2

1

2

1

2

c1

c1

Mover Class

Stayer Class c2

(c=1)

(c=2)

0.90 0.10

0.05 0.95

Time Point 1 Time Point 2


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Loglinear Modeling of Frequency Tables

c1u1 c2

c3

u3

u2


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Combinations of

Continuous and Categorical Latent Variables

• Mixture CFA, SEM

• Generalized latent class analysis

• Second-order latent class analysis (twin modeling)

• Growth mixture modeling

• Longitudinal Complier-Average Causal Effect

(CACE) modeling in randomized preventive

interventions• Non-ignorable missing data modeling


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Factor Mixture - Non-Parametric Factor Modeling

y1 y2 y3 y4

c f

y5


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Latent Class Analysis

With A Random Effect

c

u1

u2

u3

u4

f


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Twin Modeling


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Growth Mixture Modeling

• Muthén, B. & Shedden, K. (1999). Finite mixture

modeling with mixture outcomes using the EM

algorithm. Biometrics, 55, 463-469.

• Muthén, B., Brown, C.H., Masyn, K., Jo, B., Khoo,

S.T., Yang, C.C., Wang, C.P., Kellam, S., Carlin, J.,

& Liao, J. (2002). General growth mixture modeling

for randomized preventive interventions.

Biostatistics, 3, 459-475.


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y1 y2 y3 y4

i s

x c u

Growth Mixture Modeling

Outcome

Escalating

Early Onset

Normative

Age


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A Clinical Trial

of Depression Medication:

Conventional Growth Modeling

H a m i l t o n D e p r e s s i o n R a t i n g S c a

l e

0

5

1 0

1 5

2 0

2 5

3 0

B a s e l i n e

W a s h

- i n

4 8 h o u r s

1 w e e k

2 w e e k s

4 w e e k s

8 w e e k s

0

5

10

15

20

25

30

0

5

1 0

1 5

2 0

2 5

3 0

B a s e l i n e

W a s h

- i n

4 8 h o u r s

1 w e e k

2 w e e k s

4 w e e k s

8 w e e k s

0

5

10

15

20

25

30

Placebo Group Medication Group

A Clinical Trial


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A Clinical Trial

of Depression Medication:

2-Class Growth Mixture Modeling

Placebo Non-Responders, 55% Placebo Responders, 45%

H a m i l t o n D e p r e s s i o n R a t i n g S c a

l e

0

5

1 0

1 5

2 0

2 5

3 0

B a s e l i n e

W a s h - i n

4 8 h o u r s

1 w e e k

2 w e e k s

4 w e e k s

8 w e e k s

0

5

10

15

20

25

30

0

5

1 0

1 5

2 0

2 5

3 0

B a s e l i n e

W a s h

- i n

4 8 h o u r s

1 w e e k

2 w e e k s

4 w e e k s

8 w e e k s

0

5

10

15

20

25

30


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Growth Mixture Modeling in Mplus

• Mplus input for LSAY math achievement


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Growth Mixture Modeling:

LSAY Math Achievement Trajectory Classes

and the Prediction of High School Dropout

M a t h A c h i e v e m e

n t

Poor Development: 20% Moderate Development: 28% Good Development: 52%

69% 8% 1%Dropout:

7 8 9 10

4 0

6 0

8 0

1 0 0

Grades 7-107 8 9 10

4 0

6 0

8 0

1 0 0

Grades 7-107 8 9 10

4 0

6 0

8 0

1 0 0

Grades 7-10


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Longitudinal CACE,

Non-Ignorable Missing Data

• Yau & Little (2001). Inference for the complier-averagecausal effect from longitudinal data subject to noncomplianceand missing data, with application to a job training assessmentfor the unemployed. Journal of the American Statistical

Association, 96, 1232-1244.

• Frangakis & Rubin (1999). Addressing complications ofintention-to-treat analysis in the combined presence of all-or-none treatment-noncompliance and subsequent missingoutcomes. Biometrika, 86, 365-379.

• Muthén, Jo, & Brown (2003). Comment on the Barnard,Frangakis, Hill & Rubin article, Principal stratificationapproach to broken randomized experiments: A case study ofschool choice vouchers in New York City. Journal of theAmerican Statistical Association, 98, 311-314.


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Growth Mixture Modeling with Non-Ignorable

Missingness as a Function of Latent Variables

i

y2 y3 y4

s

x

c

y1

u1 u2 u3 u4

Outcome

Escalating

Early Onset

Normative

Age


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i

y2 y3 y4

s

x

c

y1

u1 u2 u3 u4

Outcome

Escalating

Early Onset

Normative

Age


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i

y2 y3 y4

s

x

c

y1

u1 u2 u3 u4

Outcome

Escalating

Early Onset

Normative

Age


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Multilevel Modeling with

Continuous and Categorical Latent Variables

• Multilevel regression

• Multilevel CFA, SEM

• Multilevel growth modeling

• Multilevel discrete-time survival analysis

• Multilevel regression mixture analysis (CACE)

• Multilevel latent class analysis

• Multilevel growth mixture modeling


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BetweenWithin

m92

s1

s2

mean_ses

catholic

per_adva

private

s1

s2

stud_ses

female

m92


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Two-Level Factor Analysis with Covariates

fb

y2

w

y1

y3

y4

y5

y6

Between

x1 fw1 y2

x2 fw2

y1

y3

y4

y5

y6

Within


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Multilevel Modeling with a

Random Slope for Latent Variables

s

ib

sb

w

School (Between)

iw sw

s

Student (Within)

y1 y2 y3 y4


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Two-Level CACE Mixture Modeling

c

x

y

c

w

y

tx

Individual level

(Within)

Cluster level

(Between)

Class-varying


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Two-Level Latent Class Analysis

c

u2 u3 u4 u5 u6u1

x

f

c#1

w

c#2

Within Between


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High SchoolDropout

Female

Hispanic

Black

Mother’s Ed.

Home Res.

Expectations

Drop Thoughts

Arrested

Expelled

c

i s

Math7 Math8 Math9 Math10

ib sb

School-Level Covariates

cb hb

Multilevel Growth Mixture Modeling


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Monte Carlo Simulations in Mplus

• Data generation, analysis, and results summariesacross replications

• Studies of tests of model fit, parameter estimation,standard errors, coverage, and power as a function of

model variations, parameter values, and sample size• Model Population, Model Missing, Model for

analysis

• Full modeling framework available: continuous and

categorical latent variables, multilevel data, differenttypes of outcomes


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References

• See the Mplus web site www.statmodel.com


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