new latent variable methods - statmodel.com€¦ · to learn more • watch the movie at mplus web...

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New Latent Variable Methods

Bengt Muthénbmuthen@ucla.edu

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To Learn More

• Watch the movie at Mplus Web Seminars: www.statmodel.com

• Attend the Thursday workshop

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General Latent Variable Modeling Framework

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General Latent Variable Modeling Framework

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General Latent Variable Modeling Framework

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

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

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

• Mplus team: Linda Muthen, Bengt Muthen, TihomirAsparouhov, Thuy Nguyen, Michelle Conn(see www.statmodel.com)

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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

mthcrs7 mthcrs8 mthcrs9 mthcrs10

math7 math8 math9 math10

f

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A Generalized Growth Model

i

s

stvc

mothed

homeres

female

mthcrs7 mthcrs8 mthcrs9 mthcrs10

math7 math8 math9 math10

f

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A Generalized Growth Model

i

s

stvc

mothed

homeres

female

mthcrs7 mthcrs8 mthcrs9 mthcrs10

math7 math8 math9 math10

fdropout

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i s

math7 math8 math9 math10

mthcrs7

Growth Modeling with a Latent Variable Interaction

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“How often have you had 6 or more drinks on one occasion during the last 30 days”

0 – Never1 – Once2 – 2 or 3 times3 – 4 or 5 times4 – 6 or 7 times5 – 8 or 9 times6 – 10 or more times

NLSY: Heavy DrinkingGeneral Population Sample

Ages 18-30 (cohort 64)

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NLSY HD89: Heavy Drinking at Age 25

0 1 2 3 4 5 6

HD89

0

50

100

150

200

250

300

350

400

450

500

550

600

650

700

750

800

Freq

uenc

y63% at zero

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Model Estimates Comparing Censored andCensored-Inflated Models of the NLSY HD

0.9320.1800.168HSDRP-2.1260.280-0.596FH123-3.0090.228-0.687ES2.9010.2200.637HISP6.5420.2251.472BLACK

-8.9680.175-1.574MALEII ON

3.3910.1950.662HSDRP1.9090.2080.398HSDRP2.0850.2590.540FH1233.0300.2830.858FH1230.8830.2130.188ES2.1730.2400.521ES

-1.0220.205-0.210HISP-2.5170.234-0.590HISP0.4140.1890.078BLACK-4.7990.203-0.973BLACK6.0330.2201.324MALE13.2840.1782.363MALE

I ONI ONEst/S.E.S.E.EstEst/S.E.S.E.Est

Censored-InflatedCensored

(prob of being zero)

N=1172, ages 18 - 30, centering at age 25

Censored logL = -7464 (30 parameters)Censored-inflated logL = -7304 (42 parameters)

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Discrete-Time Survival AnalysisWith A Random Effect (Frailty)

f

u4u3u2u1

x

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General Latent Variable Modeling Framework

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Latent Class Analysis

u1

c

u2 u3 u4

x

Item

u1

Item

u2

Item

u3

Item

u4

Class 2

Class 3

Class 4

Item Profiles

Class 1

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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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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 Trialof Depression Medication:

2-Class Growth Mixture Modeling

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

Ham

ilton

Dep

ress

ion

Rat

ing

Sca

le

05

1015

2025

30

Baseli

ne

Wash-i

n48

hours

1 wee

k

2 wee

ks

4 wee

ks

8 wee

ks

0

5

10

15

20

25

30

05

1015

2025

30

Baseli

ne

Was

h-in

48 ho

urs1 w

eek

2 wee

ks

4 wee

ks

8 wee

ks

0

5

10

15

20

25

30

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Growth Mixture Modeling: LSAY Math Achievement Trajectory Classes

and the Prediction of High School Dropout

Mat

h A

chie

vem

ent

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

69% 8% 1%Dropout:

7 8 9 10

4060

8010

0

Grades 7-107 8 9 10

4060

8010

0

Grades 7-107 8 9 10

4060

8010

0

Grades 7-10

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cx y f

y

25

f

u4

c

u3

u2

u1

u8

u7

u6

u5

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Non-Parametric Factor Analysis

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• Regular factor analysis- Normal random effects (continuous latent variables)- Numerical integration needed for ML- Normality is hard to check- Heavy computations with ML

• Non-parametric (semi-parametric) ML factor analysis- Non-parametric random effect modeling via latent

classes with class-invariant loadings and thresholds, and class-varying factor means

- Avoids distributional assumptions- Lighter computations- Better cut points (the best of both worlds: LCA-FA)

• Factor mixture analysis- Substantively meaningful classes- Class-varying parameters- Within-class variation possible

Factor Analysis Of Categorical Outcomes

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Factor Mixture - Non-Parametric Factor Modeling

y1 y2 y3 y4

c f

y5

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Damaged property Use other drugsFighting Sold marijuanaShoplifting Sold hard drugsStole < $50 “Con” someoneStole > $50 Take autoUse of force Broken into buildingSeriously threaten Held stolen goodsIntent to injure Gambling operationUse marijuana

Antisocial Behavior (ASB) Data (Continued)

Binary ItemsEFA Finds 3 factors: property offense, person offense, and drug offense

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.

F2

F1 F2

10-Class

15-Class

5-Class

Univariate Factor Distributions From Non-Parametric Factor Analysis

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10-Class Non-Parametric Factor AnalysisDistribution for F1 And F2

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Four Uses For U’s

• Categorical variable

• Indicator of being at zero in two-part modeling

• Event history indicator in survival analysis

• Missing data indicator

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Growth Mixture Modeling with Ignorable Missingness

Outcome

Escalating

Early Onset

NormativeAge

i

y2 y3 y4

s

x c

y1

u1 u2 u3 u4

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Outcome

Escalating

Early Onset

NormativeAge

Growth Mixture Modeling with Non-Ignorable Missingness as a Function of y

i

y2 y3 y4

s

x c

y1

u1 u2 u3 u4

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Outcome

Escalating

Early Onset

NormativeAge

Growth Mixture Modeling with Non-Ignorable Missingness as a Function of s

i

y2 y3 y4

s

x c

y1

u1 u2 u3 u4

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Outcome

Escalating

Early Onset

NormativeAge

Growth Mixture Modeling with Non-Ignorable Missingness as a Function of c

i

y2 y3 y4

s

x c

y1

u1 u2 u3 u4

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Outcome

Escalating

Early Onset

NormativeAge

Growth Mixture Modeling with Non-Ignorable Missingness as a Function of cu

i

y2 y3 y4

s

x cy

y1

u1 u2 u3 u4

cu

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General Latent Variable Modeling Framework

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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 sws

Student (Within)

y1 y2 y3 y4

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Twin1 Twin2

twin1covariates

twin paircovariates

twin2covariates

u1

c1

=

ACE

f1

u2

f2 c2

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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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General Latent Variable Modeling Framework

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To Learn More

• Watch the movie at Mplus Web Seminars: www.statmodel.com

• Attend the Thursday workshop