new latent variable methods - statmodel.com€¦ · to learn more • watch the movie at mplus web...
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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
10
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