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MIXTURE & MULTILEVEL MODELINGShaunna Clark & Ryne Estabrook

NIDA Workshop – October 19, 2010

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OUTLINE Mixture Models

What is mixture modeling? Growth Mixture Model Open Mx Genetic Mixture Models Other Longitudinal Mixture Models

Multilevel Models What is multilevel data? Multilevel regression model Open Mx

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HOMOGENEITY VS. HETEROGENEITY Most models assume homogeneity

i.e. Individuals in a sample all follow the same model

What have seen so far today But not always the case

Ex: Sex, Age, Alcohol Use Trajectories

12 14 16 18 21 240

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10

15

20

25

Age

Num

ber

of D

rinks

Per

Wee

k

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WHAT IS MIXTURE MODELING

Used to model unobserved heterogeneity by identifying different subgroups of individuals

Ex: IQ, Religiosity

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GROWTH MIXTURE MODELING

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GROWTH MIXTURE MODELING (GMM) Muthén & Shedden, 1999; Muthén, 2001 Setting

A single item measured repeatedly Hypothesized trajectory classes Individual trajectory variation within class

Aims Estimate trajectory shapes Estimate trajectory class probabilities

Proportion of sample in each trajectory class Estimate variation within class

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LINEAR GROWTH MODEL DIAGRAM

σ2Slope

x1 x2 x3 x4 x5

1I SmInt mSlope

σ2Int

σ2Int,Slope

1 1 111 0 1 2 3 4

σ2ε1 σ2

ε2 σ2ε3 σ2

ε4 σ2ε5

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LINEAR GMM MODEL DIAGRAM

x1 x2 x3 x4 x5

1I S

C

mInt mSlope

σ2Slopeσ2

Intσ2

Int,Slope

1 1 111 0 1 2 3 4

σ2ε1 σ2

ε2 σ2ε3 σ2

ε4 σ2ε5

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GMM EXAMPLE PROFILE PLOT

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GMM EXAMPLE PROFILE PLOT

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GROWTH MIXTURE MODEL EQUATIONS

xitk = Interceptik + λtk*Slopeik + εitk

for individual i at time t in class kεitk ~ N(0,σ)

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LCGA VS. GMM LCGA – Latent Class Growth Analysis

Nagin, 1999; Nagin & Tremblay, 1999 Same as GMM except no residual variance on

growth factors No individual variation within class (i.e. everyone

has the same trajectory LCGA is a special case of GMM

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CLASS ENUMERATION Determining the number of classes Can’t use LRT Χ2

Not distributed as Χ2 due to boundary conditions (McLachlan & Peel, 2000)

Information Criteria: AIC (Akaike, 1974), BIC (Schwartz,1978) Penalize for number of parameters and sample

size Model with lowest value

Interpretation and usefulness Profile plot Substantive theory Predictive validity

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GLOBAL VS LOCAL MAXIMUM

Log Likelihood

Parameter

GlobalLocal

Parameter

Log Likelihood

GlobalLocal

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OPEN MX EXAMPLE Take it away Ryne!

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SELECTION OF MIXTURE GENETIC ANALYSIS WRITINGS Growth Mixture Model

Wu et al., 2002; Kerner and Muthen, 2009; Gillespie et al., (submitted)

Latent Class Analysis Eaves, 1993; Muthén et al., 2006; Clark, 2010

Additional References McLachlan, Do, & Ambroise, 2004

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OTHER LONGITUDINAL MIXTURE MODELS Survival Mixture

Multiple latent classes of individuals with different survival functions

Kaplan, 2004; Masyn, 2003; Muthén & Masyn, 2005

Longitudinal Latent Class Analysis Models patterns of change over time, rather than

functional growth form Lanza & Collins, 2006; Feldman et al., 2009

Latent Transition Analysis Models transition from one state to another over

time Ex: Drinking alcohol or not over time

Graham et al., 1991; Nylund et al., 2006

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MULTILEVEL MODELS

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WHAT IS MULTILEVEL DATA . . . Most methods assume individuals are

independentResponses for one individual do not

influence another individual’s responses Multilevel, or nested data, arise when

individuals are not independentEx: Twins in a family, students in a

classroomShare common experiences

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. . .AND WHY WE SHOULD CARE When ignore nested structure, have

underestimated standard errorsCan lead to misinterpretation of the

significance of model parameters Large body of literature about how to

handle nested dataToday, focus on multilevel techniquesGeneral multilevel texts:

Raudenbush & Bryk, 2002; Snijders & Bosker, 1999

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MULTILEVEL MODEL EQUATIONFor individual i in cluster j: Level One (Individual)

yij = β0j + β1j*xij + εij

Level Two (Twin Pair\Family)β0j = γ00 + γ01*wj + μ0j

β1j = γ10 + γ11*wj + μ1j

Where εitk ~ N(0,σ), μ ~ N(0,Ψ), Cov(ε, μ) = 0

xij is an individual level covariate (age, weight)wj is a cluster level covariate (maternal smoking)

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MULTILEVEL MODEL EQUATION EXTENSIONS Can have additional levels

Ex: Individuals within nuclear families with family Can be longitudinal

Ex: Observations within individuals within families

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MIXED EFFECTS VS. MULTILEVEL MODELING They are the same thing!!Multilevel Model Equation:

Level One (L1):yij = β0j + β1j*xij + εij

Level Two (L2):β0j = γ00 + μ0j

β1j = γ10 + μ1j

Mixed Model Equation:

Plug L2 into L1, some rearranging

yij = (γ00 + μ0j) + (γ10 + μ1j) *xij + εij

yij = γ00 + γ10*xij + μ0j + μ1j*xij + εijFixed Effects

Random Effects

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MULTILEVEL VS. MULTIVARIATE MODELING OF FAMILIES Today have dealt with multivariate analyses Multivariate

Model for all variables for each family member Family members can have different parameter

values Ex: different growth trajectories for parents vs. children

Only feasible when small number of family members Ex: twins, spouses

PA

A C E

PB

A C E

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MULTILEVEL MODELING OF FAMILIES Model for variation within individual and

between family members Members of a cluster are assumed

statistically equivalent i.e. Same model for each family member

Can handle various family structures Ex: Large pedigrees, families with differing

numbers of siblings Do not have to make arbitrary assignment of

family members (and checking whether assignment impacted estimates) Ex: Assigning twins to A and B

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IMPLEMENTATION OF MULTILEVEL MODELS IN OPEN MX OpenMx Discussion

http://openmx.psyc.virginia.edu/thread/125

Discuss more tomorrow in Dynamical Systems talk

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MULTILEVEL GENETIC ARTICLES General

Discuss how to extend ACDE model to twins and larger family pedigrees

Guo & Wang, 2002; McArdle & Prescott, 2005; Rabe-Hesketh, Skrondal, Gjessing, 2008

Longitudinal McArdle, 2006

Other Inclusion of measured genotypes: Van den Oord,

2001

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DATA CONSIDERATIONS Multivariate – Wide

Multiple family members per row of data

Multilevel – Long One individual per row of data

FAMID ZYG ALC_T1 ALC_T21 1 20 102 6 15 63 4 0 5

FAMID ZYG ALC1 1 201 1 102 6 152 6 63 4 03 4 5