Mixture & Multilevel Modeling Shaunna Clark & Ryne Estabrook NIDA Workshop – October 19, 2010

Outline Mixture Models Multilevel 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

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

What is Mixture Modeling Used to model unobserved heterogeneity in a population by identifying different groups of individuals, called latent classes, that have similar observed response patterns (cross-sectional) or growth trajectory. Ex: Fish in a stream Task is to examine length of fish in a stream Two cohorts of fish – c=0 cohort of younger fish c = 1 cohort of older fish Two cohorts mix (hence mixture) to form the overall population Used to model unobserved heterogeneity by identifying different subgroups of individuals Ex: IQ, Religiosity

Growth Mixture Modeling

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

Linear Growth Model Diagram x1 x2 x3 x4 x5 1 I S mInt mSlope σ2Int σ2Int,Slope 2 3 4 σ2ε1 σ2ε2 σ2ε3 σ2ε4 σ2ε5 σ2Slope Model we saw in Nathan’s presentation. Quickly go over.

Linear GMM Model Diagram C σ2Int,Slope σ2Slope σ2Int S I 1 mInt mSlope 1 2 3 4 1 1 1 1 1 The change between the GMM and the previous model is the latent categorical or latent class variable, the circle with the c at the top. The circle with the c has arrows pointing toward the I and S indicating that the latent classes are based on difference in the intercept and slope of individuals – i.e. classifying individuals based on different trajectories. Here only have I, s, but can easily have a quadratic term or non-linear growth. x1 x2 x3 x4 x5 σ2ε1 σ2ε2 σ2ε3 σ2ε4 σ2ε5

GMM Example Profile Plot

GMM example Profile Plot

Growth Mixture Model Equations xitk = Interceptik + λtk*Slopeik + εitk for individual i at time t in class k εitk ~ N(0,σ) General GMM with intercept and slope – can add quadratic paramater

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

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

Global Vs Local Maximum Log Likelihood Parameter Global Local Log Likelihood Global Local Why need random starts mixtures can converge to local rather global solutions so try multiple sets of starting values and compare the LL, parameter estimates to make sure have the global solution Parameter

Open Mx Example Take it away Ryne!

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

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

Multilevel Models

What is Multilevel Data . . . Most methods assume individuals are independent Responses for one individual do not influence another individual’s responses Multilevel, or nested data, arise when individuals are not independent Ex: Twins in a family, students in a classroom Share common experiences

. . .And why we should Care When ignore nested structure, have underestimated standard errors Can lead to misinterpretation of the significance of model parameters Large body of literature about how to handle nested data Today, focus on multilevel techniques General multilevel texts: Raudenbush & Bryk, 2002; Snijders & Bosker, 1999

Multilevel Model Equation For 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)

Multilevel Model Equation extensions Can have additional levels Ex: Individuals within nuclear families with family Can be longitudinal Ex: Observations within individuals within families

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 + εij Fixed Effects Random Effects

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

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

Implementation of Multilevel models in open Mx OpenMx Discussion http://openmx.psyc.virginia.edu/thread/125 Discuss more tomorrow in Dynamical Systems talk

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

Data Considerations Multivariate – Wide Multilevel – Long Multiple family members per row of data Multilevel – Long One individual per row of data FAMID ZYG ALC_T1 ALC_T2 1 20 10 2 6 15 3 4 5 FAMID ZYG ALC 1 20 10 2 6 15 3 4 5