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Danielle Dean, M.A. Department of Psychology University of North Carolina, Chapel Hill Nisha Gottfredson, Ph.D. Transdisciplinary Prevention Research Center Duke University Modern Modeling Methods May 2012 Novel Applications of Mixture Models to Social Science Data

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Introduction: Indirect Mixture Applications Presented by Nisha Gottfredson

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Mixture Models Enable Analysts to Relax Parametric Assumptions Marron and Wand (1992) showed that it is possible to replicate nearly any univariate distribution using a finite mixture of normal distributions Variable means, variances, and proportions Bimodal density 2 normal distributions Strongly skewed 8 normal distributions

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Multivariate Mixture Models Each class g has a class-specific mean vector, covariance matrix, and mixing proportion Examples include Growth Mixture Models (mixtures of latent curve models; Verbeke & Lesaffre, 1996; Muthn & Shedden, 1999) and Structural Equation Mixture Models (Jedidi, Jagpal, & Desarbo, 1997; Dolan & van der Maas, 1998)

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Direct versus Indirect Applications McLachlan & Peel (2000) distinguished between direct and indirect applications of finite mixture modeling Direct applications more common in social sciences Users believe that there is qualitative heterogeneity in the population Interpret class-specific estimates Aim is to recover true groupings Indirect applications more common in statistics Analysts uncomfortable with parametric assumptions Aggregate over class-specific estimates True groupings do not exist

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Direct versus Indirect Interpretation is in the Eye of the Beholder There is no empirical way to distinguish between groups as truth and groups as statistical convenience AIC/BIC almost always suggest that classes improve model fit (Bauer & Curran, 2003; 2004; Bauer, 2007) Non-normality of variables Non-linearity When in doubt, indirect interpretation is more robust than direct interpretation (Sterba et al., 2012)

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Overview of Talks Indirect applications of mixture models Survival Mixture Models with study on multiple survival processes during transition to adulthood Shared Parameter Mixture Models to handle non-randomly missing data in longitudinal studies

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Survival Mixture Models for Simultaneously Capturing Multiple Survival Processes An Application with Data on Transitioning to Adulthood Presented by Danielle Dean

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Multiple Survival Processes How may we analyze multiple non-repeatable events which may occur at the same point in time for an individual? E.g. age of onset of different drugs E.g. age of transition to multiple roles PersonEventAge 18Age 19Age 20Age 21 1Parent001. 1Marriage001. 1College0000 2Parent00.. 2Marriage1... 2College00..

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Survival Analysis Survival or event history models Multivariate survival analysis AliveDead Never hospitalized Hospitalized due to depression Hospitalized due to anxiety Hospitalized due to other disorders MarriedSingleCohabiting Not a parent Birth of Child

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Multiple Survival Processes Non-repeatable events which may occur at the same point in time Many researchers currently run a separate survival analysis for each event process but dont analyze how the events are related Not a parentParentNot marriedMarried No College Education College Education Never worked full-time Worked full- time

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Univariate Survival Analysis One non-repeatable event Three main functions: survival, lifetime distribution, and hazard

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Individual Week 01234 Event time 100001 4 2001.. 2 3000.. ?? 401... 1 50000. Univariate Survival Analysis E.g. therapy completion: Model the hazard: Compute the lifetime distribution and survival function

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Multiple non-repeatable events How are the events related? Distribution of risk for multiple events is of unknown form Model assumes the population is composed of a finite number of latent classes in order to parsimoniously describe the underlying distribution of risk (multiple event version of model presented by Muthn & Masyn, 2005)

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Purpose of model Parsimoniously describe the underlying distribution of risk for multiple events, without assuming a specific mathematical form for the distribution Purpose is to draw attention to differences in the causes and consequences of different pathways (rather than to suggest the population is composed of literally distinct groups) In spirit of indirect application, can compute model-implied functions weighting over latent classes to evaluate the effects of covariates E.g. model implied risk for multiple events for males versus females, controlling for other covariates

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Transitions to Adulthood Life course theory Order and timing of social roles Meaning of a social role E.g., working parent Identify general structures of the life course

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Methods National Longitudinal Study of Adolescent Health N = 15,701 4 events: Parenthood, Full-time work, Marriage, College Ages 18-30

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Model Fit using Mplus 6.12 Identify pathways through the life course (Model 1) Influence of covariates on pathways (Model 2) Gender, Race, Parental Education y 1,18 y 1,30 y 4,18 y 4,30 C X...

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Model Selection

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Substantively redundant latent class in 6 class solution Lifetime Distributions:

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5 class solution (hazards)

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5 class solution (lifetime dist.)

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Order / Timing of Events Median event time Classes aggregate back to sample observed functions, average squared residual