1 / 29

Mixture & Multilevel Modeling

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

edda
Download Presentation

Mixture & Multilevel Modeling

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Mixture & Multilevel Modeling Shaunna Clark & RyneEstabrook NIDA Workshop – October 19, 2010

  2. 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

  3. 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

  4. What is Mixture Modeling Used to model unobserved heterogeneity by identifying different subgroups of individuals Ex: IQ, Religiosity

  5. Growth Mixture Modeling

  6. 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

  7. Linear Growth Model Diagram σ2Int,Slope σ2Slope σ2Int S I mInt 1 mSlope 2 0 1 3 4 1 1 1 1 1 x1 x2 x3 x4 x5 σ2ε1 σ2ε2 σ2ε3 σ2ε4 σ2ε5

  8. Linear GMM Model Diagram C σ2Int,Slope σ2Slope σ2Int S I mInt 1 mSlope 2 0 1 3 4 1 1 1 1 1 x1 x2 x3 x4 x5 σ2ε1 σ2ε2 σ2ε3 σ2ε4 σ2ε5

  9. GMM Example Profile Plot

  10. GMM example Profile Plot

  11. Growth Mixture Model Equations xitk = Interceptik + λtk*Slopeik + εitk for individual i at time t in class k εitk ~ N(0,σ)

  12. 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

  13. 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

  14. Global Vs Local Maximum Log Likelihood Log Likelihood Global Global Local Local Parameter Parameter

  15. Open Mx Example • Take it away Ryne!

  16. 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

  17. 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

  18. Multilevel Models

  19. 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

  20. . . .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

  21. 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)

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

  23. 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

  24. 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 A A C C E E PA PB

  25. 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

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

  27. 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

  28. Data Considerations • Multivariate – Wide • Multiple family members per row of data • Multilevel – Long • One individual per row of data

More Related