Finite Mixtures of Nonlinear Mixed-Effects Models for Longitudinal Data

Abstract

Thesis (Ph.D.)--University of Rochester. School of Medicine & Dentistry. Dept. of Biostatistics & Computational Biology, 2017.Finite mixture models are increasingly used to model distributions in multivariate regression settings where outcomes and covariates exhibit heterogeneous relationships. When observations are collected repeatedly within subjects, for example longitudinally, these models must be adapted to account for potential dependencies. In this thesis, we consider a class of finite mixture regression models that is suitable for longitudinal data structures. The proposed model captures inter-subject variability and intra-subject dependencies with random effects and allows both mixing proportions and component densities to depend on covariates. We propose a new version of the stochastic approximation expectation maximization (SAEM) algorithm, which we call the Newton-Raphson SAEM (NR-SAEM) algorithm for maximum likelihood (ML) estimation. The implementation of our proposed algorithm requires sampling unobservable random effects from their posterior distributions using Markov Chain Monte Carlo (MCMC) methods, and reduces stochastic variability by estimating latent group indicators unconditional on random effects. The proposed method is demonstrated using both simulated data and real data of repeated hormone levels over a key period of the menstrual cycle. We compare the NR-SAEM algorithm to the standard expectation maximization (EM) and the mixture SAEM (MSAEM) algorithms. We find that the NR-SAEM algorithm performed comparably to the standard EM algorithm that converged to the ML estimators for finite mixture regression models with linear random effects, and it exhibited better convergence than the MSAEM algorithm in simulations. Application of our proposed model to the hormone level data suggests that women’s hormone trajectories over a key part of the menstrual cycle may follow more than one type of pattern, suggesting several subpopulations which differ with respect to the distributions of BMI and PFOS exposure