A Hierarchical Spherical Radial Quadrature Algorithm for Multilevel GLMMS, GSMMS, and Gene Pathway Analysis

A HIERARCHICAL SPHERICAL RADIAL QUADRATURE ALGORITHM FOR MULTILEVEL GLMMs, GSMMs, and GENE PATHWAY ANALYSIS SEPTEMBER 2010 JACOB A. GAGNON, B.S., MASSACHUSETTS INSTITUTE OF TECHNOLOGY M.S., UNIVERSITY OF MASSACHUSETTS AT AMHERST Ph.D., UNIVERSITY OF MASSACHUSETTS AT AMHERST Directed by: Professor Anna Liu The first part of my thesis is concerned with estimation for longitudinal data using generalized semi-parametric mixed models and multilevel generalized linear mixed models for a binary response. Likelihood based inferences are hindered by the lack of a closed form representation. Consequently, various integration approaches have been proposed. We propose a spherical radial integration based approach that takes advantage of the hierarchical structure of the data, which we call the 2 SR method. Compared to Pinheiro and Chao’s multilevel Adaptive Gaussian quadrature [37], our proposed method has an improved time complexity with the number of functional evaluations scaling linearly in the number of subjects and in the dimension of random effects per level. Simulation studies show that our approach has similar to better accuracy compared to Gauss Hermite Quadrature (GHQ) and has better accuracy compared to PQL especially in the variance components.