Methods for Correlated Binomial and Multinomial Data with Multilevel Structures

Abstract

Mixed models and marginal models are commonly applied to correlated data such as longitudinal data. For non-Gaussian data with multilevel structures, generalized linear mixed models (GLMM) and generalized estimating equations (GEE) models are applicable. But these models require a high-dimensional integration over the-random effects or a complicated correlation structure. For GLMM, an efficient Adaptive Gaussian Quadrature (AGQ) algorithm is developed for multilevel data and it is consistent. AGQ solves the bias problems in Penalized Quasi-likelihood (PQL), Marginal Quasi-likelihood (MQL) and higher- order Laplace approximation. For GEE with mixed effects, the results are similar to MQL, while the advantage is the capability to model multilayer correlation structure with arbitrary dimension. Furthermore, when the fixed- effect covariate has difficulties to be modeled parametrically, kernel and spline smoothing are powerful to handle such situations. Under GEE, these approaches are equivalent. Without random effects, kernel GEE methods require one variance component and a correlation design. These can be embedded in paired estimating equations. Examples on binomial and multinomial data are presented to demonstrate these methods.