Semi‐parametric generalized linear model for binomial data with varying cluster sizes

Abstract

The semi‐parametric generalized linear model (SPGLM) proposed by Rathouz and Gao assumes that the response is from a general exponential family with unspecified reference distribution and can be applied to model the distribution of binomial event‐count data with a constant cluster size. We extend SPGLM to model response distributions of binomial data with varying cluster sizes by assuming marginal compatibility. The proposed model combines a non‐parametric reference describing the within‐cluster dependence structure with a parametric density ratio characterizing the between‐group effect. It avoids making parametric assumptions about higher order dependence and is more parsimonious than non‐parametric models. We fit the SPGLM with an expectation–maximization Newton–Raphson algorithm to the boron acid mouse data set and compare estimates with existing methods.