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

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.