Likelihood inference for exchangeable continuous data with covariates and varying cluster sizes; use of the Farlie–Gumbel–Morgenstern model

This article investigates the Farlie–Gumbel–Morgenstern class of models for exchangeable continuous data. We show how the model specification can account for both individual and cluster level covariates, we derive insights from comparisons with the multivariate normal distribution, and we discuss maximum likelihood inference when a sample of independent clusters of varying sizes is available. We propose a method for maximum likelihood estimation which is an alternative to direct numerical maximization of the likelihood that sometimes exhibits non-convergence problems. We describe an algorithm for generating samples from the exchangeable multivariate Farlie–Gumbel–Morgenstern distribution with any marginals, using the structural properties of the distribution. Finally, we present the results of a simulation study designed to assess the properties of the maximum likelihood estimators, and we illustrate the use of the FGM distributions with the analysis of a small data set from a developmental toxicity study.