Negative binomial loglinear mixed models

The Poisson loglinear model is a common choice for explaining variability in counts. However, in many practical circumstances the restriction that the mean and variance are equal is not realistic. Overdispersion with respect to the Poisson distribution can be modeled explicitly by integrating with respect to a mixture distribution, and use of the conjugate gamma mixing distribution leads to a negative binomial loglinear model. This paper extends the negative binomial loglinear model to the case of dependent counts, where dependence among the counts is handled by including linear combinations of random effects in the linear predictor. If we assume that the vector of random effects is multivariate normal, then complex forms of dependence can be modelled by appropriate specification of the covariance structure. Although the likelihood function for the resulting model is not tractable, maximum likelihood estimates (and standard errors) can be found using the NLMIXED procedure in SAS or, in more complicated examples, using a Monte Carlo EM algorithm. An alternate approach is to leave the random effects completely unspecified and attempt to estimate them using nonparametric maximum likelihood. The methodologies are illustrated with several examples.