Statistical Modelling 3 (2003), 179–191

Negative binomial loglinear mixed models

James G. Booth, George Casella,
Department of Statistics,
University of Florida,
P.O. Box 118545,
Gainesville, FL 32611-8548
U.S.A.

Herwig Friedl,
Institute of Statistics,
Technical University Graz,
Graz,
Austria

James P. Hobert,
Department of Statistics,
University of Florida,
Gainesville, FL
U.S.A.

Abstract:

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 ist not realistic. Overdispersion with respect to the Poisson distribution can be modeled explicitely 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, the 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.

Keywords:

Monte Carlo EM; NLMIXED procedure; nonparametric maximum likelihood; overdispersion; random effects
 

Downloads:

Data and SAS code in zipped archive.


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