Statistical Modelling 3 (2003), 179191
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
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