Over-dispersed Poisson versus negative binomial regression: How should we model count data

Over-dispersed Poisson and negative binomial regression models have equal numbers of parameters, and either could be used for over-dispersed count data.  While they often give similar results, there can be striking differences in estimating the effects of covariates.  We explain when and why such differences occur.  The variance of an over-dispersed Poisson model is a linear function of the mean while the variance of a negative binomial model is a quadratic function of the mean.  These variance relationships affect the weights in the iteratively reweighted least squares algorithm of fitting models to data.  Because the variance is a function of the mean, large and small counts get weighted differently in over-dispersed Poisson and negative binomial regression.  We provide an example using harbor seal counts from aerial surveys.  These counts are affected by date, time of day, and time relative to low tide.  We present results on a data set that showed a dramatic difference on the effect of date when using over-dispersed Poisson versus negative binomial regression.  This difference is described and explained in light of the different weighting used in each regression method. A general understanding of weighting can help ecologists choose between these two methods.