ASSESSING GOODNESS OF FIT FOR POISSON AND NEGATIVE BINOMIAL MODELS WITH LOW MEAN

ABSTRACT

Testing goodness of fit for a Poisson distribution is routine when the mean is sufficiently large; the scaled deviance G ² or Pearson's X ² statistic follow approximate chi-square distributions and perform the task well. When the mean is low, typically less than one, the approximations to chi-square distributions are poor. In this paper we explore the underlying reasons for this behaviour and present a practical resolution of the problem, in both single distribution and regression contexts. An extension to negative binomial models is also given. This research is motivated by a real example drawn from road accident modelling.

Keywords:

• Accident model
• Data grouping
• Pearson's X ²
• Regression
• Scaled deviance

ACKNOWLEDGMENTS

Shane Turner is thanked for his considerable encouragement and for providing the accident data set used throughout this paper. The author is also grateful to Harold Henderson and James Young for providing critical references. This research was in part funded, and fully stimulated, by Transfund New Zealand.