1 - 107 - 01416 - 9 - Regression Analysis of Count Data : Second Edition

This book is concerned with models of event counts. An event count refers to the number of times an event occurs, for example, the number of airline accidents or earthquakes. It is the realization of a nonnegative integer-valued random variable. A univariate statistical model of event counts usually specifies a probability distribution of the number of occurrences of the event known up to some parameters. Estimation and inference in such models are concerned with the unknown parameters, given the probability distribution and the count data. Such a specification involves no other variables, and the number of events is assumed to be independently identically distributed (iid). Much early theoretical and applied work on event counts was carried out in the univariate framework. The main focus of this book, however, is on regression analysis of event counts. The statistical analysis of counts within the framework of discrete parametric distributions for univariate iid random variables has a long and rich history (Johnson, Kemp, and Kotz, 2005). The Poisson distribution was derived as a limiting case of the binomial by Poisson (1837). Early applications include the classic study of Bortkiewicz (1898) of the annual number of deaths in the Prussian army from being kicked by mules. A standard generalization of the Poisson is the negative binomial distribution. It was derived by Greenwood and Yule (1920), as a consequence of apparent contagion due to unobserved heterogeneity, and by Eggenberger and Polya (1923) as a result of true contagion. The biostatistics literature of the 1930s and 1940s, although predominantly univariate, refined and brought to the forefront seminal issues that have since permeated regression analysis of both counts and durations. The development of the counting process approach unified the treatment of counts and durations. Much of the vast literature on iid counts, which addresses issues such as heterogeneity and overdispersion, true versus apparent contagion, and identifiability of Poisson mixtures, retains its relevance in the context of count data regressions. This leads to models such as the negative binomial regression model.