[Negative binomial distribution versus Poisson in the analysis of recurrent phenomena]

The aim is to unfold the difficulties likely to arise in risk calculations through aggregated database when the studied phenomenon is recurrent and to display the negative binomial distribution as a valid and simple alternative to analyse this kind of phenomenon.

When the studied phenomenon is recurrent, the analysis by means of the Poisson regression can provoke overdispersion or extra-poisson variance, what leads to underestimating the standard errors in coefficients and may divert into the statistical significance of factors which as a matter of fact are not associated with the phenomenon beforehand. The negative binomial can grasp part of the variance which the Poisson is unable to identify. In order to check this out, the fit of both distributions were compared, based on the number of hospitalizations of individuals aged between 65 and 69, during 1996. This comparison was carried out by means of two different aggregated databases: by individuals and by variables.

There were differences in the fitted models by means of both distributions in both databases. By the analysis of the residuals, when using the base by individuals, the negative binomial fits correctly 67.9% of the observations badly fitted by the Poisson. Using the aggregated variables database, the percentage is 50%. In both cases, Poisson estimates four out of the six studied variables as significant. As to the negative binomial, there are two significant based on individuals and one in the variable database.

The existence of overdispersion is frequent in recurrent-type phenomena. When this occurs, the negative binomial distribution is more appropiate than the Poisson.