A bivariate negative binomial model to explain traffic accident migration

The phenomenon of "regression to the mean" is now widely known in the study of the effectiveness of remedial treatment of traffic accident blackspots. What happens is that the criterion used for selection of sites at which treatment is to be applied gives rise to bias in the estimate of the effectiveness: the conditional expectation of the after frequency is less than the true mean, even if the treatment is totally ineffective. It has been reported in some previous studies that accident "migration" has been observed. This is the phenomenon whereby the accident rate apparently rises at sites that are untreated but that are neighbours to treated sites. If this were a genuine effect, it would have serious implications for the assessment of remedial treatments. This paper aims to explain this migration effect in purely probabilistic terms, without recourse to the concept of physical migration. The model used is a new bivariate negative binomial distribution, incorporating spatial correlation between the true mean site accident rates. As with the regression to mean effect, the migration effect can then be explained in terms of the conditioning implicit in the selection process.