Overdispersion in modelling accidents on road sections and in empirical bayes estimation

In multivariate statistical models of road safety one usually finds that the accident counts are 'overdispersed'. The extent of the overdispersion is itself subject to estimation. It is shown that the assumption one makes about the nature of overdispersion will affect the maximum likelihood estimates of model parameters. If one assumes that the same overdispersion parameter applies to all road sections in the data base, then, the maximum likelihood estimate of parameters will be unduly influenced by very short road sections and insufficiently influenced by long road sections. The same assumption about the overdispersion parameter also leads to an inconsistency when one estimates the safety of a road section by the Empirical Bayes method. A way to avoid both problems is to estimate an overdispersion parameter (phi) that applies to a unit length of road, and to set the overdispersion parameter for a road section of length L to phiL. How this would change the estimates of regression parameters for road section models now in use requires examination. Safety estimation by the Empirical Bayes method is altered substantially.