Clustering of rare events

The clustering of cases of a rare disease is considered. The number of events observed for each unit is assumed to have a Poisson distribution, the mean of which depends upon the population size and the cluster membership of that unit. Here a cluster consists of those units that are homogeneous in their rate of occurrence of the rare events under study. A sample of units is modeled by a mixture of Poisson distributions, one for each cluster, the mixing parameters being the proportions of all units represented by the components of the mixture. Maximum likelihood and Bayes approaches are employed to determine criteria for separating a sample into groups of units with homogeneous rates. A likelihood ratio test for the significance of a two-component mixture is presented as an example. The performance of the criteria is illustrated with data on the spatial occurrence of sudden infant deaths (SIDs) in North Carolina counties over a four-year period. The results suggest that the practice of dividing the counties into high- and low-risk categories on the basis of the ordered rates alone should be questioned. Tests based upon combinatorial methods are also presented to examine the significance of the number of contiguous counties among those with high rates.