Contagion distributions for defining disease clustering in time

Consider the situation in which cases of disease are recorded as frequency counts in T consecutive time intervals. In the epidemiology literature it is often of interest to determine if cases are distributed at random defined by an assumed multinomial vector for the joint distribution of the cell counts with a hypothesized known vector of probabilities for case placements. In this paper, we investigate the construction of a family of contagion type alternatives to randomness that depends on a single nonnegative clustering parameter and some function of the case placements. Numerical studies based on the use of the Gibbs sampler are used to compare the power of four tests for randomness against these contagion alternatives and are used to compare various estimators of the clustering parameter.