Upper Bounds for the P-Values of a Scan Statistic with a Variable Window

Abstract

It is asked if n independent events occurring in a given time interval are clustered or if, alternatively, the null hypothesis of a uniform distribution can be adopted. A simple scan statistic defined to be the maximum number of events within any sub-interval (or window) of given length has been used as test statistic in this context. Nagarwalla (1996) described a modification of this scan statistic, based on a generalized likelihood ratio statistic, which no longer assumes that the window width is fixed a priori. Unfortunately, the distribution of this statistic is not known and a simulation procedure had to be applied. In this paper a quite simpler statistic is proposed which can be considered as an approximation of Nagarwalla’s statistic. For this new statistic, upper bounds for the upper tail probabilities are given. Thus, the new test can be performed without recourse to a simulation. Furthermore, no restrictions on the cluster size are imposed. The procedure is illustrated by examples from epidemiology.