A scan statistic with a variable window.

Given N points or events occurring according to some probability distribution in the unit interval (0, 1), the simple scan statistic is defined to be the maximum number of points in any sub-interval of length d. In many areas, as in epidemiology, it is used to test the null hypothesis that the events are random, against the alternative that they cluster within some window of fixed width d. Since d must be chosen without snooping at the data, the test restricts the alternative to clusters of a fixed size. In this paper, we propose a scan statistic with a variable window, whose size does not need to be chosen a priori. This test is the generalized likelihood ratio test for a uniform null distribution against an alternative of non-random clustering which allows for clusters of variable width. A simple algorithm for the implementation of the method is given and applied to birth defects data previously analysed by a simple scan statistic.