Maximally Selected Chi Square Statistics for Small Samples

Two samples can be compared by selecting a cut point and forming a 2 x 2 table of the numbers of observations above and below the cut point in each sample. Different cut points correspond to the maximum chi square statistic, the maximum squared log odds ratio, and the generalized Kolmogorov-Smirnov statistic. We simulated the small-sample null distribution of these statistics. The distributions of the maximally selected chi square statistic and the squared log odds ratio statistic are compared to each other and to the large-sample results of Miller and Siegmund (1982, Biometrics 38, 1011-1016. The distributions of and agreement among the three cut points are also considered.