On the Exact Distribution of Maximally Selected Rank Statistics

The construction of simple classification rules is a frequent problem in medical research. For example a difference in overall survival time may suggest distinct types, i.e. subgroups of patients, of diffuse large B-cell lymphoma identified by gene expression profiling. Maximally selected rank statistics allow the evaluation of cutpoints, which provide the classification of different risk groups in a quantitative or ordered predictor variable. We discuss the computation of the exact distribution and we derive a new lower bound based on an extension of an algorithm for the exact distribution of a linear rank statistic. For small to moderate sample sizes and typical quantiles the lower bound of the exact distribution is a substantial improvement compared to approximations based on an improved Bonferroni inequality or based on the asymptotic Gaussian process. We use simulation results to compare our lower bound with the exact distribution. Our proposal is illustrated by two clinical studies: different prognosis for patients with diffuse large B-cell lymphoma based on gene expression profiling and the predicted value of the left ventricular ejection fraction for recurrence free survival of patients with malignant arrhythmias