Maximally Selected Rank Statistics for Dose-Response Problems

We consider the bivariate situation of some quantitative, ordinal, binary or censored response variable and some quantitative or ordinal exposure variable (dose) with a hypothetical effect on the response. Data can either be the outcome of a planned dose-response experiment with only few dose levels or of an observational study where, for example, both exposure and response variable are observed within each individual. We are interested in testing the null hypothesis of no effect of the dose variable vs. a dose-response function depending on an unknown ?threshold? parameter. The variety of dose-response functions considered ranges from no observed effect level (NOEL) models to umbrella alternatives. Here we discuss generalizations of the method of Lausen & Schumacher (Biometrics, 1992, 48, 73?85)which are based on combinations of two-sample rank statistics and rank statistics for trend. Our approach may be seen as a generalization of a proposal for change-point problems. Using the approach of Davies (Biometrika, 1987, 74, 33?43)we derive and approximate the asymptotic null distribution for a large number of thresholds considered. We use an improved Bonferroni inequality as approximation for a small number of thresholds considered. Moreover, we analyse the small sample behaviour by means of a Monte-Carlo study. Our paper is illustrated by examples from clinical research and epidemiology.