Given a sample of covariate-response pairs, the problem of subgroup selection consists of identifying a subset of the domain of the covariates for which a particular response is to be expected. For example, one might be interested in determining the subset on which the mean response is sufficiently large. In this project, novel methodology for performing subgroup selection is developed in the flexible, nonparametric setting of multivariate isotonic regression, i.e. when our only assumption on the regression function is that an increase in any explanatory variable can never lead to a decrease in the expected response.

For this setting, we present methodology and derive a non-asymptotic, uniform Type I error rate guarantee as well as a high-probability bound on the power of our procedure. We show that the performance of our proposed algorithm is minimax optimal up to log-factors and illustrate the empirical performance through simulations.

Important real-world applications of our methodology can be found in clinical trials, for example testing the efficacy of a new drug, where efficacy might be present only in a subgroup of the population and thus may be masked in a study representing the entire population. Hence and because of the uniformity of our Type I error rate guarantee, the methodology we develop can be particularly useful in the active research area of personalised medicine. Researchers: Manuel Müller, Richard Samworth, Henry Reeve (University of Bristol) and Timothy Cannings (University of Edinburgh).