Optimal designs for the development of personalized treatment rules

In the present paper, personalized treatment means choosing the best treatment for a patient while taking into account certain relevant personal covariate values. We study the design of trials whose goal is to find the best treatment for a given patient with given covariates. We assume that the subjects in the trial represent a random sample from the population, and consider the allocation, possibly with randomization, of these subjects to the different treatment groups in a way that depends on their covariates. We derive approximately optimal allocations, aiming to minimize expected regret, assuming that future patients will arrive from the same population as the trial subjects. We find that, for the case of two treatments, an approximately optimal allocation design does not depend on the value of the covariates but only on the variances of the responses. In contrast, for the case of three treatments the optimal allocation design does depend on the covariates as we show for specific scenarios. Another finding is that the optimal allocation can vary a lot as a function of the sample size, and that randomized allocations are relevant for relatively small samples, and may not be needed for very large studies.