We investigate a statistical framework for Phase I clinical trials that test the safety of two or more agents in combination. For such studies, the traditional assumption of a simple monotonic relation between dose and the probability of an adverse event no longer holds. Nonetheless, the dose toxicity (adverse event) relationship will obey an assumption of partial ordering in that there will be pairs of combinations for which the ordering of the toxicity probabilities is known. Some authors have considered how to best estimate the maximum tolerated dose (a dose providing a rate of toxicity as close as possible to some target rate) in this setting. A related, and equally interesting, problem is to partition the 2-dimensional dose space into two sub-regions: doses with probabilities of toxicity lower and greater than the target. We carry out a detailed investigation of this problem. The theoretical framework for this is the recently presented semiparametric dose finding method. This results in a number of proposals one of which can be viewed as an extension of the Product of Independent beta Priors Escalation method (PIPE). We derive useful asymptotic properties which also apply to the PIPE method when seen as a special case of the more general method given here. Simulation studies provide added confidence concerning the good behaviour of the operating characteristics.
Keywords: Bayesian method; Dose-finding design; Partial ordering; Phase I clinical trials; semiparametric method.