Two-treatment comparison based on joint toxicity and efficacy ordered alternatives in cancer trials

The primary goal of anticancer treatments is to attain efficacy, however toxicity could affect the course of the therapy. Methods have been proposed for comparing two treatments on the basis of the joint distribution for safety and efficacy outcomes, but they do not take into account the cumulative doses of drugs (chemotherapy) or radiation (radiotherapy) received by each patient. Moreover, these methods assume a parametric form for the joint distribution. In this paper we define a multi-dimensional parameter including toxicity and efficacy outcomes and the dose at which one, none or both occur. Each patient is classified into an ordered category depending on the order of occurrence of these two criteria: the sooner the patient benefits from efficacy and/or the later he/she experiences toxicity, the better is the treatment. We then apply likelihood ratio tests with ordered alternatives. This procedure requires constrained maximum likelihood estimation via isotonic regression. A large set of simulations compares the proposed tests to other more usual tests and the results show a good power and a satisfactory type I error control. Our approach is illustrated with a multi-centre randomized clinical trial involving patients with metastatic non-seminomatous germ cell tumours.