A curve-free, Bayesian decision-theoretic two-stage design is proposed to select biological efficacious doses (BEDs) for phase Ia/Ib trials in which both toxicity and efficacy signals are observed. No parametric models are assumed to govern the dose-toxicity, dose-efficacy, and toxicity-efficacy relationships. We assume that the dose-toxicity curve is monotonic non-decreasing and the dose-efficacy curve is unimodal. In the phase Ia stage, a Bayesian model on the toxicity rates is used to locate the maximum tolerated dose. In the phase Ib stage, we model the dose-efficacy curve using a step function while continuing to monitor the toxicity rates. Furthermore, a measure of the goodness of fit of a candidate step function is proposed, and the interval of BEDs associated with the best fitting step function is recommended. At the end of phase Ib, if some doses are recommended as BEDs, a cohort of confirmation is recruited and assigned at these doses to improve the precision of estimates at these doses. Extensive simulation studies show that the proposed design has desirable operating characteristics across different shapes of the underlying true toxicity and efficacy curves.
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