Sequential designs for opposing failure functions

A class of sequential urn designs is presented for operating in the presence of opposing failure functions. Consider, for example, the situation in which the probability of toxicity increases with dose, while the probability of a treatment failure given no toxicity decreases with dose. Suppose failure (no cure) from the treatment process is only observable in the absence of failure from the toxicity process because toxicity causes the trial to stop. The objective is to estimate the dose yielding the maximum probability of success (unconditional cure) while limiting the exposure of subjects to dosages at high risk of toxicity and treatment failure. For a design that changes the urn composition only when successes occur, small sample and asymptotic results are described concerning the urn composition. The maximum likelihood estimates of the cure probabilities are derived, and when standardized, are shown asymptotically to be jointly normal. A likelihood ratio test that all success probabilities are equal follows.