Optimizing up -and -down designs

Assume that the probability of success is unimodal as a function of dose. We propose a class of up-and-down designs, i.e., treatment allocation methodologies, for selecting the dose that maximizes the patients' success probability. These designs are constructed to use accruing information to limit the number of patients that are exposed to doses with high probability of failure. This treatment allocation procedure is motivated by Kiefer-Wolfwitz's stochastic approximation procedure. However, we take the response to be binary and the possible treatment space to be a lattice. The proposed procedure is shown to allocate treatments to pair of subjects in a way that causes the treatment distribution to center on the treatment with maximum success probability. The procedure defines a nonhomogeneous random walk, so well-known theory is used to explicitly characterize the treatment distribution. As an estimator of the best dose, the mode of the empirical treatment distribution converges faster than does X(I) using stochastic approximation.