Multivariate Optimizing Up and Down Design Nawar M. Shara Department of Epidemiology and Statistics, Medstar Research Institute Hyattsville, Maryland, 20783, USA nawar.shara@medstar.net Nancy Flournoy Department of Statistics, University of Missouri - Columbia Columbia, Missouri, 65211, USA nflournoy@missouri.edufl Abstract: Suppose we are interested in finding the optimal dose of two drugs (for example, Tylenol and Aspirin), that is, we are interested in determining the dose combination that maximizes the probability of patients’ success. We assume responses are binary, either failure or success, and that the treatments to be used in the study are selected from a lattice of combination drugs. We extend the univariate Optimizing Up-and-Down Design of Kpamegan (2001), using ideas from stochastic approximation, in a way that the number of subjects at each stage is independent of the number of predictor variables (e.g. drugs). keywords and phrases: Simultaneous perturbation stochastic approximation, Adaptive designs, Optimal dose, Phase II clinical trials, Markov chain, combination therapy, random walk, up-and-down designs, dose finding. 1 INTRODUCTION In univariate up-and-down designs, the next treatment is determined such that it is one level higher, one level lower, or the same level as the current treatment. Many authors have studied univariate up-and-down designs for use when the probability of response is increasing (cf. Flournoy, 2001). But suppose we are interested in determining the dose combination of two drugs (for example, Tylenol and Aspirin) that maximizes the probability of success. Assume responses are binary, either failure or success, and the treatments to be used are selected from a lattice of combination drug levels. Our technical approach is to extend the univariate Optimizing Up-and-Down Design Kpamegan (2001) using ideas from stochastic approximation procedures. Recursive 1