Convergence Results for an Adaptive Ordinal Urn Design

Consider an urn containing balls that are labeled with integers corresponding to treatment dose levels. For each consecutive trial in an experiment, treatments are determined by the draw of a ball from an urn. The observed outcomes determine a new dose level to be indicated on a replacement ball. In this manner the probability of treating at a specific level is determined by the distribution of balls in the urn which is driven by successes and failures encountered in the course of the experiment. General classes of such experimental designs are described and motivated in [4]. In the present paper, we summarize results from [1] which include a rule for changing the labels on the balls that causes the most likely treatment level to converge to a preselected quantile of a failure distribution. For the case in which failure follows a logistic distribution, we describe how to control both the mean and the variance of the limiting distribution of labels on the balls in the urn.