Up-and-down experiments of first and second order

An up-and-down (UD) experiment for estimating a given quantile of a binary response curve is a sequential procedure whereby at each step a given treatment level is used and, according to the outcome of the observations, a decision is made (deterministic or randomized) as to whether to maintain the same treatment or increase it by one level or else to decrease it by one level. The design points of such UD rules generate a Markov chain and the mode of its invariant distribution is an approximation to the quantile of interest. The main area of application of UD algorithms is in Phase I clinical trials, where it is of greatest importance to be able to attain reliable results in small-size experiments. In this paper we address the issues of the speed of convergence and the precision of quantile estimates of such procedures, both in theory and by simulation. We prove that the version of UD designs introduced in 1994 by Durham and Flournoy can in a large number of cases be regarded as optimal among all UD rules. Furthermore, in order to improve on the convergence properties of this algorithm, we propose a second-order UD experiment which, instead of making use of just the most recent observation, bases the next step on the outcomes of the last two. This procedure shares a number of desirable properties with the corresponding first-order designs, and also allows greater flexibility. With a suitable choice of the parameters, the new scheme is at least as good as the first-order one and leads to an improvement of the quantile estimates when the starting point of the algorithm is low relative to the target quantile.