Group up-and-down designs for dose-finding

Let toxicity to treatment be a Bernoulli random variable for which the probability of failure increases with dose. Consider the problem of identifying a dose $\mu$ having pre-specified probability of failure using data from groups of subjects who arrive sequentially for treatment. There is considerable theory available in this setting for fully sequential up-and-down procedures. This paper presents asymptotic and finite theoretical results for Markovian up-and-down procedures when subjects are treated in groups. Practical instructions are given on how to select the design parameters so as to cause the treatments to cluster around the unknown dose $\mu$. Examples are given to illustrate how this group procedure behaves for small sample sizes.