Optimizing a Unimodal Response Function for Binary Variables

Janis Hardwick      Quentin F. Stout
University of Michigan


Abstract: In some dose response settings in clinical trials, there are completing failure modes where low doses do not elicit a positive response, while overly large doses elicit a toxic response. Typically this occurs in trials known as phase I/II trials. In such settings, often the dose response curve which plots dose vs. success (efficacious and non-toxic), is unimodal, i.e., it is increasing and then decreasing. This then leads to the problem of finding the mode (highest point) of the curve.

In this paper, several sampling designs are examined for the problem of optimizing a response function from a set of Bernoulli populations, where the population means are assumed to have a strict unimodal structure. The designs are evaluated both on their efficiency in identifying a good population at the end of the experiment, and in their efficiency in sampling from good populations during the trial.

A new design, that adapts multiarm bandit approaches to this unimodal structure, is shown to be superior to the designs previously proposed. These earlier designs were based on Polya urns, stochastic approximation, and up and down rules (random walks, Markov chains). The bandit design utilizes approximate Gittins indices and shape constrained regression to model the dependencies between the arms. Note that our design, and some of the others, are nonparametric, so they are quite flexible and do not make strong assumptions about the shape of the curve.

Keywords: response adaptive sampling, phase I/II clinical trial, design of experiments, bandit problem, Gittins index, unimodal regression

Complete paper. This paper appears in Optimum Design 2000, A. Atkinson, B. Bogacka, and A. Zhigljavsky, eds., Kluwer, 2001, pp. 195-208.


Related Work: This paper assumes that one can only observe a success or fail outcome, without knowing which failure mode occured. We have also developed efficient designs to locate the mode when one has the additional information as to which failure mode(s) occured.

More generally, here is an explanation of response adaptive sampling, and here are our relevant papers.

Here is a description of the algorithm used to rapidly compute unimodal regression curves.

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