Implementing Asymptotically Optimal Experiments for Treatment Comparison

Sequential experiments are widely used in biomedical practice but are also highly desirable in an industrial set-up. These procedures are very flexible since the experimenter can modify the trial as it goes along, on the basis of the previous allocations and/or observations. However, response-adaptive experiments present inferential problems because observations are no longer independent. When the experimental objectives can be defined as an optimization problem, often the optimal design depends on the unknown parameters of the statistical model. For example, in the case of a binary response trial for comparing two treatments T1 and T2, with respective probabilities of success p1 and p2, if we are interested in inferring on their difference, then the proportion of assignments of T1 which minimizes the variance of the ML estimator and maximizes the power of the corresponding test is the so-called Neyman allocation, proportional to the square root of p1(1 - p1). The maximum likelihood design for comparing v treatments is based on the step-by-step updating of the target treatment allocation by ML estimates: this estimate is then used at each step for the randomized allocation of the treatments. It has been shown by Baldi Antognini and Giovagnoli (Sequential Analysis, 2005) that when the responses belong to the exponential family, for any optimality criterion the related ML design is asymptotically optimal and the MLE's of the parameters of interest retain the strong consistency and asymptotical normality properties, as if the observations were independent.. In this paper we give a computer program for implementing the ML design under some common models (binomial/normal/Poisson/exponential, etc.) and some most widely used optimality targets, and investigate its speed of convergence.