Summary There are many situations where one expects an ordering among K⩾2 experimental groups or treatments. Although there is a large body of literature dealing with the analysis under order restrictions, surprisingly, very little work has been done in the context of the design of experiments. Here, a principled approach to the design of experiments with ordered treatments is provided. In particular we propose two classes of designs which are optimal for testing different types of hypotheses. The theoretical findings are supplemented with thorough numerical experimentation and a concrete data example. It is shown that there is a substantial gain in power, or alternatively a reduction in the required sample size, when an experiment is both designed and analysed by using methods which account for order restrictions.

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This article discusses D-optimal Bayesian crossover designs for generalized linear models.Crossover trials with t treatments and p periods, for t <= p, are considered. The designs proposed in this paper minimize the log determinant of the variance of the estimated treatment effects over all possible allocation of the n subjects to the treatment sequences. It is assumed that the p observations from each subject are mutually correlated while the observations from different subjects are uncorrelated. Since main interest is in estimating the treatment effects, the subject effect is assumed to be nuisance, and generalized estimating equations are used to estimate the marginal means. To address the issue of parameter dependence a Bayesian approach is employed. Prior distributions are assumed on the model parameters which are then incorporated into the D-optimal design criterion by integrating it over the prior distribution. Three case studies, one with binary outcomes in a 4 × 4 crossover trial, second one based on count data for a 2 × 2 trial and a third one with Gamma responses in a 3 × 2 crossover trial are used to illustrate the proposed method. The effect of the choice of prior distributions on the designs is also studied.

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In this communication we examine the relationship between maxi-min, Bayes and Nash designs for some hypothesis testing problems. In particular we consider the problem of sample allocation in the standard analysis of variance framework and show that the maxi-min design is also a Bayes solution with respect to the least favourable prior, as well as a solution to a game theoretic problem, which we refer to as a Nash design. In addition, an extension to tests for order is provided. MSC2020 subject classifications: 62K05.