Suppose that we intend to perform an experiment consisting of a set of independent trials.The mean value of the response of each trial is assumed to be equal to the sum of the effect of the treatment selected for the trial, and some nuisance effects, e.g., the effect of a time trend, or blocking. In this model, we examine optimal approximate designs for the estimation of a system of treatment contrasts, with respect to a wide range of optimality criteria.We show that it is necessary for any optimal design to attain the optimal treatment proportions, which may be obtained from the marginal model that excludes the nuisance effects. Moreover, we prove that for a design to be optimal, it is sufficient that it attains the optimal treatment proportions and satisfies conditions of resistance to nuisance effects.For selected natural choices of treatment contrasts and optimality criteria, we calculate the optimal treatment proportions and give an explicit form of optimal designs. In particular, we obtain optimal treatment proportions for comparison of a set of new treatments with a set of controls. The results allow us to construct a method of calculating optimal approximate designs with a small support by means of linear programming. As a consequence, we can construct efficient exact designs by a simple heuristic.
1We show that a Φ-optimal approximate design may be obtained in two steps: (i) Calculate Φ-optimal proportions of treatment replications (treatment weights). These optimal proportions depend on the choice of contrasts of interest and on the optimality criterion Φ; however, they do not depend on the nuisance effects. (ii) Subject to keeping the optimal proportions of treatment replications, distribute the treatments to nuisance conditions such that the resulting design is resistant to nuisance effects. The designs resistant to nuisance effects are an extension of the designs orthogonal to the time trend (balanced for trend, or trend-free, cf. Cox [1951], Jacroux and Ray [1990]) to a more general class of models and treatment contrasts.The approach of first finding a design in a simpler model and then assuring that the information is retained in a finer model was used, e.g., in Schwabe [1996] and Kunert [1983].Schwabe [1996] studied optimal product designs, unlike the present paper, where optimal designs with non-product structure are provided too; Kunert [1983] studied exact designs in the case of universal optimality. Universal optimality, formulated by Kiefer [1975], means optimality for estimating a maximal system of orthonormal contrasts (which is a special case of the general system of contrasts that we consider), with respect to a wide range of criteria.For selected systems of treatment contrasts and a wide class of optimality criteria Φ, we calculate Φ-optimal treatment weights and thus obtain a class of Φ-optimal designs. For instance, for the estimation of contrasts for comparing a set of new treatments with a set of controls, we provide M V -optimal designs and optimal designs with respect to Kiefer'...
