Optimal approximate designs for estimating treatment contrasts resistant to nuisance effects

Abstract

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'...

Optimal experimental designs for treatment contrasts in heteroscedastic models with covariates

“…We would like to emphasize that the proposed support reduction method is not particularly tied to the current model (1). As noted earlier, the method was applied for a similar model by Rosa and Harman [2016]. Moreover, the proposed approach can be used for any linear regression model where an optimal design ξ * with a large support is found, either analytically or by design algorithms.…”

“…Moreover, for such A T θ, any design ξ that satisfies (9) and whose marginal treatment design is w * is Φ-optimal. The designs satisfying (9) were denoted as resistant to nuisance effects in a slightly different context by Rosa and Harman [2016]. In the present settings, designs satisfying (8) and (9) • If a Φ-optimal design ξ * is known, other Φ-optimal designs can trivially be found by solving the linear equality i,k ξ(i, k)f (i, k)f T (i, k) = M(ξ * ) that guarantees that the design ξ has the same moment matrix as ξ * .…”

“…In the experiments performed to estimate the effects of a selected set of treatments, it is common that the responses are also affected by some other experimental conditions (the covariates -e.g., time trend, block effects, the ages or the gender of the subjects in a clinical trial). The effects of the covariates are traditionally considered to be nuisance effects in the experimental design literature (e.g., Cox [1951], Majumdar and Notz [1983], Atkinson and Donev [1996], Jacroux et al [1997], Rosa and Harman [2016]). Recently, Atkinson [2015] suggested that particularly with the growing importance of personalized medicine, the covariate effects may also be of prominent interest.…”

“…Similar approach was employed by Rosa and Harman [2016] in a homoscedastic model, where the covariate effects were considered to be nuisance parameters.…”

E- and R-optimality of block designs for treatment-control comparisons

“…We provided the class of all E-optimal approximate block designs for comparisons with a control, which can be described by the simple linear constraints (2). For a strictly convex criterion, the only optimal block designs are product designs with optimal treatment proportions (see Rosa and Harman [2016]). However, since E-optimality lacks strict convexity, the class of E-optimal designs is richer and thus allows for an easier construction of efficient (or even optimal) exact designs by the rounding methods -therefore increasing the usefulness of the approximate theory.…”