This paper considers the problem of constructing optimal discriminating experimental designs for competing regression models on the basis of the T -optimality criterion introduced by Atkinson and Fedorov [Biometrika 62 (1975a) 57-70]. T -optimal designs depend on unknown model parameters and it is demonstrated that these designs are sensitive with respect to misspecification. As a solution to this problem we propose a Bayesian and standardized maximin approach to construct robust and efficient discriminating designs on the basis of the T -optimality criterion. It is shown that the corresponding Bayesian and standardized maximin optimality criteria are closely related to linear optimality criteria. For the problem of discriminating between two polynomial regression models which differ in the degree by two the robust T -optimal discriminating designs can be found explicitly. The results are illustrated in several examples.

This paper is devoted to the explicit construction of optimal designs for discrimination between two polynomial regression models of degree n − 2 and n. In a fundamental paper, Atkinson and Fedorov [Biometrika 62 (1975a) 57-70] proposed the T -optimality criterion for this purpose. Recently, Atkinson [MODA 9, Advances in Model-Oriented Design and Analysis (2010) 9-16] determined T -optimal designs for polynomials up to degree 6 numerically and based on these results he conjectured that the support points of the optimal design are cosines of the angles that divide half of the circle into equal parts if the coefficient of x n−1 in the polynomial of larger degree vanishes. In the present paper we give a strong justification of the conjecture and determine all T -optimal designs explicitly for any degree n ∈ N.In particular, we show that there exists a one-dimensional class of T -optimal designs. Moreover, we also present a generalization to the case when the ratio between the coefficients of x n−1 and x n is smaller than a certain critical value. Because of the complexity of the optimization problem, T -optimal designs have only been determined numerically so far, and this paper provides the first explicit solution of the T -optimal design problem since its introduction by Atkinson and Fedorov [Biometrika 62 (1975a) 57-70]. Finally, for the remaining cases (where the ratio of coefficients is larger than the critical value), we propose a numerical procedure to calculate the T -optimal designs. The results are also illustrated in an example.

In this paper we consider the problem of constructing T -optimal discriminating designs for Fourier regression models. We provide explicit solutions of the optimal design problem for discriminating between two Fourier regression models, which differ by at most three trigonometric functions. In general, the T -optimal discriminating design depends in a complicated way on the parameters of the larger model, and for special configurations of the parameters T -optimal discriminating designs can be found analytically. Moreover, we also study this dependence in the remaining cases by calculating the optimal designs numerically. In particular, it is demonstrated that D-and D s -optimal designs have rather low efficiencies with respect to the T -optimality criterion.

We consider the problem of estimating the derivative of the expected response in nonlinear regression models. It is demonstrated that in many cases the optimal designs for estimating the derivative have either on m or m − 1 support points, where m denotes the number of unknown parameters in the model. It is also shown that the support points and weights of the optimal designs are analytic functions, and this result is used to construct a numerical procedure for the calculation of the optimal designs. The results are illustrated in exponential regression and rational regression models.

We consider design issues for toxicology studies when we have a continuous response and the true mean response is only known to be a member of a class of nested models. This class of non-linear models was proposed by toxicologists who were concerned only with estimation problems. We develop robust and efficient designs for model discrimination and for estimating parameters in the selected model at the same time. In particular, we propose designs that maximize the minimum of D-or D1-efficiencies over all models in the given class. We show that our optimal designs are efficient for determining an appropriate model from the postulated class, quite efficient for estimating model parameters in the identified model and also robust with respect to model misspecification. To facilitate the use of optimal design ideas in practice, we have also constructed a website that freely enables practitioners to generate a variety of optimal designs for a range of models and also enables them to evaluate the efficiency of any design. This is an electronic reprint of the original article published by the ISI/BS in Bernoulli, 2010, Vol. 16, No. 4, 1164-1176. This reprint differs from the original in pagination and typographic detail. 1350-7265 c 2010 ISI/BSOptimal designs for discriminating between dose-response models 1165 include how to select the number of dose levels to observe the continuous outcome, where these levels are and how many repeated observations to take at each of these levels. This work assumes, for the sake of simplicity, that there is only one independent variable, the dose level and only non-sequential designs are considered.When we have competing models, a design should be able to discriminate among these models and select the most appropriate ones. Dette [5][6][7] found optimal discrimination designs for polynomial regression models, and Dette and Roeder [9] and Dette and Haller [10] found optimal discrimination designs for trigonometric and Fourier regression models, respectively. T -optimal designs are usually used to discriminate between homoscedastic models with normal errors [1-3, 12]. For discriminating non-linear models, only numerical results are possible; Lopez-Fidalgo et al. [16] investigated optimal designs maximizing a weighted average of two T -criterion functions and Lopez-Fidalgo et al. [17] constructed T -optimal designs for Michaelis-Menten-like models. When the design problem involves model discrimination and another optimality criterion, the problem is more complicated. Hill et al. [12] was among the first to consider studies with two goals: model discrimination and estimation of model parameters. Dette et al. [8] gave a concrete example where they wanted to discriminate between the Michaelis-Menten-model and the Emax model and estimate model parameters in an enzyme-kinetic study. A key reason for there having been so little research into such design problems for non-linear models is that there are serious technical difficulties. The motivation for this work comes from recent proposals by Pie...