T-optimal designs formulti-factor polynomial regressionmodelsvia a semidefinite relaxation method

We consider T-optimal experiment design problems for discriminating multi-factor polynomial regression models wherethe design space is defined by polynomial inequalities and the regression parameters are constrained to given convex sets.Our proposed optimality criterion is formulated as a convex optimization problem with a moment cone constraint. When theregression models have one factor, an exact semidefinite representation of the moment cone constraint can be applied to obtainan equivalent semidefinite program.When there are two or more factors in the models, we apply a moment relaxation techniqueand approximate the moment cone constraint by a hierarchy of semidefinite-representable outer approximations. When therelaxation hierarchy converges, an optimal discrimination design can be recovered from the optimal moment matrix, and itsoptimality can be additionally confirmed by an equivalence theorem. The methodology is illustrated with several examples

Robust T-optimal discriminating designs

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 (1975) 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.Comment: Published in at http://dx.doi.org/10.1214/13-AOS1117 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Optimal discrimination designs

We consider the problem of constructing optimal designs for model discrimination between competing regression models. Various new properties of optimal designs with respect to the popular $T$-optimality criterion are derived, which in many circumstances allow an explicit determination of $T$-optimal designs. It is also demonstrated, that in nested linear models the number of support points of $T$-optimal designs is usually too small to estimate all parameters in the extended model. In many cases $T$-optimal designs are usually not unique, and in this situation we give a characterization of all $T$-optimal designs. Finally, $T$-optimal designs are compared with optimal discriminating designs with respect to alternative criteria by means of a small simulation study.Comment: Published in at http://dx.doi.org/10.1214/08-AOS635 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Optimal discrimination designs

We consider the problem of constructing optimal designs for model discrimination between competing regression models. Various new properties of optimal designs with respect to the popular T-optimality criterion are derived, which in many circumstances allow an explicit determination of T-optimal designs. It is also demonstrated, that in nested linear models the number of support points of T-optimal designs is usually too small to estimate all parameters in the extended model. In many cases T-optimal designs are usually not unique, and we give a characterization of all T-optimal designs. Finally, T-optimal designs are compared with optimal discriminating designs with respect to alternative criteria by means of a small simulation study. --Model discrimination,optimal design,T-optimality,Ds-optimality,nonlinear approximation

Bayesian T-optimal discriminating designs

The problem of constructing Bayesian optimal discriminating designs for a class of regression models with respect to the T-optimality criterion introduced by Atkinson and Fedorov (1975a) is considered. It is demonstrated that the discretization of the integral with respect to the prior distribution leads to locally T-optimal discrimination designs can only deal with a few comparisons, but the discretization of the Bayesian prior easily yields to discrimination design problems for more than 100 competing models. A new efficient method is developed to deal with problems of this type. It combines some features of the classical exchange type algorithm with the gradient methods. Convergence is proved and it is demonstrated that the new method can find Bayesian optimal discriminating designs in situations where all currently available procedures fail.Comment: 25 pages, 3 figure

TT-optimal designs for discrimination between two polynomial models

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 $n\in\mathbb{N}$. 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.Comment: Published in at http://dx.doi.org/10.1214/11-AOS956 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

A Metaheuristic Adaptive Cubature Based Algorithm to Find Bayesian Optimal Designs for Nonlinear Models

Finding Bayesian optimal designs for nonlinear models is a difficult task because the optimality criteriontypically requires us to evaluate complex integrals before we perform a constrained optimization. Wepropose a hybridized method where we combine an adaptive multidimensional integration algorithm anda metaheuristic algorithm called imperialist competitive algorithm to find Bayesian optimal designs. Weapply our numerical method to a few challenging design problems to demonstrate its efficiency. Theyinclude finding D-optimal designs for an item response model commonly used in education, Bayesianoptimal designs for survivalmodels, and Bayesian optimal designs for a four-parameter sigmoid Emax doseresponse model. Supplementary materials for this article are available online and they contain an R packagefor implementing the proposed algorithm and codes for reproducing all the results in this paper