Standardized Maximin Research Articles

Standardized maximin criterion for discrimination and parameter estimation of nested models

A new criterion for approximate designs called the standardized maximin criterion suited for both model discrimination and parameter estimation based on D- and Ds -optimality criteria is introduced and studied. It is proved that the computation of an experimental design which is optimal with respect to this criterion can be reduced to the computation of multiple experimental designs which are optimal with respect to the simpler weighted criterion. Several numerical examples that describe the efficiency of the proposed criterion are provided.

Robust Designs for Discriminating between Trigonometric Regression Models

This study is devoted to the issue of constructing robust T-optimal discriminating designs for two competing trigonometric regression models, which differ by three trigonometric functions at most. To solve this problem, we propose using a Bayesian and standardized maximin approaches. The robust T-optimal discriminating designs were found explicitly in a number of particular cases. In the general case, on account of the complexity of the optimization problem, the corresponding optimal designs are not easy to find and have to be determined numerically. The results are illustrated by means of several examples.

Standardized maximim D-optimal designs for enzyme kinetic inhibition models

Locally optimal designs for nonlinear models require a single set of nominal values for the unknown parameters. An alternative is the maximin approach that allows the user to specify a range of values for each parameter of interest. However, the maximin approach is difficult because we first have to determine the locally optimal design for each set of nominal values before maximin types of optimal designs can be found via a nested optimization process. We show that particle swarm optimization (PSO) techniques can solve such complex optimization problems effectively. We demonstrate numerical results from PSO can help find, for the first time, formulae for standardized maximin D-optimal designs for nonlinear model with 3 or 4 parameters on the compact and nonnegative design space. Additionally, we show locally and standardized maximin D-optimal designs for inhibition models are not necessarily supported at a minimum number of points. To facilitate use of such designs, we create a web-based tool for practitioners to find tailor-made locally and standardized maximin optimal designs.

Application of imperialist competitive algorithm to find minimax and standardized maximin optimal designs

Finding optimal designs for nonlinear models is complicated because the design criterion depends on the model parameters. If a plausible region for these parameters is available, a minimax optimal design may be used to remove this dependency by minimizing the maximum inefficiency that may arise due to misspecification in the parameters. Minimax optimal designs are often analytically intractable and are notoriously difficult to find, even numerically. A population-based evolutionary algorithm called imperialist competitive algorithm (ICA) is applied to find minimax or nearly minimax D-optimal designs for nonlinear models. The usefulness of the algorithm is also demonstrated by showing it can hybridize with a local search to find optimal designs under a more complicated criterion, such as standardized maximin optimality.

Minimax optimal designs via particle swarm optimization methods

Particle swarm optimization (PSO) techniques are widely used in applied fields to solve challenging optimization problems but they do not seem to have made an impact in mainstream statistical applications hitherto. PSO methods are popular because they are easy to implement and use, and seem increasingly capable of solving complicated problems without requiring any assumption on the objective function to be optimized. We modify PSO techniques to find minimax optimal designs, which have been notoriously challenging to find to date even for linear models, and show that the PSO methods can readily generate a variety of minimax optimal designs in a novel and interesting way, including adapting the algorithm to generate standardized maximin optimal designs.

On Standardized Maximin Designs for Discrimination Between Two Polynomial Models

This article is devoted to standardized maximin designs for discrimination between two polynomial models that differ in degree by two. A relation of such designs to the designs that are optimal for estimating two elder coefficients of a polynomial model was established in a recent paper (Dette et al. 2012b). Under the condition that the ratio of the coefficients belongs to an arbitrary given compact set that is symmetric around zero, the problem was reduced to a much easier maximin problem that can be of some independent interest. Here we present some results on this problem that allow finding standardized maximin discriminating designs explicitly for many cases of theoretical and practical interest. A table of such designs is given.

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.

Optimal Designs for Quantile Regression Models

Despite their importance, optimal designs for quantile regression models have not been developed so far. In this article, we investigate the D-optimal design problem for nonlinear quantile regression analysis. We provide a necessary condition to check the optimality of a given design and use it to determine bounds for the number of support points of locally D-optimal designs. The results are illustrated, determining locally, Bayesian and standardized maximin D-optimal designs for quantile regression analysis in the Michaelis–Menten and EMAX model, which are widely used in such important fields as toxicology, pharmacokinetics, and dose–response modeling.

Optimal design for additive partially nonlinear models

We develop optimal design theory for additive partially nonlinear regression models, showing that Bayesian and standardized maximin D-optimal designs can be found as the products of the corresponding optimal designs in one dimension. A sufficient condition under which analogous results hold for Ds-optimality is derived to accommodate situations in which only a subset of the model parameters is of interest. To facilitate prediction of the response at unobserved locations, we prove similar results for Q-optimality in the class of all product designs. The usefulness of this approach is demonstrated through an application from the automotive industry, where optimal designs for least squares regression splines are determined and compared with designs commonly used in practice.

Bayesian and maximin optimal designs for heteroscedastic regression models

The authors consider the problem of constructing standardized maximin D-optimal designs for weighted polynomial regression models. In particular they show that by following the approach to the construction of maximin designs introduced recently by Dette, Haines & Imhof (2003), such designs can be obtained as weak limits of the corresponding Bayesian q-optimal designs. They further demonstrate that the results are more broadly applicable to certain families of nonlinear models. The authors examine two specific weighted polynomial models in some detail and illustrate their results by means of a weighted quadratic regression model and the Bleasdale–Nelder model. They also present a capstone example involving a generalized exponential growth model.

Robust and efficient design of experiments for the Monod model

In this paper the problem of designing experiments for the Monod model, which is frequently used in microbiology, is studied. The model is defined implicitly by a differential equation and has numerous applications in microbial growth kinetics, environmental research, pharmacokinetics, and plant physiology. The designs presented so far in the literature are local optimal designs, which depend sensitively on a preliminary guess of the unknown parameters, and are for this reason in many cases not robust with respect to their misspecification. Uniform designs and maximin optimal designs are considered as a strategy to obtain robust and efficient designs for parameter estimation. In particular, standardized maximin D- and E-optimal designs are determined and compared with uniform designs, which are usually applied in these microbiological models. It is demonstrated that maximin optimal designs are substantially more efficient than uniform designs. Parameter variances can be decreased by a factor of two by simply sampling at optimal times during the experiment. Moreover, the maximin optimal designs usually provide the possibility for the experimenter to check the model assumptions, because they have more support points than parameters in the Monod model.