Weng Kee Wong

On the Equivalence of Constrained and Compound Optimal Designs

O n the Equivalence o f C o n s t r a i n e d and C o m p o u n d O p t i m a l Designs Author(s): R. Dennis Cook and W e n g Kee W o n g Source: Journal of the American Statistical Association, V o l . 89, No. 426 (Tun., 1994), pp. 687¬ P u b l i s h e d b y : American Statistical Association Stable U R L : http://www.jstor.org/stable/2290872 Accessed: 2 3 / 0 5 / 2 0 1 1 Your use of the JSTOR archive indicates your acceptance of JSTORs Terms and Conditions of Use, available at http://www.jstor.org/page/info/about/policies/terms.jsp. JSTORs Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at http://www.jstor.org/action/showPublisher?publisherCode=astata. Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org. H1 American Statistical Association is collaborating with JSTOR to digitize, preserve and extend access to Journal of the American Statistical Association. m S T O R http://www.jstor.org

Optimal Design for Goodness-of-Fit of the Michaelis–Menten Enzyme Kinetic Function

We construct efficient designs for the Michaelis–Menten enzyme kinetic model capable of checking model assumptions. An extended model called EMAX is also considered for this purpose. This model is widely used in pharmacokinetics and reduces to the Michaelis–Menten model for a specific choice of parameter settings. Our strategy is to find efficient designs for estimating the parameters in the EMAX model and at the same time test the validity of the Michaelis–Menten model against the EMAX model by maximizing a minimum of the D or D1 efficiencies taken over a range of values for the nonlinear parameters. In particular, we show that such designs are (a) efficient for estimating parameters in the EMAX model, (b) about 70% efficient for estimating parameters in the Michaelis–Menten model, (c) efficient for testing the Michaelis–Menten model against the EMAX model, and (d) robust with respect to misspecification of the unknown parameters in the nonlinear model.

E-optimal designs for the Michaelis–Menten model

We construct locally E-optimal designs for the two parameter Michaelis-Menten model under various assumptions on the error structure. Illustrative examples are given and the performance of these designs are compared with D-optimal designs.

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.

A Modified Particle Swarm Optimization Technique for Finding Optimal Designs for Mixture Models

Particle Swarm Optimization (PSO) is a meta-heuristic algorithm that has been shown to be successful in solving a wide variety of real and complicated optimization problems in engineering and computer science. This paper introduces a projection based PSO technique, named ProjPSO, to efficiently find different types of optimal designs, or nearly optimal designs, for mixture models with and without constraints on the components, and also for related models, like the log contrast models. We also compare the modified PSO performance with Fedorovs algorithm, a popular algorithm used to generate optimal designs, Cocktail algorithm, and the recent algorithm proposed by [1].

A semi-infinite programming based algorithm for finding minimax optimal designs for nonlinear models

Minimax optimal experimental designs are notoriously difficult to study largely because the optimality criterion is not differentiable and there is no effective algorithm for generating them. We apply semi-infinite programming (SIP) to solve minimax design problems for nonlinear models in a systematic way using a discretization based strategy and solvers from the General Algebraic Modeling System (GAMS). Using popular models from the biological sciences, we show our approach produces minimax optimal designs that coincide with the few theoretical and numerical optimal designs in the literature. We also show our method can be readily modified to find standardized maximin optimal designs and minimax optimal designs for more complicated problems, such as when the ranges of plausible values for the model parameters are dependent and we want to find a design to minimize the maximal inefficiency of estimates for the model parameters.

Using animal instincts to design efficient biomedical studies via particle swarm optimization

Particle swarm optimization (PSO) is an increasingly popular metaheuristic algorithm for solving complex optimization problems. Its popularity is due to its repeated successes in finding an optimum or a near optimal solution for problems in many applied disciplines. The algorithm makes no assumption of the function to be optimized and for biomedical experiments like those presented here, PSO typically finds the optimal solutions in a few seconds of CPU time on a garden-variety laptop. We apply PSO to find various types of optimal designs for several problems in the biological sciences and compare PSO performance relative to the differential evolution algorithm, another popular metaheuristic algorithm in the engineering literature.

Optimizing Two-Level Supersaturated Designs Using Swarm Intelligence Techniques

Supersaturated designs (SSDs) are often used to reduce the number of experimental runs in screening experiments with a large number of factors. As more factors are used in the study, the search for an optimal SSD becomes increasingly challenging because of the large number of feasible selection of factor level settings. This article tackles this discrete optimization problem via an algorithm based on swarm intelligence. Using the commonly used E(s2) criterion as an illustrative example, we propose an algorithm to find E(s2)-optimal SSDs by showing that they attain the theoretical lower bounds found in previous literature. We show that our algorithm consistently produces SSDs that are at least as efficient as those from the traditional CP exchange method in terms of computational effort, frequency of finding the E(s2)-optimal SSD, and also has good potential for finding D3-, D4-, and D5-optimal SSDs. Supplementary materials for this article are available online.

On the equivalence of D and G-optimal designs in heteroscedastic models

Conditions are derived for the Kiefer-Wolfowitzs theorem (KWT) to apply to linear models with heteroscedastic errors. It is shown that both D- and G-optimal designs remain equivalent only under very stringent conditions. Examples are constructed and their efficiencies are compared when they are not equivalent. For the simple linear regression model with a symmetric efficiency function, a relationship between the efficiencies of the optimal designs and the support points of the D-optimal design is noted.

Optimal designs for dose finding experiments in toxicity studies

We construct optimal designs for estimating fetal malformation rate, prenatal death rate and an overall toxicity index in a toxicology study under a broad range of model assumptions. We use Weibull distributions to model these rates and assume that the number of implants depend on the dose level. We study properties of the optimal designs when the intra-litter correlation coefficient depends on the dose levels in different ways. Locally optimal designs are found, along with robustified versions of the designs that are less sensitive to mis-specification in the nominal values of the model parameters. We also report e?ciencies of commonly used designs in toxicological experiments and efficiencies of the proposed optimal designs when the true rates have non-Weibull distributions. Optimal design strategies for ?nding multiple-objective designs in toxicology studies are outlined as well.