This paper reviews the literature on Bayesian experimental design. A unified view of this topic is presented, based on a decision-theoretic approach. This framework casts criteria from the Bayesian literature of design as part of a single coherent approach. The decision-theoretic structure incorporates both linear and nonlinear design problems and it suggests possible new directions to the experimental design problem, motivated by the use of new utility functions. We show that, in some special cases of linear design problems, Bayesian solutions change in a sensible way when the prior distribution and the utility function are modified to allow for the specific structure of the experiment. The decision-theoretic approach also gives a mathematical justification for selecting the appropriate optimality criterion.

/pdf/bayesian-experimental-design-a-review-1z4pqf25ac.pdf

Bayesian Experimental Design: A Review

The purpose of this article is to provide a survey of the various stages in the development of response surface methodology RSM. The coverage of these stages is organized in three parts that describe the evolution of RSM since its introduction in the early 1950s. Part I covers the period, 1951-1975, during which the so-called classical RSM was developed. This includes a review of basic experimental designs for fitting linear response surface models, in addition to a description of methods for the determination of optimum operating conditions. Part II, which covers the period, 1976-1999, discusses more recent modeling techniques in RSM, in addition to a coverage of Taguchi's robust parameter design and its response surface alternative approach. Part III provides a coverage of further extensions and research directions in modern RSM. This includes discussions concerning response surface models with random effects, generalized linear models, and graphical techniques for comparing response surface designs. Copyright © 2010 John Wiley & Sons, Inc.

Response Surface Methodology.

Response surface methodology

A tutorial on linear and bilinear matrix inequalities

The original work in response surface methodology (RSM) has been widely used in the chemical and process industries. Recent years have seen more widespread use and new developments in RSM. RMS activities since 1989 are reviewed, and areas of current and..

Response Surface Methodology: A Retrospective and Literature Survey

This article reports the application of the annealing algorithm to the construction of exact D−, I−, and G-optimal designs for polynomial regression of degree 5 on the interval [—1, l] and for the second-order model in two factors on the design space [—1, l] × [—1, 1]. Details of the perturbation scheme and the annealing schedules used are given, and the method of implementation is illustrated by means of a simple example. The algorithm is assessed by comparing its performance, in terms of computer time and effkiency, with the modified Fedorov procedure, and it is shown to be particularly effective in finding G-optimal designs. The salient features of the exact designs constructed in this study are also summarized.

The application of the annealing algorithm to the construction of exact optimal designs for linear-regression models

Phase I clinical trials are typically small, uncontrolled studies designed to determine a maximum tolerated dose of a drug which will be used in further testing. Two divergent schools have developed in designing phase I clinical trials. The first defines the maximum tolerated dose as a statistic computed from data, and hence it is identified, rather than estimated. The second defines the maximum tolerated dose as a parameter of a monotonic dose-response curve, and hence is estimated. We review techniques from both philosophies. The goal is to present these methods in a single package, to compare them from philosophical and statistical grounds, to hopefully clear up some common misconceptions, and to make a few recommendations. This paper is not a review of simulation studies of these designs, nor does it present any new simulations comparing these designs.

Competing designs for phase I clinical trials: a review.

A broad approach to the design of Phase I clinical trials for the efficient estimation of the maximum tolerated dose is presented. The method is rooted in formal optimal design theory and involves the construction of constrained Bayesian c- and D-optimal designs. The imposed constraint incorporates the optimal design points and their weights and ensures that the probability that an administered dose exceeds the maximum acceptable dose is low. Results relating to these constrained designs for log doses on the real line are described and the associated equivalence theorem is given. The ideas are extended to more practical situations, specifically to those involving discrete doses. In particular, a Bayesian sequential optimal design scheme comprising a pilot study on a small number of patients followed by the allocation of patients to doses one at a time is developed and its properties explored by simulation.

Bayesian optimal designs for Phase I clinical trials.

A statistical approach to the analytic hierarchy process with interval judgements. (I). Distributions on feasible regions

Publisher Summary This chapter presents several designs for nonlinear and generalized linear models. Optimum experimental designs for nonlinear models depend on the values of the unknown parameters and the problem of their construction is therefore necessarily more complicated than that for linear models. One approach to the problem of design for nonlinear models is to adopt a best guess for the parameters. The simplicity of optimum design for linear models is then recovered and, in particular, locally D -optimum designs for the precise estimation of all the parameters in the nonlinear model, and locally c -optimum designs for the precise estimation of specified linear combinations of these parameters, are readily constructed and their optimality confirmed. An alternative and, in a sense, a more realistic approach to this design problem is to introduce a prior distribution on the parameters and to incorporate this prior into appropriate design criteria, usually by integrating D - or c -optimality criteria over the prior distribution. Designs optimizing these criteria are termed “Bayesian optimum designs” and are usually constructed numerically.

Linda M. Haines

Papers

The application of the annealing algorithm to the construction of exact optimal designs for linear-regression models

Competing designs for phase I clinical trials: a review.

Bayesian optimal designs for Phase I clinical trials.

A statistical approach to the analytic hierarchy process with interval judgements. (I). Distributions on feasible regions

14 Designs for nonlinear and generalized linear models