Optimal Design of Dose Response Experiments: A Model-Oriented Approach
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0092-86 I5/200I Copyright 0 2001 Drug Information Association Inc.
Printed in the USA. All rights reserved.
OPTIMAL DESIGN OF DOSE RESPONSE EXPERIMENTS: A MODEL-ORIENTED APPROACH* VALERIIV. FEDOROV,PHD, AND SERGEIL. LEONOV,PHD Biomedical Data Sciences, GlaxoSmithKline Pharmaceuticals, Collegeville, Pennsylvania
We discuss optimal experimental design issues f o r nonlinear models arising in dose response studies. The optimization is pegormed with respect to various criteria which depend on the Fisher information matrix. Special attention is given to models with a variance component that depends on unknown parameters. Key Words: Dose response; Nonlinear regression models; Optimal design of experiments; Variance depending on unknown parameters; Locally optimal designs
INTRODUCTION IN THIS PAPER WE present a compact overview of the classic optimal design methods, together with new extensions, which can be applied in dose response studies. For the purposes of this paper, it is convenient to introduce optimal design concepts via quantal dose response models. In the next section, we start with the simple examples of binary dose response models as a preamble to the more formal approach to optimal experimental designs. In the section on continuous models, we present examples of continuous dose response models, and formulate the optimal design problem for a general nonlinear regression model. Following that, we extend the classic results to models with a variance function that depends on unknown parameters. In this case, direct substitution of the estimated variance into the information matrix obtained from the simpler regression framework leads, in general, to nonoptimal designs. We show how to properly take the information contained in the variance component into account.
QUANTAL MODELS Often dose response studies can be considered within the framework of binary response experiments such as success-failure or dead-alive in toxicology studies. For such outcomes, quantal models provide a popular example. For a quantal model, a binary variable Y that depends on dose x is introduced,
*Presented at the FDMndustry Workshop “Statistically Sound Decision Making,” September 14-15, 2000, Bethesda, Maryland. Reprint address: Sergei Leonov, GlaxoSmithKline Pharmaceuticals, 1250 So. Collegeville Rd., Po Box 5089, Collegeville, PA, 194264989, E-mail: [email protected].
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Valerii V. Fedorov and Sergei L. Leonov
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Y = Y(x) =
1, if there is a response on dose x,
no response,
and the probability of response is modeled as
, I 1 is a given function, and 8 = (O0, where 0 I ~ ( x8) parameters. It is often assumed that
el,. . . ,
is a vector of unknown
andJ(x) are given functions of dose x. In many applications, including our examples below, ~ ( x 8) , is taken as a probability distribution function. However, there are cases when ~ ( x , 8) is not monotonic with respect to x (eg, in some oncology studies). In this paper, we
restrict ourselves t
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