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ToxSci Advance Access published online on September 14, 2005
Toxicological Sciences, doi:10.1093/toxsci/kfi320
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© The Author 2005. Published by Oxford University Press on behalf of the Society of Toxicology. All rights reserved. For Permissions, please email: journals.permissions@oxfordjournals.org
Received May 17, 2005
Accepted September 7, 2005

Neurotoxicology

D-Optimal Experimental Designs to Test for Departure From Additivity in a Fixed-Ratio Mixture Ray

Todd Coffey 1, Chris Gennings 1*, Jane Ellen Simmons 2, and David W. Herr 2

1 Department of Biostatistics, Virginia Commonwealth University, Richmond, VA 23298 USA
2 NHEERL, ORD, U.S. E.P.A., Research Triangle Park, NC 27711 USA

* To whom correspondence should be addressed.
Chris Gennings, E-mail: gennings{at}hsc.vcu.edu


  Abstract

Traditional factorial designs for evaluating interactions among chemicals in a mixture may be prohibitive when the number of chemicals is large. Using a mixture of chemicals with a fixed ratio (mixture ray) results in an economical design that allows estimation of additivity or non-additive interaction for a mixture of interest. This methodology is extended easily to a mixture with a large number of chemicals. Optimal experimental conditions can be chosen that result in increased power to detect departures from additivity. Although these designs are used widely for linear models, optimal designs for nonlinear threshold models are less well known. In the present work, the use of D-optimal designs is demonstrated for nonlinear threshold models applied to a fixed-ratio mixture ray. For a fixed sample size, this design criterion selects the experimental doses and number of subjects per dose level that result in minimum variance of the model parameters and thus increased power to detect departures from additivity. An optimal design is illustrated for a 2:1 ratio (chlorpyrifos: carbaryl) mixture experiment. For this example, and in general, the optimal designs for the nonlinear threshold model depend on prior specification of the slope and dose threshold parameters. Use of a D-optimal criterion produces experimental designs with increased power, while standard non-optimal designs with equally-spaced dose groups may result in low power if the active range or threshold is missed.

Keywords: additivity; non-additivity; nonlinear threshold models; optimal designs.
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