 2003 Royal Statistical Society 0035–9254/03/52261 Appl. Statist. (2003) 52, Part 3, pp. 261–278 Horwitz’s rule, transforming both sides and the design of experiments for mechanistic models Anthony C. Atkinson London School of Economics and Political Science, UK [Received November 2001. Revised February 2003] Summary. The paper develops methods for the design of experiments for mechanistic models when the response must be transformed to achieve symmetry and constant variance.The power transformation that is used is partially justified by a rule in analytical chemistry. Because of the nature of the relationship between the response and the mechanistic model, it is necessary to transform both sides of the model. Expressions are given for the parameter sensitivities in the transformed model and examples are given of optimum designs, not only for single-response models, but also for experiments in which multivariate responses are measured and for exper- iments in which the model is defined by a set of differential equations which cannot be solved analytically.The extension to designs for checking models is discussed. Keywords: Analytical chemistry; Box–Cox transformation; Chemical kinetics; Direct method; D-optimum design; Model checking; Parameter sensitivities 1. Introduction Thispaperisconcernedwiththedesignofexperimentsforthemechanisticmodelswhichtyp- icallyariseinpharmacokineticsandchemicalkinetics.Itiscustomarytoconvertsuchmodels intostatisticalmodelsbyincludingadditiveindependenterrorsofconstantvariance.Thispaper presentsevidencethatthisisofteninappropriate,thevarianceincreasingwiththemean. Thevariancecanbestabilized,andtheerrordistributionmadesymmetrical,bytransform- ingtheresponse.But,withthemechanisticmodelsthatareconsideredhere,itisnecessaryto transformbothsidesofthemodel.Theoptimumdesignwillthendependonthetransformation thatisappropriateaswellasonthemodel. Theinformationmatrixoftheparameterestimatesisafunctionoftheparametersensitivi- ties,i.e.thederivativesoftheresponsewithrespecttotheparameters.Asimpleexpressionis obtainedforthesensitivitiesforthetransformationmodel.However,manymechanisticmodels are sets of differential equations which cannot be solved analytically. Differential equations forthesensitivitiescan,however,bederived,whicharethensolvednumericallyalongwiththe equationsforthemodel. Thepaperstartswiththesimpleexampleofdesignsforestimatingthesingleparameterin exponential decay and shows how the design changes if, because of the error structure, it is appropriatetoworkwiththelogarithmsoftheobservations.Itisshownthatthelog-transfor- mation,whichissometimesusedinpharmacokinetics,e.g.Mentr´ e etal. (1997)orStroud etal. (2001),page354,givesasillydesigninthisexample.Section3presentsevidencethatconstant varianceissometimesaninappropriatemodelfortheerrors.PartofthisevidenceisHorwitz’s Address for correspondence: Anthony C. Atkinson, Department of Statistics, London School of Economics andPoliticalScience,HoughtonStreet,London,WC2A2AE,UK. E-mail:a.c.atkinson@lse.ac.uk