Optimal designs for inverse prediction in univariate nonlinear calibration models Nancy Franc ßois a, * , Bernadette Govaerts a , Bruno Boulanger b a Universite ´ catholique de Louvain, Institut de Statistique, Voie du Roman Pays 20, B-1348 Louvain-la-Neuve, Belgium b Eli Lilly, rue Granbonpre ´ 11, B-1348 Mont-Saint-Guibert, Belgium Received 3 December 2003; received in revised form 5 May 2004; accepted 18 May 2004 Available online 13 August 2004 Abstract Univariate calibration models are intended to link a quantity of interest X (e.g. the concentration of a chemical compound) to a value Y obtained from a measurement device. In this context, a major concern is to build calibration models that are able to provide precise (inverse) predictions for X from measured responses Y . This paper aims at answering the following question: which experiments should be run to set up a (linear or nonlinear) calibration curve that maximises the inverse prediction precisions? The well known class of optimal designs is presented as a possible solution. The calibration model setup is first reviewed in the linear case and extended to the heteroscedastic nonlinear one. In this general case, asymptotic variance and confidence interval formulae for inverse predictions are derived. Two optimality criteria are then introduced to quantify a priori the quality of inverse predictions provided by a given experimental design. The V I criterion is based on the integral of the inverse prediction variance over the calibration domain and the G I criterion on its maximum value. Algorithmic aspects of the optimal design generation are discussed. In a last section, the methodology is applied to four possible calibration models (linear, quadratic, exponential and four parameter logistic). V I and G I optimal designs are compared to classical D, V and G optimal ones. Their predictive quality is also compared to the one of simple traditional equidistant designs and it is shown that, even if these last designs have very different shapes, their predictive quality are not far from the optimal ones. Finally, some simulations evaluate small sample properties of asymptotic inverse prediction confidence intervals. D 2004 Elsevier B.V. All rights reserved. Keywords: Calibration; Inverse prediction; Nonlinear models; Optimal designs 1. Introduction In many scientific and industrial fields (e.g. biology, chemistry, pharmacy), univariate calibration models are developed to link a quantity of interest X (e.g. the concen- tration of a chemical compound in a product sample) to its measured value Y obtained from an analytical device (e.g. an optical density value). There are two steps in the calibration process. First, for a series of samples for which the value of X is known, the response Y is measured and a statistical model, the calibra- tion curve, is fitted to the data. Then, at a second stage, samples with unknown X values are analysed with the test method and the values of X’s are predicted from the responses Y’s by inverting the calibration curve equation. Univariate calibration has received a wide interest in the literature and the reader may refer to the general review of Osborne [1]. In many applications, the functional relating X and Y is linear over the calibration domain of interest and a simple linear statistical model is chosen as calibration curve. For more complicated analytical systems, the linear model may not be appropriate and nonlinear models are needed to approximate the phenomenon. It is for example the case in ligand-binding essays intended to support pharmacoki- netic and toxicokinetic assessments of macromolecules in pharmaceutical applications. The validation guideline paper of DeSilav et al. [2] present the four parameter logistic model as a common calibration curve in this area. The main concern of the analyst in this context is to get ‘‘good’’ inverse predictions at the second step of the cali- bration process and to be able to quantify their precision. If the model equation is supposed to be correct, the quality of the inverse prediction may be quantified by building a 0169-7439/$ - see front matter D 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.chemolab.2004.05.005 * Corresponding author. Tel.: +32-10-47-43-14; fax: +32-10-47-30-32. E-mail address: francois@stat.ucl.ac.be (N. Franc ßois). www.elsevier.com/locate/chemolab Chemometrics and Intelligent Laboratory Systems 74 (2004) 283 – 292