Optimal designs for inverse prediction in univariate nonlinear calibration models

摘要

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.