D-optimal design for a model with interaction between a qualitative and a quantitative factor in the presence of random block effects

Abstract

Optimal experimental designs are developed for linear models with a qualitative and a quantitative factor when there is a random block effect and the regression parameters of the quantitative predictor may depend on the level of the qualitative factor. In particular, we present a characterization of D-optimal designs for randomly blocked polynomial regression experiments with measurements at baseline (zero) where the experimental settings and the corresponding weights are determined analytically. Surprisingly, the optimal designs within the quantitative predictor are not affected by the intraclass correlation (or, equivalently, the ratio of the variance components), while the proportion of baseline measurements increases smoothly from the optimal weight in a model without block effects to the optimal weights under the assumption that all block effects are nonrandom. Despite this dependence, we show that those limiting designs that are obtained when the intraclass correlation tends to zero or one (or, equivalently, the variance ratio tends to zero or infinity), respectively, are quite robust.