Optimal Designs for Multi-Group Linear Models with Measurement Errors

Multi-group linear models with measurement errors are frequently employed in situations where covariates cannot be precisely measured, thereby compromising the validity of between-group comparisons. However, the study of experimental design theory for these models is currently at an underdeveloped stage. This paper is concerned with the problem of constructing locally c-, DA- and D-optimal designs of multi-group linear models with measurement errors for estimating parameters or contrasts in the model parameters. Equivalence theorems are established to confirm the optimality of the designs for such models under each criterion, and the generalization of Elfving’s theorem is proved to describe the geometrical characterization of locally c-optimal designs for such models. Furthermore, the locally D-optimal designs for a class of multi-group linear models with measurement errors can be explicitly determined. It is shown that the locally D-optimal design for such models is given by the product of the locally D-optimal designs for linear measurement error models corresponding to those groups. To illustrate these concepts, several examples are provided.