Optimal designs for multiple treatments with unequal variances

The response of a patient in a clinical trial usually depends on both the selected treatment and some latent covariates, while its variance varies across the treatment groups. A general heteroscedastic linear additive model incorporating the treatment effect and the covariate effects is often used in such studies. In this paper, under D- and D_A-optimality criteria, it is shown that the product of an optimal treatment allocation and an optimal design for covariates is also optimal among all possible designs for this linear additive model. Moreover, the optimal treatment allocation is characterized by a unique set of solutions to a system of equations. The connection between D- and D_A-optimal designs is also revealed. Several examples are presented to illustrate the applications of the above results to some selected models.