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In clinical trials, the response of a given subject often depends on theselected treatment as well as on some covariates. We study optimal approximatedesigns of experiments in the models with treatment and covariate effects. Weallow for the variances of the responses to depend on the chosen treatments,which introduces heteroscedasticity into the models. For estimating systems oftreatment contrasts and linear functions of the covariates, we extend knownresults on D-optimality of product designs by providing product designs thatare optimal with respect to general eigenvalue-based criteria. In particular,A- and E-optimal product designs are obtained. We then formulate a method basedon linear programming for constructing optimal designs with smaller supportsfrom the optimal product designs. The sparser designs can be more easilyconverted to practically applicable exact designs. The provided results and theproposed sparsification method are demonstrated on some examples.

Optimal experimental designs for treatment contrasts in heteroscedastic models with covariates