On optimal designs and complete class theorems for experiments with continuous and discrete factors of influence

Abstract Linear models with one discrete factor (treatment) and several continuous factors are considered. Restriction to product designs, i.e. designs with equal regression design measure for all treatment levels facilitates the optimal design problem considerably. It is shown that the product designs form an essentially complete class with respect to D-optimality for inference on (i) the treatment effects (ii) the regression parameters (iii) the combined parameter vector. These theorems yield characterizations of optimal designs in terms of optimal designs for pure regression setups. Applications of the results for special regression functions are included as well as open problems concerning A-optimality.