On optimal designs and complete class theorems for experiments with continuous and discrete factors of influence
Consider a linear model: \(EX(i,t)=\beta_ j(1)+a'(t)\beta (2)\), where \(i\in \{1,...,I\}=:T^{(1)}\) denotes the discrete factor level, \(t\in T^{(2)}\subset R^ K\) the continuous factor combination, \(a'=(a_ 1,...,a_ p)\) the vector of known regression functions, \(\beta (1)\in R^ I\) the unknown vector of treatment effects and \(\beta\) (2) the unknown vector of regression parameters. \textit{V. Kurotschka} [A general approach to optimum designs of experiments with qualitative and quantitative factors. Proc. Indian Stat. Inst. Golden Jubilee Int. Conf. Stat., Appl. New Directions, Calcutta, 353-368 (1981)] dealt with the optimum design problem for experiments with qualitative and quantitative factors in terms of generalized designs. There the case of complete interaction between discrete and continuous factors has been solved by reducing it in some sense to optimum design problems for the linear regression term, whereas for the case of no interaction between discrete and continuous factors he uses heuristical arguments to restrict attention to the class of product designs. In this paper, for the D-optimality criterion the heuristical arguments are replaced by rigorous proofs. 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 an open problem concerning A- optimality.
experiments with qualitative and quantitative factors
complete interaction between discrete and continuous factors
scientific article; zbMATH DE number 3986467
scientific article