Optimal Designs for Estimation of Optimum Mixture Under Darroch–Waller and Log-Contrast Models
01 Jan 2014-Vol. 1028, pp 137-147
TL;DR: In this article, the problem of finding optimum designs for the estimation of optimum mixture combination when the mean response is defined by (i) the additive quadratic mixture model due to Darroch and Waller (1985), and (ii) the quad ratic log-contrast model due by Aichison and Bacon-Shoane (1984).
Abstract: This chapter addresses the problem of finding optimum designs for the estimation of optimum mixture combination when the mean response is defined by (i) the additive quadratic mixture model due to Darroch and Waller (1985) and (ii) the quadratic log-contrast model due to Aichison and Bacon-Shoane (1984). Both the models have some advantage over Scheffe quadratic mixture model, in specific situations.
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TL;DR: For general optimality criteria, this article obtained criteria equivalent to $\Phi$-optimality under various conditions on ''Phi'' and showed that such equivalent criteria are useful for analytic or machine computation of ''phi''-optimum designs.
Abstract: For general optimality criteria $\Phi$, criteria equivalent to $\Phi$-optimality are obtained under various conditions on $\Phi$. Such equivalent criteria are useful for analytic or machine computation of $\Phi$-optimum designs. The theory includes that previously developed in the case of $D$-optimality (Kiefer-Wolfowitz) and $L$-optimality (Karlin-Studden-Fedorov), as well as $E$-optimality and criteria arising in response surface fitting and minimax extrapolation. Multiresponse settings and models with variable covariance and cost structure are included. Methods for verifying the conditions required on $\Phi$, and for computing the equivalent criteria, are illustrated.
736 citations
TL;DR: In this article, the problem of multilinear regression on the simplex has been studied and a sufficient condition for optimality is given, and a corrected version is given to the condition which Karlin and Studden (1966a) state as equivalent to optimality.
Abstract: This paper consists of new results continuing the series of papers on optimal design theory by Kiefer (1959), (1960), (1961), Kiefer and Wolfowitz (1959), (1960), Farrell, Kiefer and Walbran (1965) and Karlin and Studden (1966a). After disposing of the necessary preliminaries in Section 1, we show in Section 2 that in several classes of problems an optimal design for estimating all the parameters is supported only on certain points of symmetry. This is applied to the problem (introduced by Scheffe (1958)) of multilinear regression on the simplex. In Section 3 we consider optimality when nuisance parameters are present. A new sufficient condition for optimality is given. A corrected version is given to the condition which Karlin and Studden (1966a) state as equivalent to optimality, and we prove the natural invariance theorem involving this condition. These results are applied to the problem of multilinear regression on the simplex when estimating only some of the parameters. Section 4 consists primarily of a number of bounds on the efficiency of designs; these are summarized at the beginning of that section.
186 citations
TL;DR: In this article, a new form of expected response function involving log contrasts of the proportions is introduced for experiments with mixtures, and the advantages and disadvantages of log contrast models are discussed and illustrated in applications.
Abstract: SUMMARY A new form of expected response function involving log contrasts of the proportions is introduced for experiments with mixtures. The advantages and disadvantages of log contrast models are discussed and illustrated in applications. In particular, the parameters of the model are related to certain useful hypotheses about mixtures and some relevant contrasts are identified.
143 citations
TL;DR: In this article, a survey article on known results about analytic solutions and numerical solutions of optimal designs for various regression models for experiments with mixtures is presented, including polynomial models, models with homogeneous functions, models containing inverse terms and ratios, log contrast models, and models with quantitative variables, and mod els containing the amount of mixture.
Abstract: This is a survey article on known results about analytic solutions and numerical solutions of optimal designs for various regression models for experiments with mixtures. The regression models include polynomial models, models containing homogeneous functions, models containing inverse terms and ratios, log contrast models, models with quantitative variables, and mod els containing the amount of mixture, Optimality criteria considered include D-, A-, E-,φp- and Iλ-Optimalities. Uniform design and uniform optimal design for mixture components, and efficiencies of the {q,2} simplex-controid design are briefly discussed.
46 citations