# D-optimal design for estimation of optimum mixture in a three-component mixture experiment with two responses

TL;DR: In this article, the primary response function was assumed to be quadratic in the mixing proportions, while the secondary response function is assumed to have a non-quadratic response function.

Abstract: The paper studies a mixture experiment with two responses – a primary response and a secondary response. The primary response function is assumed to be quadratic in the mixing proportions, while th...

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TL;DR: In this paper, the authors present the theory and development of an algorithm associated with the exploration of a dual response surface system, where a user can generate simple two dimensional plots to determine the conditions of constrained maximum primary response regardless of the number of independent variables in the system.

Abstract: The purpose of this paper is to present the theory and develop an algorithm associated with the exploration of a dual response surface system. The approach is to find conditions on a set of independent or “design” variables which maximize (or minimize) a “primary response” function subject to the condition that a “constraint response” function takes on some specified or desirable value. A method is outlined whereby a user can generate simple two dimensional plots to determine the conditions of constrained maximum primary response regardless of the number of independent variables in the system. He thus is able to reduce to simple plotting the complex task of exploring the dual response system. The procedure that is used to generate the plots depends on the nature of the individual univariate response functions. In certain situations it becomes necessary to apply the additional constraint that the located operating conditions are a certain “distance” from the origin of the independent variables (or the cent...

280 citations

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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.

172 citations

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TL;DR: In this paper, the authors considered quadratic regression with mixtures of nonnegative components and compared the designs that are optimum with respect to the D-, A-, and E-optimality criteria in their performance relative to these and other criteria.

Abstract: Designs for quadratic regression are considered when the possible values of the controlable variable are mixtures x = (x 1, x 2, …, x q + 1) of nonnegative components x i with Σ q + 1 1 x i = 1. The designs that are optimum with respect to the D-, A-, and E-optimality criteria are compared in their performance relative to these and other criteria. Computational routines for obtaining these designs are developed, and the geometry of optimum structures is discussed. Except when q = 2, the A-optimum design is supported by the vertices and midpoints of edges of the simplex, as is the case for the previously known D-optimum design. Although the E-optimum design requires more observation points, it is more robust in its efficiency, under variation of criterion: but all three designs perform reasonably well in this sense.

63 citations

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TL;DR: In this paper, conditions under which an exact D-optimal design for uncorrelated observations with common variance is also D -optimal for correlated observations were established for regression models with multiple response.

Abstract: In the general linear model we set conditions under which an exact D-optimal design for uncorrelated observations with common variance is also D-optimal for correlated observations. Further we determine conditions under which approximate D-optimal designs can be considered as approximate D-optimal designs for correlated observations. Then these results are applied to a regression model with multiple response generalizing Theorem 1 of Krafft and Schaefer (J. Multivariate Anal. 42, 1992). In the above context, however, a serious problem may arise if the covariance matrix is not known; for the Gauss-Markov estimator with respect to a D-optimal design does not need to be calculable for the correlated case. This leads to D-optimal-invariant designs introduced by Bischoff (Ann. Inst. Statist. Math., 44, 1992); such a design τ∗ remains D-optimal when the covariance matrix is changed, and additionally the Gauss-Markov estimator with respect to the design τ∗ stays fixed. For regression models with multiple response we determine classes of covariance matrices for which a D-optimal design for uncorrelated observations with common variance is D-optimal-invariant. As examples we consider linear models where each response belongs to a regression model with intercept term.

43 citations

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TL;DR: In this paper, the authors try to find optimum designs for the estimation of optimum mixture combination on the assumption that the response function is quadratic concave over the simplex region.

Abstract: In a mixture experiment, the measured response is assumed to depend only on the relative proportion of ingredients or components present in the mixture. Scheffe [1958. Experiments with mixtures. J. Roy. Statist. Soc. B 20, 344–360; 1963. Simplex—centroid design for experiments with mixtures. J. Roy. Statist. Soc. B 25, 235–263] first systematically considered this problem and introduced different models and designs suitable in such situations. Optimum designs for the estimation of parameters of different mixture models are available in the literature. However, in a mixture experiment, often one is more interested in the optimum proportion of ingredients. In this paper, we try to find optimum designs for the estimation of optimum mixture combination on the assumption that the response function is quadratic concave over the simplex region.

28 citations