Mixture Experiment Design Study Of A Number Of Issues

Regression models for experiments with mixtures can be classified according to whether theresponse depends only on the relative proportions of the mixture components but not the actualamount of the mixture, or depends on both. The first type of model is called a mixture model,while the second type is called a mixture amount model. Literature on classical mixture experi-ment is assumed the exact correctness of relationship between the response variable and the designvariables. In practice, the assumed model is likely to be only a reasonable approximation to thetrue model for the response. So, a design should be chosen such that the fitted model is robust tothe exact form of the true model in some sense. In chapter 2, we will consider the case in whichthe true response is given by the mixture model with an unknown random bias, and derive thecriteria to obtain the robust optimal designs. Few results are available on multi- response modelsfor experiments with mixtures, other than Smith and Cornell (1993), so in chapter 3 and 4, weconsider this problem.In chapter 2, we assume the true response for mixture experiments is given by f(x) =βTg(x) + h(x), whereβTg(x) is the mixture model and the bias h(x) is an unknown randomfunction with mean zero and covarianceτ2K(x,t) between h(x) and h(t). The estimate of theparameters are obtained by the standard least squares method. Taking the average mean squarederror of the LSEβ? as the loss function, we develop a design criterion for getting robust designs.The numerical results of scheffe′'s linear model with misspecification show that the design pointslook like more and more uniformly scattered on the simplex as r decreases from 1 to 0. We alsofind that the efficiencies of all-bias designs and uniform designs have the similar tendency and theScheffe′'s simplex-lattice designs(all-variance designs) have a high efficiency when the true modelis approximately Scheffe′'s linear model. In the last section of this chapter we deal with the axialdesigns when there is bias from the assumed Scheffe′'s linear mixture model. We demonstratethat if the correlation function satisfies a special condition then the optimal axial design resultsin a Scheffe′'s {q, 1}-lattice, and three types of correlation functions satisfying the condition areillustrated.In chapter 3, the multiresponse models E(Yi(x)) =ηi(x) = fiT (x)βi (i = 1, 2) are consid-ered, whereηi is the Scheffe′'s linear model and quadratic model respectively. D- and A-optimaldesigns are investigated in this chapter. More precisely, we restrict ourselves to a particular typeof designξ? whose support points are x ? (1, 0,···, 0) and x ? (21, 21, 0,···, 0), and obtainthe weights r1? and r2? according to the D- and the A-optimal design criteria when the covariancematrixΣis known. The numerical results about the efficiencies of these optimal designs show that D- and A-optimal designs do not differ too much in some mixture experiments.In chapter 4, we investigate the mixture amount experiment in the spirit of Hilgers & Bauer(1995) and Heiligers & Hilgers (2003), who consider the mixture-amount design in so calledcomponent amount models, which are obtained from classical mixture setups by including termscapturing the total amount, simultaneously dropping the sidecondition on the proportions to sumup to one. While design optimality usually depends sensitively on the underlying model, we estab-lish a close relation between admissible mixture designs and admissible mixture amount designsin two response situations when the covariance matrixΣis known. The responses are additive andhomogeneous following Becker (1968).