Some aspects of mixture and response surface designs.

Abstract

Many products are manufactured by mixing two or more ingredients together. If the proportion of the i-th component is $x\sb{i}$ (i = 1, 2, ..., q), then they have to satisfy the constraints $x\sb{i} \ge$ 0 and $x\sb1$ + $x\sb2$ + ... + $x\sb{q}$ = 1. Gasoline blends, paints, cake mixes, concrete textile fiber blends and drugs are some examples of such processes. This constraint makes mixture designs and their analysis unique. This dissertation develops topics adding to the theoretical basis of mixture designs including response surface designs. Primarily, the Scheffe model of mixture designs is extended to include block effects. The analysis of mixture experiments in incomplete block with and without recovery of interblock information is developed. A solution to the problem of combining two independent sets of estimates, with one set providing estimates of all parameters while the other set provides estimates of only a few linear functions of the parameters, in the context of mixture designs, is provided. We also consider robustness of mixture designs with respect to missing values and remedial techniques such as reinforcement. The optimality property known as rotatability, which is very attractive in cases of response surface designs, is not necessarily so in the case of mixture designs. This is investigated by introducing a new measure of rotatability which is simpler and easier to use than an existing measure due to Khuri. This measure is related to the moment generating function of the design matrix.