A response surface approach to the mixture problem when the mixture components are categorized
A method is developed for experiments with mixtures where the mixture components are categorized (acids, bases, etc.), and each category of components contributes a fixed proportion to the total mixture. The number of categories of mixture components is general and each category will be represented in every mixture by one or more of its member components.
The purpose of this paper is to show how standard response surface designs and polynomial models can be used for estimating the response to mixtures of the k mixture components. The experimentation is concentrated in an ellipsoidal region chosen by the experimenter, subject to the constraints placed on the components. The selection of this region, the region of interest, permits the exclusion of work in areas not of direct interest.
The transformation from a set of linearly dependent mixture components to a set of linearly independent design variables is shown. This transformation is accomplished with the use of an orthogonal matrix. Since we want the properties of the predictor ŷ at a point w to be invariant to the arbitrary elements of the transformation matrix, we choose to use rotatable designs.
Frequently, there are underlying sources of variation in the experimental program whose effects can be measured by dividing the experimentation into stages, that is, blocking the observations. With the use of orthogonal contrasts of the observations, it is shown how these effects can be measured. This concept of dividing the program of experiments into stages is extended to include second degree designs.
The radius of the largest sphere, in the metric of the design variables, that will fit inside the factor space is derived. This sphere provides an upper bound on the size of an experimental design. This is important when one desires to use a design to minimize the average variance of ŷ only for a first-degree model. It is also shown with an example how with the use of the largest sphere, one can cover almost all combinations of the mixture components, subject to the constraints.