A Comparison between Two Ten-Point Designs for Studying Three-Component Mixture Systems

Abstract

Three-component mixture experiments are very common. Reasons for their popularity are: (i) ease in selecting the component combinations to run experiments on, (ii) ease in fitting empirical models to the data collected, as well as interpreting the coefficients in the fitted models, and (iii) ease in displaying the surface shape characteristics of the particular response of interest by plotting surface contours or the response surface itself. In this paper we discuss the selection of a design for a three-component mixture experiment (reason (i) above). Two ten-point design arrangements are suggested. One is the {3, 3} simplex-lattice arrangement consisting of nine points equi-spaced on the perimeter of the triangular composition space with one point at the centroid, while the other design is the simplex-centroid design augmented with three interior points. The two design patterns are compared using the properties (variance and bias) of fitted mixture models, starting with the Scheffé first-degree model and building up to the special cubic model. As anticipated, if the shape of the response surface is best described by the use of quadratic or cubic terms in the model involving pairs of components, then the {3, 3} simplex-lattice is preferable. On the other hand, if the shape of the response surface inside the triangle (the response to complete blends) is vastly different from the shape of the surface near the boundaries of the triangle, higher degree terms such as β₁₂₃x₁x₂x₃ or γ₁₂₂₃x₁x²₂x₃ are needed in the model, and the augmented simplex-centroid design must be used. Data from an artificial sweetener experiment are used to illustrate the differences between the types of response surfaces just mentioned, and summary recommendations are made for selecting one design over the other for particular mixture systems.

• Binary Mixtures
• Complete Mixtures
• Nonlinear Blending
• Polynomial Model
• Response Surface
• Simplex Factor Space

Additional information

Notes on contributors

John A. Cornell

Dr. Cornell is a Professor in the Department of Statistics. He is a Member of ASQC.