Mixture Component-Amount Designs via Projections, Including Orthogonally Blocked Designs

Abstract

Traditional mixture experiments are designed to investigate the effects of changing the relative proportions of mixture ingredients using the same total amount, usually coded to 1, or to 100%. In some studies however, such as those on fertilizers, the total amount used also affects the response and needs to be investigated as well. For such component-amount experiments, several levels of total amount are needed. Two ways of obtaining suitable designs are given here. Both ways use the idea of projecting known mixture designs into fewer dimensions.

First, standard designs of the simplex-lattice and simplex-centroid types are projected to obtain sets of points at various levels of total amount for two or more ingredients. The sets occur with varying multiplicities, which can be varied to produce D-optimal choices, if desired. Second, orthogonally blocked (for standard second order mixtures models) designs that have previously been constructed by combining Latin squares are projected down into lower dimensions. These projected designs are also orthogonal to specific component-amount models suggested in the literature for fitting data from component-amount designs, with proper choice of blocking variables. Moreover, the specific designs of this type that are D-optimal in four dimensions for a second-order Scheffé model remain D-optimal for the corresponding second-order component-amount model after projection into three dimensions.

A previously published bread-making experiment that employed an orthogonally blocked design for four mixture ingredients provides data that are re-analyzed. We show a component-amount equation that performs well can be fitted in only two components, while the orthogonal blocking feature for the original design and mixture model is preserved for the projected design and the mixture-amount model.

• Latin Squares
• Mixture Experiments
• Orthogonal Blocks
• Quadratic Response Surface Models
• Scheffé Quadratic Model

Additional information

Notes on contributors

Philip Prescott

Dr. Prescott is Professor of Statistics at the University of Southampton, UK. His email address is [email protected].

Norman R. Draper

Dr. Draper is Emeritus Professor of Statistics at the University of Wisconsin, Madison, Wisconsin, USA and is a Fellow of the American Society for Quality.