Discussion of “Extreme Vertices Design of Mixture Experiments” by R. A. McLean and V. L. Anderson

Abstract

The extreme vertices designs are an important contribution to experiments with mixtures. The algorithm for locating the extreme vertices is not only most welcome but also a practical necessity for designs with more than four components. Even with four components, where it is still possible to visualize odd regions inside the basic tetrahedron, the rapid identification of all vertices generated by the constraints is very helpful. Because the constraints on the individual factors determine the design, the choice of constraints is crucial. Often in exploratory work the constraints cannot be set precisely and, in fact, may have to be estimated experimentally. In these cases, the initial observations might be limited to extreme vertices and the (a 1) dimensional centroid to see whether a feasible region of composition space has been established. In this way one might avoid taking too much data in unprofitable regions due to a poor choice of constraints. When the number of extreme vertices equals the number of components, the constrained factor space itself is a (n 1) simplex (if the vertex mixtures are considered as pseudocomponents). This occurs, for example, when the constraints on each component are only lower bounds. For these regions Scheffe’s(‘*‘) quadratic, special cubic, or simplex centroid lattices, which use the midpoints of all edges, will specify enough points for at least a quadratic fit and will space the points uniformly over the region. If some of the extreme vertex points tend to cluster (for example, as measured by the authors’ normalized distances) it might be expedient to see whether slight alterations of the constraints will cause the cluster to collapse to a single point. For example, the four-component region with constraints: