A new design for experiments with mixtures of q components comprises the 2" -1 points representing the mixtures consisting of all possible subsets of the q components, present in equal proportions. A suitable regression equation is proposed and fitted. Designs and regression equations permitting the introduction of n "process variables" in addition to the q "mixture variables" are considered, and fractional replication of the designs in the case where the process variables are all at two levels.
THE SIMPLEX-CENTROID DESIGN. AN ASSOCIATED POLYNOMIAL
REGRESSION FUNCTIONIn the simplex-centroid design 2 q -l observations are taken, one on each of the following: the q pure components, the (~) binary mixtures with equal proportions, the (g) ternary mixtures with equal proportions, ... , and the q-nary mixture with equal proportions. This corresponds to the points (Xl' X2' ... , x q ) of the simplex (Ll) obtained by making the q permutations of (1,0,0, ,0), the (~) permutations of (t, t, 0, ... ,0), the (~) permutations of{t, t, t, 0, ... ,0), , and the point (1lq, llq, ... , llq).The design thus consists of the centroid of the simplex (Ll) and the centroids of all the lower-dimensional simplexes it contains.t Conjectured by Scheffe (1958), who proved it for m= 1,2,3; since proved in general by Savage (see footnote 6 in Kiefer reference) and Kiefer (1961, pp. 319-320). 1963] SCHEFFE -Simplex-Centroid Design for Experiments with Mixtures 237