Philip Prescott

In practical situations involving mixtures formed from several ingredients, interest is sometimes centered on the response in an ellipsoidal neighborhood around a standard formulation. We show that standard, orthogonally blocked, response surface designs, defined on a q − 1 dimensional unit sphere, may be transformed into similarly orthogonally blocked q-ingredient mixture designs defined within an ellipsoid centered at the standard formulation. The method is illustrated using several examples of mixture experiments with three, four, and five ingredients, arranged in two, three, or four orthogonal blocks, obtained by projecting standard central composite designs and Box–Behnken designs into the ellipsoidal mixture region. Rotations of the resulting designs within the ellipsoidal regions are also considered.

In mixture experiments, the settings of the predictor variables are limited by the fact that the ingredients are dependent. There may also be other, non-mixture, variables (process variables) to consider. Here, designs are considered for two types of second-order models in mixture and process variables, including certain mixture/process interactions. Several new balanced D-optimal (or nearly D-optimal) designs are proposed and are compared with quasi D-optimal designs and other balanced designs previously suggested. The properties of these new balanced designs are examined for the extended model with linear mixture by second-order process variable interactions; the designs are either D-optimal or nearly D-optimal and have diagonally partitioned information matrices, simplifying the model building process. An example from the bread industry illustrates the value of such designs.

Quadratic models, fitted to data arising from mixture experiments with highly constrained ingredient proportions, often have large estimated coefficients that bear little resemblance to the size of the observations. Many practitioners find this disconcerting. Various transformations have been suggested to overcome this problem, although the predictions are unaffected by re-parameterizing the original terms in the model. We propose a new transformation of the coordinate system to the largest simplex that can be fitted inside the restricted region defined by the mixture constraints. In some cases the largest simplex will not be unique, in which case the one containing as many design points as possible is recommended. Then, the estimated coefficients will be much closer to the observations and thus more acceptable to the experimenter.

Nearly uniform mixture designs using the centers of inscribed hyper-spheres within the mixture simplex are considered. The uniformity properties of these designs compare favorably with designs produced using more complicated methods. This is particularly so when the design region is the full or reduced mixture simplex produced when bounds are placed on the mixture ingredients. When the restricted region is not too different from a simplex, these designs may be projected into this restricted region to produce nearly uniform designs. When the bounds create an elongated restricted mixture region, an alternative method of generating nearly uniform mixture designs is described.