Optimality criteria applied to certain response surface designs

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Virginia Polytechnic Institute and State University

The estimation of a particular matrix of coefficients of a second-order polynomial model was shown to be important in Response Surface Methodology (RSM). This led naturally to designing RSM experiments for best estimation of these coefficients as a primary goal. A design criterion, DS-optimality, was applied to several classes of RSM designs to find optimal choices of design parameters. Further, previous results on D-optimal RSM designs were extended. The designs resulting from the use of the two criteria were compared.

Two other design criteria were also studied. These were IV, the prediction variance of ŷ integrated over a region R, and IV*, sum of the variances of ∂ŷ/∂α again integrated over R. Three different choices of the region R were used. The object of the study was not only to identify optimal choices of design parameters, but also to compare the resulting designs with those obtained using the determinantal criteria.

An extension of a method for constructing D-optimal designs was used to construct DS-optimal central composite designs. This involved viewing the design points as having continuous weights. DS-best central composite designs were constructed either analytically or numerically for a fixed axial point distance. The results of previous work by other authors were extended for D-optimality by varying the axial point distance. Other design classes studied were Box-Behnken, equiradial, and some small composite designs.

The novel study of IV and the extended IV, called IV*, was done for each of the four design classes mentioned previously. The results of the study were presented graphically, or tabularly. The best designs according to IV and IV* were compared with the DS-best designs.

Composite designs performed well in all criteria, with the central composite designs performing best. The Box-Behnken and equiradial seemed to suffer from a lack of flexibility. The DS-best designs agreed well with the designs suggested by the IV* criteria.