For general optimality criteria $\Phi$, criteria equivalent to $\Phi$-optimality are obtained under various conditions on $\Phi$. Such equivalent criteria are useful for analytic or machine computation of $\Phi$-optimum designs. The theory includes that previously developed in the case of $D$-optimality (Kiefer-Wolfowitz) and $L$-optimality (Karlin-Studden-Fedorov), as well as $E$-optimality and criteria arising in response surface fitting and minimax extrapolation. Multiresponse settings and models with variable covariance and cost structure are included. Methods for verifying the conditions required on $\Phi$, and for computing the equivalent criteria, are illustrated.

/pdf/general-equivalence-theory-for-optimum-designs-approximate-4vewaduwfi.pdf

General Equivalence Theory for Optimum Designs (Approximate Theory)

Response sarfxe methodology (RSM) is a collection of tools developed in the 1950s for the purpose of determining optimum operating conditions in applications in the chemical industry. This article reviews the progrrss of RSM in the general areas of experimental design and analysis and indicates how its role has been affected by advanccs in other fields of applied statistics. Current areas of research in RSM are highlighted. and areas for future research are discussed.

https://www.jstor.org/stable/pdfplus/1268813.pdf

Response surface methodology: 1966–1988

After stating the model and the design problem, we briefly present the results for regression design prior to the work of Kiefer and Wolfowitz. We then review the major results of Kiefer and Wolfowitz, particularly those on the theory of design, as well as the way the criterion has been extended to non-linear models. Finally, we discuss algorithms for constructing D-optimum designs.

D-Optimality for Regression Designs: A Review

Qualitative and quantitative experiment design for phenomenological models—a survey

A review of response surface methodology from a biometric viewpoint.

This paper consists of new results continuing the series of papers on optimal design theory by Kiefer (1959), (1960), (1961), Kiefer and Wolfowitz (1959), (1960), Farrell, Kiefer and Walbran (1965) and Karlin and Studden (1966a). After disposing of the necessary preliminaries in Section 1, we show in Section 2 that in several classes of problems an optimal design for estimating all the parameters is supported only on certain points of symmetry. This is applied to the problem (introduced by Scheffe (1958)) of multilinear regression on the simplex. In Section 3 we consider optimality when nuisance parameters are present. A new sufficient condition for optimality is given. A corrected version is given to the condition which Karlin and Studden (1966a) state as equivalent to optimality, and we prove the natural invariance theorem involving this condition. These results are applied to the problem of multilinear regression on the simplex when estimating only some of the parameters. Section 4 consists primarily of a number of bounds on the efficiency of designs; these are summarized at the beginning of that section.

/pdf/optimal-and-efficient-designs-of-experiments-b6t0kfmb4h.pdf

Corwin L. Atwood

Papers

Optimal and Efficient Designs of Experiments