Designs For Quadratic Regression Research Articles

Restricted second-order designs on one on two concentric balls

Designs for quadratic regression on one and two concentric balls are considered in this paper. The balls are defined by The 2 k factorial structure with additional 2 axial points provides the basic framework for the experiment. Restrictions such as orthogonality, rotatability, etc. which induce desirable properties are imposed on the design and their effects on the variance of the quadratic model compared.

Φp-optimal second order designs for symmetric regions

Abstract Designs for quadratic regression on a cube, on a cube with truncated vertices and on a ball are studied in terms of a family of criteria, introduced by Kiefer (1974, 1975), that includes A-, D- and E-optimality. Both theoretical and numerical results on structure and performance are presented. In particular, D- and E-optimal designs are described and a procedure of construction of nearly robust (under variation of criterion) integer designs is suggested. Some examples are given for dimensions 4, 5 and 6.

Comparison of Box-Draper and D-Optimum Designs for Experiments with Mixtures

In continuation of the study of [10] and [5], designs for quadratic regression are considered when the possible values of the controlable variable are mixtures x = (x 1, x 2, …, x q + 1) of nonnegative components x i with Σ q + 1 1 x i = I. The “all-boas” design of Box and Draper [1] for guarding against cubic bias, and the design that is optimum with respect to the D-optimality criterion ignoring the posaibility of such bias, are compared in terms of the average and the maximum of the variance and bias functions of the fitted repression. The D-optimum design performs well in terms of the Box-Draper criterion unless the sample size is fairly large, and is superior in terms of maximum variance and bias.

Comparison of Box-Draper and D-Optimum Designs for Experiments with Mixtures

In continuation of the study of [10] and [5], designs for quadratic regression are considered when the possible values of the controlable variable are mixtures x = (x 1, x 2, …, x q + 1) of nonnegative components x i with Σ q + 1 1 x i = I. The “all-boas” design of Box and Draper [1] for guarding against cubic bias, and the design that is optimum with respect to the D-optimality criterion ignoring the posaibility of such bias, are compared in terms of the average and the maximum of the variance and bias functions of the fitted repression. The D-optimum design performs well in terms of the Box-Draper criterion unless the sample size is fairly large, and is superior in terms of maximum variance and bias.

Comparison of Simplex Designs for Quadratic Mixture Models

Designs for quadratic regression are considered when the possible values of the controlable variable are mixtures x = (x 1, x 2, …, x q + 1) of nonnegative components x i with Σ q + 1 1 x i = 1. The designs that are optimum with respect to the D-, A-, and E-optimality criteria are compared in their performance relative to these and other criteria. Computational routines for obtaining these designs are developed, and the geometry of optimum structures is discussed. Except when q = 2, the A-optimum design is supported by the vertices and midpoints of edges of the simplex, as is the case for the previously known D-optimum design. Although the E-optimum design requires more observation points, it is more robust in its efficiency, under variation of criterion: but all three designs perform reasonably well in this sense.

Optimal design: Variation in structure and performance under change of criterion

SUMMARY The advisability of comparing designs on the basis of several different criteria of goodness is discussed. As an illustration, designs for quadratic regression on a simplex are compared in terms of a family of criteria that includes those of A-, D- and E-optimality, as well as one of Box & Draper. Efficiency robustness under change of region, as well as of criterion, is considered. The dependence of structure of good designs on dimension is determined.