Use of Doehlert Designs for Second-order Polynomial Models

L.Rob Verdooren

doi:10.13189/ms.2017.050202

Abstract

The most popular designs for fitting the second-order polynomial model are the central composite designs of Box and Wilson [2] and the designs of Box and Behnken [1]. For k = 2, 4, 6 and 8, the uniform shell designs of Doehlert [4] require fewer experimental runs than the central composite or Box-Behnken designs. In analytic chemistry the Doehlert designs are widely used. The uniform shell designs are based on a regular simplex, this is the geometric figure formed by k + 1 equally spaced points in a k -dimensional space; an equilateral triangle is a two-dimensional regular simplex. The shell designs are used for fitting a response surface to k independent factors over a spherical region. Doehlert (1930 - 1999) proposed in 1970 the design for k = 2 factors starting from an equilateral triangle with sides of length 1, to construct a regular hexagon with a centre point at (0, 0). The n = 7 experimental points are (1, 0), (0.5, 0.866), (0, 0), (-0.5, 0.866), (-1, 0), (-0.5, -0.866) and (0.5, -0.866).The 6 outer points lie on a circle with a radius 1 and centre (0, 0). This Doehlert design has an equally spaced distribution of points over the experimental region, a so-called uniform space filler, where the distances between neighboring experiments are equal. Response surface designs are usually applied by scaling the coded factor ranges to the ranges of the experimental factors. The first factor covers the interval [-1, + 1], the second factor covers the interval [-0.866, + 0.866]. Doehlert design for four factors needs only 21 trials. Doehlert and Klee [5] show how to rotate the uniform shell designs to minimize the number of levels of the factors. Most of the rotated uniform shell designs have no more than five levels of any factor; the central composite design has five levels of every factor. The D-Optimality determinant criterion of the variance matrix of Doehlert designs will be compared with central composite designs and Box-Behnken designs, see Rasch et al. [6].

Highlights

The most popular designs for fitting the second-order polynomial model are the central composite designs of Box and Wilson [2] and the designs of Box and Behnken [1].For k = 2, 4, 6 and 8, the uniform shell designs of Doehlert [4] require fewer experimental runs than the central composite or Box-Behnken designs
The uniform shell designs are based on a regular simplex, this is the geometric figure formed by k + 1 spaced points in a k– dimensional space; an equilateral triangle is a two-dimensional regular simplex
The shell designs are used for fitting a response surface to k independent factors over a spherical region

Summary

Introduction

The most popular designs for fitting the second-order polynomial model are the central composite designs of Box and Wilson [2] and the designs of Box and Behnken [1]. The second-order polynomial model for k = 2 factors and n experimental units is: E(yi ) = β0 + β1 X1i + β2 X2i + β11 X1i2 + + β22 X2i2 + β12 X1i X2i (i = 1, 2, ..., n). Doehlert (1930 – 1999) proposed in 1970 the design for k = 2 factors X1 and X2, starting from an equilateral triangle with sides of length 1, to construct a regular hexagon with a centre point at (0,0). The 6 outer points lie on a circle with a radius 1 and the inner point is the centre (0, 0) This Doehlert design has an spaced distribution of points over the experimental region, a so-called uniform space filler, where the distances between neighboring experimental units are equal. All the points in the Doehlert design for k=2 factors are on a unit circle (in centered and scaled units).

Rotation of Doehlert Designs

Comparison of Number of Runs for Second Order Polynomial Designs

D-Optimality Criterion Comparison for Designs with 2 or 3 Factors