Use of Doehlert Designs for Second-order Polynomial Models

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].


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].
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 kdimensional 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.
There are npar = 6 parameters and to obtain the least squares estimators for the parameters, the factors X 1 and X 2 must have at least three different levels for the second-order polynomial and the number of experimental units n must be larger than 6, the number of parameters npar. Doehlert (1930Doehlert ( -1999 proposed in 1970 the design for k = 2 factors X 1 and X 2 , starting from an equilateral triangle with sides of length 1, to construct a regular hexagon with a centre point at (0,0). David H. Doehlert (1930Doehlert ( -1999 The n = 7 experimental units (points) of the Doehlert design with k=2 factors. X 1 X 2 X 1 X 2 cos (0) sin(0) 1 0 cos(π/3) sin(π/3) 0.5 0.866 cos(2π/3) sin(2π/3) -0.5 0.866 cos(π) sin(π) -1 0 cos(4π/3) sin(4π/3) -0.5 -0.866 cos(5π/3) sin(5π/3) 0.5 -0.866 0 0 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 equally spaced distribution of points over the experimental region, a so-called uniform space filler, where the distances between neighboring experimental units 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].
If the desired results are not found in the first study domain (A), the domain can be extended in the direction where the most of the desired points are likely to be. Just add three experimental units to find a new Doehlert design (B) which forms again a regular hexagon, and further add again three experimental units to find another new Doehlert design (C).
All the points in the Doehlert design for k=2 factors are on a unit circle (in centered and scaled units). The domain defined by Doehlert designs is spherical: a circle in two dimensions, a sphere in three dimensions for k=3 factors, a hypersphere in more than three dimensions for k>3 factors.

Rotation of Doehlert Designs
The Doehlert designs can be rotated to change the design region. Furthermore the symmetry of the design matrix can be changed by a rotation of the design.
The rotation will be demonstrated for k = 2 factors X 1 and X 2 with a regular hexagon design. We rotate over an angle of π/12 clockwise. This means that the coordinates (X 1 , X 2 ) are multiplied with the matrix Afterwards we want that the range of the new coordinates is [-1, +1], hence we must multiply the rotated coordinates with 1/0.96593 = 1.035272 .

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Use of Doehlert Designs for Second-order Polynomial Models 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. Note that the central composite design has five levels of every factor.

Comparison of Number of Runs for Second Order Polynomial Designs
In the following table the number of factors k=2, 3, 4 and 5 is given with the number of parameters npar of the second-order polynomial, the minimum number n of the experimental units for the Doehlert design, the Central Composite Design (CCD) when it is rotatable and non-orthogonal (CCD1) and for the Central Composite Design when it is rotatable and orthogonal (CCD2). (If the variance of the estimator of E(y) at all points from the centre point is constant then the design is called rotatable. If the columns of the design matrix X in E(y) = Xβ are orthogonal, the design is called orthogonal.) Further for k = 3, 4 and 5 the minimal n is given for the Box

D-Optimality Criterion Comparison for Designs with 2 or 3 Factors
To investigate optimality criterion of the design, the 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]. The criterion of the D-optimality requires the minimization of the determinant of the variance-covariance matrix of the estimator of the vector of regression coefficient, which equals σ 2 (X T X) -1 . We put without loss of generality σ 2 = 1 and use besides critI = det[(X T X) -1 ] which has to be minimized, a modified criterion critII = {det[(X T X) -1 ]} 1/k which has to be maximized. The resulting design is of course the same in both cases; the second criterion is easier to handle. We use however a criterion proposed by Draper and Lin [3]: critIII = critII/n = {det [(X T X) -1 ]} 1/k /n.
When we compare for k = 2 factors the CCD1 (n = 9) with a Doehlert design, where we add two extra centre points (0, 0) to get also n = 9 experimental units, we have for this Doehlert design with n = 9 critI = 1/91.10362 = 0.010977 and critIII = 1.060536. This Doehlert design is rotatable.