Exchange Algorithm for Generating D-optimal RS Split-

RS Split-plot Designs

Several papers have been published in recent years advocating the use of the D-optimal designs and presenting exchange algorithms for generating D- optimal RS split-plot designs. For a split-plot design the HTC factor levels need to be constant for several experimental runs while the ETC factor lev- els may change from run to run. This creates a difference when developing a search algorithm for generating a RS split-plot design although the basic technique is the same as described in the last section. Another issue is that for a RS split-plot design the D-optimality criterion depends on the model matrix X and the variance-covariance matrix Σ. This requires that the ex- perimenter must have determined the RS model to be fitted and should also have some knowledge about the values for the variance components. These values for the variance components can be obtained if some experiments were executed previously for the same research problem or by personal experience or judgement of the experimenter. As a result, it is quite possible to obtain different optimal designs for different values of the variance components.

Goos & Vandebroek (2001)[14] proposed a point exchange algorithm for generating a RS split-plot design with n experimental runs without impos- ing any restriction on the numbers of whole-plots or a fixed whole-plot size although the authors claimed that such conditions can easily be included in their algorithm. They proposed a search algorithm which takes into account the presence of the variance-covariance matrix Σ in the optimality criterion but it does not enforce the constant HTC factor levels for some consecu- tive runs. The authors showed that, for a split-plot design, the information matrix M can be written as

M = X0Σ−1X = 1 σ2  wp X i=1 ki X j=1 f (wi, sij)f0(wi, sij) − wp X i=1 η 1 + kiη (X0_i1ki)(X 0 i1ki) 0 ! , (3.11) where ki is the ith whole-plot size, η = σγ2/σ2 is the ratio of the variance

components, Xi is the model matrix that belongs to the whole-plot i, and

whole-plot and the HTC factor level settings are expressed by wi whereas

the ETC factor level settings are expressed by sij. The expression in (3.11)

makes it easier to update the information matrix after each exchange of the design points. Because this algorithm does not have any restriction on the number of whole-plots and sub-plots within each whole-plot, the generated design does not necessarily look like a split-plot design with the levels of the HTC factors constant for some consecutive runs. But different treatments having the same combinations of the HTC factor levels can then be grouped together to form the whole-plots of the experiment.

Later Goos & Vandebroek (2003)[15] gave another point exchange algo- rithm for generating a RS split-plot design with n experimental runs with given numbers of whole-plots and sub-plots within each whole-plot. The first stage of their algorithm is to generate a starting design. The starting design is generated by first randomly assigning the different settings, at least as many as the number of the pure whole-plot model parameters, of the HTC factors to the whole-plots. Then a starting design is generated by using a forward procedure i.e. first a random number of design points are selected at random from the set of candidate points and each of these design points is allocated to the whole-plots with the corresponding whole-plot factor set- tings and then the candidate points with the largest prediction variances are sequentially added to complete the design. The improvement is made using three different techniques. First by exchanging the design points with the candidate points that have the same whole-plot factor settings. Then design points are exchanged between whole-plots having same settings of the whole- plot factors. And finally improvement is made by exchanging the whole-plot factor settings between whole-plots. If an improvement is recorded, these exchange processes are repeated until there is no further improvement in the D-criterion value. A design generated by this approach is given in table 3.4. Jones & Goos (2007)[18] presented a coordinate exchange algorithm for generating a RS split-plot design. An advantage of their approach is that it does not require a pre-specified set of candidate points instead it generates a starting design by randomly choosing the required levels for the HTC and the ETC factors. Since our algorithm is based on this approach, we will discuss this algorithm in more detail in the next chapter. Arnouts & Goos (2010)[1] gave mathematical formula for fast updating the determinant of the information matrix for coordinate exchange algorithm for generating split-

Table 3.4: A D-optimal design for 1 HTC and 4 ETC factors given by Goos & Vandebroek (2003). Here W represents the HTC factor and Si denotes

the ith ETC factor.

Whole plot W1 S1 S2 S3 S4 Whole plot W1 S1 S2 S3 S4

1 -1 1 -1 -1 1 12 0 0 -1 1 1 1 -1 1 1 1 -1 12 0 1 1 1 -1 2 -1 -1 0 -1 -1 13 1 -1 -1 1 -1 2 -1 1 1 -1 1 13 1 1 1 -1 -1 3 -1 -1 1 1 1 14 1 -1 0 0 1 3 -1 1 1 -1 -1 14 1 1 1 1 1 4 -1 -1 -1 1 1 15 1 -1 -1 -1 -1 4 -1 1 -1 -1 -1 15 1 0 1 1 -1 5 -1 -1 -1 1 -1 16 1 -1 -1 1 1 5 -1 0 1 0 0 16 1 1 1 -1 1 6 -1 -1 1 -1 1 17 1 0 -1 -1 1 6 -1 1 -1 1 0 17 1 1 -1 1 0 7 -1 -1 -1 -1 1 18 1 -1 1 -1 -1 7 -1 -1 1 1 -1 18 1 1 0 0 1 8 -1 -1 1 -1 -1 19 1 -1 0 0 -1 8 -1 1 -1 1 -1 19 1 1 -1 -1 1 9 -1 -1 -1 0 0 20 1 -1 1 1 1 9 -1 1 0 1 1 20 1 1 -1 1 -1 10 0 0 0 -1 0 21 1 -1 1 -1 1 10 0 1 -1 0 1 21 1 1 -1 -1 -1 11 0 0 -1 0 -1 11 0 1 0 -1 0

plot designs. But none of the above mentioned algorithms for generating the D-optimal RS split-plot designs allows one to pre-specify the numbers of degrees of freedom for estimating the pure error variance components.