A candidate-set-free algorithm for generating D-optimal split-plot designs
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Introduction
Split-plot designs arise in experimental studies when a completely randomized run order
is structurally impossible, expensive or inconvenient. So, a statistician’s recommendation
to randomize the order of the runs is often ignored in practice. Instead the experimenter
rearranges the runs of the design so that hard-to-change factors only need to be changed a
few times over the course of the study. This rearrangement, though seemingly innocuous,
creates a split-plot structure. That is, the experiment is performed in groups of runs
where the hard-to-change factors stay constant within each group.



Statistical model and analysis
For a split-plot experiment with sample size n and b whole plots, the model can be written
as
y = Xβ + Zγ + ε, (1)
where X represents the n×p model matrix containing the settings of both the whole-plot
factors w and the sub-plot factors s and their model expansions, β is a p-dimensional
vector containing the p fixed effects in the model, Z is an n × b matrix of zeroes and
ones assigning the n runs to the b whole plots, γ is the b-dimensional vector containing
the random effects of the b whole plots, and ε is the n-dimensional vector containing the
random errors. It is assumed that


Theoretical example
Consider a problem with two whole-plot factors and five sub-plot factors. Suppose that
there are resources for 24 runs t