Optimal Split Plot Overview

g = Xb + Zg + ε

where X represents the n×p fixed design matrix containing the settings of both the whole-plot factors w and the sub-plot factors s and their model expansions, b 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, g 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.

The following assumptions are made:

E(ε) = 0n and cov(ε) = s2ε ln,

E(g) = 0b and cov(g) = s2g lb,

cov(g,ε) = 0bxn

Under these assumptions, the covariance matrix of the responses, var(g), is:

V = s 2ε ln + s2g ZZ'

The generalized least squares estimator is:

= (X'V-1X)-1X'V-1g,

with covariance matrix:

var( ) = (X'V-1 X)-1

The information matrix on the unknown fixed parameters b is given by

M = X'V -1X

The D-optimality criterion seeks to maximize the determinant of this information matrix.

This is the criterion has been used for constructing split-plot designs by STATISTICA’s optimal split plot design module. For more detailed information see Bradley and Goos (2006).