R or G-structures for split-plot designs?
by
Mario D'Antuono
Department of Agriculture, Western Australia

y=Xt+Zu+e,

where y denotes the (n ×1) vector of yield observations, t is a (p ×1) vector of fixed effects, X is an (n ×p) design matrix, u is a (q ×1) vector of random effects, Z is an (n ×q) design matrix which associates the yield observations with the appropriate combination of random effects, and e is the (n ×1) vector of residual errors. It is assumed that the distribution of the random effects is normal and takes the form

é ë u e ù û ~ N æ è é ë 0 0 ù û , \sigma2 é ë G(\gamma) 0 0 R(\phi) ù û ö ø ,

where the matrices G and R and are functions of the parameters \gamma and \phi respectively, with \sigma² a variance or scale parameter.

There are a number of ways of thinking of the split-plot model and how we model it using the above specifications. The correlations between the sub-plots and the main plots influence the form of R or G structure we need to consider. Case studies from agricultural field trials in Western Australia are presented to illustrate some of the practical considerations that may arise from modelling spatial correlations. Diagnostics for spatial correlations will also be discussed.

Date received: September 6, 2001