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15 Examples 26115.6 Spatial analysis of a field experiment - BarleyIn this section we illustrate the ASReml syntax for performing spatial and incompleteblock analysis of a field experiment. There has been a large amountof interest in developing techniques for the analysis of spatial data both in thecontext of field experiments and geostatistical data (see for example, Cullis andGleeson, 1991; Cressie, 1991; Gilmour et al., 1997). This example illustrates theanalysis of ’so-called’ regular spatial data, in which the data is observed on alattice or regular grid. This is typical of most small plot designed field experiments.Spatial data is often irregularly spaced, either by design or because ofthe observational nature of the study. The techniques we present in the followingcan be extended for the analysis of irregularly spaced spatial data, though, largerspatial data sets may be computationally challenging, depending on the degreeof irregularity or models fitted.The data we consider is taken from Gilmour et al. (1995) and involves a fieldexperiment designed to compare the performance of 25 varieties of barley. Theexperiment was conducted at Slate Hall Farm, UK in 1976, and was designed asa balanced lattice square with replicates laid out as shown in Table 15.6. Thedata fields were Rep, RowBlk, ColBlk, row, column and yield. Lattice rowand column numbering is typically within replicates and so the terms specified inthe linear model to account for the lattice row and lattice column effects wouldbe Rep.latticerow Rep.latticecolumn. However, in this example lattice rowsand columns are both numbered from 1 to 30 across replicates (see Table 15.6).The terms in the linear model are therefore simply RowBlk ColBlk. Additionalfields row and column indicate the spatial layout of the plots.The ASReml input file is presented below. Three models have been fitted to thesedata. The lattice analysis is included for comparison in PATH 3. In PATH 1 weuse the separable first order autoregressive model to model the variance structureof the plot errors. Gilmour et al. (1997) suggest this is often a useful model tocommence the spatial modelling process. The form of the variance matrix for theplot errors (R structure) is given byσ 2 Σ = σ 2 (Σ c ⊗ Σ r ) (15.5)where Σ c and Σ r are 15 × 15 and 10 × 10 matrix functions of the column (φ c )and row (φ r ) autoregressive parameters respectively.Gilmour et al. (1997) recommend revision of the current spatial model basedon the use of diagnostics such as the sample variogram of the residuals (from