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2 Some theory 9Variance structures for the errors: R structuresThe vector e will in some situations be a series of vectors indexed by a factoror factors. The convention we adopt is to refer to these as sections. Thuse = [e ′ 1 , e′ 2 , . . . , e′ s] ′ and the e j represent the errors of sections of the data. For example,these sections may represent different experiments in a multi-environmenttrial (MET), or different trials in a meta analysis. It is assumed that R is thedirect sum of s matrices R j , j = 1 . . . s, that is,⎡⎤R 1 0 . . . 0 00 R 2 . . . 0 0R = ⊕ s j=1R j =. ⎢ . . .. . .,⎥⎣ 0 0 . . . R s−1 0 ⎦0 0 . . . 0 R sso that each section has its own variance structure which is assumed to be independentof the structures in other sections.A structure for the residual variance for the spatial analysis of multi-environmenttrials (Cullis et al., 1998) is given byR j = R j (φ j )= σj 2 (Σ j (ρ j )).Each section represents a trial and this model accounts for between trial errorvariance heterogeneity (σj 2 ) and possibly a different spatial variance model foreach trial.In the simplest case the matrix R could be known and proportional to an identitymatrix. Each component matrix, R j (or R itself for one section) is assumed tobe the direct product (see Searle, 1982) of one, two or three component matrices.The component matrices are related to the underlying structure of the data. If thestructure is defined by factors, for example, replicates, rows and columns, thenthe matrix R can be constructed as a direct product of three matrices describingthe nature of the correlation across replicates, rows and columns. These factorsmust completely describe the structure of the data, which means that1. the number of combined levels of the factors must equal the number of datapoints,2. each factor combination must uniquely specify a single data point.These conditions are necessary to ensure the expression var (e) = θR is valid.The assumption that the overall variance structure can be constructed as a direct