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2 Some theory 19NewThe sample variogram reported by ASReml has two forms depending on whetherthe spatial coordinates represent a complete rectangular lattice (as typical of afield trial) or not. In the lattice case, the sample variogram is calculated from thetriple (l ij1 , l ij2 , v ij ) where l ij1 = s i1 −s j1 and l ij2 = s i2 −s j2 are the displacements.As there will be many v ij with the same displacements, ASReml calculates themeans for each displacement pair l ij1 , l ij2 either ignoring the signs (default) orseparately for same sign and opposite sign (!TWOWAY), after grouping the largerdisplacements: 9-10, 11-14, 15-20, .... The result is displayed as a perspectiveplot (see page 205) of the one or two surfaces indexed by absolute displacementgroup. In this case, the two directions may be on different scales.Otherwise ASReml forms a variogram √ based on polar coordinates. It calculatesthe distance between points d ij = lij1 2 + l2 ij2 and an angle θ ij (−180 < θ ij < 180)subtended by the line from (0, 0) to (l ij1 , l ij2 ) with the x-axis. The angle can becalculated as θ ij = tan −1 (l ij1 /l ij2 ) choosing (0 < θ ij < 180) if l ij2 > 0 and(−180 < θ ij < 0) if l ij2 < 0. Note that the variogram has angular symmetryin that v ij = v ji , d ij = d ji and |θ ij − θ ji | = 180. The variogram presentedaverages the v ij within 12 distance classes and 4, 6 or 8 sectors (selected usinga !VGSECTORS qualifier) centred on an angle of (i − 1) ∗ 180/s (i = 1, ...s). Afigure is produced which reports the trends in ¯v ij with increasing distance foreach sector.ASReml also computes the variogram from predictors of random effects whichappear to have a variance structures defined in terms of distance. The variogramdetails are reported in the .res file.2.6 Inference: Fixed effectsIntroductionNewInference for fixed effects in linear mixed models introduces some difficulties.In general, the methods used to construct F -tests in analysis of variance andregression cannot be used for the diversity of applications of the general linearmixed model available in ASReml. One approach would be to use likelihood ratiomethods (see Welham and Thompson, 1997) although their approach is not easilyimplemented.Wald-type test procedures are generally favoured for conducting tests concerningτ . The traditional Wald statistic to test the hypothesis H 0 : Lτ = l for given