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2.5 Inference: Fixed effects
The variogram has been suggested as a useful diagnostic for assisting with the identification
of appropriate variance models for spatial data (Cressie, 1991). Gilmour et al. (1997)
demonstrate its usefulness for the identification of the sources of variation in the analysis
of field experiments. If the elements of the data vector (and hence the residual vector) are
indexed by a vector of spatial coordinates, si , i = 1, . . . , n, then the ordinates of the sample
variogram are given by
1
vij = [ẽi (si ) − ẽj (sj )]2 , i, j = 1, . . . , n; i 6= j
2
The sample variogram reported by ASReml has two forms depending on whether the spatial
coordinates represent a complete rectangular lattice (as typical of a field trial) or not. In
the lattice case, the sample variogram is calculated from the triple (lij1 , lij2 , vij ) where lij1 =
si1 − sj1 and lij2 = si2 − sj2 are the displacements. As there will be many vij with the
same displacements, ASReml calculates the means for each displacement pair lij1 , lij2 either
ignoring the signs (default) or separately for same sign and opposite sign (!TWOWAY), after
grouping the larger displacements: 9-10, 11-14, 15-20, .... The result is displayed as a
perspective plot (see page 235) of the one or two surfaces indexed by absolute displacement
group. In this case, the two directions may be on different scales.
Otherwise ASReml forms
q a variogram based on polar coordinates. It calculates the distance
2
2
+ lij2
and an angle θij (−180 < θij < 180) subtended by the line
between points dij = lij1
from (0, 0) to (lij1 , lij2 ) with the x-axis. The angle can be calculated as θij = tan−1 (lij1 /lij2 )
choosing (0 < θij < 180) if lij2 > 0 and (−180 < θij < 0) if lij2 < 0. Note that the variogram
has angular symmetry in that vij = vji , dij = dji and |θij − θji | = 180. The variogram
presented averages the vij within 12 distance classes and 4, 6 or 8 sectors (selected using a
!VGSECTORS qualifier) centred on an angle of (i − 1) ∗ 180/s (i = 1, ...s). A figure is produced
which reports the trends in v̄ij with increasing distance for each sector.
ASReml also computes the variogram from predictors of random effects which appear to have
a variance structures defined in terms of distance. The variogram details are reported in the
.res file.
2.5
2.5.1
Inference: Fixed effects
Introduction
Inference for fixed effects in linear mixed models introduces some difficulties. In general,
the methods used to construct F -tests in analysis of variance and regression cannot be
used for the diversity of applications of the general linear mixed model available in ASReml.
One approach would be to use likelihood ratio methods (see Welham and Thompson, 1997)
although their approach is not easily implemented.
Wald-type test procedures are generally favoured for conducting tests concerning τ . The
traditional Wald statistic to test the hypothesis H0 : Lτ = l for given L, r × p, and l, r × 1,
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