Spatial Statistics and Digital Image Analysis (1991) / Chapter Skim
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6. Spatial Statistics in the Analysis of Agricultural Field Experiments
6. Spatial Statistics in the Analysis of Agricultural Field Experiments
Pages 109-128
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From page 109... ...
The most common measurement is that of plot yield at harvest, which in an ideal world would provide a direct assessment of the corresponding treatment or variety eject. However, yield is influenced by external factors such as weather and plot fertility.
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From page 110... ...
Perhaps the first was the use of check plots (e.g., Wiancko, 1914~; that is, plots interspersed regularly at frequent intervals throughout the experiment and containing a standard treatment. The yields from these plots can be used to calculate a local fertility index for each experimental plot and to adjust its yield accordingly; the assumption is that variations in adjacent plots are relatively small.
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From page 111... ...
and the validity of the associated standard errors within the inferential framework. Since this framework is not at all obvious, we provide some brief discussion in the particular context of the simplest common example, namely, the randomized complete blocks design, for which blocks and replicates coincide.
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From page 112... ...
S) but it turns out that the calculation of treatment estimates and associated standard errors coincides exactly with an ordinary least-squares analysis of the corresponding linear mode!
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From page 113... ...
where T denotes treatment effects, T is the corresponding fur-rank treatment design matrix, x represents the (fixed) fertility effects measured about zero, and z is residual error.
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From page 114... ...
has any particular merit, or that the ~—~ induced by typical Papadakis adjustment holds appeal as inverse covariance matrices in a random field formulation. Finally, note that, where in our discussion ~—~ is singular, estimates of treatment contrasts rather than T*
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From page 115... ...
6.3.2 One-Dimensional Adjustment In discussing specification of the fertility process X for layouts that consist of a single column of n plots, it proves convenient initially to consider an ostensibly infinite column, with plots labelled i = 0,+1,..., according to their positions with respect to a reference plot 0. Let Xi denote the random fertility in plot i, measured about zero.
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From page 116... ...
§6.3.3~. Of course, for real experiments with finite numbers of plots, it holds only as an approximation because of edge effects.
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From page 117... ...
Besag and Kempton, 1986~. Thus, for any intermediate value of a, i,' provides a compromise between the ordinary least-squares estimates of ~ based on u and on y, respectively; this resembles the combination of intra- and interblock information in the classical analysis of incomplete block designs but here using the notion of a moving block of size two.
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From page 118... ...
118 TABLE 6.1: Layout and Yields (t/ha) for 62 Varieties of Winter Wheat Repl icat e 1 Variety Yield 10 7.32 50 6.34 18 7.44 58 8.54 42 7.26 26 7.50 2 6.92 34 8.46 56 7.22 32 8.22 16 7.15 8 6.90 24 7.48 48 7.02 40 8.16 41 7.92 9 7.56 33 8.57 49 8.57 1 7.35 25 8.85 57 8.06 17 8.10 45 8.64 61 8.41 13 8.95 37 7.39 5 7.30 29 8.31 53 8.87 21 7.83 20 8.45 36 7.07 52 8.00 44 8.28 4 7.48 60 8.69 28 6.88 12 8.17 62 7.99 14 7.48 54 7.17 22 7.67 6 7.60 38 7.40 46 8.55 30 7.50 39 7.75 15 4.82 55 8.40 31 9.02 47 7.57 23 9.12 7 9.96 11 8.51 19 8.55 59 9.14 3 7.70 43 8.43 27 7.98 51 7.66 35 8.24 Repl icat e 2 Variety Yield 58 7.52 31 7.50 40 6.61 41 7.00 4 5.59 22 6.01 13 7.52 51 7.07 27 7.63 18 7.28 8 6.32 45 7.43 55 7.31 62 8.36 36 7.04 9 7.23 56 8.16 28 6.86 46 8.21 19 8.54 10 7.56 37 7.41 1 7.35 44 8.50 26 9.01 17 8.60 54 7.25 61 8.39 35 8.45 16 7.70 7 9.46 48 7.28 57 7.71 3 7.46 30 7.63 50 6.32 21 7.29 12 8.29 39 7.09 49 8.30 2 7.49 20 8.94 11 8.09 29 7.99 47 8.05 38 7.38 32 9.00 52 9.24 59 9.60 33 9.13 42 8.20 14 7.90 23 9.26 5 7.80 15 6.28 43 8.95 53 8.96 25 9.32 24 8.24 34 9.15 60 9.40 6 8.17 Replicate 3 Variety Yield 32 7.29 14 6.70 23 7.57 51 7.38 33 7.71 2 6.78 60 8.44 45 7.61 48 6.93 9 7.23 36 6.76 5 6.19 27 7.86 18 7.82 54 6.69 34 8.35 25 8.35 24 7.69 52 8.25 3 7.77 15 4.38 61 8.31 46 8.45 11 8.30 42 7.43 7 9.51 38 6.87 20 8.69 57 7.69 56 7.84 29 7.96 6 7.75 19 7.98 55 7.82 28 6.58 37 7.51 10 7.31 41 7.90 35 8.60 62 8.22 16 7 ~ 55 53 8.52 26 8.58 47 7.15 4 8.37 17 7.89 59 9.16 13 9.04 1 7.72 40 8.15 31 8.98 50 7.10 44 8.20 22 8.20 58 -8.92 39 7.68 8 6.78 21 7.67 49 8.57 12 8.14 43 7.95 30 7.67
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From page 119... ...
Here we illustrate first-differences analysis on an official United Kingdom trial for winter wheat, carried out by the East of Scotland College of Agriculture and involving final assessment of 62 different varieties. The layout of the experiment, in three physically separated complete replicates, and the corresponding yields (t/ha)
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From page 120... ...
The fairly close agreement with spatial statistical analysis seems typical but of course the latter does not require a sophisticated design and applies equally to the simple layouts encountered more commonly in practice. 6.3.3 Two-Dimensional Formulation In this section, we seek to generalize the previous one-dimensional formulation.
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From page 121... ...
This generalizes to arbitrary bilateral autoregressions and, if X is Gaussian, is equivalent to asymptotic maximumlikelihood estimation. However, here we are concerned with observation on Y in (6.5)
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From page 122... ...
Such a formulation is of course entirely consistent with the one-dimensional development in §6.3.2; again, we may expect that Z will often be negligible, which if assumed from the start would lead to entirely straightforward estimation although we do not wish to exclude the possibility of a non-zero or. ~ v Certainly the problems of estimation are not insuperable but they require further research, especially as regards standard errors for treatment contrasts; these must retain approximate validity somewhat outside the narrow confines of the model itself.
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From page 123... ...
He then extends this approach to two-dimensional adjustment by considering only the class of processes that, after row or column differencing, are separable; that is, are stationary and have interplot autocorrelations Pk satisfying PI ,k2 = Pki,0 Polka (6.15) Separability leads to a considerable simplification in the computation of parameter estimates, though the advantage is diminished with the inclusion of superimposed random error; see Martin (1989)
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From page 124... ...
The analyses also include approximate standard errors for treatment contrasts ant! graphs of estimated treatment, fertility, and residual effects across each of the experimental areas.
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From page 125... ...
Predicted standard errors can also be compared with actual variability of estimates. Unfortunately, in a random field framework, each trial provides but a single assessment and many sets of uniformity data are required for a proper evaluation.
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From page 126... ...
Kunsch, Edge effects and efficient parameter estimation for stationary random fields, Biometrika 74 (1987)
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From page 127... ...
A Hunter, The efficiency of incomplete block designs in National List and Recommended List cereal variety trials, ]
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