Download spatial analysis of an agricultural field trial for five time periods

S PATIAL ANALYSIS OF AN AGRICULTURAL FIELD TRIAL FOR FIVE TIME
PERIODS
1,2
1,2
1,2
3
1,2
Margaret Donald , Chris Strickland , Clair Alston , Rick Young , Kerrie Mengersen
1
Discipline of Mathematical Sciences, Queensland University of Technology, Brisbane, Australia
2
Centre for Data Analysis, Modelling and Computation, Brisbane, Australia
3
Tamworth Agricultural Institute, Industry & Investment NSW, Calala NSW 2340, Australia
Introduction
Results
Bayesian Conditional Autoregressive models are used
to account for local site correlations in an agricultural
dataset. Conditional autoregressive models have not
been widely used in analyses of agricultural data. Here we
illustrate the usefulness of these models for agricultural
field data. Models were fitted using pyMCMC, a newly
developed general purpose-built batch updating Markov
chain Monte Carlo program [1].
Table 1: Summary of DICs
The data motivating the methods involve four dimensions
and comprise agricultural field trial data, from the Liverpool Plains, NSW, eastern Australia. These data were collected to determine a cropping system to maximise water
use for grain production while minimising leakage below
the crop root zone. We consider 5 separate days of trial
data, taken roughly six months apart, which therefore represented two years of the trial. The innovative approach
here was to fit separate conditional autoregressive models for each depth layer, while simultaneously estimating
depth profile functions for each site treatment, i.e., modelling the response as a function of depth, orthogonal to
the layers. The questions of interest were whether response cropping would lead to less moist soils in the root
zone and beyond, and whether this was consistent across
the days.
pD
DIC
Day 1 K3
K5
135
1032
1019
1065
-5915 *
-5875
-5850
Day 2 K3
K5
D220
135
1048
1039
1054
1061
-6049 *
-6038
-5995
-5952
Day 3 K3
K5
135
1044
1034
1058
-5996 *
-5961
-5890
Day 4 K3
K5
135
1064
1070
1092
-6619
-6623 *
-6570
Day 5 K3
K5
135
1024
1024
1053
-6396 *
-6378
-6321
Figure 1: Long Fallow - Response Contrast
K3: 3-knot linear spline.
K5: 5-knot linear spline.
135: Saturated model, 9 × 15 terms.
Table 2: Sign of slopes for depth 200cm-300cm where the 95% credible interval does not span zero).
Day
Treatment 1 2 3 4 5
Model
For site i at depth d, the full model is
yi,d = µj(i),d + ψi,d + ²i,d
where µj(i),d is the effect of treatment, j, at site, i, ψi,d is
the spatial residual at (i, d), ²i,d is the non-spatial residual,
with variance σd2, specific to each depth layer, d.
1
2
3
4
5
6
7
8
9
+ +
- - - - -
The conditional probability of the spatial residual, ψi,d,
given its neighbours, ψk,d, is given by the CAR proper of
[2]
From Table 1, the best model for most days is the 3-knot linear spline,
and for the day when it is not the best model, it is almost equivalent
to the best model. Hence, this is the model used for the contrasts
displayed in Figures 1-3.
´
³ P
wik ψk,d
2w )
,
1/(γ
ψi,d|ψk,d, k ∈ ∂i ∼ N ρ
wi+
d i+
where ∂i is the set of neighbours for site i. Thus, there
are 15 identical neighbourhood matrices, with each depth
layer having a potentially different variance, 1/(γd2wi+).
From Table 2, we see that for most days and treatments the linear segment from 200 cm to 300 cm is effectively flat, with the exception of
one of the lucerne mixtures (8), which is still exerting an influence on
the moisture and apparently causing a consistent decrease in moistures at these depths, and of the Long fallowing (2), where there is
increasing moisture in this segment for two of the five days.
The treatment effect was seen as a continuous (and initially as a smooth) function of depth. However, earlier
modelling showed that the data were well modelled with
a linear spline with five knots. Thus, µj(i),d = µj(i)(d), with
the curve being modelled using linear splines, with a varying number of knots. The spline models were compared to
a ‘saturated’ model with each treatment effect being fitted
at each depth (135 parameters).
The linear splines for each P
treatment were:
µj(i)(d) = β0 + ψj + β1,j d + K
1 uj,k zk (d)
where
zk (d) = (d − κk )+ for some knot sequence κ1, ...κK .
We considered a linear spline with 5 equally spaced knots,
a linear spline with 3 knots with equal spacing up to 200
cm, with the final knot at 200cm and finally, the ‘saturated’ model, a model with 9 × 15 parameters, where each
treatment by depth had its own parameter. The criterion
for model choice was the Deviance Information Criterion
(DIC) of [3].
Figure 2: Cropping - Pastures Contrast
Discussion
Graphs of the contrasts (Figures 1-3) show that there is very little consistency between days in the contrasts at the shallower depths. However, for the strongly positive contrast of cropping versus pastures,
where cropping treatments show markedly more moisture than pasture treatments, this difference starts to decrease consistently over
days once a depth of 100 cm is reached. By 200 cm, the difference
remains constant but becomes more variable with depth.
For the contrast of long fallowing versus response cropping, the differences start to become consistent over days once the depth of 200
cm is reached, where the contrast becomes largely positive, but with
increasing variability with depth. If this result is consistent for all days
for which moisture measurements were taken, then this would mean
that response cropping is less likely than long fallowing to lead to salination.
Finally, for the contrast of lucerne mixtures versus native pastures, differences become consistent across days at a depth of about 140cm
where the markedly lower moistures in the soils with lucerne mixtures
in comparison with those for the native pastures remains true but becomes less marked. In the final segment the differences increase with
lucerne treatments showing decreasing moisture values in comparison with the native pasture treatment.
Figure 3: Lucerne mixtures - Native Contrast
The moister the contrast, the higher the value of the y-axis for the
graphs showing the contrast.
References
[1] Strickland, C.: n.d., pymcmc: a statistical package for bayesian
mcmc analysis., Journal of Computational and Graphical Statistics
pp. submitted August, 2010.
[2] Gelfand, A. E. and Vounatsou, P.: 2003, Proper multivariate conditional autoregressive models for spatial data analysis, Biostatistics
4(1), 11–25.
[3] Spiegelhalter, D. J., Best, N. G., Carlin, B. P. and Linde, A. v. d.:
2002, Bayesian measures of model complexity and fit, Journal
of the Royal Statistical Society. Series B (Statistical Methodology)
64(4), 583–639.
[4] Besag, J., York, J. and Mollie, A.: 1991, Bayesian image restoration with applications in spatial statistics (with discussion), Annals
of the Institute of Mathematical Statistics 43, 1–59.
[5] Besag, J. and Kooperberg, C.: 1995, On conditional and intrinsic
autoregressions, Biometrika 82(4), 733–746.
[6] Besag, J. and Higdon, D.: 1999, Bayesian analysis of agricultural
field experiments, Journal of the Royal Statistical Society Series BStatistical Methodology 61, 691–717. Part 4.
[7] Lunn, D. J., A. Thomas, N. Best, and D. Spiegelhalter (2000). Winbugs - a bayesian modelling framework: Concepts, structure, and
extensibility. Statistics and Computing 10(4), 325–337.
[8] Ngo, L. and Wand, M.: 2004, Smoothing with mixed model software, Journal of Statistical Software 9, 1–56.
The authors would like to thank QUT for financial support.