Unreplicated early generation variety trial - Wheat
Introduction
To further illustrate the approaches presented in the previous
section, we consider an unreplicated field experiment conducted at
Tullibigeal situated in south-western NSW. The trial was an S1 (early
stage) wheat variety evaluation trial and consisted of 525 test lines
which were randomly assigned to plots in a 67 by 10 array. There was a
check plot variety every 6 plots within each column. That is the check
variety was sown on rows 1,7,13,...,67 of each column. This variety
was numbered 526. A further 6 replicated commercially available
varieties (numbered 527 to 532) were also randomly assigned to plots
with between 3 to 5 plots of each. The aim of these trials is to
identify and retain the top, say 20% of lines for further
testing. Cullis et al. (1989) considered the analysis of early
generation variety trials, and presented a one-dimensional spatial
analysis which was an extension of the approach developed by Gleeson
and Cullis (1987). The test line effects are assumed random, while the
check variety effects are considered fixed. This may not be sensible
or justifiable for most trials and can lead to inconsistent
comparisons between check varieties and test lines. Given the large
amount of replication afforded to check varieties there will be very
little shrinkage irrespective of the realised heritability.
We consider an
initial analysis with spatial correlation in one
direction and fitting
the variety effects (check, replicated and unreplicated lines) as random.
We present three further spatial models for
comparison. The ASReml input file is
Tullibigeal trial
linenum
yield
weed
column 10
row 67
variety 532 # testlines 1:525, check lines 526:532
wheat.asd !skip 1 !DOPATH 1
!PATH 1 # AR1 x I
y ~ mu weed mv !r variety
residual ar1(row).column
!PATH 2 # AR1 x AR1
y ~ mu weed mv !r variety
residual ar1(row).ar1(column)
!PATH 3 # AR1 x AR1 + column trend
y ~ mu weed pol(column,-1) mv !r variety
residual ar1(row).ar1(column)
!PATH 4 # AR1 x AR1 + Nugget + column trend
y ~ mu weed pol(column,-1) mv !r variety units
residual ar1(row).ar1(column)
predict var
The data fields represent the factors variety, row
and column, a covariate weed and the plot
yield ( yield). There are three paths in the ASReml file. We begin with
the one-dimensional spatial model, which assumes the variance model
for the plot effects within columns is described by a first order
autoregressive process. The abbreviated output file is
1 LogL=-4280.75 S2= 0.12850E+06 666 df 0.1000 1.000 0.1000
2 LogL=-4268.57 S2= 0.12138E+06 666 df 0.1516 1.000 0.1798
3 LogL=-4255.89 S2= 0.10968E+06 666 df 0.2977 1.000 0.2980
4 LogL=-4243.76 S2= 88033. 666 df 0.7398 1.000 0.4939
5 LogL=-4240.59 S2= 84420. 666 df 0.9125 1.000 0.6016
6 LogL=-4240.01 S2= 85617. 666 df 0.9344 1.000 0.6428
7 LogL=-4239.91 S2= 86032. 666 df 0.9474 1.000 0.6596
8 LogL=-4239.88 S2= 86189. 666 df 0.9540 1.000 0.6668
9 LogL=-4239.88 S2= 86253. 666 df 0.9571 1.000 0.6700
10 LogL=-4239.88 S2= 86280. 666 df 0.9585 1.000 0.6714
Final parameter values 0.9592 1.0000 0.6721
- - - Results from analysis of yield - - -
Akaike Information Criterion 8485.76 (assuming 3 parameters).
Bayesian Information Criterion 8499.26
Model_Term Gamma Sigma Sigma/SE % C
variety IDV_V 532 0.959184 82758.6 8.98 0 P
ar1(row).column 670 effects
Residual SCA_V 670 1.000000 86280.2 9.12 0 P
row AR_R 1 0.672052 0.672052 16.04 1 P
Wald F statistics
Source of Variation NumDF DenDF F-inc P-inc
7 mu 1 83.6 9799.20 <.001
3 weed 1 477.0 109.33 <.001
Notice: The DenDF values are calculated ignoring fixed/boundary/singular
variance parameters using algebraic derivatives.
Solution Standard Error T-value T-prev
3 weed
1 -217.481 20.7995 -10.46
7 mu
1 2893.05 29.9404 96.63
8 mv_estimates 2 effects fitted
6 variety 532 effects fitted
Residual [section 11, column 10 (of 10), row 13 (of 67)] is -4.26 SD
Finished: 24 Jan 2014 15:06:51.854 Warning: LogL not converged
The iterative sequence converged, the REML estimate of the
autoregressive parameter indicating substantial within column
heterogeneity.
The abbreviated output from the two-dimensional AR1 cross AR1 spatial
model is
1 LogL=-4277.99 S2= 0.12850E+06 666 df
2 LogL=-4266.13 S2= 0.12097E+06 666 df
3 LogL=-4253.05 S2= 0.10777E+06 666 df
4 LogL=-4238.72 S2= 83156. 666 df
5 LogL=-4234.53 S2= 79868. 666 df
6 LogL=-4233.78 S2= 82024. 666 df
7 LogL=-4233.67 S2= 82725. 666 df
8 LogL=-4233.65 S2= 82975. 666 df
9 LogL=-4233.65 S2= 83065. 666 df
10 LogL=-4233.65 S2= 83100. 666 df
- - - Results from analysis of yield - - -
Akaike Information Criterion 8475.29 (assuming 4 parameters).
Bayesian Information Criterion 8493.30
Model_Term Gamma Sigma Sigma/SE % C
variety IDV_V 532 1.06038 88117.5 9.92 0 P
ar1(row).ar1(column) 670 effects
Residual SCA_V 670 1.000000 83100.1 8.90 0 P
row AR_R 1 0.685387 0.685387 16.65 0 P
column AR_R 1 0.285909 0.285909 3.87 0 P
Wald F statistics
Source of Variation NumDF DenDF F-inc Prob
7 mu 1 41.7 6248.65 <.001
3 weed 1 491.2 85.84 <.001
The change in REML LogL is significant (χ21= 12.46, p<.001) with
the inclusion of the autoregressive parameter for
columns. The Figure presents the sample variogram of the
residuals for the AR1 cross AR1 model. There is an indication that
a linear drift from column 1 to column 10 is present. We include a
linear regression coefficient pol(column,-1) in the model to
account for this. Note we use the '-1' option in the pol term to
exclude the overall constant in the regression, as it is already
fitted. The linear regression of column number on yield is significant
(t=-2.96). The sample variogram (Figure 2 ) is more satisfactory,
though interpretation of variograms is often difficult, particularly
for unreplicated trials. This is an issue for further research.
Figure 1. Sample variogram of the residuals from the AR1 cross AR1 model for the Tullibigeal data
Figure 2. Sample variogram of the residuals from the AR1 cross AR1 + pol(column,-1) model for the Tullibigeal data
The abbreviated output for this model and the final model in which a
nugget effect has been included is
#AR1xAR1 + pol(column,-1)
1 LogL=-4270.99 S2= 0.12730E+06 665 df
2 LogL=-4258.95 S2= 0.11961E+06 665 df
3 LogL=-4245.27 S2= 0.10545E+06 665 df
4 LogL=-4229.50 S2= 78387. 665 df
5 LogL=-4226.02 S2= 75375. 665 df
6 LogL=-4225.64 S2= 77373. 665 df
7 LogL=-4225.60 S2= 77710. 665 df
8 LogL=-4225.60 S2= 77786. 665 df
9 LogL=-4225.60 S2= 77806. 665 df
Source Model terms Gamma Component Comp/SE % C
variety 532 532 1.14370 88986.3 9.91 0 P
Variance 670 665 1.00000 77806.0 8.79 0 P
Residual AR=AutoR 67 0.671436 0.671436 15.66 0 U
Residual AR=AutoR 10 0.266088 0.266088 3.53 0 U
Wald F statistics
Source of Variation NumDF DenDF F-inc Prob
7 mu 1 42.5 7073.70 <.001
3 weed 1 457.4 91.91 <.001
8 pol(column,-1) 1 50.8 8.73 0.005
#
#AR1xAR1 + units + pol(column,-1)
#
1 LogL=-4272.85 S2= 0.11684E+06 665 df
2 LogL=-4265.70 S2= 83872. 665 df : 1 components restrained
3 LogL=-4240.99 S2= 80942. 665 df
4 LogL=-4227.44 S2= 53712. 665 df
5 LogL=-4221.09 S2= 52201. 665 df
6 LogL=-4220.94 S2= 54803. 665 df
7 LogL=-4220.94 S2= 54935. 665 df
8 LogL=-4220.94 S2= 54934. 665 df
- - - Results from analysis of yield - - -
Akaike Information Criterion 8451.88 (assuming 5 parameters).
Bayesian Information Criterion 8474.37
Model_Term Gamma Sigma Sigma/SE % C
variety IDV_V 532 1.32827 72967.0 6.99 0 P
units IDV_V 670 0.562308 30889.9 3.78 0 P
ar1(row).ar1(column) 670 effects
Residual SCA_V 670 1.000000 54934.0 5.15 0 P
row AR_R 1 0.835396 0.835396 18.38 0 P
column AR_R 1 0.375499 0.375499 3.25 0 P
Wald F statistics
Source of Variation NumDF DenDF F-inc P-inc
7 mu 1 13.6 4272.13 <.001
3 weed 1 470.3 86.31 <.001
8 pol(column,-1) 1 27.4 3.69 0.065
The change in LogL from adding units is not large but is significant.
However, adding units reduces the significance of the linear column trend,
as that is now picked up better by the ar1(column) term.
Warning: mv_estimates is ignored for prediction
Warning: units is ignored for prediction
---- ---- ---- ---- ---- ---- ---- 1 ---- ---- ---- ---- ---- ---- ---- ----
column evaluated at 5.5000
weed is evaluated at average value of 0.4597
Predicted values of yield
variety Predicted_Value Standard_Error Ecode
1.0000 2917.1782 179.2881 E
2.0000 2957.7405 178.7688 E
3.0000 2872.7615 176.9880 E
4.0000 2986.4725 178.7424 E
. . .
522.0000 2784.7683 179.1541 E
523.0000 2904.9421 179.5383 E
524.0000 2740.0330 178.8465 E
525.0000 2669.9565 179.2444 E
526.0000 2385.9806 44.2159 E
527.0000 2697.0670 133.4406 E
528.0000 2727.0324 112.2650 E
529.0000 2699.8243 103.9062 E
530.0000 3010.3907 112.3080 E
531.0000 3020.0720 112.2553 E
532.0000 3067.4479 112.6645 E
SED: Overall Standard Error of Difference 245.8
Note that the (replicated) check lines have lower SE than the (unreplicated)
test lines. There will also be large diffeneces in SEDs.
Rather than obtaining the large table of all SEDs, you could do the
prediction in parts
predict var 1:525 column 5.5
predict var 526:532 column 5.5 !SED
to examine the matrix of pairwise prediction errors of
variety differences.
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