Split plot design - Oats
Introduction
The first example involves the analysis of a split plot design
originally presented by Yates (1935). The experiment was conducted to
assess the effects on yield of three oat varieties (Golden Rain,
Marvellous and Victory) with four levels of nitrogen application (0,
0.2, 0.4 and 0.6 cwt/acre). The field layout consisted of six blocks
(labelled I, II, III, IV, V and VI) with three whole-plots per block,
each split into four sub-plots. The three varieties were randomly
allocated to the three whole-plots while the four levels of nitrogen
application were randomly assigned to the four sub-plots within each
whole-plot. The data is
nitrogen
block variety 0.0cwt 0.2cwt 0.4cwt 0.6cwt
GR 111 130 157 174
I M 117 114 161 141
V 105 140 118 156
GR 61 91 97 100
II M 70 108 126 149
V 96 124 121 144
GR 68 64 112 86
III M 60 102 89 96
V 89 129 132 124
GR 74 89 81 122
IV M 64 103 132 133
V 70 89 104 117
GR 62 90 100 116
V M 80 82 94 126
V 63 70 109 99
GR 53 74 118 113
VI M 89 82 86 104
V 97 99 119 121
A standard analysis of these data recognises the two basic elements
inherent in the experiment. These two aspects are firstly the
stratification of the experiment units, that is the blocks, whole-plots and sub-plots,
and secondly, the treatment structure that is superimposed on the
experimental material. The latter is of prime interest, in the
presence of stratification. Thus the aim of the analysis is to examine
the importance of the treatment effects while accounting for the
stratification and restricted randomisation of the treatments to the
experimental units.
The ASReml input file is presented below.
split plot example
blocks 6 # Coded 1...6 in first data field of oats.asd
nitrogen !A 4 # Coded alphabetically
subplots * # Coded 1...4
variety !A 3 # Coded alphabetically
wplots * # Coded 1...3
yield
oats.asd !SKIP 2
yield ~ mu variety nitrogen variety.nitrogen !r blocks blocks.wplots
predict nitrogen # Print table of predicted nitrogen means
predict variety
predict variety nitrogen !SED
The data fields were blocks, wplots, subplots, variety, nitrogen
and yield. The first five variables are factors that describe the stratification or
experiment design and treatments. The standard split plot
analysis is achieved by fitting the model terms blocks
and blocks.wplots as random effects. The blocks.wplots.subplots
term is not listed in the model because this interaction corresponds
to the experimental units and is automatically included as the
residual term. The fixed effects include the main effects of both
variety and nitrogen and their interaction. The tables of
predicted means and associated standard errors of differences ( SEDs)
have been requested. These are reported in the .pvs file.
Abbreviated output is shown below.
- - - Results from analysis of yield - - -
Approximate stratum variance decomposition
Stratum Degrees-Freedom Variance Component Coefficients
blocks 5.00 3175.06 12.0 4.0 1.0
blocks.wplots 10.00 601.331 0.0 4.0 1.0
Residual Variance 45.00 177.083 0.0 0.0 1.0
Source Model terms Gamma Component Comp/SE % C
blocks 6 6 1.21116 214.477 1.27 0 P
blocks.wplots 18 18 0.598937 106.062 1.56 0 P
Variance 72 60 1.00000 177.083 4.74 0 P
Wald F statistics
Source of Variation NumDF DenDF F-inc Prob
7 mu 1 5.0 245.14 <.001
4 variety 2 10.0 1.49 0.272
2 nitrogen 3 45.0 37.69 <.001
8 variety.nitrogen 6 45.0 0.30 0.932
For simple variance component models such as the above, the default
parameterisation for the variance component parameters is as the ratio
to the residual variance. Thus ASReml prints the variance component ratio and variance component
for each term in the random model in the columns labelled Gamma and
Component respectively.
The analysis of variance ( ANOVA) is printed below this summary. The
usual decomposition has three strata, with treatment effects separating
into different strata as a consequence of the balanced design and the
allocation of variety to whole-plots. In this balanced case, it is
straightforward to derive the ANOVA estimates of the stratum variances
from the REML estimates of the variance components. That is
blocks = 12 S2b + 4S2w +
S2e = 3175.1
blocks.wplots = 4S2w + S2e = 601.3
residual =S2e = 177.1
The default output for testing fixed effects used by ASReml is a
table of so-called
incremental F-statistics.
The statistics are simply the appropriate Wald
test statistics divided by the number of estimable effects for that
term. In this example there are four terms included in the
summary. The overall mean (denoted by mu)
is of no interest for these data. The tests are sequential, that is
the effect of each term is assessed by the change in sums of squares
achieved by adding the term to the current model, defined by the model
which includes those terms appearing above the current term given the variance parameters. For
example, the test of nitrogen is calculated from the change in
sums of squares for the two models mu variety nitrogen and
mu variety. No refitting occurs, that is the variance parameters
are held constant at the REML estimates obtained from the currently
specified fixed model.
The incremental Wald statistics have an asymptotic χ2 distribution,
with degrees of freedom (df) given by the number of estimable effects
(the number in the DF column). In this example, the incremental
F-statistics are numerically the same as the ANOVA F-statistics, and
ASReml has calculated the
appropriate denominator df for testing fixed effects. This is a simple
problem for balanced designs, such as the split plot design, but it is
not straightforward to determine the relevant denominator df in
unbalanced designs, such as the rat data set described in the next
section.
Tables of predicted means are presented for the nitrogen, variety, and
variety by nitrogen tables in the .pvs file. The qualifier
!SED has been used on the third predict statement
and so the matrix of SEDs for the variety by
nitrogen table is printed. For the first two predictions, the average SED
is calculated from the average variance of differences. Note also that
the order of the predictions (e.g. 0.6cwt, 0.4cwt 0.2cwt 0cwt for nitrogen)
is simply the order those treatment labels were discovered in the data file.
Split plot analysis - oat Variety.Nitrogen 14 Apr 2008 16:15:49
oats
Ecode is E for Estimable, * for Not Estimable
The predictions are obtained by averaging across the hypertable
calculated from model terms constructed solely from factors
in the averaging and classify sets.
Use !AVERAGE to move ignored factors into the averaging set.
---- ---- ---- ---- ---- ---- ---- 1 ---- ---- ---- ---- ---- ---- ----
Predicted values of yield
The averaging set: variety
The ignored set: blocks wplots
nitrogen PredictedValue StandardError Ecode
0.6cwt 123.3889 7.1747 E
0.4cwt 114.2222 7.1747 E
0.2cwt 98.8889 7.1747 E
0cwt 79.3889 7.1747 E
SED: Overall Standard Error of Difference 4.436
---- ---- ---- ---- ---- ---- ---- 2 ---- ---- ---- ---- ---- ---- ----
Predicted values of yield
The averaging set: nitrogen
The ignored set: blocks wplots
variety PredictedValue StandardError Ecode
Marvellous 109.7917 7.7975 E
Victory 97.6250 7.7975 E
GoldenRain 104.5000 7.7975 E
SED: Overall Standard Error of Difference 7.079
---- ---- ---- ---- ---- ---- ---- 3 ---- ---- ---- ---- ---- ---- ----
Predicted values of yield
The ignored set: blocks wplots
nitrogen variety PredictedValue StandardError Ecode
0.6cwt Marvellous 126.8333 9.1070 E
0.6cwt Victory 118.5000 9.1070 E
0.6cwt GoldenRain 124.8333 9.1070 E
0.4cwt Marvellous 117.1667 9.1070 E
0.4cwt Victory 110.8333 9.1070 E
0.4cwt GoldenRain 114.6667 9.1070 E
0.2cwt Marvellous 108.5000 9.1070 E
0.2cwt Victory 89.6667 9.1070 E
0.2cwt GoldenRain 98.5000 9.1070 E
0cwt Marvellous 86.6667 9.1070 E
0cwt Victory 71.5000 9.1070 E
0cwt GoldenRain 80.0000 9.1070 E
Predicted values with SED(PV)
126.833
118.500 9.71503
124.833 9.71503 9.71503
117.167 7.68295 9.71503 9.71503
110.833 9.71503 7.68295 9.71503 9.71503
114.667 9.71503 9.71503 7.68295 9.71503
9.71503
108.500 7.68295 9.71503 9.71503 7.68295
9.71503 9.71503
89.6667 9.71503 7.68295 9.71503 9.71503
7.68295 9.71503 9.71503
98.5000 9.71503 9.71503 7.68295 9.71503
9.71503 7.68295 9.71503 9.71503
86.6667 7.68295 9.71503 9.71503 7.68295
9.71503 9.71503 7.68295 9.71503 9.71503
71.5000 9.71503 7.68295 9.71503 9.71503
7.68295 9.71503 9.71503 7.68295 9.71503
9.71503
80.0000 9.71503 9.71503 7.68295 9.71503
9.71503 7.68295 9.71503 9.71503 7.68295
9.71503 9.71503
SED: Standard Error of Difference: Min 7.6830 Mean 9.1608 Max 9.7150
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