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 121A 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 !SEDThe 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 0.932For 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
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