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66 5 Designs with several error terms5.1 Split-plot designV3 N3 V3 N2 V3 N2 V3 N3V3 N1 V3 N0 V3 N0 V3 N1V1 N0 V1 N1 V2 N0 V2 N2V1 N3 V1 N2 V2 N3 V2 N1V2 N0 V2 N1 V1 N1 V1 N2V2 N2 V2 N3 V1 N3 V1 N0V3 N2 V3 N0 V2 N3 V2 N0V3 N1 V3 N3 V2 N2 V2 N1V1 N3 V1 N0 V1 N2 V1 N3V1 N1 V1 N2 V1 N0 V1 N1V2 N1 V2 N0 V3 N2 V3 N3V2 N2 V2 N3 V3 N1 V3 N0V2 N1 V2 N2 V1 N2 V1 N0V2 N3 V2 N0 V1 N3 V1 N1V3 N3 V3 N1 V2 N3 V2 N2In the split-plot design shown here,the treatments are three varieties ofoats (Victory, Golden rain andMarvellous) and four levels ofnitrogen (0, 0.2, 0.4 and 0.6 cwt).As it is feasible to work withsmaller plots for fertiliser than forvarieties, the six blocks wereinitially split into three whole-plotsand then each whole-plot was splitinto four subplots. The varietieswere allocated (at random) to thewhole-plots within each block, andthe nitrogen levels (at random) tothe subplots within eachwhole-plot. In a randomized-blockdesign, we have a hierarchicalstructure with blocks and then plotswithin blocks.Results from the experiment arein spreadsheet file Oats.gsh inthe Data folder.V3 N2 V3 N0 V2 N0 V2 N1V1 N0 V1 N3 V3 N0 V3 N1V1 N1 V1 N2 V3 N2 V3 N3The split-plot is anotherdesign with a customized settingin the general Analysis ofVariance menu, as shown inFigure 5.1. The treatmentstructure is a factorial with twofactors, and is specified by amodel formula as described inChapter 3. The block structure isset up automatically by GenStatfrom the factors specified in the Figure 5.1Blocks, Whole Plots and Sub-plotsfields.The analysis-of-variance table shows that we now have three strata in the hierarchy:blocks, whole-plots within blocks, and subplots within whole plots (within blocks).Moreover, the analysis has more than one residual: in the split-plot design we need to
5.1 Split-plot design 67consider the random variability of the whole-plots as well as the variability of thesubplots. The sum of squares for Variety (which was applied to complete whole-plots)can correctly be compared with a residual which represents the random variability of thewhole-plots. Conversely, Nitrogen (which was applied to subplots) and theVariety.Nitrogen interaction are compared with the residual for subplots withinwhole-plots.Analysis of varianceVariate: yieldSource of variation d.f. s.s. m.s. v.r. F pr.blocks stratum 5 15875.3 3175.1 5.28blocks.wplots stratumvariety 2 1786.4 893.2 1.49 0.272Residual 10 6013.3 601.3 3.40blocks.wplots.subplots stratumnitrogen 3 20020.5 6673.5 37.69
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Anova and Design
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ContentsIntroduction 11 From t-test
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1 From t-test to one-way anovaIn th
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4 1 From t-test to one-way anovaIf
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6 1 From t-test to one-way anovaWe
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8 1 From t-test to one-way anova1.2
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10 1 From t-test to one-way anovawi
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12 1 From t-test to one-way anovaAl
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14 1 From t-test to one-way anovaYo
Page 20 and 21: 16 1 From t-test to one-way anovaFi
Page 22 and 23: 18 1 From t-test to one-way anovaTa
Page 24 and 25: 20 1 From t-test to one-way anovaTh
Page 26 and 27: 2 Blocking structuresThe blocking s
Page 28 and 29: 24 2 Blocking structuresBlock 1D4.6
Page 30 and 31: 26 2 Blocking structuresNow we repe
Page 32 and 33: 28 2 Blocking structureswould give
Page 34 and 35: 30 2 Blocking structuresAnalysis of
Page 36 and 37: 3 Treatment structureSo far we have
Page 38 and 39: 34 3 Treatment structureTables of m
Page 40 and 41: 36 3 Treatment structureFigure 3.3
Page 42 and 43: 38 3 Treatment structureThere is a
Page 44 and 45: 40 3 Treatment structureblock.*Unit
Page 46 and 47: 42 3 Treatment structureTables of c
Page 48 and 49: 44 3 Treatment structureN * Swas us
Page 50 and 51: 46 3 Treatment structureFumigant Am
Page 52 and 53: 48 3 Treatment structureFumigated S
Page 54 and 55: 50 3 Treatment structureCovariate 1
Page 56 and 57: 4 Checking the assumptionsIn this c
Page 58 and 59: 54 4 Checking the assumptions4.2 No
Page 60 and 61: 56 4 Checking the assumptions4.5 Tr
Page 62 and 63: 58 4 Checking the assumptionsFigure
Page 64 and 65: 60 4 Checking the assumptionsThe pr
Page 66 and 67: 62 4 Checking the assumptionsTotal
Page 68 and 69: 64 4 Checking the assumptionsMessag
Page 72 and 73: 68 5 Designs with several error ter
Page 78 and 79: 74 6 Design and sample size6.1 Desi
Page 80 and 81: 76 6 Design and sample sizeClicking
Page 82 and 83: 78 6 Design and sample sizeFigure 6
Page 84 and 85: 80 6 Design and sample sizeFigure 6
Page 86 and 87: 82 7 Balance and non-orthogonality7
Page 88 and 89: 84 7 Balance and non-orthogonalityA
Page 90 and 91: 86 7 Balance and non-orthogonalityT
Page 92 and 93: 88 7 Balance and non-orthogonalityF
Page 94 and 95: 90 7 Balance and non-orthogonalityf
Page 96 and 97: 92 7 Balance and non-orthogonality7
Page 98 and 99: 94 7 Balance and non-orthogonalityM
Page 100 and 101: 96 7 Balance and non-orthogonalityM
Page 102 and 103: 8 REML analysis of unbalanced desig
Page 104 and 105: 100 8 REML analysis of unbalanced d
Page 118 and 119: 9 Commands for analysis of variance
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116 9 Commands for analysis of vari
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118 9 Commands for analysis of vari
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IndexA2DISPLAY procedure 117A2WAY p
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122 IndexTransformation 56logarithm
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