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34 3 Treatment structureTables of meansVariate: yieldGrand mean 1.104N 0 180 2300.601 1.313 1.398S 0 10 20 400.829 1.155 1.167 1.266N S 0 10 20 400 0.560 0.770 0.524 0.552180 0.894 1.289 1.525 1.545230 1.032 1.404 1.454 1.700Standard errors of differences of meansTable N S NSrep. 12 9 3d.f. 22 22 22s.e.d. 0.0864 0.0998 0.1729GenStat has represented the grain yield y, recorded on the experimental plots, by themodely = ì + â + n + s + ns + åijk i j k jk ijkThis model is an extension of the one-way analysis discussed earlier except that now wehave a termâito represent the effect of blocks (block stratum in the aov table),and three terms to represent the effects of the treatments. The parametersnjrepresent the main effect of nitrogen (N)skrepresent the main effect of sulphur (S), andnsjkrepresent the interaction between nitrogen and sulphur (N.S).Just as in the one-way analysis, the analysis of variance essentially fits each term inturn, to allow you decide how complicated a model is required to describe the results ofthe experiment. The analysis-of-variance table has a line for each of these, to allow youto assess whether the corresponding parameters are needed in the model. The full model,above, will estimate the fitted values for sulphur and nitrogen (the values predicted by themodel) as
3.1 Factorial designs with two treatment factors 35S×NmeansN0 N180 N230S0 0.560 0.894 1.032S10 0.770 1.289 1.404S20 0.524 1.525 1.454S40 0.552 1.545 1.700=ì + S + N: N0 N180 N230 + N.S N0 N180 N2301.104 S0 0.276 0.503 0.209 0.294 S0 0.234 0.144 0.090S10 0.051 S10 0.118 0.075 0.044S20 0.063 S20 0.141 0.148 0.007S40 0.162 S40 0.211 0.071 0.141A model like this, where you are fitting factors and their interactions, is called a factorialmodel. Here we have a 4×3 factorial.It will be much easier to describe what is happening if there is no interaction. Themodel will then bey = ì + â + n + s + åijk i j k ijkleading to fitted valuesN×SmeansN0 N180 N230 = ì + S + N: N0 N180 N230S0 0.326 1.038 1.122 1.104 S0 0.276 0.503 0.209 0.294S10 0.652 1.364 1.448 S10 0.051S20 0.665 1.377 1.461 S20 0.063S40 0.763 1.475 1.559 S40 0.162and you will see that we can decide on the best level of nitrogen without needing toconsider how much sulphur is to be applied, and on the best level of sulphur withoutneeding to think about the level of nitrogen on the plot. This is what we mean by sayingthat the two factors do not interact: the interaction assesses the way in which the changesin yield caused by the various levels of nitrogen differ according to the amount of sulphuror, equivalently, the way in which the response to amount of sulphur differs accordingto the level of nitrogen. Figure 3.3 plots the means for the model with an interaction, andFigure 3.4 plots those for the model with no interaction. When there is no interaction thelines are "parallel".
Page 1 and 2: Anova and Design
Page 4 and 5: ContentsIntroduction 11 From t-test
Page 6 and 7: 1 From t-test to one-way anovaIn th
Page 8 and 9: 4 1 From t-test to one-way anovaIf
Page 10 and 11: 6 1 From t-test to one-way anovaWe
Page 12 and 13: 8 1 From t-test to one-way anova1.2
Page 14 and 15: 10 1 From t-test to one-way anovawi
Page 16 and 17: 12 1 From t-test to one-way anovaAl
Page 18 and 19: 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 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 70 and 71: 66 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
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84 7 Balance and non-orthogonalityA
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86 7 Balance and non-orthogonalityT
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88 7 Balance and non-orthogonalityF
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90 7 Balance and non-orthogonalityf
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92 7 Balance and non-orthogonality7
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94 7 Balance and non-orthogonalityM
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96 7 Balance and non-orthogonalityM
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8 REML analysis of unbalanced desig
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100 8 REML analysis of unbalanced d
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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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