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88 7 Balance and non-orthogonalityFACTOR [LABELS=!T('OOO','OOD','OKO','OKD',\'NOO','NOD','NKO','NKD')] TThe main effect of T would not be balanced: the comparison of levels{'OOO' 'OOD' 'OKO' 'OKD'}with {'NOO' 'NOD' 'NKO' 'NKD'}has efficiency factor one in the Blocks.Plots stratum and zero in the Blocks stratum(this contrast is equivalent to the main effect of N in the original specification); but thecomparison of levels{'NOO' 'NOD' 'OKO' 'OKD'}with {'OOO' 'OOD' 'NKO' 'NKD'}has efficiency 0.25 in the Blocks stratum and 0.75 in the Blocks.Plots stratum (thisis equivalent to N.K in the original specification). Thus the main effect of T is notbalanced, since in the Block.Plots stratum some of its contrasts have efficiency factorone, while others have efficiency factor 0.75. GenStat can detect unbalanced designs likethis, and will give you an error diagnostic.Fault 23, code AN 1, statement 1 on line 78Command: ANOVADesign unbalanced - cannot be analysed by ANOVA.Model term T (non-orthogonal to term Blocks) is unbalanced, in the Blocks.Plots stratum.It is still possible to analyse this particular design by ANOVA, by defining pseudo-factors(see Guide to the GenStat Command Language, Part 2, Section 4.7.3). However, thisrequires extra skill for the specification, and it may not be feasible in many cases. So, ifyou have a single error term, you can use the Unbalanced ANOVA menu (Section 7.4).Alternatively, if you have several error terms you can use the REML menus (Chapter 8).7.3 PracticalFactorial designs with interactionsconfounded with blocks can beconstructed using the GenerateFactorial Designs in Blocks menu,which can be opened by clicking onthe Generate a Factorial Design inBlocks sub-option of the Designoption of the Stats menu (Figure7.3).Figure 7.3
7.4 Unbalanced designs with two treatment factors 89Use the menu, as shown in Figure7.4, to construct a design for asingle replicate of a 2×2×2×2 designin blocks of size 8.Figure 7.47.4 Unbalanced designs with two treatment factorsMost of the designs covered by the Analysis of Variance menus are balanced and, in fact,all of those discussed so far in the earlier chapters have been orthogonal. Essentially thismeans that the order in which the treatment terms are fitted is unimportant (other thanthat each main effect must be fitted before any of its interactions). So we could havespecified sulphur as the first treatment factor and nitrogen as the second treatment factorin the menus in Figures 3.2 and 3.5, and still have obtained the same sums of squares andeffects. This contrasts with the situation in multiplelinear regression (see e.g. Section 5.2 of theIntroduction to GenStat for Windows), where the x-variates are usually correlated (i.e. non-orthogonal),and so different regression coefficients are obtainedfor each x-variate according to which other x-variates had been fitted beforehand.GenStat spreadsheet file Foster.gsh (Figure7.5) contains the results of an experiment to studythe effect of foster feeding of rats (Scheffe, 1959,The Analysis of Variance; also see McConway,Jones & Taylor, 1999, Statistical Modelling usingGENSTAT, Example 7.6). The rats were from fourdifferent genotypes (A, B, I or J), the experimentalunit was a litter of four rats, and the responsevariate was the weight of the litter after a period offeeding. The interest was in whether the genotypeof a foster mother would affect the weight. So thereare two treatment factors, each with four levels, thegenotype of the mother and the genotype of theFigure 7.5
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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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2 Blocking structuresThe blocking s
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24 2 Blocking structuresBlock 1D4.6
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26 2 Blocking structuresNow we repe
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28 2 Blocking structureswould give
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30 2 Blocking structuresAnalysis of
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3 Treatment structureSo far we have
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34 3 Treatment structureTables of m
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36 3 Treatment structureFigure 3.3
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Page 48 and 49: 44 3 Treatment structureN * Swas us
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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
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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
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Page 118 and 119: 9 Commands for analysis of variance
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Page 124 and 125: IndexA2DISPLAY procedure 117A2WAY p
Page 126: 122 IndexTransformation 56logarithm
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