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4 Checking the assumptionsIn this chapter you will learn• what assumptions are needed to ensure the validity of an analysis of variance• why the variance must be homogeneous (for example the variability of theresiduals should be the same at high values of the response variable as at lowvalues)• how to assess whether the variance is homogeneous• that the residuals should come from identical and independent Normal distributions• how to assess the Normality of the residuals• why the model must be additive (that is, differences between treatment effects mustremain the same however large or small the underlying size of the variablemeasured)• how to identify outliers• how transforming the response variate may correct for failures in the assumptions• how to print back-transformed tables of means • how to do a permutation or exact test Note: the topics marked are optional.
4.1 Homogeneity of variance4.1 Homogeneity of variance 53It is assumed that the variance is homogeneous, that is, the size of the random variationis similar over all the units. Homogeneity of variance can easily be assessed by plottingthe residuals (estimates of the random error) against the fitted values: if the variance ishomogeneous, the residuals should lie within a uniform band as in Figure 4.1 below.Figure 4.1 Figure 4.2It is quite common, especially with count data, to find that the variation of the residualsincreases as the value of the response increases, as in Figure 4.2. In this case, the standarderrors of differences between treatments will be over-estimated for differences betweentreatments with low means, and under-estimated for differences between larger means,causing incorrect conclusions to be drawn. If a plot of residuals against fitted valuesindicates non-homogeneity of variances, a transformation of the data should beconsidered, as we show in Section 4.5.One situation where unequal variances can occur, but where a transformation may nothelp, is when analyses are performed on data collected in different years or at differentlocations. It is then important to check that the variances within the years (or at eachlocation) are homogeneous. Otherwise a weighted analysis will be required, with the datafrom each year being weighted by the reciprocal of the variance at that year. (This can bedone automatically by using the Multiple Experiments / Meta Analysis (REML) menu,although we do not cover that here.)
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Anova and Design
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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
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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 58 and 59: 54 4 Checking the assumptions4.2 No
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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
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Page 90 and 91: 86 7 Balance and non-orthogonalityT
Page 92 and 93: 88 7 Balance and non-orthogonalityF
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Page 102 and 103: 8 REML analysis of unbalanced desig
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102 8 REML analysis of unbalanced d
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9 Commands for analysis of variance
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IndexA2DISPLAY procedure 117A2WAY p
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122 IndexTransformation 56logarithm
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