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8 REML analysis of unbalanced designsThe Analysis of Variance menus, described in the earlier chapters, deal mainly withbalanced designs. This ideal situation, however, is not always achievable. Therandomized-block design in Section 2.2 is balanced because every block contained oneof each treatment combination. However, there may sometimes be so many treatmentsthat the blocks would become unrealistically large. Designs where each block containsless than the full set of treatments include cyclic designs and Alpha designs (both ofwhich can be generated within GenStat by clicking Stats on the menu bar, selectingDesign and then Select Design), neither of which tend to be balanced. In experiments onanimals, some subjects may fail to complete the experiment for reasons unconnected withthe treatments. So even an initially balanced experiment may not yield a balanced set ofdata for analysis. The Mixed Models (REML) menus, which use the GenStat REML directive,are designed to handle these situations. They also allow you to fit models to the complexcorrelation structures that occur in repeated measurements or in spatially-correlated datafrom field experiments.In this chapter you will learn• how to use the Linear Mixed Models menu• what output is given by a GenStat REML analysis, and how it compares to GenStatANOVA• how to assess fixed terms using Wald and F statistics• how effects and means can be produced by GenStat ANOVA, combining all theavailable information when treatment terms that are estimated in several strata Note: the topics marked are optional.
8.1 Linear mixed models: split-plot design 998.1 Linear mixed models: split-plot designWe start by reanalysing the split-plot data (Oats.gsh) in Section 5.1, to highlight thedifferences and similarities between REML and ANOVA.Figure 8.1 shows the LinearMixed Models menu, obtained byclicking Stats on the menu barand selecting Mixed Models(REML), followed by Linear MixedModels. The Fixed Model boxcorresponds to the TreatmentStructure box in the split-plotmenu, and specifies the termsdefining the fixed effects in themodel to be fitted. The LinearMixed Models menu providesgeneral facilities covering anyFigure 8.1type of design, and so the random effects are defined explicitly by the contents of theRandom Model box, instead of being defined automatically as in the split-plot menu. Themodel is the same though, namelyblocks/wplots/subplotswhich expands to give the three (random) terms; see Section 3.4.block + block.wplot + block.wplot.subplotSimilarly, the fixed modelvariety * nitrogenexpands as before tovariety + nitrogen + nitrogen.varietyto request that GenStat fits the main effects of nitrogen and variety, and their interaction.(The Interactions box, which operates just like the one in the Analysis of Variance menu, hasrequested all interactions in the fixed model to be included.)
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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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20 1 From t-test to one-way anovaTh
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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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38 3 Treatment structureThere is a
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40 3 Treatment structureblock.*Unit
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42 3 Treatment structureTables of c
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44 3 Treatment structureN * Swas us
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46 3 Treatment structureFumigant Am
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Page 56 and 57: 4 Checking the assumptionsIn this c
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Page 78 and 79: 74 6 Design and sample size6.1 Desi
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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
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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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