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100 8 REML analysis of unbalanced designsThe Options button producesthe Linear Mixed Model Optionsmenu, shown in Figure 8.2. Thestandard model options (asshown in the figure) are fine forthis design, so we need onlyselect the output to display (andthen click OK).Returning to the main menu(Figure 8.1): initial values areseldom required for simple REMLanalyses like this, and the SplineModel box is not relevant (this ismainly useful with repeatedmeasurements), so we can clickon Run and generate the outputshown below.Figure 8.2REML variance components analysisResponse variate: yieldFixed model:Constant + variety + nitrogen + variety.nitrogenRandom model: blocks + blocks.wplots + blocks.wplots.subplotsNumber of units: 72blocks.wplots.subplots used as residual termSparse algorithm with AI optimisationEstimated variance componentsRandom term component s.e.blocks 214.5 168.8blocks.wplots 106.1 67.9Residual variance modelTerm Factor Model(order) ParameterEstimate s.e.blocks.wplots.subplots Identity Sigma2 177.137.3Tests for fixed effectsSequentially adding terms to fixed modelFixed term Wald statistic n.d.f. F statistic d.d.f. F prvariety 2.97 2 1.49 10.0 0.272nitrogen 113.06 3 37.69 45.0
8.1 Linear mixed models: split-plot design 101variety.nitrogen 1.82 6 0.30 45.0 0.932Dropping individual terms from full fixed modelFixed term Wald statistic n.d.f. F statistic d.d.f. F prvariety.nitrogen 1.82 6 0.30 45.0 0.932Message: denominator degrees of freedom for approximate F-tests arecalculated using algebraic derivatives ignoring fixed/boundary/singularvariance parameters.The output first lists the terms in the fixed and random model, and indicates the residualterm. The residual term is a random term with a parameter for every unit in the design.Here we have specified a suitable term, blocks.wplots.subplots, explicitly.However, if we had specified only blocks and blocks.wplots as the Random Model(for example by putting blocks/wplots), GenStat would have added an extra term*units* to act as residual. (*units* would be a private factor with a level for everyunit in the design.)GenStat estimates a variance component for every term in the random model, apartfrom the residual. The variance component measures the inherent variability of the term,over and above the variability of the sub-units of which it is composed. Generally, thisis positive, indicating that the units become more variable the larger they become. So herethe whole-plots are more variable than the subplots, and the blocks are more variable thanthe whole-plots within the blocks. (This is the same conclusion that you would draw fromthe analysis-of-variance table in Section 5.1 and, in fact, you can also produce thevariance components as part of the stratum variances output from the Analysis of Variancemenu.) However, the variance component can sometimes be negative, indicating that thelarger units are less variable than you would expect from the contributions of the subunitsof which they are composed. This could happen if the sub-units were negativelycorrelated.The section of output summarising the residual variance model indicates that we havenot fitted any specialized correlation model on this term (see the column headed Model),and gives an estimate of the residual variance; this is the same figure as is given by themean square in the residual line in the blocks.wplots.subplots stratum in the splitplotanalysis-of-variance table.The next section, however, illustrates a major difference between the two analyses.When the design is balanced, GenStat is able to partition the variation into strata with anappropriate random error term (or residual) for each treatment term (see Section 5.1). Nosuch partitioning is feasible for the unbalanced situations that REML is designed to handle.Instead GenStat produces a Wald statistic to assess each fixed term.If the design is orthogonal, the Wald statistic is equal to the treatment sum of squaresdivided by the stratum residual mean square. So under the usual assumption that theresiduals come from Normal distributions, the Wald statistic divided by its degrees offreedom will have an F distribution, F m,n, where m is the number of degrees of freedomof the fixed term, and n is the number of residual degrees of freedom for the fixed term.By default, unless the design is large or complicated, GenStat estimates n, and prints itin the column headed “d.d.f.” (i.e. denominator degrees of freedom); m is shown the
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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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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
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16 1 From t-test to one-way anovaFi
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18 1 From t-test to one-way anovaTa
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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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48 3 Treatment structureFumigated S
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Page 56 and 57: 4 Checking the assumptionsIn this c
Page 58 and 59: 54 4 Checking the assumptions4.2 No
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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
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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
Page 92 and 93: 88 7 Balance and non-orthogonalityF
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Page 98 and 99: 94 7 Balance and non-orthogonalityM
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Page 102 and 103: 8 REML analysis of unbalanced desig
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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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