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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