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44 3 Treatment structureN * Swas used by GenStat to specify the two-way analysis of variance introduced in Section3.1. This is expanded to become the formulaN + S + N.Swhich has three terms: N for the nitrogen main effect, S for the main effect of sulphur,and N.S for the nitrogen by sulphur interaction. Higher-order terms like N.S representall the joint effects of the factors N and S that have not been removed by earlier terms inthe formula. Thus here it represents the interaction between nitrogen and sulphur as bothmain effects have been removed.The other most-commonly used operator is the nesting operator (/). This occurs mostoften in block formulae. For example, the formulablock / plotis expanded to become the formulablock + block.plotThis specification assumes that there is no special similarity between the plot numbered1, for example, in block 1 and plot 1 in any other block. So the formula contains no"main effect" for plot, and the term block.plot thus represents plot-within-blockeffects (that is the differences between individual plots after removing any overallsimilarity between plots that belong to the same block). This is similar to the block modelfor the randomized design in Section 2.2 except that we have the factor plot instead of*Units*.Treatments can be nested too. For example, in a study of potential energy crops, wemay want to study two varieties of Miscanthus (M 1 ... M 2) and three of Reed CanaryGrass (R 1 ... R 3). We will certainly be interested in assessing overall differences betweenMiscanthus and Reed Canary Grass. We may also be interested in how much variationthere is between Mp 1 and Mp 2, and amongst {R 1, R 2 and R 3}; that is whether there isvariability of the varieties beyond the variability of the individual plants of each variety.The model of interest (assuming that there is no blocking) would then bey = ì + s + sv + åijk i ij ijkwhere parameterss represent the effects of the species (i = 1, 2), andisvijrepresent the variety within species effects (j = 1,2 for i=1, j = 1...3 for i=2).Notice that we do not have any term for a variety main effect the actual numberallocated to each variety does imply any special similarity for example between the strainnumbered 2 for Miscanthus and the strain numbered 2 for Reed Canary Grass.A formula can contain more than one of these operators. The three-factor factorialmodelbecomesA * B * CA + B + C + A.B + A.C + B.C + A.B.CThe interaction A.B.C then assesses whether the joint effects of factors A and B differaccording to the level of C (or, equivalently, whether the joint effects of A and C differ