Download pdf guide - VSN International

82 7 Balance and non-orthogonality7.1 Confounding and efficiency factorsIn the split-plot design it is the main effect of one of the treatment factors that isestimated in the higher stratum. Statistically, we would say that this main effect isconfounded with whole plots within blocks. For the factor Variety in Section 5.1, thisis completely acceptable; the main interest in the trial was to look at the Nitrogen factorand the interaction between Nitrogen and Variety. However, on other occasions, wemay want all the main effects to be estimated with the extra precision that should beavailable in the bottom stratum, and so we may want the interactions to be estimated inthe higher strata instead.n 0 0 k n 0 0 k 0 0 0 0 n k n kThe plan above shows a design in which the interaction between the factors N and K isconfounded with blocks. The definition of the N × K interaction is that it is the differencebetween the effect of N estimated at the different levels of K. Here we have factors at twolevels 0 and n for N, and 0 and k for K. For the 0 level of K, the effect of adding N is givenby the mean of the plots with the combination (n, 0) minus the mean of the plots with(0, 0); while for K at level k, it is given by the mean of the plots with (n, k) minus themean of the plots with (0, k). So the difference between the two estimates (which givesthe interaction contrast) is{ mean of plots with (n, 0) + mean of plots with (0, k) } { mean of plots with (0, 0) + mean of plots (n, k) }The left-hand block above contains only combinations (n, 0) and (0, k), while the righthandblock contains only combinations (0, 0) and (n, k). Consequently the differencebetween the means of the plots in the two blocks also estimates the interaction: that is,the N × K interaction is confounded with blocks.Usually, in a situation like this, youwould have more than two blocks. Infact, the two blocks above are part of adesign with eight blocks, each with fourplots, that was used to study factors N, Kand D (see Yates, 1937, Design andAnalysis of Factorial Experiments, page21; also John, 1972, Statistical Designand Analysis of Experiments, page 135).The left-hand block in the plan is block3 of the design, and the right-hand blockis block 4. If we analyse just those twoblocks with treatment model N*K, theanalysis of variance table belowconfirms that the interaction isestimated in the Blocks stratum (and,Figure 7.1as we have analysed only these two blocks, there are no degrees of freedom left over forthe residual).