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7.4 Unbalanced designs with two treatment factors 89Use the menu, as shown in Figure7.4, to construct a design for asingle replicate of a 2×2×2×2 designin blocks of size 8.Figure 7.47.4 Unbalanced designs with two treatment factorsMost of the designs covered by the Analysis of Variance menus are balanced and, in fact,all of those discussed so far in the earlier chapters have been orthogonal. Essentially thismeans that the order in which the treatment terms are fitted is unimportant (other thanthat each main effect must be fitted before any of its interactions). So we could havespecified sulphur as the first treatment factor and nitrogen as the second treatment factorin the menus in Figures 3.2 and 3.5, and still have obtained the same sums of squares andeffects. This contrasts with the situation in multiplelinear regression (see e.g. Section 5.2 of theIntroduction to GenStat for Windows), where the x-variates are usually correlated (i.e. non-orthogonal),and so different regression coefficients are obtainedfor each x-variate according to which other x-variates had been fitted beforehand.GenStat spreadsheet file Foster.gsh (Figure7.5) contains the results of an experiment to studythe effect of foster feeding of rats (Scheffe, 1959,The Analysis of Variance; also see McConway,Jones & Taylor, 1999, Statistical Modelling usingGENSTAT, Example 7.6). The rats were from fourdifferent genotypes (A, B, I or J), the experimentalunit was a litter of four rats, and the responsevariate was the weight of the litter after a period offeeding. The interest was in whether the genotypeof a foster mother would affect the weight. So thereare two treatment factors, each with four levels, thegenotype of the mother and the genotype of theFigure 7.5