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15 Examples 275namely sqrt(rootwt) in the presence/absence of bloodworms. Thus to minimiseselection bias the variety main effects and thence the tmt.variety interactionsare taken as random. The main effect of treatment is fitted as fixed to allow forthe likely scenario that rather than a single population of treatment by varietyeffects there are in fact two populations (control and treated) with a differentmean for each. There is evidence of this prior to analysis with the large differencein mean sqrt(rootwt) for the two groups (14.93 and 8.23 for control andtreated respectively). The inclusion of tmt as a fixed effect ensures that BLUPsof tmt.variety effects are shrunk to the correct mean (treatment means ratherthan an overall mean).The model for the data is given byy = Xτ + Z 1 u 1 + Z 2 u 2 + Z 3 u 3 + Z 4 u 4 + Z 5 u 5 + e (15.7)where y is a vector of length n = 264 containing the sqrt(rootwt) values, τcorresponds to a constant term and the fixed treatment contrast and u 1 . . . u 5correspond to random variety, treatment by variety, run, treatment by run andvariety by run effects. The random effects and error are assumed to be independentGaussian variables with zero means and variance structures var (u i ) = σ 2 i I b i(where b i is the length of u i , i = 1 . . . 5) and var (e) = σ 2 I n .The ASReml code for this analysis isBloodworm data Dr M Stevenspair 132rootwtrun 66tmt 2 !Aidvariety 44 !Arice.asd !skip 1 !DOPATH 1!PATH 1sqrt(rootwt) ~ mu tmt !r variety variety.tmt run pair run.tmt0 0 0!PATH 2sqrt(rootwt) ~ mu tmt !r variety tmt.variety run pair tmt.run,uni(tmt,2)0 0 2tmt.variety 22 0 DIAG .1 .1 !GU44 0 0tmt.run 22 0 DIAG .1 .1 !GU66 0 0