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regression over Els is obtained for each treatment and used as a basis for determining "recommendation domains," a notion loosely similar (but not identical) to the mixed model concept of prediction space. In terms of ANOVA, this could be expressed by modifying model (2):
Yiik = Y + f, + r(f), + v + ((ElI) + vfik + eiik (3)
where El, is the index of the il environment, and
G. is the linear regression coefficient for the kth treatment.
In essence, El in model (3) replaces type in model (2). Also, f, in model (3) is equivalent to t, + f(t),, in model (2) and vfik in model (3) is equivalent to vf(t)iil in model (2). Since environment (represented by "F') aside from El, is a random effect, the ANOVA is:
SOURCE OF VARIATION df EXPECTED MEAN SQUARE
F F-1 a2 + Ro'2 + Vaf2 + RVo 2
R(F) F(R-1) a + Vo,2
V V-1 oa2 + Ra,2 + FRO,
V x El V-1 a2 + Ra',2 + FRee
V x F (V-1)(F-2) a2 + Rao2
residual F(V-1 )(R-1) o2
Equality of the Gk can be tested using MS[V x EI]/MS[V x Fl. A "significant" F-ratio would imply that treatments respond unequally to El (and thus to whatever environmental types +he El imply). This would in. turn provide formal justification for predicting that different treatments are optimal for various "recommendation domains."
There is no reason why the use of environmental indices need be limited to linear regression. For example, model (3) can easily be extended to
Y= + f, + r(f), + Vk + MIk(EI) + Gn(EI)2 + vfk + eGik, (4)
where 13, is the linear regression coefficient for the kd' treatment, and
M~k is the quadratic regression coefficient for the kd' treatment.
The ANOVA for model (4) would be identical to the ANOVA for model (3) except that an additional line for V x Ell (or V x El x El) with V-1 degrees of freedom would appear immediately after V x El and the remaining V x F term would have (V-1)(F-3) degrees of freedom.
The F-ratio MS[V x E12]/MSIV x F] tests the equality of quadratic regression over El for the various treatments. Pictorially, this can be visualized as in Figure 4. Note that the quadratic regressions are quite different for the treatments, although their linear components are similar. Several authors have noted the limitations of linear-only regression over El, e.g. Westcott (1985). However, model (4) should make it clear that this restriction is unnecessary. Indeed, model (4) can be extended to more complex forms of regression over El.
If there is only one "replication" per farm (a discussion of the advantages and
disadvantages of this appears below) then the R(F) and residual terms in the ANOVA have no degrees of freedom and the result is the following simplified form: