SOURCE OF VARIATION df EXPECTED MEAN SQUARE
F F-1 o + Vo 2
V V-1 o2 + Fo,
V x El V-1 o'Qf2 + Fog
V x E12 V-1 o'w + FoE,2
V x F (now the residual) (V-1)(F-3) alt,2
Note that this has no impact on the F-ratio used.
The use of El in stability analysis has been widely criticized because the independent variable El is in fact a function of the dependent variable. Westcott (1985) makes a case for greater use of independently determined "environmental variables.' He also notes that "environmental measurements are very seldom available.* Theoretical objections aside, the onfarm researcher often has but two alternatives: using El or being unable to make useful recommendations within a reasonable period of time. And, as McCullagh and Nelder (1989) point out, "A first, though at first sight, not a very helpful principle, is that all models are wrong; some, though, are more useful than others and we should seek those." Critics often point to the weaknesses in formal statistical properties of analysis using El. These difficulties clearly exist; however, a more compelling point is that the researcher often has the El as the ONLY objective guide to environmental quality. These criticisms would be severe problems if formal. definitive statistical inference were the objective. It is not. The more important use of this type of analysis is to obtain preliminary insight 'egarding the consistency of treatment performance, which fields, farms or groups of farms appear to be troublesome, what recommendations appear to be reasonable eto Thissort ot analysisis. alwaysa starting point, never an end in itself.
For the researcher to make the jump from finding a significant El x treatment interaction from a model such as (3) or (4) to associating El with predictable future environments or "recommendation domains" and making reliable treatment recommendations for them obviously requires a great deal of thought and care (and involves, to a large extent, non-statistical questions, i.e. why are some El low and others high). Predicted treatment performance for farms included in the trial can be made using well known best linear unbiased prediction methods. The Els have no intrinsic meaning, so predictons for fields or farms not included in the trial are only as good as the researcher's ability to predict which fields or farms will be in which recommendation domain. The on-farm trial will not by itself generate data suitable for this purpose.
AN IMPORTANT NOTE ON DESIGNING-ON-FARM TRIALS
Note that neither MS[R(FT)] nor MS[residl are ever used in the analysis of the "usual" onfarm trial, i.e. one described by some variation an model (2). The appropriate denominator term for all tests of interest is MSIV x F(T)]. Why is this important? Both MS[R(FT)] and MS[residl require that R, the number of "replications" per farm, be at least two. However, neither of these terms has any role in the analysis of the standard on-farm trial. What would happen if only one replication per farm were observed? Neither. MS[R(FT)1 nor MS[resid] could be calculated. However, since neither term plays any role in the analysis, this is no real disadvantage.
It IS important to have as many farms per type as possible. This maximizes the degrees of freedom for MS[V x Fm]; since this is the denominator term for all F-ratios of interest, this will maximize power and, consequently, the usable information available. Thus, it is the FARM that is the true replication in an on-farm trial, not the "replication" within a farm (hence the motivation for the quotation marksl). This is important because on-farm researchers often have been advised to replicate within a farm, even for examplein Hildebrand and Poey (1985)l From an ANOVA