8
which interaction implies that relative differences among treatments are affected by systematic, identifiable and repeatable farm characteristics (i.e. the characteristics that motivated the choice of the farms in the first place). In fact, the test for interaction is not particularly interesting if farms are random: if o,, is not greater than zero, then the assumption of random farms is probably defective. Of interest is the treatment main effect. This is evaluated using the F-ratio MS(V)/MS(V x F). Its purpose is to verify that differences among treatment means, substantial and consistent enough to be seen through the population of inference, over and above random differences among treatment by farm, actually exist.
To summarize, if farms are fixed, the F-ratio of primary interest is that for the V x F
interaction, MS(V x F)/MS(resid), or, if the V x F interaction is negligible then the V main effect is assessed by MS(V)IMS(resid). If farms are random, the V x F test is of little intrinsic interest (except to verify the validity of the assumptions); of primary interest is the V main effect, which in this case has an F-ratio MS(V)/MS(V x F).
In on-farms trials as they are actually conducted, farms are rarely purely fixed or purely random effects. The above ANOVAs, therefore, are useful as academic exercises to illustrate issues the farming systems researcher needs to understand, but neither, unmodified, is likely to be of much use in practice.
PARTITIONING THE FARM BY TREATMENT INTERACTION
In most on-farm trials, the population of inference includes a set of *types of
environments,* that the researcher wants to be represented. In the extreme fixed effects case, the number of types, would be F, and thus only.one environment per type would be observed. In the extreme random effects case, there- would be exactly one type of environment. (or so little would be known about the environments that typing could not be done prior to conducting the trial) and F randomly sampled environments per type. Usually on-farm researchers would reject either extreme; a more realistic design would be to randomly sample a number of environments from each of the several types of in the population.
If the types of 'farms* are very well defined, model (1) could be modified as follows:
y~i, = u + t, + f(t) + r(ff)iik + v, + vt + vf(t),, + e., (2)
where t, is the effect of farm type,
f(t)q is the effect of farm within type,
vtv is the farm type by treatment interaction,
and other terms follow by extension from model (1).
In model (2) type and treatment would be considered fixed, farm and replication random,
and analysis would proceed accordingly based on thefollowing ANOVA:
SOURCE OF VARIATION Sa EXPECTED MEAN SQUARE
T T-1 o2 + Ro',u2+ Vo'a2 + RVo'f2 + FRV0p
FM T(F-1) a2 + Rau2+ Voa, + RVoft2
R(TF) TF(R-1) oa2 + Vo,2
V V-1 e + Ro,.2 + TFRO,,
VxT (T-1)(V-1) oz + Ra,2 + FRo,,
V x F(T) T(V-1)(F-1) oa + Ro,,2
residual. TF(R-1 )(V-1) a2