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represented, but some form of random sampling is done within each condition. Essentially, this amounts to stratified random sampling.
It follows that farms are not easily categorized as fixed or random. Usually, in fact, the .correct" analysis of the on-farm trial will involve some compromise between the analysis with farm as a fixed effect and the analysis with farms as random. Before examining this "compromise,* it is instructive to look at the appropriate analyses with farms strictly fixed or strictly random.
If farms are fixed then the only random components of model (1) are r(f)j and eljk. Denote the variance of r(f),, by a,2 and the variance of elik by .2. Then the expected values of the mean squares of the ANOVA are as follows:
SOURCE OF VARIATION EXPECTED MEAN SQUARE .
F a2 + FVa,2 +RVo,
RR a2 + FVa,2
V o2 + FRO,,
VxF o2 + RO
residual a 2
where 0,, 0, and Of denote variation attributable to the fixed effects fi, vk, and vfik,
respectively.
If farms are random. then the components f, and-vflk from model (1) are also random. Denote their variances by a,2 and o2, respectively. Then the expected mean squares are:
SOURCE OF VARIATION EXPECTED MEAN SQUARE
F a2 + Ra2 + Vaf2 + RVo,2
R(F) aZ + Voa,2
V a2 + RO-2. + FRO,,
VxF o2 + Ro.2
residual a2
These two ANOVA tables imply very different approaches to inference. When farms are fixed the data analyst's first concern must be the farm by treatment interaction (V x F), the magnitude of which is assessed by the F-ratio MS(V x F)/MS(resid). If this F-ratio indicates the existence of interaction, then effort must be focused on understanding its nature. Even if the interaction F-ratio appears to be negligible, the data analyst would do Well to partition MS(V x F) into meaningful components, e.g. using contrasts, since important interactions often are masked by the large number of degrees of freedom associated with the V x F effect (see Snedecor and Cochran 1980), pp. 304-307).
When farms are considered fixed, the treatment main effect is of interest only if the interaction effects are negligible, i.e. if it is clear that the same relationships among treatment means appear to hold for every farm in the population of inference. This is generally not true, but if it is then the treatment main effect can be evaluated using the F-ratio MS(V)/MS(resid).
When farms are considered random then test of farm by treatment interaction, which uses the same F-ratio as above, has a far different interpretation. Specifically, it means that differences among-treatments vary at random by farm. This is quite.distinctfrom thefixed. effect case, in