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Recently, there has been a great deal of interest in applications of mixed model theory in agriculture. Henderson ( 1975) developed best inear unbiased predictors (or BLUPs) as an alternative to more enumerative-type estimators. Perceived at first as an ad hoc procedure, Harville (1976) put BLUP on sound theoretical footing. A regional publication of the southern Research and Information Exchange Group in statistics (Southern Regional Bulletin, 1989) contained several examples of mixed model applications in agriculture. This publication also contained articles by McLean (1989) and Stroup (1 989a) describing mixed model theory and methods.
In the following section, mixed linear models appropriate for on-farm trials are discussed. These models superficially resemble models used to evaluate on-station data. The goal of this section is to show how to use mixed model theory to understand the distinction between the various assumptions that can be made about these models, their effect on the resulting analysis, and their implications for the on-farm researcher. The larger objective is to empower the on-farm researcher with a relevant statistical perspective so that design and analysis choices appropriate to on-farm trials can be made.
THE "TYPICAL" ON-FARM TRIAL
On-farm trials are conducted in a variety of ways, but most have a common basic structure. The following is a generic description of the essential elements:
Suppose a number of treatments, V, are to be evaluated. Each treatment is observed at F different farms where the specific biophysical and socioeconomic characteristics of the specific site on. the farm -will be characterized. At. each farm site, each treatment is "replicated" R times the word "replicated* appears in quotes here because, as will become apparent later in this discussion, multiple observations on treatments within a farm site may not be true replications. Note that the term 'farm%, to be designated in what follows by the letter "F* is generic. The term more specifically should be interpreted as "environment'. In specific trials, 'field", "location', 'village", etc. may apply equally.
Schematically, this trial can be represented as in Figure 1. As a starting point for analysis of this trial, the following mathematical model can be used:
yw= yi + f, + r(f), + v,, + vfk+ e,,1, (1)
where y~k is the observation on the ji' replication of the ii" farm for the kd" treatment,
# is the overall mean,
f, is the effect of the ii" farm,
r(f), is the effect of the j1h replication in the il' farm,
vk is the effect of the W"' treatment,
vf, is the interaction between the il farm and k1 treatment, -and
e~k is residual variation not accounted for by the above effects.
The analysis of variance (ANO VA) implied by this model has the following general form: