6
SOURCE OF VARIATION degrees of freedom
FARM F F-1
REP(FARM) R(F) F(R-1)
TREATMENT V V-1
FARM X TREATMENT V x F (F-1)(V-1)
RESIDUAL resid F(R-1)(V-1)
TOTAL FRV 1
This ANOVA has several possible interpretations, depending on the specific objectives of a given on-farm trial and how the effects in the model are defined as a consequence. In order to make appropriate use of this ANOVA table, the researcher must be clear about the objectives of the trial and the nature of the effects being observed. Some useful definitions follow.
Population of inference: The set of elements (e.g. farms) to which the results of the study are to be applied. 'his is similar to the concept of a research domain.
Prediction space: Applications of study results from on-farm trials often take the form of
recommendations. Recommendations are based on the predicted behavior of the treatments, either for the entire population or for various sub-populations. The set of elements (e.g. farms or environments) to which a prediction is intended to be applicable is called the prediction space. This is similar to the concept of a recommendation domain.
Random .and-Fixed. Effects:.. Effects irmthe-study treatments, farms, "replications" can be considered as fixed or random depending on 1) how they are chosen and 2) what prediction space is appropriate to the objectives of the study. An effect is considered fixed if the levels of a particular factor are chosen deliberately in advance of the study. In this case identical levels would be used again were the study to be repeated based on the same prior knowledge, and prediction is limited to only those levels actually represented in the study. Typically, treatments such as tillage methods or fertilizer levels in a variety trial would be considered fixed effects. An effect is considered random if the levels actually observed in the study result from a random sample of a larger population identical levels in a repeat of the study would be exceedingly unlikely. Prediction in this case is intended to apply to the population of which the levels observed are onlyrepresentatives. The most blatant example of a random effect would be the effect of "replication" or of residual variation. Many effects are not clearly fixed or random the effect of farm site or environment, for example. Whether an effect is fixed or random has a major impact on the analysis, as will be demonstrated below.
Most statistical methods texts, e.g. Steel and Torri (1980) or Snedecor and Cochran
(19801, contain discussions of fixed and random. effects. Many texts on the design or planning of experiments discuss the population of inference, e.g. Cox (1958) or Mead (1988). The reader is referred to these texts for more detail.
IMPACT OF FIXED OR RANDOM EFFECTS ON ANOVA
In the ANOVA for the on-farm trial given above, it is usually fairly clear that *treatments* are fixed effects and "replications' are random. Farms, however, are not so easily categorized. Different farms may have been selected quite intentionally based on certain criteria: size, income, technology level, soil type, climatic characteristics, etc. Or they may have been selected at random from a target population. Actually, these are extremes usually, farms are selected using a combination of fixed and random effect tactics.- That is, a spectrum of-defmed conditions must be
r r