9
The type by treatment (V x T) interaction would be of initial primary interest. Its F-ratio is MS[V x T]/MS[V x F(T)].
As before, partitioning MS[V x TI into meaningful contrasts would be strongly advisable. For example, suppose the farm types are:
1. higher rainfall, mechanized
2. higher rainfall, non-mechanized
3. lower rainfall, mechanized
4. lower rainfall, non-mechanized
and the treatments are:
1. standard variety, no fertilizer
2. standard variety, with fertilizer
3. resistant variety, no fertilizer
4. resistant variety, with fertilizer
The type main effect could be partitioned into rainfall and mechanization main effects and a rainfall by mechanization interaction. The treatment main effect could be partitioned into variety and fertilizer main effects* and a variety by fertilizer interaction. Then the interaction of any of the three type effects with any of the three treatment effects could be evaluated. For example, a rainfall by variety effect could be examined to see if the resistant variety is equally advantageous at lower and higher rainfall. In the unusual case that type by treatment interactions are negligible, the
-treatment -main.effect, could be tested using. MS[VIIMS[V x FM].
Predicted performance of treatments for particular farm types can be obtained using confidence intervals for the treatment x farm type means. Care should be taken to base the confidence interval on the correct standard error. Most statistical software packages are poorly suited to work with mixed linear models such as model (2) without special attention. For a complete discussion of this issue, see McLean (1989) and Stroup 01.989a). Predicted performance of specific farms within a given farm type for a particular treatment can be obtained by calculating best linear unbiased predictors (Henderson, 1975). These are not the same as usual sample means. Again, see McLean (1989) and Stroup (1 989a and 1 989b) for a full discussion of best linear unbiased prediction.
STABIUTY ANALYSIS
A special case of the above analysis occurs when 'environmental types' and their potential interactions with~ treatment are not well understood prior to conducting the on-farm trial. In such cases, the researcher makes an attempt toa represent as wide a spectrum of types as possible within the-population of inference. but a cleano partition of the variability among environments- into types and environments within types may not be possible. Indeed. one objective of the research may be to provide insight concerning which environments favor or disfavor certain treatments and what features are common to these environments. Various forms of *stability analysis' are important examples of this approach.
Excellent review articles on stability analysis are available (see Freeman (1973), Hill (1975), Westcott (1985)). Hildebrand (1984) has adapted the approach for on-farm trials and its use is demonstrated in the following section. This discussion will be restricted to pointing out its relation to model (2) above. In Hildebrand's modified stability analysis (MSA), an index for a given environment (El) is defined -as- the mean response over all treatments at that farm site. A linear