Full and reduced models for yield trials

Empirical results routinely demonstrate that the reduced Additive Main effects and Multiplicative Interaction (AMMI) model achieves better predictive accuracy for yield trials than does the full treatment means model. It may seem mysterious that treatment means are not the most accurate estimates, but rather that the AMMI model is often more accurate than its data. The statistical explanation involves the Stein effect, whereby a small sacrifice in bias can produce a large gain in accuracy. The corresponding agricultural explanation is somewhat complex, beginning with a yield trial's design and ending with its research purposes and applications. In essence, AMMI selectively recovers pattern related to the treatment design in its model, while selectively relegating noise related to the experimental design in its discarded residual. For estimating the yield of a particular genotype in a particular environment, the AMMI model uses the entire yield trial, rather than only the several replications of this particular trial, as in the treatment means model. This use of more information is the source of AMMI's gain in accuracy.