Statistical Theory

Two publications explain the statistical theory underlying AMMIWINS. Chapter 6 of Gauch (1992), especially pages 220-230, first explored the application of AMMI to mega-environment analysis, using fictitious graphs rather than any actual data. Gauch and Zobel (1997) developed these concepts much more fully, using a corn yield trial. This paper also introduced some new kinds of graphs that use AMMI to understand and delineate mega-environments. This documentation for AMMIWINS merely sketches the relevant theory, leaving a fuller account to these two publications. For those not already familiar with the AMMI model used by AMMIWINS, AMMI is explained in Bradu and Gabriel (1978), Kempton (1984), Gauch (1992), and Gauch and Zobel in Kang and Gauch (1996:85-122), and the MATMODEL software is introduced in Gauch and Furnas (1991).

AMMI offers a model family, not just a single model, so a decision is required about which AMMI model to use for a given dataset, that is, how many IPCA axes to include in the model (before relegating higher axes to a discarded residual). Ordinarily, the primary consideration for model choice is to optimize predictive accuracy by including those lower axes capturing mostly signal while discarding those higher axes capturing mostly noise (Gauch 1988, 1993, Gauch and Zobel 1988). For replicated data, the validation mode of MATMODEL can diagnose this most predictively accurate model. Otherwise, various heuristics and general experience can provide fairly reasonable guidance. Frequently, AMMI-1 or AMMI-2 is most accurate, but sometimes a higher model is best. Occasionally, the merely additive model, AMMI-0, is best, which means that a single genotype wins everywhere, so no mega-environment subdivision is needed.

Sometimes a secondary consideration for model choice is to restrict the allowable number of mega-environments to a small number, perhaps three to five, because of practical constraints regarding testing locations, seed production, and so on. In this case, a lower-order AMMI model than that indicated by predictive accuracy alone could be chosen to produce the required smaller roster of winners. However, it is probably much better to stay with the model chosen by the criterion of predictive accuracy, but then eliminate those winners with the fewest wins until a small enough roster of winners remains. This procedure minimizes the yield losses caused by the constraint of allowing just several mega-environments.

For the specified AMMI model, and for each environment, AMMIWINS solves the AMMI equation for every genotype and determines the winner. (The AMMI model is used to determine winners, not the raw data, because the AMMI model has been chosen for having greater predictive accuracy than the raw data.) Each genotype with one or more wins defines a mega-environment. AMMIWINS identifies each mega-environment by its winning genotype, counts its number of wins, and calculates the average expected yield over those environments included in that mega-environment. If desired, winners with few wins can be eliminated to reduce the number of mega-environments rather substantially while decreasing the crop yields only slightly.

The low-order models AMMI-1 and AMMI-2 permit some illuminating graphs that have no analogue for higher-order models, at least not on two-dimensional paper. These matters are explained in the next section of this documentation. Also see Gauch and Zobel (1997).