Motivation

To simplify testing programs and to streamline seed production and sales, breeders have attempted to produce cultivars that are as stable and widely adapted as possible. When wide adaptation is the goal, it is helpful and efficient to identify mega-environments and then place a testing station in each mega-environment. On the other hand, there are limits to this goal of wide adaptation because there is never one genotype that wins every time in every place; rather, genotype-environment (GE) interaction is ubiquitous. To exploit these interactions to increase yields, it is necessary to subdivide a crop's growing region into several mega-environments. Anyway, note that identifying mega-environments is strategic regardless whether a breeder's goal is wide adaptation or narrow adaptation (or both).

Various statistical methods have been used to identify mega-environments, especially various classification or ordination methods (or both, which is called pattern analysis). But AMMI offers and combines four substantial advantages.

(1) AMMI can focus on relevant variation in the data and ignore irrelevant variation. AMMI partitions the total variation into the genotype main effects, environment main effects, and GE interaction effects that are further partitioned into several interaction principal component analysis (IPCA) axes and a residual. Often the environment main effects have the largest sum of squares (SS), and yet they are irrelevant for genotype rankings within environments and hence are irrelevant for breeders' selections. AMMI can ignore this irrelevant variation, whereas some other analyses combine and confound genotype, environment, and interaction effects and thereby put most of their emphasis on features of the data that are irrelevant for identifying mega-environments. Furthermore, AMMI can reduce noise. Because the interaction has a large number of degrees of freedom, most of the noise goes into the interaction. Noise makes yield-trial results appear more complicated than they really are. Noise also makes weaker entries appear to win sometimes and thereby swells the roster of winners unreasonably. AMMI can concentrate the interaction's real signal in early IPCA axes while relegating most of the interaction's spurious noise to a discarded residual, thereby reducing noise and gaining accuracy. Hence, AMMI can concentrate on genotype main effects and real interaction effects that are relevant for mega-environments while disregarding environment main effects and most of the interaction noise that are irrelevant. Often this relevant variation is only 10% to 40% of the overall variation, so AMMI's clear focus promotes effective results. Unfortunately, some analyses, such as clustering based on the original data (rather than these data minus the environment main effects), pay greatest attention to irrelevant features of the data, so mega-environments are delineated poorly.

(2) AMMI mega-environment analysis is relevant for agronomists' and breeders' research questions, especially "What wins where?" An answer to this question provides a basis for grouping locations with identical (or at least similar) winners into a mega-environment and for targeting suitable genotypes for each mega-environment. Accordingly, AMMIWINS focuses on rank changes among the winners while disregarding changes among the losers. That is, AMMIWINS focuses on just that fraction of the interaction that involves just those genotypes winning in one or more environments. Regrettably, many other analyses place equal emphasis on all interactions, regardless whether they involve changes among winners or losers, so the resulting mega-environments are excessively numerous and have little bearing on variety recommendations to maximize yields. Again, attending to extraneous features of the data results in poor delineation of mega-environments.

(3) AMMI analysis provides an integrated understanding of the genotypes and environments. The underlying concept of GE interaction, as well as the derived concept of mega-environments, are fundamentally dual, involving both genotypes and environments. Hence, the objectives of identifying mega-environments and targeting genotypes are inherently and deeply interrelated and must be pursued jointly. By contrast, many other analyses treat only genotypes or else only environments, or perhaps treat both with two separate and independent analyses. For example, a popular approach has been to cluster environments by algorithms that provide no information on genotypes whatsoever. Unfortunately, such analyses cannot possibly provide insight on GE interaction, even though interaction is relevant for identifying mega-environments.

(4) AMMI is flexible regarding the structure of available yield-trial data. The data may have a two-way genotypes by environments structure or a three-way genotypes by locations by years structure (which is analyzed by treating location-year combinations as environments). Also, the trial may be replicated or not. Furthermore, MATMODEL implements an Expectation Maximization version of AMMI that can tolerate and impute missing data. So, MATMODEL analysis is flexible, accepting data over this whole range of possibilities. Naturally, more extensive data allow more extensive results, but smaller experiments also merit effective analysis.

Because of this combination of four important advantages, AMMI can focus selectively on those particular features of the data that are relevant for delineating mega-environments. Furthermore, several kinds of graphs can readily convey a tremendous amount of information about even very large and complex yield trials. Such results promote insight into each genotype's responses to the various environments. Most of all, AMMI analysis shows how many and which genotypes are needed for every environment to achieve optimal (or nearly optimal) yield. It also quantifies the yield penalty for not subdividing a crop's growing region into two or more mega-environments when needed; that is, the penalty for recommending only a single genotype, the main-effect winner, throughout a crop's growing region when this one genotype is not winning everywhere.