SUMMARY The additive main effects and multiplicative interaction (AMMI) model first applies the additive analysis of variance (ANOVA) model to two-way data, and then applies the multiplicative principal components analysis (PCA) model to the residual from the additive model, that is, to the interaction. AMMI analysis of yield trial data is a useful extension of the more familiar ANOVA, PCA, and linear regression procedures, particularly given a large genotype-by-environment interaction. Model selection and validation are considered from both predictive and postdictive perspectives, using data splitting and F-tests, respectively. A New York soybean yield trial serves as an example. Yield trials generate observations of yield, ordinarily replicated, for a number of genotypes grown in a number of environments (site-year combinations). Often the data are rather noisy, with a standard deviation for plot yields in excess of 25 % of the mean. An additional challenging feature of these data is the frequent presence of important and complex genotype-by-environment (GE) interactions. Plant breeders use yield trials to identify promising genotypes, and agronomists use them to make recommendations for farmers. The level of success in meeting these goals depends critically on two factors: (i) the accuracy of yield estimates, and (ii) the magnitudes of genotype-by-site, genotype-by-year, and genotype-by-site-by-year interactions (Talbot, 1984). In essence, these two factors reflect within-trial accuracy and between-trial predictability. This paper addresses only the first concern. Nothing is said regarding between-trial predictability, or agrotechnology transfer, other than to observe that success with withintrial accuracy is a necessary prelude to success with between-trial predictability. The within-trial accuracy of a statistical model may be assessed by two fundamentally different criteria: Postdictive success concerns a model's fit to its own data, whereas predictive success concerns the fit between a model constructed using part of the data and validation data not used in modelling. In either case, the statistical setting is that of an incompletely specified model, where empirical considerations enter into the decision to include a given' potential source in the model, or alternatively to relegate it to the model's residual (Bancroft, 1964). Whenever the data are noisy, postdiction and prediction are different tasks, and in general the model chosen by predictive criteria will be different and simpler than the model chosen by postdictive criteria. Because the several replicates of a yield trial constitute a noisy sample, a model chosen by predictive criteria may be expected to provide yield estimates that are closer to the true means (and to validation observations) than will a model chosen by postdictive criteria. This claim is readily subjected to experimental test. In what follows, the relative performances of statistical models chosen

Model selection and validation for yield trials with interaction

The analysis of series of crop variety trials has a long history with the earliest approaches being based on ANOVA methods. Kempton (1984) discussed the inadequacies of this approach, summarized the alternatives available at that time and noted that all of these approaches could be classified as multiplicative models. Recently, mixed model approaches have become popular for the analysis of series of variety trials. There are numerous reasons for their use, including the ease with which incomplete data (not all varieties in all trials) can be handled and the ability to appropriately model within-trial error variation. Currently, the most common mixed model approaches for series of variety trials are mixed model versions of the methods summarized by Kempton (1984). In the present paper a general formulation that encompasses all of these methods is described, then individual methods are considered in detail.

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The analysis of crop cultivar breeding and evaluation trials: an overview of current mixed model approaches

The recommendation of new plant varieties for commercial use requires reliable and accurate predictions of the average yield of each variety across a range of target environments and knowledge of important interactions with the environment. This information is obtained from series of plant variety trials, also known as multi-environment trials (MET). Cullis, Gogel, Verbyla, and Thompson (1998) presented a spatial mixed model approach for the analysis of MET data. In this paper we extend the analysis to include multiplicative models for the variety effects in each environment. The multiplicative model corresponds to that used in the multivariate technique of factor analysis. It allows a separate genetic variance for each environment and provides a parsimonious and interpretable model for the genetic covariances between environments. The model can be regarded as a random effects analogue of AMMI (additive main effects and multiplicative interactions). We illustrate the method using a large set of MET data from a South Australian barley breeding program.

Analyzing variety by environment data using multiplicative mixed models and adjustments for spatial field trend.

The accuracy of a yield trial can be increased by improved experimental techniques, more replicates, or more efficient statistical analyses. The third option involves nominal fixed costs, and is therefore very attractive. The statistical analysis recommended here combines the Additive main effects and multiplicative interaction (AMMI) model with a predictive assessment of accuracy. AMMI begins with the usual analysis of variance (ANOVA) to compute genotype and environment additive effects. It then applies principal components analysis (PCA) to analyze non-additive interaction effects. Tests with a New York soybean yield trial show that the predictive accuracy of AMMI with only two replicates is equal to the predictive accuracy of means based on five replicates. The effectiveness of AMMI increases with the size of the yield trial and with the noisiness of the data. Statistical analysis of yield trials with the AMMI model has a number of promising implications for agronomy and plant breeding research programs.

Predictive and postdictive success of statistical analyses of yield trials.

Breeding of temperate forage grasses is aimed at improving the economic and environmental sustainability of production from cattle and sheep. The primary objective is to ensure that forage can be the main source of feed for ruminants. This requires consistent production of herbage with a high feeding value, usually under nitrogen-limiting conditions. The most important traits affecting the feeding value of herbage are in vitro dry matter digestibility (DMD), the ratios of crude protein, water-soluble carbohydrate and fibre, and the concentration of alkaloid toxins. Improvements in these primary quality traits must be combined with good plant persistency, adequate tolerance to a range of environmental stresses, adequate resistance to a large number of different pathogens and invertebrate pests, and adequate seed yield. Forage grasses also have considerable potential to produce material for refining, to provide protein extracts for feeding to monogastric animals and carbohydrate for fermentation into fuel or into feedstocks for other industries.

Progress in breeding perennial forage grasses for temperate agriculture

This paper provides estimates of varieties × environments and plot error variances based on more than 1000 trials of varieties of oats, barley, wheat, perennial and Italian ryegrass, timothy, cocksfoot, potato, sugar beet, swedes, autumn kale, forage maize and field beans. Tables of critical differences and acceptance probabilities are presented to guide in planning future series of trials.

M. Talbot

Papers

Yield variability of crop varieties in the U.K.