Publisher Summary This chapter presents the statistical analysis of multilocation trials. Multilocation trials play an important role in plant breeding and agronomic research. Data from such trials have three main agricultural objectives—to accurately estimate and predict yield based on limited experimental data; to determine yield stability and the pattern of response of genotypes or agronomic treatments across environments; and to provide reliable guidance for selecting the best genotypes or agronomic treatments for planting in future years and at new sites. Agronomists use multilocation trials to compare combinations of agronomic factors, such as fertilizer levels and plant density, and on this basis make recommendations for farmers. Breeders compare different improved genotypes to identify the superior ones. Data collected in multilocation trials are intrinsically complex, having three fundamental aspects—namely, structural patterns; nonstructural noise; and relationships among genotypes, environments, and genotypes and environments considered jointly. Pattern implies that a number of genotypes respond to certain environments in a systematic, significant, and interpretable manner, whereas noise suggests that the responses are unpredictable and uninterpretable.

Statistical analyses of multilocation trials

The Additive Main effects and Multiplicative Interaction (AMMI) model, Genotype main effects and Genotype X Environment interaction (GGE) model, and Principal Components Analysis (PCA) are singular value decomposition (SVD) based statistical analyses often applied to yield-trial data. This paper presents a systematic comparison, using both statistical theory and empirical investigations, while considering both current practices and best practices. Agricultural researchers using these analyses face two inevitable choices. First is the choice of a model for visualizing data. AMMI is decidedly superior, not for statistical reasons, but rather for agricultural reasons. AMMI partitions the overall variation into genotype main effects, environment main effects, and genotype X environment interactions. These three sources of variation present agricultural researchers with different challenges and opportunities, so it is best to handle them separately, while still considering all three in an integrated manner. Second is the choice of a member of a given model family for gaining predictive accuracy. AMMI, GGE, and other SVD-based model families are essentially equivalent, but best practices require model diagnosis for each individual dataset to determine which member is most predictively accurate. Making these two choices well allows researchers to extract more usable information from their data, thereby increasing efficiency and accelerating progress.

Statistical Analysis of Yield Trials by AMMI and GGE

This is largely a review paper, describing various statistical methods for analysing interactions in general and genotype-environment interactions in particular, and giving nearly 100 references to previous work. The joint regression analysis approach introduced by Yates and Cochran (1938) is considered in some detail; alternatives to regression are discussed, as are various stability parameters. Much work has been done on statistical methods for testing for interactions in general, and this also is reviewed, from Tukey's (1949) one degree of freedom for non-additivity to Milliken and Graybill's (1970) generalisation to testing for various types of possible interaction. The difficulties of testing and inference in the presence of interaction are discussed. Data from a two-way table may be regarded as a multivariate set, as first shown by Williams (1952) and later extended by others, particularly Mandel (1969b and 1971 ). These methods are only just beginning to be used in studies of genotype-environment interactions, and several recent references are given. External measurements may be used to measure the environment and these may be either physical or biological. Again, the appropriate methods of analysis are fairly new. The interpretation of interactions is considered in relation to the use to be made of the results. It is suggested that various multivariate techniques may be used to assist in the elucidation of interactions, especially when these are not easy to explain by simpler methods of analysis.

Statistical methods for the analysis of genotype-environment interactions.

The recent rise of interest in the use of nonparametric tests stems from two main sources. One is the concern about the use of parametric tests when the underlying assumptions are not met. The other is the problem of whether or not the measurement scale is suitable for application of parametric procedures. On both counts parametric tests are generally more in danger than nonparametric tests. Because of this, and because of a natural enthusiasm for a new technique, there has been a sometimes uncritical acceptance of nonparametric procedures. By now a certain degree of agreement concerning the more practical aspects involved in the choice of tests appears to have been reached. However, the measurement theoretical issue has been less clearly resolved. The principal purpose of this article is to discuss this latter issue further. For the sake of completeness, a brief overview of practical statistical considerations will also be included. A few preliminary comments are needed in order to circumscribe the subsequent discussion. In the first place, it is assumed throughout that the data at hand arise from some sort of measuring scale which gives numerical results. This restriction is implicit in the proposal to compare parametric and nonparametric tests

Scales and statistics: parametric and nonparametric.

Least Squares Approximation of Matrices by Additive and Multiplicative Models

Where the joint effects of two or more factors are not additive, a simple model is proposed for representing the effects. The effects of one factor are assumed to be proportional, rather than equal, at different levels of the other factors. The main effects of the first factor are given as weighted averages of the simple effects at the different levels of the other factors, the weights being the estimated factors of proportionality. The weights are given as the latent vector of a matrix of sums of squares and products corresponding to the largest latent root of the matrix; the sum of squares for the weighted main effect is a multiple of this latent root, and the other latent roots correspond to a partition of the interaction sum of squares. The analysis is closely related to the canonical analysis of a set of variates. Tests of significance of (a) the residual interactions and (b) the adequacy of a proposed set of weights are discussed. For the case where the matrix has only two non-vanishing latent roots, the approach of the joint distribution of the roots to its limiting form is discussed. The joint probability density is expanded as a series of Bessel functions of imaginary argument. Asymptotic formulae for the moments and product-moments of the roots are derived. Exact tests for the adequacy of a proposed set of weights, when there are only two non-vanishing latent roots, are presented. The methods of analysis are illustrated with a numerical example.

E. J. Williams

Papers

The interpretation of interactions in factorial experiments