COMPOSIÇÕES SISPL
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Theor Appl Genet (1992) 85: 89-t00 Springer-Verlag 1992 Interpreting genotype-by-environment interaction using redundancy analysis E A. van Eeuwijk DLO-Center for Plant Breeding and Reproduction Research (CPRO-DLO), P.O. Box 16, 6700 AA Wageningen, The Netherlands Received January 3, 1992; Accepted March 24, 1992 Communicated by A. R. Hallauer Summary. Methods for the interpretation of genotype- by-environment interaction in the presense of explicitly measured environmental variables can be divided into two groups. Firstly, methods that extract environmental characterizations from the data itself, which are sub- sequently related to measured environmental variables, e.g., regression on the mean or singular value decomposi- tion of the matrix of residuals from additivity, followed by correlation, or regression, methods. Secondly, meth- ods that incorporate measured environmental variables directly into the model, e.g., multiple regression of indi- vidual genotypical responses on environmental variables, or factorial regression in which a genotype-by-environ- ment matrix is modelled in terms of concomitant vari- ables for the environmental factor. In this paper a redun- dancy analysis is presented, which can be derived from the singular-value decomposition of the residuals from additivity by imposing the restriction on the environmen- tal scores of having to be linear combinations of environ- mental variables. At the same time, redundancy analysis is derivable from factorial regression by rotation of the axes in the space spanned by the fitted values of the factorial regression, followed by a reduction of dimen- sionality through discarding the least explanatory axes. Redundancy analysis is a member of the second group of methods, and can be an important tool in the interpreta- tion of genotype-by-environment interaction, especially with reference to concomitant environmental informa- tion. A theoretical treatise of the method is given, fol- lowed by a practical example in which the results of the method are compared to the results of the other methods mentioned. Key words: Genotype-by-environment interaction - Fac- torial regression - AMMI analysis - Multiple regression - Redundancy analysis Lettuce Introduction In plant breeding, genotype-by-environment interaction typically refers to non-additivity in two-way tables of genotypes by environments. The data consist of evalua- tions of genotypes collected in a number of environments. The environments usually are made up of combinations of years and locations, but they may also involve different treatments. Environments may be characterized by a number of variables, e.g., soil, climatological, and treat- ment variables. The classical approach of Yates and Cochran (1938), revived by Finlay and Wilkinson (1963), uses the follow- ing model for an observation on genotype j (j = 1, 2.... , m) in environment i (i = 1, 2..... n) Yi3 = # + Gj + Bj E i + eij , (i) in which # stands for the overall mean, Gj for the geno- typical main effect, Bj for the slope of the linear regression of the response of genotype j on the environmental main effect Ei, and e~j is an error term. Bj is often interpreted as some kind of genotypical stability or sensitivity to the complex of environmental variables embodied in the environmental main effect E i. Hence the environment effectively is reduced to the mean performance of the genotypes in that environment, and the genotype-by- environment interaction is subsequently described as the heterogeneity of the slopes of the regressions of the indi- vidual genotypical responses on this mean. For obvious reasons this model is referred to as the regression-on-the mean model. The environment is modelled in terms of the observations of the matrix; no use is made of explicitly measured environmental variables. Regression-on-the mean provides modelling opportu- nities for interactions describable in one dimension. For modelling higher dimensional interaction, recourse can
Transcript of COMPOSIÇÕES SISPL
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