Resumo: O objetivo deste trabalho foi propor um esquema de ponderação para o modelo de efeitos principais aditivos e interação multiplicativa (AMMI), bem como avaliar a utilidade deste modelo W-AMMI no estudo da interação genótipo x ambiente (GxA) e da interação de locos associados a caracteres quantitativos x ambiente (QxA) para dados não replicados. Utilizou-se a população de cevada (Hordeum vulgare) 'Harrington' x TR306, com 141 genótipos avaliados em 25 ambientes, para comparar os resultados do modelo AMMI com os de duas versões propostas do modelo W-AMMI: pesos iguais por linha e pesos iguais por coluna. O algoritmo W-AMMI de colunas proposto é viável para analisar informação com heterogeneidade de variâncias, quando não há repetições disponíveis. O uso dos modelos AMMI e W-AMMI, nos casos indicados, melhora a detecção de QTLs, além de propiciar uma intepretação adequada da GxA e um melhor entendimento da QxA, o que possibilita a obtenção de informações importantes para o aumento da produtividade em diferentes ambientes.

Abstract: The objective of this work was to propose a weighting scheme for the additive main effects and multiplicative interactions (AMMI) model, as well as to assess the usefulness of this W-AMMI model in the study of genotype x environment interaction (GxE) and quantitative trait locus x environment interaction (QxE) for nonreplicated data. Data from the 'Harrington' x TR306 barley (Hordeum vulgare) mapping population, with 141 genotypes evaluated in 25 environments, were used to compare the results from the AMMI model with those of two proposed versions of the W-AMMI model: equal weights per row and equal weights per column. The proposed W-AMMI columns algorithm is viable to analyze data with heterogeneous variance, when there are no replicates available. The use of the AMMI and W-AMMI models, in the indicated cases, improves QTL detection, besides providing a sound interpretation of GxE and a better understanding of QxE, which allows obtaining valuable information on increasing productivities in different environments.

In the context of multi-environment trials, where a series of experiments is conducted across different environmental conditions, the analysis of the structure of genotype-by-environment interaction is an important topic. This paper presents a generalization of the joint regression analysis for the cases where the response (e.g. yield) is not linear across environments and can be written as a second (or higher) order polynomial or another non-linear function. After identifying the common form regression function for all genotypes, we propose a selection procedure based on the adaptation of two tests: (i) a test for parallelism of regression curves; and (ii) a test of coincidence for those regressions. When the hypothesis of parallelism is rejected, subgroups of genotypes where the responses are parallel (or coincident) should be identified. The use of the Scheffé multiple comparison method for regression coefficients in second-order polynomials allows to group the genotypes in two types of groups: one with upward-facing concavity (i.e. potential yield growth), and the other with downward-facing concavity (i.e. the yield approaches saturation). Theoretical results for genotype comparison and genotype selection are illustrated with an example of yield from a non-orthogonal series of experiments with winter rye (Secalecereale L.). We have deleted 10 % of that data at random to show that our meteorology is fully applicable to incomplete data sets, often observed in multi-environment trials.

This paper joins the main properties of joint regression analysis (JRA), a model based on the Finlay-Wilkinson regression to analyse multi-environment trials, and of the additive main effects and multiplicative interaction (AMMI) model. The study compares JRA and AMMI with particular focus on robustness with increasing amounts of randomly selected missing data. The application is made using a data set from a breeding program of durum wheat (Triticum turgidum L., Durum Group) conducted in Portugal. The results of the two models result in similar dominant cultivars (JRA) and winner of mega-environments (AMMI) for the same environments. However, JRA had more stable results with the increase in the incidence rates of missing values.