Weighted Estimation of AMMI and GGE Models

  • S. Hadasch
  • J. Forkman
  • W. A. Malik
  • H. P. Piepho


The AMMI/GGE model can be used to describe a two-way table of genotype–environment means. When the genotype–environment means are independent and homoscedastic, ordinary least squares (OLS) gives optimal estimates of the model. In plant breeding, the assumption of independence and homoscedasticity of the genotype–environment means is frequently violated, however, such that generalized least squares (GLS) estimation is more appropriate. This paper introduces three different GLS algorithms that use a weighting matrix to take the correlation between the genotype–environment means as well as heteroscedasticity into account. To investigate the effectiveness of the GLS estimation, the proposed algorithms were implemented using three different weighting matrices, including (i) an identity matrix (OLS estimation), (ii) an approximation of the complete inverse covariance matrix of the genotype–environment means, and (iii) the complete inverse covariance matrix of the genotype–environment means. Using simulated data modeled on real experiments, the different weighting methods were compared in terms of the mean-squared error of the genotype–environment means, interaction effects, and singular vectors. The results show that weighted estimation generally outperformed unweighted estimation in terms of the mean-squared error. Furthermore, the effectiveness of the weighted estimation increased when the heterogeneity of the variances of the genotype–environment means increased.


Generalized least squares Genotype–environment interaction Jacobi iterative method Multi-environment analysis Multi-environment trial 

Supplementary material

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Copyright information

© International Biometric Society 2018

Authors and Affiliations

  • S. Hadasch
    • 1
  • J. Forkman
    • 2
  • W. A. Malik
    • 1
  • H. P. Piepho
    • 1
  1. 1.Biostatistics Unit, Institute of Crop ScienceUniversity of HohenheimStuttgartGermany
  2. 2.Department of Crop Production EcologySwedish University of Agricultural SciencesUppsalaSweden

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