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

13253_2018_323_MOESM1_ESM.docx (1.7 mb)
Supplementary material 1 (docx 1749 KB) (11 kb)
Supplementary material 2 (zip 10 KB)


  1. Besag, J., and Higdon, D. (1999), “Bayesian analysis of agricultural field experiments,” Journal of the Royal Statistical Society. Series B (Statistical Methodology), 61, 691–746. MathSciNetCrossRefzbMATHGoogle Scholar
  2. Bro, R., Kjeldahl, K., Smilde, A.K., and Kiers, H.A.L. (2008), “Cross-validation of component models: A critical look at current methods,” Analytical and Bioanalytical Chemistry, 390, 1241–1251. CrossRefGoogle Scholar
  3. Caliński, T., Czajka, S., Denis, J.B., and Kaczmarek, Z. (1992), “EM and ALS algorithms applied to estimation of missing data in series of variety trials,” Biuletyn Oceny Odmian, 24–25, 8–31.Google Scholar
  4. Cornelius, P.L., Seydsadr, M., and Crossa, J. (1992), “Using the shifted multiplicative model to search for separability in crop cultivar trials,” Theoretical and Applied Genetics, 84, 161–172. CrossRefGoogle Scholar
  5. Damesa, M.T., Möhring, J., Worku, M., and Piepho, H.P. (2017), “One step at a time: Stage wise analysis of a series of experiments,” Agronomy Journal, 109, 845–857. CrossRefGoogle Scholar
  6. Dias, C.T.d.S., and Krzanowski, W.J. (2003), “Model selection and cross validation in additive main effect and multiplicative interaction models,” Crop Science, 43, 865–873. CrossRefGoogle Scholar
  7. Digby, P.G.N., and Kempton, R.A. (1987), “Multivariate analysis of ecological communities,” Chapman & Hall, London.CrossRefGoogle Scholar
  8. Forkman, J., and Piepho, H.P. (2014), “Parametric bootstrap methods for testing multiplicative terms in GGE and AMMI models,” Biometrics, 70, 639–647. MathSciNetCrossRefzbMATHGoogle Scholar
  9. Gabriel, K.R., and Zamir, S. (1979), “Lower rank approximation of matrices by least squares with any choice of weights,”Technometrics, 21, 489–498.CrossRefzbMATHGoogle Scholar
  10. Gauch, H.G. Jr. (1988), “Model selection and validation for yield trials with interaction,” Biometrics, 44, 705–715. CrossRefzbMATHGoogle Scholar
  11. Gauch, H.G. Jr., and Zobel, R.W. (1990), “Imputing missing yield trial data,” Theoretical and Applied Genetics, 79, 753–761.CrossRefGoogle Scholar
  12. Gauch, H.G. Jr. (1992), “Statistical analysis of regional yield trials,” Elsevier Science Publishers, Amsterdam.Google Scholar
  13. Gollob, H.F. (1968), “A Statistical model which combines features of factor analytic and analysis of variance techniques,” Psychometrika, 33, 73–115. MathSciNetCrossRefzbMATHGoogle Scholar
  14. Green, B.F. (1952), “The orthogonal approximation of an oblique structure in factor analysis,” Psychometrika, 17, 429–440.MathSciNetCrossRefzbMATHGoogle Scholar
  15. Hadasch, S., Forkman, J., Piepho, H.P. (2016), “Cross-validation in AMMI and GGE models: A comparison of methods,” Crop Science, 57, 264–274.
  16. Josse, J., van Eeuwijk, F.A., and Piepho, H.P., Denis, J.B. (2014), “Another look at Bayesian analysis of AMMI models for genotype–environment data,” Journal of Agricultural, Biological, and Environmental Statistics, 19, 240–257.MathSciNetzbMATHGoogle Scholar
  17. Kotz, S., Balakrishnan, N., Read, C.B., Vidakovic, B., and Johnson, N.L. (2006), “Encyclopedia of statistical sciences, second edition ”John Wiley & Sons, John Wiley & Sons, Hoboken, New Jersey.zbMATHGoogle Scholar
  18. Kruskal, J. B. (1965), “Analysis of factorial experiments by estimating monotonic transformations of the data,” Journal of the Royal Statistical Society, 27, 251–263.Google Scholar
  19. Mandel, J. (1969), “The partitioning of interaction in analysis of variance,” Technometrics, 73, 309–328.MathSciNetzbMATHGoogle Scholar
  20. Meng, X.L., and Rubin, D.B. (1993), “Maximum likelihood estimation via the ECM algorithm: A general framework,” Biometrika, 80, 267–278.MathSciNetCrossRefzbMATHGoogle Scholar
  21. Perez-Elizalde, S., Jarquin, D., and Crossa, J. (2012), “A general Bayesian estimation method of linear–bilinear models applied to plant breeding trials with genotype x environment interaction,” Journal of Agricultural, Biological, and Environmental Statistics, 17, 15–37. MathSciNetCrossRefzbMATHGoogle Scholar
  22. Piepho, H.P. (1994), “Best linear unbiased prediction (BLUP) for regional yield trials: a comparison to additive main effects and multiplicative interaction (AMMI) analysis,” Theoretical and Applied Genetics, 89, 647–654. CrossRefGoogle Scholar
  23. Piepho, H.P. (1997), “Analyzing genotype-environment data by mixed models with multiplicative effects” Biometrics, 53, 761–766. MathSciNetCrossRefzbMATHGoogle Scholar
  24. Piepho, H.P. (2004), “An algorithm for a letter-based representation of all-pairwise comparisons,” Journal of Computational and Graphical Statistics, 13, 456–466.MathSciNetCrossRefGoogle Scholar
  25. Piepho, H.P., Möhring, J., Schulz-Streeck, T., and Ogutu, J.O. (2012), “A stage-wise approach for the analysis of multi-environment trials,”Biometrical Journal, 54, 844–860. MathSciNetCrossRefzbMATHGoogle Scholar
  26. Rodrigues, P.C., Malosetti, M., Gauch, H.G. Jr., and van Eeuwijk, F.A. (2014), “A weighted AMMI algorithm to study genotype-by-environment interaction and QTL-by-environment interaction,”Crop Science, 54, 1555–1569. CrossRefGoogle Scholar
  27. Schönemann, P.H. (1966), “A generalized solution to the orthogonal procrustes problem,” Psychometrica.
  28. Searle, S.R, Casella, G., and McCulloch, C.E. (1992), “Variance components,”John Wiley & Sons, New York.CrossRefzbMATHGoogle Scholar
  29. Smith, A., Cullis, B., and Gilmour, A. (2001a), “The analysis of cop variety evaluation data in Australia,” Australian and New Zealand Journal of Statistics, 43, 129–145. MathSciNetCrossRefzbMATHGoogle Scholar
  30. Smith, A., Cullis, B.R., and Thompson, R. (2001b), “Analysing variety by environment data using multiplicative mixed models and adjustment for spatial field trend,” Biometrics, 57, 1138–1147. MathSciNetCrossRefzbMATHGoogle Scholar
  31. Srebro, N., and Jaakkola, T. (2003), “Weighted low-rank approximations,” Proceedings of the Twentieth International Conference on Machine Learning, Washington DC.Google Scholar
  32. Yan, W., Hunt, L.A., Sheng, Q., and Szlavnics, Z. (2000), “Cultivar evaluation and mega-environment investigation based on the GGE Biplot,”Crop Science, 40, 597–605. CrossRefGoogle Scholar
  33. Yan, W., and Kang, M.S. (2002), “GGE biplot analysis,” CRC Press, Boca Raton.CrossRefGoogle Scholar

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

Personalised recommendations