Testing multiplicative terms in AMMI and GGE models for multienvironment trials with replicates

  • Waqas Ahmed MalikEmail author
  • Johannes Forkman
  • Hans-Peter Piepho
Original Article


Key message

For analysing multienvironment trials with replicates, a resampling-based method is proposed for testing significance of multiplicative interaction terms in AMMI and GGE models, which is superior compared to contending methods in robustness to heterogeneity of variance.


The additive main effects and multiplicative interaction model and genotype main effects and genotype-by-environment interaction model are commonly used for the analysis of multienvironment trial data. Agronomists and plant breeders are frequently using these models for cultivar trials repeated across different environments and/or years. In these models, it is crucial to decide how many significant multiplicative interaction terms to retain. Several tests have been proposed for this purpose when replicate data are available; however, all of them assume that errors are normally distributed with a homogeneous variance. Here, we propose resampling-based methods for multienvironment trial data with replicates, which are free from these distributional assumptions. The methods are compared with competing parametric tests. In an extensive simulation study based on two multienvironment trials, it was found that the proposed methods performed well in terms of Type-I error rates regardless of the distribution of errors. The proposed method even outperforms the robust \( F_{R} \) test when the assumptions of normality and homogeneity of variance are violated.



Additive main effects and multiplicative interaction


Genotype and genotype environment interaction


Multienvironment trial


Singular value decomposition



This work was supported by the German Research Foundation (DFG), Grant No. PI 377/17-1. The authors acknowledge support by the Baden-Württemberg high-performance computing (bwHPC) cluster.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.

Ethical standards

The authors declare that the experiments comply with the current laws of the countries in which the experiments were performed.

Supplementary material

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Supplementary material 1 (TXT 19 kb)
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Supplementary material 2 (TXT 4 kb)
122_2019_3339_MOESM3_ESM.txt (15 kb)
Supplementary material 3 (TXT 14 kb)


  1. Cornelius PL (1993) Statistical tests and retention of terms in the additive main effects and multiplicative interaction model for cultivar trials. Crop Sci 33:1186–1193CrossRefGoogle Scholar
  2. Dias CTS, Krzanowski WJ (2003) Model selection and cross validation in additive main effect and multiplicative interaction models. Crop Sci 43:865–873CrossRefGoogle Scholar
  3. Dias CTS, Krzanowski WJ (2006) Choosing components in the additive main effect and multiplicative interaction models. Sci Agric 63:169–175CrossRefGoogle Scholar
  4. Forkman J, Piepho HP (2014) Parametric bootstrap methods for testing multiplicative terms in GGE and AMMI models. Biometrics 70:639–647CrossRefGoogle Scholar
  5. Gauch HG (1988) Model selection and validation for yield trials with interaction. Biometrics 44:705–715CrossRefGoogle Scholar
  6. Gauch HG (1992) Statistical analysis of regional yield trials. AMMI analysis of factorial designs. Elsevier, New YorkGoogle Scholar
  7. Gauch HG (2006) Statistical analysis of yield trials by AMMI and GGE. Crop Sci 46:1488–1500CrossRefGoogle Scholar
  8. Gauch HG (2013) A simple protocol for AMMI analysis of yield trials. Crop Sci 53:1860–1869CrossRefGoogle Scholar
  9. Gauch HG, Zobel RW (1988) Predictive and postdictive success of statistical analyses of yield trials. Theor Appl Genet 76:1–10CrossRefGoogle Scholar
  10. Hadasch S, Forkman J, Piepho HP (2017) Cross-validation in AMMI and GGE models: a comparison of methods. Crop Sci 57:264–274CrossRefGoogle Scholar
  11. Hadasch S, Forkman J, Malik WA, Piepho HP (2018) Weighted estimation of AMMI and GGE models. J Agric Biol Environ Stat 23(2):255–275CrossRefGoogle Scholar
  12. Hu X, Yan S, Shen K (2013) Heterogeneity of error variance and its influence on genotype comparison in multi-location trials. Field Crops Res 149:322–328CrossRefGoogle Scholar
  13. Malik WA, Hadasch S, Forkman J, Piepho HP (2018) Nonparametric resampling methods for testing multiplicative terms in AMMI and GGE models for multienvironment trials. Crop Sci 58:752–761CrossRefGoogle Scholar
  14. Perez-Elizalde S, Jarquin D, Crossa J (2012) A general Bayesian estimation method of linear–bilinear models applied to plant breeding trials with genotype x environment interaction. J Agric Biol Environ Stat 17:15–37CrossRefGoogle Scholar
  15. Piepho HP (1992) Vergleichende Untersuchungen der statistischen Eigenschaften verschiedener Stabilitätsmasse mit Anwendungen auf Hafer, Winterraps, Ackerbohnen sowie Futter- und Zuckerrüben. Doctoral thesis, University of Kiel, GermanyGoogle Scholar
  16. Piepho HP (1994) Best linear unbiased prediction (BLUP) for regional yield trials: a comparison to additive main effects and multiplicative interaction (AMMI) analysis. Theor Appl Genet 89:647–654CrossRefGoogle Scholar
  17. Piepho HP (1995) Robustness of statistical tests for multiplicative terms in the additive main effects and multiplicative interaction model for cultivar trials. Theor Appl Genet 90:438–443CrossRefGoogle Scholar
  18. Yan W, Hunt LA, Sheng Q, Szlavnics Z (2000) Cultivar evaluation and mega-environment investigation based on the GGE biplot. Crop Sci 40:597–605CrossRefGoogle Scholar
  19. Yang RC, Crossa J, Cornelius PL, Burgueño J (2009) Biplot analysis of genotype × environment interaction: proceed with caution. Crop Sci 49:1564–1576CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  • Waqas Ahmed Malik
    • 1
    Email author
  • Johannes Forkman
    • 2
  • Hans-Peter 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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