Another Look at Bayesian Analysis of AMMI Models for Genotype-Environment Data

  • Julie Josse
  • Fred van Eeuwijk
  • Hans-Peter Piepho
  • Jean-Baptiste Denis
Article
  • 403 Downloads

Abstract

Linear–bilinear models are frequently used to analyze two-way data such as genotype-by-environment data. A well-known example of this class of models is the additive main effects and multiplicative interaction effects model (AMMI). We propose a new Bayesian treatment of such models offering a proper way to deal with the major problem of overparameterization. The rationale is to ignore the issue at the prior level and apply an appropriate processing at the posterior level to be able to arrive at easily interpretable inferences. Compared to previous attempts, this new strategy has the great advantage of being directly implementable in standard software packages devoted to Bayesian statistics such as WinBUGS/OpenBUGS/JAGS. The method is assessed using simulated datasets and a real dataset from plant breeding. We discuss the benefits of a Bayesian perspective to the analysis of genotype-by-environment interactions, focusing on practical questions related to general and local adaptation and stability of genotypes. We also suggest a new solution to the estimation of the risk of a genotype not exceeding a given threshold.

Key Words

Adaptation AMMI models Bayesian inference Genotype-by-environment interaction Overparameterization Singular-value decomposition Stability 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Adams, E., Walczak, B., Vervaet, C., Risha, P. G., and Massart, D. L. (2002), “Principal Component Analysis of Dissolution Data with Missing Elements,” International Journal of Pharmaceutics, 234, 169–178. CrossRefGoogle Scholar
  2. Chikuse, Y. (2003), Statistics on Special Manifolds, New York: Springer. CrossRefMATHGoogle Scholar
  3. Cornelius, P., and Crossa, J. (1999), “Prediction Assessment of Shrinkage Estimators of Multiplicative Models for Multi-environment Cultivar Trials,” Crop Science, 39, 998–1009. CrossRefGoogle Scholar
  4. Cornelius, P., Crossa, J., and Seyedsadr, M. (1996), “Statistical Tests and Estimators of Multiplicative Models for Genotype by Environment Interaction,” in Genotype by Environment Interaction, eds. M. S. Kang and H. G. Gauch, Boca Raton, FL: CRC Press, pp. 199–234. Google Scholar
  5. Crossa, J., Perez-Elizalde, S., Jarquin, D., Miguel Cotes, J., Viele, K., Liu, G., and Cornelius, P. (2011), “Bayesian Estimation of the Additive Main Effects and Multiplicative Interaction Model,” Crop Science, 51, 1468–1469. CrossRefGoogle Scholar
  6. Denis, J. B., and Gower, J. C. (1994), “Asymptotic Covariances for the Parameters of Biadditive Models,” Utilitas Mathematica, 46, 193–205. MATHMathSciNetGoogle Scholar
  7. — (1996), “Asymptotic Confidence Regions for Biadditive Models: Interpreting Genotype-Environment Interactions,” Applied Statistics, 45 (4), 479–493. CrossRefGoogle Scholar
  8. Dias, S., and Krzanowski, W. (2003), “Model Selection and Cross Validation in Additive Main Effect and Multiplicative Interaction Models,” Crop Science, 43 (3), 865–873. CrossRefGoogle Scholar
  9. Edwards, J. W., and Jannink, J. L. (2006), “Bayesian Modeling of Heterogeneous Error and Genotype X Environment Interaction Variances,” Crop Science, 46, 820–833. CrossRefGoogle Scholar
  10. Eskridge, K., and Mumm, R. (1992), “Choosing Plant Cultivars Based on the Probability of Outperforming a Check,” Theoretical and Applied Genetics, 84, 494—500. Google Scholar
  11. Gabriel, K. R., and Zamir, S. (1979), “Lower Rank Approximation of Matrices by Least Squares with Any Choice of Weights,” Technometrics, 21 (4), 236–246. CrossRefGoogle Scholar
  12. Gauch, H. (1990), “Using Interaction to Improve Yield Estimates,” in Genotype by Environment Interaction, ed. M. S. Kang, Boca Raton, FL: CRC Press, pp. 141–150. Google Scholar
  13. Gauch, H., and Zobel, R. (1990), “Imputing Missing Yield Trial Data,” Theoretical and Applied Genetics, 79, 753–761. CrossRefGoogle Scholar
  14. — (1996), “AMMI Analysis of Yield Trials,” in Genotype by environment interaction, eds. M. S. Kang, and H. G. Gauch, Boca Raton, FL: CRC Press, pp. 141–150. CrossRefGoogle Scholar
  15. Gelman, A. (2006), “Prior Distributions for Variance Parameters in Hierarchical Models,” Bayesian Analysis, 3, 515–533. MathSciNetGoogle Scholar
  16. Hastie, T., Tibshirani, R., and Friedman, J. (2009), The Elements of Statistical Learning. Data Mining, Inference and Prediction. Springer Series in Statistics (2nd ed.). MATHGoogle Scholar
  17. Hoff, P. D. (2007), “Model Averaging and Dimension Selection for the Singular Value Decomposition,” Journal of the American Statistical Association, 102 (478), 674–685. CrossRefMATHMathSciNetGoogle Scholar
  18. — (2009), “Simulation of the Matrix Bingham–von Mises–Fisher Distribution, with Applications to Multivariate and Relational Data,” Journal of Computational and Graphical Statistics, 18 (2), 438–456. CrossRefMathSciNetGoogle Scholar
  19. — (2012), “rstiefel: Random Orthonormal Matrix Generation on the Stiefel Manifold,” available at http://CRAN.R-project.org/package=rstiefel, R Package Version 0.9.
  20. Jolliffe, I. T. (2002), Principal Component Analysis, New York: Springer. MATHGoogle Scholar
  21. Josse, J., and Denis, J. (2012), “Inferring Biadditive Models Within the Bayesian Paradigm,” Tech. Rep., INRA, MIA. Google Scholar
  22. Josse, J., and Husson, F. (2011), “Multiple Imputation in PCA,” Advances in Data Analysis and Classification, 5 (3), 231–246. CrossRefMATHMathSciNetGoogle Scholar
  23. — (2012), “Selecting the Number of Components in PCA Using Cross-Validation Approximations,” Computational Statistics & Data Analysis, 56 (6), 1869–1879. CrossRefMATHMathSciNetGoogle Scholar
  24. — (2013), “Handling Missing Values in Exploratory Multivariate Data Analysis Methods,” Journal de la Société Française de Statistique, 153 (2), 79–99. MathSciNetGoogle Scholar
  25. Khatri, C. G., and Mardia, K. V. (1977), “The von Mises–Fisher Matrix Distribution in Orientation Statistics,” Journal of the Royal Statistical Society, Series B, 39 (1), 95–106. MATHMathSciNetGoogle Scholar
  26. Kiers, H. A. L. (1997), “Weighted Least Squares Fitting Using Ordinary Least Squares Algorithms,” Psychometrika, 62 (2), 251–266. CrossRefMATHMathSciNetGoogle Scholar
  27. Mandel, J. (1969), “The Partitioning of Interaction in Analysis of Variance,” Journal of Research of the National Bureau of Standards. B, Mathematical Sciences, 73, 309–328. CrossRefMATHMathSciNetGoogle Scholar
  28. Martyn, P. (2003), “Jags: A Program for Analysis of Bayesian Graphical Models Using Gibbs Sampling,” in Proceedings of the 3rd International Workshop on Distributed Statistical Computing (DSC 2003), March 20–22, Vienna, Austria. Google Scholar
  29. Moreno-Gonzalez, J., Crossa, J., and Cornelius, P. L. (2003), “Additive Main Effects and Multiplicative Interaction Model: II. Theory on Shrinkage Factors for Predicting Cell Means,” Crop Science, 43, 1976–1982. CrossRefGoogle Scholar
  30. Nelder, J. A. (1994), “The Statistics of Linear Models: Back to Basics,” Statistics and Computing, 4 (4), 221–234. CrossRefGoogle Scholar
  31. Perez-Elizalde, S., Jarquin, D., and Crossa, J. (2011), “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 (1), 15–37. CrossRefMathSciNetGoogle Scholar
  32. Piepho, H. (1996), “A Simplified Procedure for Comparing the Stability of Cropping Systems,” Biometrics, 52, 315–320. CrossRefMATHGoogle Scholar
  33. — (1997), “Analyzing Genotype-Environment Data by Mixed Models with Multiplicative Effects,” Biometrics, 53, 761–766. CrossRefMATHMathSciNetGoogle Scholar
  34. — (1998), “Empirical Best Linear Unbiased Prediction in Cultivar Trials Using Factor Analytic Variance-Covariance Structures,” Theoretical and Applied Genetics, 97, 195–201. CrossRefGoogle Scholar
  35. R Core Team (2013), “R: A Language and Environment for Statistical Computing,” R Foundation for Statistical Computing, Vienna, Austria, available at http://www.R-project.org/.
  36. Robinson, G. K. (1991), “That BLUP Is a Good Thing: The Estimation of Random Effects,” Statistical Science, 6 (1), 15–51. CrossRefMATHMathSciNetGoogle Scholar
  37. Royo, C., Rodriguez, A., and Romagosa, I. (1993), “Differential Adaptation of Complete and Substitute Triticale,” Plant Breeding, 111, 113–119. CrossRefGoogle Scholar
  38. Smidl, V., and Quinn, A. (2007), “On Bayesian Principal Component Analysis,” Computational Statistics & Data Analysis, 51, 4101–4123. CrossRefMATHMathSciNetGoogle Scholar
  39. Smith, A., Cullis, B., and Thompson, R. (2001), “Analyzing Variety by Environment Data Using Multiplicative Mixed Models and Adjustments for Spatial Field Trend,” Biometrics, 57 (4), 1138–1147. CrossRefMATHMathSciNetGoogle Scholar
  40. Theobald, C. M., Talbot, M., and Nabugoomu, F. (2002), “A Bayesian Approach to Regional and Local-Area Prediction from Crop Variety Trials,” Journal of Agricultural, Biological, and Environmental Statistics, 7 (3), 403–419. CrossRefGoogle Scholar
  41. Viele, K., and Srinivasan, C. (2000), “Parsimonious Estimation of Multiplicative Interaction in Analysis of Variance Using Kullback–Leibler Information,” Journal of Statistical Planning and Inference, 84, 201–219. CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© International Biometric Society 2014

Authors and Affiliations

  • Julie Josse
    • 1
  • Fred van Eeuwijk
    • 2
  • Hans-Peter Piepho
    • 3
  • Jean-Baptiste Denis
    • 4
  1. 1.Applied Mathematics DepartmentAgrocampus OuestRennesFrance
  2. 2.Plant Sciences DepartmentWageningen UniversityWageningenThe Netherlands
  3. 3.Crop Science InstituteHohenheim UniversityHohenheimGermany
  4. 4.Applied Mathematics and Informatics UnitINRAJouy-en-JosasFrance

Personalised recommendations