A General Bayesian Estimation Method of Linear–Bilinear Models Applied to Plant Breeding Trials With Genotype × Environment Interaction

  • Sergio Perez-Elizalde
  • Diego Jarquin
  • Jose CrossaEmail author
Open Access

Statistical analyses of two-way tables with interaction arise in many different fields of research. This study proposes the von Mises–Fisher distribution as a prior on the set of orthogonal matrices in a linear–bilinear model for studying and interpreting interaction in a two-way table. Simulated and empirical plant breeding data were used for illustration; the empirical data consist of a multi-environment trial established in two consecutive years. For the simulated data, vague but proper prior distributions were used, and for the real plant breeding data, observations from the first year were used to elicit a prior for parameters of the model for data of the second year trial. Bivariate Highest Posterior Density (HPD) regions for the posterior scores are shown in the biplots, and the significance of the bilinear terms was tested using the Bayes factor. Results of the plant breeding trials show the usefulness of this general Bayesian approach for breeding trials and for detecting groups of genotypes and environments that cause significant genotype × environment interaction. The present Bayes inference methodology is general and may be extended to other linear–bilinear models by fixing certain parameters equal to zero and relaxing some model constraints.

Key Words:

Bayesian inference Bilinear interaction terms Two-way table with interaction von Mises–Fisher 


  1. Burgueño, J., Crossa, J., Cornelius, P. L., and Yang, R.-C. (2008), “Using Factor Analytic Models for Joining Environments and Genotypes Without Crossover Genotype × Environment Interaction,” Crop Science, doi: 10.2135/cropsci2007.11.0632. Google Scholar
  2. Chib, S. (1995), “Marginal Likelihood From the Gibbs Output,” Journal of the American Statistical Association, 90, 1313–1321. MathSciNetzbMATHCrossRefGoogle Scholar
  3. Cornelius, P. L., and Seyedsadr, M. S. (1997), “Estimation of General Linear–Bilinear Models for Two-Way Tables,” Journal of Statistical Computation and Simulation, 58, 287–322. MathSciNetzbMATHCrossRefGoogle Scholar
  4. Cornelius, P. L., Seyedsadr, M. S., 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. Cornelius, P. L., Crossa, J., and Seyedsadr, M. S. (1994), “Tests and Estimators of Multiplicative Models for Variety Trials,” in Proceedings of the 1993 Kansas State University Conference on Applied Statistics in Agriculture, Manhattan, Kansas. Google Scholar
  6. Cornelius, P. L., Crossa, J., and Seyedsadr, M. S. (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: CRC Press, pp. 199–234. Google Scholar
  7. Crossa, J., Yang, R.-C., and Cornelius, P. L. (2004), “Studying Crossover Genotype × Environment Interaction Using Linear–Bilinear Models and Mixed Models,” Journal of Agricultural, Biological, and Environmental Statistics, 9, 362–380. CrossRefGoogle Scholar
  8. Crossa, J., Perez-Elizalde, S., Jarquin, D., Cotes, J. M., Viele, K., Liu, G., and Cornelius, P. L. (2011), “Bayesian Estimation of the Additive Main Effects and Multiplicative Interaction (AMMI) Model,” Crop Science, 51, 1458–1469. CrossRefGoogle Scholar
  9. Gabriel, K. R. (1978), “Least Squares Approximation of Matrices by Additive and Multiplicative Models,” Journal of the Royal Statistical Society. Series B, Statistical Methodology, 40, 186–196. MathSciNetzbMATHGoogle Scholar
  10. Gauch, H. G. (1988), “Model Selection and Validation for Yield Trials With Interaction,” Biometrics, 44, 705–715. zbMATHCrossRefGoogle Scholar
  11. Gelman, A., and Rubin, D. B. (1992), “Inference From Iterative Simulation Using Multiple Sequences,” Statistical Science, 7, 457–511. CrossRefGoogle Scholar
  12. Gollob, H. F. (1968), “A Statistical Model Which Combines Features of Factor Analytic and Analysis of Variance,” Psychometrika, 33, 73–115. MathSciNetzbMATHCrossRefGoogle Scholar
  13. Han, C., and Carlin, B. P. (2001), “Markov Chain Monte Carlo Methods for Computing Bayes Factors: A Comparative Review,” Journal of the American Statistical Association, 96, 1122–1132. CrossRefGoogle Scholar
  14. Herz, C. S. (1955), “Bessel Functions of Matrix Argument,” Annals of Mathematics, 61, 474–523. MathSciNetzbMATHCrossRefGoogle Scholar
  15. Hoff, P. D. (2009), “Simulation of the Matrix Bingham–von Mises–Fisher Distribution, With Applications to Multivariate and Relational Data,” Journal of Computational and Graphical Statistics, 18, 438–456. MathSciNetCrossRefGoogle Scholar
  16. James, A. T. (1964), “Distributions of Matrix Variates and Latent Roots Derived From Normal Samples,” Annals of Mathematical Statistics, 35, 475–501. MathSciNetzbMATHCrossRefGoogle Scholar
  17. Jeffreys, H. (1961), Theory of Probability, Oxford: Clarendon Press. zbMATHGoogle Scholar
  18. Johnson, D. E., and Graybill, G. A. (1972), “An Analysis of a Two-Way Model With Interaction and No Replication,” Journal of the American Statistical Association, 67, 862–868. MathSciNetzbMATHCrossRefGoogle Scholar
  19. Kass, R. E., and Raftery, A. (1995), “Bayesian Factors,” Journal of the American Statistical Association, 90, 773–795. zbMATHCrossRefGoogle Scholar
  20. Kempton, R. A. (1984), “The Use of Biplots in Interpreting Variety by Environment Interactions,” Journal of Agriculture Science, 103, 123–135. CrossRefGoogle Scholar
  21. 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. Methodological, 39 (1), 95–106. MathSciNetzbMATHGoogle Scholar
  22. Koev, P., and Edelman, A. (2006), “The Efficient Evaluation of the Hypergeometric Function of a Matrix Argument,” Mathematics of Computation, 75, 833–845. MathSciNetzbMATHCrossRefGoogle Scholar
  23. Liu, G. (2001). “Bayesian Computation for General Linear–Bilinear Models,” Ph.D. dissertation, Department of Statistics, University of Kentucky, Lexington, KY. Google Scholar
  24. Mandel, J. (1969), “The Partitioning of Interaction in Analysis of Variance,” Journal of Research of the National Bureau of Standards, Series B, 73, 309–328. MathSciNetzbMATHGoogle Scholar
  25. Mandel, J. (1971), “A New Analysis of Variance Models for Non-Additive Data,” Technometrics, 13, 1–18. zbMATHCrossRefGoogle Scholar
  26. Marasinghe, M. G. (1985), “Asymptotic Tests and Monte Carlo Studies Associated With the Multiplicative Interaction Model,” Communications in Statistics. Theory and Methods, 14, 2219–2231. CrossRefGoogle Scholar
  27. Mardia, K. V., Kent, J. T., and Bibbi, J. B. (1979), Multivariate Analysis, London: Academic Press. zbMATHGoogle Scholar
  28. Raftery, A. E., and Lewis, S. M. (1995), “The Number of Iterations, Convergence Diagnostics and Generic Metropolis Algorithms,” in Practical Markov Chain Monte Carlo, eds. W. R. Gilks, D. J. Spiegelhalter, and S. Richardson, London: Chapman and Hall. Google Scholar
  29. Schott, J. R. (1986), “A Note on the Critical Value in Stepwise Tests of Multiplicative Components of Interactions,” Communications in Statistics. Theory and Methods, 15, 1561–1570. MathSciNetzbMATHCrossRefGoogle Scholar
  30. Seyedsadr, M. S., and Cornelius, P. L. (1992), “Shifted Multiplicative Models for Non-additive Two-Way Tables,” Communications in Statistics. Simulation and Computation, 21, 807–822. MathSciNetzbMATHCrossRefGoogle Scholar
  31. 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. MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© The Author(s) 2011

Authors and Affiliations

  • Sergio Perez-Elizalde
    • 1
  • Diego Jarquin
    • 1
  • Jose Crossa
    • 2
    Email author
  1. 1.Programa de Posgrado en Socieconomía, Estadística e InformáticaColegio de PostgraduadosMontecillosMexico
  2. 2.Biometrics and Statistics UnitInternational Maize and Wheat Improvement Center (CIMMYT)Mexico DFMexico

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