Bayesian AMMI analysis
11 bilinear term(s): lambda_1 = 3,665, lambda_2 = 3,282; the chains have
converged.
Findings:
- The chains have converged by the criteria of Crossa et al. (2011) (largest
potential scale reduction factor 1.007); the smallest effective sample size
of the main parameters is 1,040.
- 2 of 200 Heidelberger-Welch stationarity and half-width tests failed (about
5% fail by chance at level 0.05).
- Overall mean 3,848 (95% HPD 3,789 to 3,910); error variance 1,705,000, a
coefficient of variation of 33.9%.
- Genotypes significantly above average (HPD of the effect excludes zero): 10
and 19; below average: 38, 40 and 42.
- Environments with significantly high means: 3, 5, 8, 18, 22 and 3 more;
low: 2, 4, 7, 9, 10 and 8 more.
- Singular values with 95% HPD intervals: lambda_1 = 3,665 (2,886 to 4,436);
lambda_2 = 3,282 (2,601 to 3,972); lambda_3 = 2,819 (2,210 to 3,437);
lambda_4 = 2,500 (1,925 to 3,075); lambda_5 = 2,221 (1,665 to 2,853);
lambda_6 = 1,852 (1,258 to 2,454); lambda_7 = 1,529 (944.7 to 2,105);
lambda_8 = 1,233 (669.7 to 1,784); lambda_9 = 933.1 (398.3 to 1,496);
lambda_10 = 642.7 (147.2 to 1,172); lambda_11 = 330.8 (0.2213 to 768.8).
- On the first bilinear term, genotypes with significantly positive scores:
10, 20 and 42; negative: 2, 12 and 19.
- Environments with significantly positive first scores: 5 and 25; negative:
4 and 9.
- Genotypes and environments with significant first scores of the same sign
interact positively: the genotype yields more there than its main effect
predicts; of opposite sign, less.
Recommendations:
- Recommend the genotypes with significant first scores for the environments
whose scores have the same sign (bayes_ammi_biplot() shows the regions).
- Draw bayes_ammi_biplot() to see which genotypes and environments form
homogeneous groups (overlapping HPD regions).
Cautions:
- The HPD interval of lambda_11 reaches down to near zero: the term is weakly
supported and its scores are uncertain.
Sources:
- 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(1), 15-37.
doi:10.1007/s13253-011-0063-9
- 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 model. Crop Science, 51(4),
1458-1469. doi:10.2135/cropsci2010.06.0343
- 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(2), 438-456.
doi:10.1198/jcgs.2009.07177
- Spiegelhalter, D. J., Best, N. G., Carlin, B. P. and van der Linde, A.
(2002). Bayesian measures of model complexity and fit. Journal of the Royal
Statistical Society B, 64(4), 583-639. doi:10.1111/1467-9868.00353
- Watanabe, S. (2010). Asymptotic equivalence of Bayes cross validation and
widely applicable information criterion in singular learning theory.
Journal of Machine Learning Research, 11, 3571-3594.
