Bayesian Estimation of the Additive Main Effects and Multiplicative Interaction Model

Bayesian estimation of the additive main effects and multiplicative interaction (AMMI) model of multi-environment trials by the Gibbs sampler of Crossa, Perez-Elizalde, Jarquin, Cotes, Viele, Liu and Cornelius (2011) : posterior distributions of the overall mean, genotype and environment effects, singular values, genotype and environment scores (von Mises-Fisher conditionals on the constrained spheres) and error variance; several chains with burn-in and thinning; posterior summaries with highest posterior density intervals; the convergence diagnostics of Raftery-Lewis, Heidelberger-Welch and Gelman-Rubin; the AMMI biplot with highest posterior density regions for inference on genotypes and environments; least-squares AMMI; and a plain-language interpretation of every result, computed from its numbers. Beyond the article: unbalanced trials with missing plots and cells (data augmentation), environment-specific error variances, the Bayesian GGE (sites regression) model of Perez-Elizalde, Jarquin and Crossa (2012) , probabilities of superior performance and stability (Dias et al. 2022 ), posterior ranks, Bayesian stability measures, predictions and the winner of every environment, model choice by DIC and WAIC with posterior predictive checks, draws for the 'posterior', 'bayesplot' and 'coda' packages, a compiled sampler, and a built-in 'shiny' application. Bayesian stability analysis: the Finlay-Wilkinson, Eberhart-Russell and Shukla models by Gibbs sampling, and the posterior distributions of classical stability indices (Wricke, Shukla, Lin-Binns, Annicchiarico, Eskridge's safety-first) and of AMMI-based indices (ASV, SIPC, EV, Za, WAAS, WAASY). Every result prints the references of its method.

bayesammi: Bayesian Estimation of the Additive Main Effects and Multiplicative Interaction Model

Overview

bayesammi estimates the additive main effects and multiplicative interaction (AMMI) model of multi-environment trials by the Gibbs sampler of Crossa et al. (2011). Every parameter has a posterior distribution, so the AMMI biplot gets statistical inference: highest posterior density (HPD) regions show which genotypes and environments interact significantly and which form homogeneous groups.

• bayes_ammi(): the Gibbs sampler (overall mean, genotype and environment effects, singular values, scores on the constrained spheres, error variance), with several chains, burn-in, thinning, priors from bam_priors(), posterior summaries with HPD intervals and the convergence diagnostics of Raftery-Lewis, Heidelberger-Welch and Gelman-Rubin;
• biplots(): the biplot with HPD regions of the genotype and environment coordinates;
• ge_ammi(), ge_mean(), ge_eff() and friends: least-squares AMMI and the tables it is built on;
• bam_interpret(): a plain-language interpretation of every result, computed from its numbers.

Beyond the article:

• unbalanced trials (missing plots and empty cells, by data augmentation) and environment-specific error variances (.het = TRUE);
• the Bayesian GGE (sites regression) model (.model = "GGE") and bam_winners(), the probability that each genotype wins in each environment;
• decisions for breeders: bam_superiority() (probabilities of superior performance and stability of Dias et al. 2022), bam_ranks(), bam_stability() (Bayesian AMMI stability value and interaction variance) and bam_predict() (every cell, observed or not);
• Bayesian stability analysis: bayes_stability() fits the Finlay-Wilkinson, Eberhart-Russell and Shukla models by Gibbs sampling (slopes, deviation and stability variances, Eskridge’s safety-first index, all with HPD intervals); bam_indices() and bam_ammi_indices() give the posterior distributions of Wricke’s, Shukla’s, Lin and Binns’, Annicchiarico’s and Eskridge’s indices and of ASV, SIPC, EV, Za, WAAS and WAASY;
• model choice: DIC and WAIC with bam_compare(), posterior predictive checks with bam_ppc(), and the draws for posterior, bayesplot and coda;
• bam_app(): an application that runs every analysis without code, with results that copy cleanly.

Version 1.0.0 rewrites the sampler of version 0.3.0, which, among other faults, never updated the singular values and did not return the error variance (see NEWS). The new sampler runs in C++ and reproduces Tables 1 and B1 of Crossa et al. (2011). Every function keeps its name, arguments and result components; fifteen vignettes come in HTML and PDF. Every printed result ends with the sources of its method; bam_cite() gives them as BibTeX.

Installation

install.packages("bayesammi")
# development version
# install.packages("remotes")
remotes::install_github("myaseen208/bayesammi")

Quick start

library(bayesammi)
data(Maiz)
fit <-
  bayes_ammi(
      .data   = Maiz
    , .y      = y
    , .gen    = entry
    , .env    = site
    , .rep    = rep
    , .nIter  = 3000
    , .k      = 2
    , .burnin = 500
    , .thin   = 1
    , .chains = 2
    , .priors = bam_priors(
          mu_mean     = 0
        , mu_var      = 1e+8
        , tau_mean    = 0
        , tau_var     = 1e+8
        , delta_mean  = 0
        , delta_var   = 1e+8
        , lambda_mean = 0
        , lambda_var  = 1e+8
        , nu          = 1
        , s2          = NULL
        )
    , .level  = 0.95
    , .seed   = 1
    , .align  = TRUE
    , .model  = "AMMI"
    , .het    = FALSE
    , .waic   = TRUE
    )
fit$summary[block %in% c("mu", "sigma2", "lambda")]

   parameter  block label axis       mean         sd       mc_se        lower
1:        mu     mu         NA   4858.071    31.8860   0.4116465   4795.71823
2:    sigma2 sigma2         NA 762170.002 43547.0839 737.6230979 680191.01282
3: lambda[1] lambda          1   5459.383   462.2078   6.5519088   4526.53731
4: lambda[2] lambda          2   1695.053   770.6989  30.0725905     59.23157
        upper       ess      rhat
1:   4920.000 6000.0000 1.0000264
2: 849256.062 3485.3720 1.0004346
3:   6305.934 4976.6695 1.0006729
4:   2891.761  656.7918 0.9999747

bp <-
  biplots(
      model         = fit
    , burnin        = 0
    , thin          = 0.5
    , pb            = c(0.05, 0.01)
    , plot_stable   = TRUE
    , plot_unstable = TRUE
    , ncolors       = 5
    , .which        = "both"
    , .grid         = 100
    )
bp$biplot

Which genotypes are likely to be among the best two, and stable?

sp <-
  bam_superiority(
      fit           = fit
    , .s            = 0.2
    , higher_better = TRUE
    )
sp$marginal

   Gen     mean prob_performance prob_stability   prob_joint
1:   6 5331.377        0.8958333     0.16483333 0.1476631944
2:   5 5247.055        0.6376667     0.09366667 0.0597281111
3:   9 4927.836        0.0025000     0.05733333 0.0001433333
4:   4 5217.619        0.4640000     0.00000000 0.0000000000
5:   1 4615.315        0.0000000     0.02033333 0.0000000000
6:   2 4604.674        0.0000000     0.24033333 0.0000000000
7:   3 4824.549        0.0000000     0.71666667 0.0000000000
8:   7 4670.736        0.0000000     0.70583333 0.0000000000
9:   8 4283.482        0.0000000     0.00100000 0.0000000000

plot(x = sp, which = "marginal")

Adaptability and stability by the Bayesian Eberhart-Russell model:

fs <-
  bayes_stability(
      .data   = Maiz
    , .y      = y
    , .gen    = entry
    , .env    = site
    , .rep    = rep
    , .model  = "ER"
    , .nIter  = 3000
    , .burnin = 500
    , .thin   = 1
    , .chains = 2
    , .priors = bam_stab_priors(
          mu_var = NULL
        , g_var  = NULL
        , h_var  = NULL
        , b_var  = 100
        , nu     = 1
        , s2     = NULL
        , nu_d   = 1
        , s2_d   = NULL
        )
    , .level  = 0.95
    , .alpha  = 0.05
    , .seed   = 1
    , .waic   = TRUE
    )
plot(x = fs, which = "slopes")

The same analyses without code:

bam_app(
    launch.browser = TRUE
  )

Learn more

• Vignettes: browseVignettes("bayesammi"), starting with bayesammi_01_getting_started; PDF versions on the site.
• Site: https://myaseen208.com/bayesammi/
• Notation: bam_notation()
• References of the methods behind a result: bam_cite()

Citation

Yaseen M, Crossa J, Perez-Elizalde S, Jarquin D, Cotes JM, Viele K, Liu
G, Cornelius PL, Abadillo Velasco JG (2026). _bayesammi: Bayesian
Estimation of the Additive Main Effects and Multiplicative Interaction
Model_. R package version 1.0.0 (first released 2018),
<https://CRAN.R-project.org/package=bayesammi>.

BibTeX:

@Manual{bayesammi,
  title = {{bayesammi}: Bayesian Estimation of the Additive Main Effects and Multiplicative Interaction Model},
  author = {Muhammad Yaseen and Jose Crossa and Sergio Perez-Elizalde and Diego Jarquin and Jose Miguel Cotes and Kert Viele and Genzhou Liu and Paul L. Cornelius and Julian Garcia {Abadillo Velasco}},
  year = {2026},
  note = {R package version 1.0.0 (first released 2018)},
  url = {https://CRAN.R-project.org/package=bayesammi},
}

Please also cite the methods you use: citation("bayesammi") lists them with the package, and bam_cite(x) gives the references of a single result.