--- title: "The Bayesian GGE (SREG) model" author: "Muhammad Yaseen" date: last-modified format: html: self-contained: true number-sections: true toc: true toc-depth: 3 highlight-style: pygments code-tools: true html-math-method: mathjax vignette: > %\VignetteIndexEntry{The Bayesian GGE (SREG) model} %\VignetteEngine{quarto::html} %\VignetteEncoding{UTF-8} execute: echo: true warning: false message: false --- {{< include _bayesammi-macros.txt >}} ```{r setup, include = FALSE} # HTML figures are drawn as PNG and kept as WebP: lossless (the same pixels in about half the space of PNG), or # lossy at quality 90 when that is less than half the size, as for traces and densities of thousands of draws. The # PDF figures stay vector graphics. Any failure of magick (not installed, no WebP, or a broken installation whose # system libraries are missing) keeps the PNG webp <- function(path) { if (knitr::is_latex_output() || !grepl(pattern = "[.]png$", x = path)) return(path) tryCatch( { if (!requireNamespace("magick", quietly = TRUE) || !isTRUE(magick::magick_config()$webp)) return(path) img <- magick::image_read(path) out <- sub(pattern = "[.]png$", replacement = ".webp", x = path) tmp <- tempfile(fileext = ".webp") magick::image_write( image = img , path = out , format = "webp" , quality = 100 , defines = c(`webp:lossless` = "true") ) magick::image_write(image = img, path = tmp, format = "webp", quality = 90) if (file.size(tmp) < file.size(out) / 2) file.copy(from = tmp, to = out, overwrite = TRUE) unlink(c(path, tmp)) out } , error = function(e) path ) } knitr::opts_chunk$set(collapse = FALSE, comment = "", fig.width = 7, fig.height = 5.5, dev = if (knitr::is_latex_output()) "cairo_pdf" else "ragg_png", dpi = 96, fig.process = webp) if (requireNamespace("bayesammi", quietly = TRUE)) library(bayesammi) else pkgload::load_all(export_all = FALSE, helpers = FALSE, quiet = TRUE) ``` # Abstract The GGE model, also called the sites regression (SREG) model, puts the genotype main effects into the bilinear terms. Its biplot shows directly which genotype wins where (Yan et al. 2000). `bayes_ammi(.model = "GGE")` fits it by the Gibbs sampler of Perez-Elizalde et al. (2012), and `bam_winners()` turns the draws into probabilities that each genotype wins in each environment. # Model $$y_{ijl} = \mu + \delta_j + \sum_{k=1}^{K}\lambda_k\alpha_{ik}\gamma_{jk} + e_{ijl}.$$ There are no genotype effects $\tau_i$. The genotype scores $\va_k$ still sum to zero, but the environment scores $\vg_k$ are only orthonormal: they need not sum to zero, because the bilinear terms now carry the genotype main effects. The conditional of $\vg_k$ is therefore von Mises-Fisher on the sphere orthogonal to the other environment vectors only. # The fit ```{r gge} fg <- 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 = 5 , .align = TRUE , .model = "GGE" , .het = FALSE , .waic = TRUE ) fg ``` # The GGE biplot with HPD regions ```{r gge-bip, fig.width = 9} bg <- biplots( model = fg , burnin = 0 , thin = 1 , pb = 0.05 , plot_stable = TRUE , plot_unstable = TRUE , ncolors = 5 , .which = "both" , .grid = 100 ) bg$biplot ``` On a GGE biplot the first axis mostly reflects the genotype means. A genotype region that lies away from the origin on the first axis therefore marks a genotype that is above or below average, rather than one with interaction. # Which genotype wins where The winner of environment $j$ is the genotype with the largest $\eta_{ij}$. Over the draws, $P_{ij}$ is the posterior probability that genotype $i$ wins in environment $j$: ```{r winners} wg <- bam_winners( fit = fg , higher_better = TRUE ) wg ``` ```{r winners-plot, fig.height = 4.8} plot(x = wg) ``` ```{r mega, include = FALSE} mg <- wg$mega ``` The environments split into `r nrow(mg)` groups by their most probable winner. In a GGE analysis such groups are the candidates for mega-environments. The probability column shows how sure each assignment is: a winner with probability near one half has a close rival, given in `Second`. # AMMI or GGE? The two models describe the same cell means with different constraints, so they fit almost equally well: ```{r gge-ammi} fa <- 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 = 5 , .align = TRUE , .model = "AMMI" , .het = FALSE , .waic = TRUE ) bam_compare( AMMI = fa , GGE = fg ) ``` Choose AMMI to study interaction separately from the genotype means, and GGE to study which genotype wins where. # Interpretation ```{r gge-int} bam_interpret( x = wg , higher_better = TRUE , top = 5 ) ``` # References 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, 15--37. doi:10.1007/s13253-011-0063-9 Yan, W., Hunt, L. A., Sheng, Q. and Szlavnics, Z. (2000). Cultivar evaluation and mega-environment investigation based on the GGE biplot. *Crop Science*, 40(3), 597--605. 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 Wood, A. T. A. (1994). Simulation of the von Mises Fisher distribution. *Communications in Statistics - Simulation and Computation*, 23(1), 157--164.