| Type: | Package |
| Title: | Bayesian Stability Analysis of Genotype by Environment Interaction (GEI) |
| Version: | 0.2.0 |
| Maintainer: | Muhammad Yaseen <myaseen208@gmail.com> |
| Description: | Performs general Bayesian estimation method of linear–bilinear models for genotype × environment interaction. The method is explained in Perez-Elizalde, S., Jarquin, D., and Crossa, J. (2011) (<doi:10.1007/s13253-011-0063-9>). |
| Depends: | R (≥ 3.1) |
| Imports: | dplyr, ggplot2, lme4, MASS, rstiefel, rlang, scales, stats, tibble, tidyr |
| License: | GPL-2 |
| Encoding: | UTF-8 |
| LazyData: | true |
| RoxygenNote: | 7.3.2 |
| NeedsCompilation: | no |
| Packaged: | 2024-10-15 02:30:56 UTC; myaseen208 |
| Author: | Muhammad Yaseen [aut, cre], Diego Jarquin [aut, ctb], Sergio Perez-Elizalde [aut, ctb], Juan Burgueño [aut, ctb], Jose Crossa [aut, ctb] |
| Repository: | CRAN |
| Date/Publication: | 2024-10-15 04:40:03 UTC |
Bayesian Estimation of Genotype by Environment Interaction Model
Description
Bayesian estimation method of linear–bilinear models for Genotype by Environment Interaction Model
Usage
## Default S3 method:
bayes_ammi(.data, .y, .gen, .env, .rep, .nIter)
Arguments
.data |
data.frame |
.y |
Response Variable |
.gen |
Genotypes Factor |
.env |
Environment Factor |
.rep |
Replication Factor |
.nIter |
Number of Iterations |
Value
Genotype by Environment Interaction Model
Author(s)
Muhammad Yaseen (myaseen208@gmail.com)
Diego Jarquin (diego.jarquin@gmail.com)
Sergio Perez-Elizalde (sergiop@colpos.mx)
Juan Burgueño (j.burgueno@cgiar.org)
Jose Crossa (j.crossa@cgiar.org)
References
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 × Environment Interaction. Journal of Agricultural, Biological, and Environmental Statistics, 17, 15–37. (doi:10.1007/s13253-011-0063-9)
Examples
## Not run:
library(baystability)
library(dplyr)
data(cultivo2008)
fm1 <-
ge_ammi(
.data = cultivo2008
, .y = y
, .gen = entry
, .env = site
, .rep = rep
)
r0 <- fm1$g
c0 <- fm1$e
n0 <- fm1$Rep
k0 <- fm1$k
mu0 <- fm1$mu
sigma20 <- fm1$sigma2
tau0 <- fm1$tau
tao0 <- fm1$tao
delta0 <- fm1$delta
lambdas0 <- fm1$lambdas
alphas0 <- fm1$alphas
gammas0 <- fm1$gammas
ge_means0 <- fm1$ge_means$ge_means
data(cultivo2008)
fm2 <-
ge_ammi(
.data = cultivo2009
, .y = y
, .gen = entry
, .env = site
, .rep = rep
)
k <- fm2$k
alphasa <- fm2$alphas
gammasa <- fm2$gammas
alphas1 <-
tibble::as_tibble(fm2$alphas, .name_repair = "unique") %>%
setNames(paste0("V", 1:ncol(.)))
gammas1 <-
tibble::as_tibble(fm2$gammas, .name_repair = "unique") %>%
setNames(paste0("V", 1:ncol(.)))
# Biplots OLS
library(ggplot2)
BiplotOLS1 <-
ggplot(data = alphas1, mapping = aes(x = V1, y = V2)) +
geom_point() +
geom_hline(yintercept = 0) +
geom_vline(xintercept = 0) +
geom_text(aes(label = 1:nrow(alphas1)), vjust = "inward", hjust = "inward") +
scale_x_continuous(
limits = c(-max(abs(c(range(alphas1[, 1:2]))))
, max(abs(c(range(alphas1[, 1:2])))))) +
scale_y_continuous(
limits = c(-max(abs(c(range(alphas1[, 1:2]))))
, max(abs(c(range(alphas1[, 1:2])))))) +
labs(title = "OLS", x = expression(u[1]), y = expression(u[2])) +
theme_bw() +
theme(plot.title = element_text(hjust = 0.5))
print(BiplotOLS1)
BiplotOLS2 <-
ggplot(data = gammas1, mapping = aes(x = V1, y = V2)) +
geom_point() +
geom_hline(yintercept = 0) +
geom_vline(xintercept = 0) +
geom_text(aes(label = 1:nrow(gammas1)), vjust = "inward", hjust = "inward") +
scale_x_continuous(
limits = c(-max(abs(c(range(gammas1[, 1:2]))))
, max(abs(c(range(gammas1[, 1:2])))))) +
scale_y_continuous(
limits = c(-max(abs(c(range(gammas1[, 1:2]))))
, max(abs(c(range(gammas1[, 1:2])))))) +
labs(title = "OLS", x = expression(v[1]), y = expression(v[2])) +
theme_bw() +
theme(plot.title = element_text(hjust = 0.5))
print(BiplotOLS2)
BiplotOLS3 <-
ggplot(data = alphas1, mapping = aes(x = V1, y = V2)) +
geom_point() +
geom_hline(yintercept = 0) +
geom_vline(xintercept = 0) +
geom_text(aes(label = 1:nrow(alphas1)), vjust = "inward", hjust = "inward") +
geom_point(data = gammas1, mapping = aes(x = V1, y = V2)) +
geom_segment(data = gammas1, aes(x = 0, y = 0, xend = V1, yend = V2),
arrow = arrow(length = unit(0.2, "cm")), alpha = 0.75, color = "red") +
geom_text(data = gammas1,
aes(x = V1, y = V2, label = paste0("E", 1:nrow(gammasa)))
, vjust = "inward", hjust = "inward") +
scale_x_continuous(
limits = c(-max(abs(c(range(alphas1[, 1:2], gammas1[, 1:2]))))
, max(abs(c(range(alphas1[, 1:2], gammas1[, 1:2])))))) +
scale_y_continuous(
limits = c(-max(abs(c(range(alphas1[, 1:2], gammas1[, 1:2]))))
, max(abs(c(range(alphas1[, 1:2], gammas1[, 1:2])))))) +
labs(title = "OLS", x = expression(PC[1]), y = expression(PC[2])) +
theme_bw() +
theme(plot.title = element_text(hjust = 0.5))
print(BiplotOLS3)
data(cultivo2009)
fm3 <-
bayes_ammi(
.data = cultivo2009
, .y = y
, .gen = entry
, .env = site
, .rep = rep
, .nIter = 200
)
Mean_Alphas <- fm3$Mean_Alphas %>% setNames(paste0("V", 1:ncol(.)))
Mean_Gammas <- fm3$Mean_Gammas %>% setNames(paste0("V", 1:ncol(.)))
# Biplots Bayesian
BiplotBayes1 <-
ggplot(data = Mean_Alphas, mapping = aes(x = V1, y = V2)) +
geom_point() +
geom_hline(yintercept = 0) +
geom_vline(xintercept = 0) +
geom_text(aes(label = 1:nrow(Mean_Alphas)),
vjust = "inward"
, hjust = "inward") +
scale_x_continuous(
limits = c(-max(abs(c(range(Mean_Alphas[, 1:2]))))
, max(abs(c(range(Mean_Alphas[, 1:2])))))) +
scale_y_continuous(
limits = c(-max(abs(c(range(Mean_Alphas[, 1:2]))))
, max(abs(c(range(Mean_Alphas[, 1:2])))))) +
labs(title = "Bayes", x = expression(u[1]), y = expression(u[2])) +
theme_bw() +
theme(plot.title = element_text(hjust = 0.5))
print(BiplotBayes1)
BiplotBayes2 <-
ggplot(data = Mean_Gammas, mapping = aes(x = V1, y = V2)) +
geom_point() +
geom_hline(yintercept = 0) +
geom_vline(xintercept = 0) +
geom_text(aes(label = 1:nrow(Mean_Gammas)), vjust = "inward", hjust = "inward") +
scale_x_continuous(
limits = c(-max(abs(c(range(Mean_Gammas[, 1:2]))))
, max(abs(c(range(Mean_Gammas[, 1:2])))))) +
scale_y_continuous(
limits = c(-max(abs(c(range(Mean_Gammas[, 1:2]))))
, max(abs(c(range(Mean_Gammas[, 1:2])))))) +
labs(title = "Bayes", x = expression(v[1]), y = expression(v[2])) +
theme_bw() +
theme(plot.title = element_text(hjust = 0.5))
print(BiplotBayes2)
BiplotBayes3 <-
ggplot(data = Mean_Alphas, mapping = aes(x = V1, y = V2)) +
geom_point() +
geom_hline(yintercept = 0) +
geom_vline(xintercept = 0) +
geom_text(aes(label = 1:nrow(Mean_Alphas)),
vjust = "inward", hjust = "inward") +
geom_point(data = Mean_Gammas, mapping = aes(x = V1, y = V2)) +
geom_segment(data = Mean_Gammas,
aes(x = 0, y = 0, xend = V1, yend = V2),
arrow = arrow(length = unit(0.2, "cm"))
, alpha = 0.75, color = "red") +
geom_text(data = Mean_Gammas,
aes(x = V1, y = V2,
label = paste0("E", 1:nrow(Mean_Gammas))),
vjust = "inward", hjust = "inward") +
scale_x_continuous(
limits = c(-max(abs(c(range(Mean_Alphas[, 1:2], Mean_Gammas[, 1:2]))))
, max(abs(c(range(Mean_Alphas[, 1:2], Mean_Gammas[, 1:2])))))) +
scale_y_continuous(
limits = c(-max(abs(c(range(Mean_Alphas[, 1:2], Mean_Gammas[, 1:2]))))
, max(abs(c(range(Mean_Alphas[, 1:2], Mean_Gammas[, 1:2])))))) +
labs(title = "Bayes", x = expression(PC[1]), y = expression(PC[2])) +
theme_bw() +
theme(plot.title = element_text(hjust = 0.5))
print(BiplotBayes3)
Plot1Mu <-
ggplot(data = fm3$mu1, mapping = aes(x = 1:nrow(fm3$mu1), y = mu)) +
geom_line(color = "blue") +
scale_x_continuous(labels = scales::comma) +
scale_y_continuous(labels = scales::comma) +
labs(y = expression(mu), x = "Iterations") +
theme_bw()
print(Plot1Mu)
Plot2Mu <-
ggplot(data = fm3$mu1, mapping = aes(mu)) +
geom_histogram() +
scale_x_continuous(labels = scales::comma) +
scale_y_continuous(labels = scales::comma) +
labs(y = "Frequency", x = expression(mu)) +
theme_bw()
print(Plot2Mu)
Plot1Sigma2 <-
ggplot(data = fm3$tau1, mapping = aes(x = 1:nrow(fm3$tau1), y = tau)) +
geom_line(color = "blue") +
scale_x_continuous(labels = scales::comma) +
scale_y_continuous(labels = scales::comma) +
labs(y = expression(sigma^2), x = "Iterations") +
theme_bw()
print(Plot1Sigma2)
Plot2Sigma2 <-
ggplot(data = fm3$tau1, mapping = aes(tau)) +
geom_histogram() +
scale_x_continuous(labels = scales::comma) +
scale_y_continuous(labels = scales::comma) +
labs(y = "Frequency", x = expression(sigma^2)) +
theme_bw()
print(Plot2Sigma2)
# Plot of Alphas
Plot1Alpha1 <-
ggplot(data = fm3$tao1, mapping = aes(x = 1:nrow(fm3$tao1), y = tao1)) +
geom_line(color = "blue") +
scale_x_continuous(labels = scales::comma) +
scale_y_continuous(labels = scales::comma) +
labs(y = expression(alpha[1]), x = "Iterations") +
theme_bw()
print(Plot1Alpha1)
Plot2Alpha1 <-
ggplot(data = fm3$tao1, mapping = aes(tao1)) +
geom_histogram() +
scale_x_continuous(labels = scales::comma) +
scale_y_continuous(labels = scales::comma) +
labs(y = "Frequency", x = expression(alpha[1])) +
theme_bw()
print(Plot2Alpha1)
Plot1Alpha2 <-
ggplot(data = fm3$tao1, mapping = aes(x = 1:nrow(fm3$tao1), y = tao2)) +
geom_line(color = "blue") +
scale_x_continuous(labels = scales::comma) +
scale_y_continuous(labels = scales::comma) +
labs(y = expression(alpha[2]), x = "Iterations") +
theme_bw()
print(Plot1Alpha2)
Plot2Alpha2 <-
ggplot(data = fm3$tao1, mapping = aes(tao2)) +
geom_histogram() +
scale_x_continuous(labels = scales::comma) +
scale_y_continuous(labels = scales::comma) +
labs(y = "Frequency", x = expression(alpha[2])) +
theme_bw()
print(Plot2Alpha2)
# Plot of Betas
Plot1Beta1 <-
ggplot(data = fm3$delta1, mapping = aes(x = 1:nrow(fm3$delta1), y = delta1)) +
geom_line(color = "blue") +
scale_x_continuous(labels = scales::comma) +
scale_y_continuous(labels = scales::comma) +
labs(y = expression(beta[1]), x = "Iterations") +
theme_bw()
print(Plot1Beta1)
Plot2Beta1 <-
ggplot(data = fm3$delta1, mapping = aes(delta1)) +
geom_histogram() +
scale_x_continuous(labels = scales::comma) +
scale_y_continuous(labels = scales::comma) +
labs(y = "Frequency", x = expression(beta[1])) +
theme_bw()
print(Plot2Beta1)
Plot1Beta2 <-
ggplot(data = fm3$delta1, mapping = aes(x = 1:nrow(fm3$delta1), y = delta2)) +
geom_line(color = "blue") +
scale_x_continuous(labels = scales::comma) +
scale_y_continuous(labels = scales::comma) +
labs(y = expression(beta[2]), x = "Iterations") +
theme_bw()
print(Plot1Beta2)
Plot2Beta2 <-
ggplot(data = fm3$delta1, mapping = aes(delta2)) +
geom_histogram() +
scale_x_continuous(labels = scales::comma) +
scale_y_continuous(labels = scales::comma) +
labs(y = "Frequency", x = expression(beta[2])) +
theme_bw()
print(Plot2Beta2)
Plot1Beta3 <-
ggplot(data = fm3$delta1, mapping = aes(x = 1:nrow(fm3$delta1), y = delta3)) +
geom_line(color = "blue") +
scale_x_continuous(labels = scales::comma) +
scale_y_continuous(labels = scales::comma) +
labs(y = expression(beta[3]), x = "Iterations") +
theme_bw()
print(Plot1Beta3)
Plot2Beta3 <-
ggplot(data = fm3$delta1, mapping = aes(delta3)) +
geom_histogram() +
scale_x_continuous(labels = scales::comma) +
scale_y_continuous(labels = scales::comma) +
labs(y = "Frequency", x = expression(beta[3])) +
theme_bw()
print(Plot2Beta3)
## End(Not run)
Data for Genotypes by Environment Interaction (GEI)
Description
cultivo2008 is used for performing Genotypes by Environment Interaction (GEI) Analysis.
Usage
data(cultivo2008)
Format
A data.frame 1320 obs. of 6 variables.
Details
Gen Genotype
Institute Institute
Rep Replicate
Block Block
Env Environment
Yield Yield Response
Author(s)
Muhammad Yaseen (myaseen208@gmail.com)
Diego Jarquin (diego.jarquin@gmail.com)
Sergio Perez-Elizalde (sergiop@colpos.mx)
Juan Burgueño (j.burgueno@cgiar.org)
Jose Crossa (j.crossa@cgiar.org)
References
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 × Environment Interaction. Journal of Agricultural, Biological, and Environmental Statistics, 17, 15–37. (doi:10.1007/s13253-011-0063-9)
Examples
data(cultivo2008)
Data for Genotypes by Environment Interaction (GEI)
Description
cultivo2009 is used for performing Genotypes by Environment Interaction (GEI) Analysis.
Usage
data(cultivo2009)
Format
A data.frame 1320 obs. of 6 variables.
Details
Gen Genotype
Institute Institute
Rep Replicate
Block Block
Env Environment
Yield Yield Response
Author(s)
Muhammad Yaseen (myaseen208@gmail.com)
Diego Jarquin (diego.jarquin@gmail.com)
Sergio Perez-Elizalde (sergiop@colpos.mx)
Juan Burgueño (j.burgueno@cgiar.org)
Jose Crossa (j.crossa@cgiar.org)
References
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 × Environment Interaction. Journal of Agricultural, Biological, and Environmental Statistics, 17, 15–37. (doi:10.1007/s13253-011-0063-9)
Examples
data(cultivo2009)
Environment Effects
Description
Calcuates Environment Effects
Usage
## Default S3 method:
e_eff(.data, .y, .gen, .env)
Arguments
.data |
data.frame |
.y |
Response Variable |
.gen |
Genotypes Factor |
.env |
Environment Factor |
Value
Environment Effects
Author(s)
Muhammad Yaseen (myaseen208@gmail.com)
References
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 × Environment Interaction. Journal of Agricultural, Biological, and Environmental Statistics, 17, 15–37. (doi:10.1007/s13253-011-0063-9)
Examples
data(cultivo2008)
e_eff(
.data = cultivo2008
, .y = y
, .gen = entry
, .env = site
)
Genotype Effects
Description
Calcuates Genotype Effects
Usage
## Default S3 method:
g_eff(.data, .y, .gen, .env)
Arguments
.data |
data.frame |
.y |
Response Variable |
.gen |
Genotypes Factor |
.env |
Environment Factor |
Value
Genotype Effects
Author(s)
Muhammad Yaseen (myaseen208@gmail.com)
References
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 × Environment Interaction. Journal of Agricultural, Biological, and Environmental Statistics, 17, 15–37. (doi:10.1007/s13253-011-0063-9)
Examples
data(cultivo2008)
g_eff(
.data = cultivo2008
, .y = y
, .gen = entry
, .env = site
)
AMMI of Genotype by Environment Interaction Model
Description
Performs Additive Main Effects and Multiplication Interaction Analysis of Genotype by Environment Interaction Model
Usage
ge_ammi(.data, .y, .gen, .env, .rep)
## Default S3 method:
ge_ammi(.data, .y, .gen, .env, .rep)
Arguments
.data |
data.frame |
.y |
Response Variable |
.gen |
Genotypes Factor |
.env |
Environment Factor |
.rep |
Replication Factor |
Value
Genotype by Environment Interaction Model
Author(s)
Muhammad Yaseen (myaseen208@gmail.com)
References
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 × Environment Interaction. Journal of Agricultural, Biological, and Environmental Statistics, 17, 15–37. (doi:10.1007/s13253-011-0063-9)
Examples
data(cultivo2008)
fm1 <-
ge_ammi(
.data = cultivo2008
, .y = y
, .gen = entry
, .env = site
, .rep = rep
)
data(cultivo2009)
fm2 <-
ge_ammi(
.data = cultivo2009
, .y = y
, .gen = entry
, .env = site
, .rep = rep
)
Genotype by Environment Interaction Effects
Description
Calcuates Genotype by Environment Interaction Effects
Usage
## Default S3 method:
ge_eff(.data, .y, .gen, .env)
Arguments
.data |
data.frame |
.y |
Response Variable |
.gen |
Genotypes Factor |
.env |
Environment Factor |
Value
Genotype by Environment Interaction Effects
Author(s)
Muhammad Yaseen (myaseen208@gmail.com)
References
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 × Environment Interaction. Journal of Agricultural, Biological, and Environmental Statistics, 17, 15–37. (doi:10.1007/s13253-011-0063-9)
Examples
data(cultivo2008)
ge_eff(
.data = cultivo2008
, .y = y
, .gen = entry
, .env = site
)
Genotype by Environment Interaction Means
Description
Calcuates Genotype by Environment Interaction Means
Usage
## Default S3 method:
ge_mean(.data, .y, .gen, .env)
Arguments
.data |
data.frame |
.y |
Response Variable |
.gen |
Genotypes Factor |
.env |
Environment Factor |
Value
Genotype by Environment Interaction Means
Author(s)
Muhammad Yaseen (myaseen208@gmail.com)
References
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 × Environment Interaction. Journal of Agricultural, Biological, and Environmental Statistics, 17, 15–37. (doi:10.1007/s13253-011-0063-9)
Examples
data(cultivo2008)
ge_mean(
.data = cultivo2008
, .y = y
, .gen = entry
, .env = site
)
Genotype by Environment Interaction Model
Description
Calcuates Genotype by Environment Interaction Model
Usage
ge_model(.data, .y, .gen, .env, .rep)
## Default S3 method:
ge_model(.data, .y, .gen, .env, .rep)
Arguments
.data |
data.frame |
.y |
Response Variable |
.gen |
Genotypes Factor |
.env |
Environment Factor |
.rep |
Replication Factor |
Value
Genotype by Environment Interaction Model
Author(s)
Muhammad Yaseen (myaseen208@gmail.com)
References
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 × Environment Interaction. Journal of Agricultural, Biological, and Environmental Statistics, 17, 15–37. (doi:10.1007/s13253-011-0063-9)
Examples
data(cultivo2008)
fm1 <-
ge_model(
.data = cultivo2008
, .y = y
, .gen = entry
, .env = site
, .rep = rep
)
Genotype by Environment Interaction Variances
Description
Calcuates Genotype by Environment Interaction Variances
Usage
## Default S3 method:
ge_var(.data, .y, .gen, .env)
Arguments
.data |
data.frame |
.y |
Response Variable |
.gen |
Genotypes Factor |
.env |
Environment Factor |
Value
Genotype by Environment Interaction Variances
Author(s)
Muhammad Yaseen (myaseen208@gmail.com)
References
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 × Environment Interaction. Journal of Agricultural, Biological, and Environmental Statistics, 17, 15–37. (doi:10.1007/s13253-011-0063-9)
Examples
data(cultivo2008)
ge_var(
.data = cultivo2008
, .y = y
, .gen = entry
, .env = site
)
k Matrix
Description
Gives k matrix
Usage
matrix_k(n)
## Default S3 method:
matrix_k(n)
Arguments
n |
Number of columns |
Value
Matrix
Author(s)
Muhammad Yaseen (myaseen208@gmail.com)
References
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 × Environment Interaction. Journal of Agricultural, Biological, and Environmental Statistics, 17, 15–37. (doi:10.1007/s13253-011-0063-9)
Orthogonal Normalization
Description
Perform Orthogonal Normalization of a matrix
Usage
orthnorm(u = NULL, basis = TRUE, norm = TRUE)
## Default S3 method:
orthnorm(u = NULL, basis = TRUE, norm = TRUE)
Arguments
u |
Matrix |
basis |
Logical argument by default TRUE |
norm |
Logical argument by default TRUE |
Value
Matrix
Author(s)
Muhammad Yaseen (myaseen208@gmail.com)
References
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 × Environment Interaction. Journal of Agricultural, Biological, and Environmental Statistics, 17, 15–37. (doi:10.1007/s13253-011-0063-9)