2026-11-09
\[ \newcommand{\vy}{\mathbf{y}} \newcommand{\vx}{\mathbf{x}} \newcommand{\vb}{\mathbf{b}} \newcommand{\ve}{\mathbf{e}} \newcommand{\vone}{\mathbf{1}} \newcommand{\vbeta}{\boldsymbol{\beta}} \newcommand{\vmu}{\boldsymbol{\mu}} \newcommand{\mX}{\mathbf{X}} \newcommand{\mZ}{\mathbf{Z}} \newcommand{\mV}{\mathbf{V}} \newcommand{\mI}{\mathbf{I}} \newcommand{\mJ}{\mathbf{J}} \newcommand{\mSigma}{\boldsymbol{\Sigma}} \newcommand{\mY}{\mathbf{Y}} \newcommand{\mK}{\mathbf{K}} \newcommand{\mH}{\mathbf{H}} \newcommand{\va}{\boldsymbol{\alpha}} \newcommand{\vg}{\boldsymbol{\gamma}} \newcommand{\vt}{\boldsymbol{\tau}} \newcommand{\vd}{\boldsymbol{\delta}} \newcommand{\vtheta}{\boldsymbol{\theta}} \newcommand{\vv}{\mathbf{v}} \newcommand{\E}{\mathrm{E}} \newcommand{\Var}{\mathrm{Var}} \newcommand{\Cov}{\mathrm{Cov}} \newcommand{\tr}{\operatorname{tr}} \]
The joint regression of Finlay and Wilkinson (1963) and Eberhart and Russell (1966) and the stability variance of Shukla (1972) are the most used measures of adaptability and stability. Their classical estimates come with tests, not with the uncertainty a breeder needs to choose. bayes_stability() fits the three models by Gibbs sampling, so every slope, deviation variance and stability variance has a posterior distribution. This vignette gives the model, the sampler, an analysis of Maiz, a comparison of the three models and the safety-first index of Eskridge (1990).
For genotype \(i\), environment \(j\) and plot \(l\), \[y_{ijl} = \mu + g_i + (1 + b_i)h_j + d_{ij} + e_{ijl}, \qquad d_{ij} \sim N(0, \sigma^2_{d,i}), \quad e_{ijl} \sim N(0, \sigma^2),\] with \(\sum_i g_i = \sum_j h_j = \sum_i b_i = 0\). The environmental index \(h_j\) is a parameter, estimated jointly with the others, as in the Bayesian Finlay-Wilkinson regression of Lian and de los Campos (2016). The classical analysis plugs in the environment means instead and so ignores their error. The slope of genotype \(i\) is \(1 + b_i\), with average 1.
.model |
Slopes | Deviation variance | Classical counterpart |
|---|---|---|---|
"ER" |
yes | \(\sigma^2_{d,i}\) for every genotype | Eberhart and Russell (1966); Bayesian version of Oliveira et al. (2018) |
"FW" |
yes | one \(\sigma^2_d\) | Finlay and Wilkinson (1963) |
"Shukla" |
no (\(b_i = 0\)) | \(\sigma^2_{d,i}\) for every genotype: the stability variance | Shukla (1972); Edwards and Jannink (2006) |
The replicates separate the deviations \(d_{ij}\) from the plot error \(e_{ijl}\), so the data must have replicates.
bam_stab_priors() sets normal priors on \(\mu\) and on the contrasts \(\mK_G\mathbf{g}\), \(\mK_E\mathbf{h}\) and \(\mK_G\mathbf{b}\) (so the sums are zero), and scaled inverse chi-squared priors on \(\sigma^2\) and \(\sigma^2_{d,i}\). The defaults are vague and in the unit of the trait: \(10^4\) times the variance of the response for the effects, and the least-squares variances as scales with one degree of freedom.
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
)Priors of the Bayesian stability models
mu ~ N(0, 1e4 x var(y))
K_G g ~ N(0, 1e4 x var(y) I)
K_E h ~ N(0, 1e4 x var(y) I)
K_G b ~ N(0, 100 I) (slopes 1 + b_i)
sigma^2 ~ Inv-chi^2(1, within-cell mean square)
sigma_d^2 ~ Inv-chi^2(1, least-squares deviation variance)Every conditional distribution is normal or scaled inverse chi-squared. Drawing the deviations \(d_{ij}\) one at a time next to the genotype effects would mix slowly, because the mean of the deviations of a genotype and its effect \(g_i\) trade off. The sampler therefore draws \(\mu\), \(\mathbf{g}\), \(\mathbf{h}\) and \(\mathbf{b}\) with the deviations integrated out: each observed cell mean then has variance \(\sigma^2_{d,i} + \sigma^2/n_{ij}\). The deviations come next, then the variances. This is a partially collapsed Gibbs sampler: each sweep is still an exact draw from the posterior. Unbalanced data need nothing more, as the weights use the number of plots of every cell, and an empty cell has weight zero. The sampler is compiled; vignette 13 checks it against a plain R version.
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
)
fsBayesian Eberhart-Russell stability model: 9 genotypes, 20 environments, 4 replicates
2 chain(s) x 3000 kept draws (iterations 3000 after a burn-in of 500, thinning 1); 1.4 s
Genotypes: posterior means and 95% HPD intervals; safety-first index at alpha = 0.05
Gen mean mean_lower mean_upper slope slope_lower slope_upper
<char> <num> <num> <num> <num> <num> <num>
1: 1 4618 4259 4982 0.9493 0.6548 1.2320
2: 2 4603 4330 4875 0.9366 0.7046 1.1640
3: 3 4826 4574 5042 0.9357 0.7329 1.1240
4: 4 5218 4938 5465 1.3520 1.1460 1.5800
5: 5 5248 5007 5499 1.2470 1.0490 1.4590
6: 6 5330 5110 5562 1.0820 0.8950 1.2740
7: 7 4673 4407 4926 0.8538 0.6461 1.0630
8: 8 4287 3973 4579 0.4975 0.2446 0.7457
9: 9 4932 4696 5193 1.1470 0.9280 1.3370
prob_slope_above_1 dev_var dev_var_lower dev_var_upper R2 safety_first
<num> <num> <num> <num> <num> <num>
1: 0.3590000 483200 85600 1036000 0.7349 3477
2: 0.2830000 193900 14610 464200 0.8689 3875
3: 0.2487000 108600 11620 273700 0.9216 4268
4: 0.9973000 167800 13440 450100 0.9408 4267
5: 0.9900000 124100 10230 302700 0.9478 4499
6: 0.8060000 77570 9611 192600 0.9559 4835
7: 0.0811700 152400 11200 364100 0.8760 3975
8: 0.0003333 288100 31390 618100 0.5723 2978
9: 0.9222000 130800 10310 345400 0.9364 4278
safety_first_lower safety_first_upper prob_best_safety
<num> <num> <num>
1: 2757 4116 0.0000000
2: 3339 4383 0.0001667
3: 3820 4675 0.0103300
4: 3737 4759 0.0216700
5: 4038 4925 0.1113000
6: 4464 5202 0.8407000
7: 3482 4439 0.0005000
8: 2393 3514 0.0000000
9: 3775 4722 0.0153300
Overall mean 4859, error variance 756700; largest potential scale reduction factor 1.017
All parameters: x$summary; environmental index: x$environments
Sources: Eberhart and Russell (1966); Oliveira et al. (2018); Lian and de los
Campos (2016); Eskridge (1990); Spiegelhalter et al. (2002); Watanabe (2010).
Full references: bam_cite(x).The slopes of genotypes 4 and 5 lie above 1 with 95% posterior probability: they respond more than the average genotype to good environments. The slope of 8 lies below 1. The posterior probability that a slope exceeds 1 is in the column prob_slope_above_1. A breeder can use it directly, which a test of \(b_i = 1\) does not allow.
plot(x = fs, which = "slopes")
plot(x = fs, which = "lines")
plot(x = fs, which = "deviations")
The deviation variances are poorly determined from 20 environments, so their HPD intervals are wide and overlap. The Bayesian analysis shows this directly; a classical F test of \(\sigma^2_{d,i} = 0\) would hide it.
data.table::data.table(
Gen = s$Gen
, slope_ls = round(x = fs$ls$slope, digits = 3)
, slope = round(x = s$slope, digits = 3)
, dev_var_ls = round(fs$ls$dev_var)
, dev_var = round(s$dev_var)
) Gen slope_ls slope dev_var_ls dev_var
<char> <num> <num> <num> <num>
1: 1 0.981 0.949 298673 483210
2: 2 0.965 0.937 49012 193865
3: 3 0.954 0.936 18897 108601
4: 4 1.314 1.352 250809 167772
5: 5 1.228 1.247 96052 124129
6: 6 1.057 1.082 23383 77575
7: 7 0.858 0.854 93153 152443
8: 8 0.524 0.498 212947 288101
9: 9 1.118 1.147 145199 130778The posterior means of the slopes are close to the least-squares slopes (correlation 0.997). The deviation variances differ more. The posterior of a variance estimated from 20 environments is skewed to the right, so its mean lies above its mode. The prior, with its scale at the pooled least-squares value, also pulls the small variances up. The order of the genotypes changes less (rank correlation 0.73).
Eskridge (1990) ranks genotypes by a lower quantile of their yield across environments, \(\mu + g_i - z_{1-\alpha}\sqrt{V_i}\), where \(V_i\) is the variance of genotype \(i\) across environments under the model:
| Model | \(V_i\) |
|---|---|
| Eberhart-Russell | \(b_i^2 s^2_h (1 - 1/E) + \sigma^2_{d,i}\) |
| Finlay-Wilkinson | \(b_i^2 s^2_h (1 - 1/E)\) |
| Shukla | \(s^2_h + \sigma^2_{d,i}\) |
Here \(s^2_h\) is the variance of the environmental index. The index is computed on every draw, so it has an HPD interval and every genotype has a probability of being the best on it. Here 6 is most probably the best (0.84).
s[order(-safety_first), list(
Gen
, mean
, safety_first
, safety_first_lower
, safety_first_upper
, prob_best_safety
)] Gen mean safety_first safety_first_lower safety_first_upper
<char> <num> <num> <num> <num>
1: 6 5330.120 4834.788 4464.444 5202.405
2: 5 5248.139 4499.144 4037.793 4924.656
3: 9 4931.551 4278.128 3775.390 4722.382
4: 3 4826.080 4268.433 3820.211 4675.155
5: 4 5217.723 4266.802 3737.334 4759.201
6: 7 4672.868 3974.792 3482.001 4439.187
7: 2 4602.566 3875.111 3338.771 4382.980
8: 1 4618.040 3477.059 2757.315 4115.888
9: 8 4286.651 2978.271 2393.288 3514.087
prob_best_safety
<num>
1: 0.8406666667
2: 0.1113333333
3: 0.0153333333
4: 0.0103333333
5: 0.0216666667
6: 0.0005000000
7: 0.0001666667
8: 0.0000000000
9: 0.0000000000ff <-
bayes_stability(
.data = Maiz
, .y = y
, .gen = entry
, .env = site
, .rep = rep
, .model = "FW"
, .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
)
fh <-
bayes_stability(
.data = Maiz
, .y = y
, .gen = entry
, .env = site
, .rep = rep
, .model = "Shukla"
, .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
)
cp <-
bam_compare(
ER = fs
, FW = ff
, Shukla = fh
)
cpComparison of Bayesian models (smaller is better)
model terms variances DIC pD WAIC p_waic
<char> <int> <char> <num> <num> <num> <num>
1: ER NA deviation per genotype 11893.02 104.1660 11897.91 97.2984
2: FW NA one deviation variance 11896.06 101.2211 11901.92 95.4625
3: Shukla NA interaction per genotype 11897.47 108.9504 11903.37 101.7497
se lambda_last_lower lambda_last_mean dWAIC dWAIC_se
<num> <num> <num> <num> <num>
1: 46.1831 NA NA 0.0000 0.0000
2: 45.9799 NA NA 4.0128 5.0903
3: 45.4214 NA NA 5.4558 7.7438
Sources: Spiegelhalter et al. (2002); Watanabe (2010); Vehtari et al. (2017).
Full references: bam_cite(x).WAIC prefers the ER model; the next, FW, is 4 above it (0.8 standard errors of the difference). The standard errors do not separate the three models here; the choice can follow the question, the slopes for adaptability and the variances for stability. The models are fitted to the same plots, so they can also be compared with the AMMI and GGE fits of bayes_ammi() (vignette 9).
The draws of the cell means \(\mu + g_i + (1 + b_i)h_j + d_{ij}\) feed the decision tools of vignette 9:
bam_superiority(
fit = fs
, .s = 0.2
, higher_better = TRUE
)$marginal Gen mean prob_performance prob_stability prob_joint
<char> <num> <num> <num> <num>
1: 6 5329.591 0.8831666667 0.584666667 0.5163581111
2: 5 5248.083 0.6336666667 0.048500000 0.0307328333
3: 4 5218.175 0.4790000000 0.002500000 0.0011975000
4: 9 4929.700 0.0038333333 0.179333333 0.0006874444
5: 3 4824.993 0.0003333333 0.676333333 0.0002254444
6: 1 4617.564 0.0000000000 0.007166667 0.0000000000
7: 2 4602.071 0.0000000000 0.311000000 0.0000000000
8: 7 4672.272 0.0000000000 0.190500000 0.0000000000
9: 8 4281.845 0.0000000000 0.000000000 0.0000000000bam_interpret(
x = fs
, higher_better = TRUE
, top = 5
)Bayesian Eberhart-Russell stability model
Best by the safety-first index (alpha = 0.05): 6, 5, 9, 3 and 4; most
probably the best: 6 (0.84).
Findings:
- The chains have converged (largest potential scale reduction factor 1.017;
smallest effective sample size of the main parameters 574).
- Slopes whose HPD interval lies above 1 (respond more than average to good
environments): 4 and 5; below 1 (respond less, relatively better in poor
environments): 8.
- Smallest deviation variance: 6 (77,600); largest: 1 (483,000).
- Share of each genotype's variation explained by the regression (R2): 1
0.73, 2 0.87, 3 0.92, 4 0.94, 5 0.95, 6 0.96, 7 0.88, 8 0.57, 9 0.94.
Recommendations:
- For a wide recommendation choose genotypes with a high mean, a slope near 1
and a small deviation variance; for favourable environments a slope above 1
with a high mean.
- The safety-first index combines the two: 6, 5 and 9 lead.
Cautions:
- The environmental index is estimated from the same trial; with few
environments the slopes are uncertain (see the HPD intervals).
Sources:
- Eberhart, S. A. and Russell, W. A. (1966). Stability parameters for
comparing varieties. Crop Science, 6(1), 36-40.
doi:10.2135/cropsci1966.0011183X000600010011x
- Oliveira, T. R. A., Carvalho, H. W. L., Nascimento, M., Costa, E. F. N.,
Amaral Junior, A. T., Gravina, G. A. and Carvalho Filho, J. L. S. (2018).
The Eberhart and Russell's Bayesian method used as an instrument to select
maize hybrids. Euphytica, 214, 64. doi:10.1007/s10681-018-2146-y
- Lian, L. and de los Campos, G. (2016). FW: an R package for
Finlay-Wilkinson regression that incorporates genomic/pedigree information
and covariance structures between environments. G3: Genes, Genomes,
Genetics, 6(3), 589-597. doi:10.1534/g3.115.026328
- Eskridge, K. M. (1990). Selection of stable cultivars using a safety-first
rule. Crop Science, 30, 369-374.
- 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.bam_cite(
x = fs
, style = c("text", "bibtex")[1]
)Eberhart, S. A. and Russell, W. A. (1966). Stability parameters for comparing varieties. Crop Science, 6(1), 36-40. doi:10.2135/cropsci1966.0011183X000600010011x
Oliveira, T. R. A., Carvalho, H. W. L., Nascimento, M., Costa, E. F. N., Amaral Junior, A. T., Gravina, G. A. and Carvalho Filho, J. L. S. (2018). The Eberhart and Russell's Bayesian method used as an instrument to select maize hybrids. Euphytica, 214, 64. doi:10.1007/s10681-018-2146-y
Lian, L. and de los Campos, G. (2016). FW: an R package for Finlay-Wilkinson regression that incorporates genomic/pedigree information and covariance structures between environments. G3: Genes, Genomes, Genetics, 6(3), 589-597. doi:10.1534/g3.115.026328
Eskridge, K. M. (1990). Selection of stable cultivars using a safety-first rule. Crop Science, 30, 369-374.
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. Finlay, K. W. and Wilkinson, G. N. (1963). The analysis of adaptation in a plant-breeding programme. Australian Journal of Agricultural Research, 14(6), 742–754. doi:10.1071/AR9630742
Eberhart, S. A. and Russell, W. A. (1966). Stability parameters for comparing varieties. Crop Science, 6(1), 36–40. doi:10.2135/cropsci1966.0011183X000600010011x
Shukla, G. K. (1972). Some statistical aspects of partitioning genotype-environmental components of variability. Heredity, 29(2), 237–245.
Eskridge, K. M. (1990). Selection of stable cultivars using a safety-first rule. Crop Science, 30, 369–374.
Lian, L. and de los Campos, G. (2016). FW: an R package for Finlay-Wilkinson regression that incorporates genomic/pedigree information and covariance structures between environments. G3: Genes, Genomes, Genetics, 6(3), 589–597. doi:10.1534/g3.115.026328
Oliveira, T. R. A., Carvalho, H. W. L., Nascimento, M., Costa, E. F. N., Amaral Junior, A. T., Gravina, G. A. and Carvalho Filho, J. L. S. (2018). The Eberhart and Russell’s Bayesian method used as an instrument to select maize hybrids. Euphytica, 214, 64. doi:10.1007/s10681-018-2146-y
Edwards, J. W. and Jannink, J.-L. (2006). Bayesian modeling of heterogeneous error and genotype x environment interaction variances. Crop Science, 46(2), 820. doi:10.2135/cropsci2005.0164