2026-11-09
\[ \newcommand{\vy}{\mathbf{y}} \newcommand{\vx}{\mathbf{x}} \newcommand{\vb}{\mathbf{b}} \newcommand{\ve}{\mathbf{e}} \newcommand{\vone}{\mathbf{1}} \newcommand{\vbeta}{\boldsymbol{\beta}} \newcommand{\vmu}{\boldsymbol{\mu}} \newcommand{\mX}{\mathbf{X}} \newcommand{\mZ}{\mathbf{Z}} \newcommand{\mV}{\mathbf{V}} \newcommand{\mI}{\mathbf{I}} \newcommand{\mJ}{\mathbf{J}} \newcommand{\mSigma}{\boldsymbol{\Sigma}} \newcommand{\mY}{\mathbf{Y}} \newcommand{\mK}{\mathbf{K}} \newcommand{\mH}{\mathbf{H}} \newcommand{\va}{\boldsymbol{\alpha}} \newcommand{\vg}{\boldsymbol{\gamma}} \newcommand{\vt}{\boldsymbol{\tau}} \newcommand{\vd}{\boldsymbol{\delta}} \newcommand{\vtheta}{\boldsymbol{\theta}} \newcommand{\vv}{\mathbf{v}} \newcommand{\E}{\mathrm{E}} \newcommand{\Var}{\mathrm{Var}} \newcommand{\Cov}{\mathrm{Cov}} \newcommand{\tr}{\operatorname{tr}} \]
Real trials lose plots, and some genotypes are not grown in every environment. The error variance also differs from one site to the next. This vignette shows how bayes_ammi() handles both: missing plots and empty cells by data augmentation, and a separate error variance for every environment with .het = TRUE. It ends with a comparison of the two variance models on Maiz.
With \(B\) the largest number of plots in a cell, a cell \((i, j)\) with \(n_{ij} < B\) plots has \(B - n_{ij}\) missing plots. A cell with \(n_{ij} = 0\) is an empty cell. At every iteration the sampler draws the missing plots from the model, \[y^{\mathrm{mis}}_{ijl} \mid \vtheta \sim N(\eta_{ij}, \sigma^2_j), \qquad \eta_{ij} = \mu + \tau_i + \delta_j + \sum_{k=1}^{K}\lambda_k\alpha_{ik}\gamma_{jk},\] and then draws the parameters from the conditional distributions of the balanced design (vignette 2) given the completed data. This is the data augmentation of Tanner and Wong (1987). The draws of the parameters come from their posterior given the observed plots only, so nothing is invented: a missing plot adds no information, and an empty cell is predicted from the row and column structure of the model. The missing plots are assumed to be missing at random (Little and Rubin 2019).
The sampler needs only the sums, sums of squares and counts of the observed plots in every cell, so the completed data are never stored.
From Maiz we remove 60 plots at random and every plot of three genotype-environment combinations:
set.seed(seed = 21)
mz <- Maiz[-sample(x = nrow(Maiz), size = 60), ]
gone <- (mz$entry == 2 & mz$site == 5) | (mz$entry == 7 & mz$site == 11) | (mz$entry == 9 & mz$site == 18)
mz <- mz[!gone, ]
table(plots = table(factor(mz$entry):factor(mz$site)))plots
0 1 2 3 4
3 1 5 47 124 fu <-
bayes_ammi(
.data = mz
, .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 = 3
, .align = TRUE
, .model = "AMMI"
, .het = FALSE
, .waic = TRUE
)
fuBayesian AMMI with 2 multiplicative term(s): 9 genotypes, 20 environments, 648 plots (3 empty cells)
2 chain(s) x 3000 kept draws (iterations 3000 after a burn-in of 500, thinning 1); 1.7 s
Posterior means, standard deviations and 95% HPD intervals
parameter mean sd lower upper ess rhat
<char> <num> <num> <num> <num> <num> <num>
1: mu 4865.378 34.85 4.797e+03 4931.90 4639 1.0023
2: sigma2 771133.151 45862.81 6.884e+05 866667.90 3493 1.0004
3: tau[1] -235.197 95.45 -4.318e+02 -56.12 5393 1.0003
4: tau[2] -287.087 99.35 -4.902e+02 -100.13 4427 1.0000
5: tau[3] -16.772 97.76 -2.130e+02 165.23 5358 1.0002
6: tau[4] 368.080 96.72 1.815e+02 557.44 5813 1.0006
7: tau[5] 395.278 95.27 2.256e+02 600.17 5604 1.0004
8: tau[6] 439.592 99.69 2.415e+02 628.05 4629 1.0000
9: tau[7] -197.581 105.05 -4.019e+02 10.22 3681 1.0001
10: tau[8] -528.930 100.93 -7.224e+02 -328.54 4393 1.0000
11: tau[9] 62.618 99.87 -1.305e+02 260.72 4541 1.0007
12: delta[1] -1246.974 147.62 -1.550e+03 -973.11 5313 0.9998
13: delta[2] -603.360 155.02 -9.078e+02 -307.74 4487 1.0008
14: delta[3] 287.881 144.60 -8.507e+00 548.59 5619 0.9999
15: delta[4] 436.272 151.48 1.404e+02 727.13 4672 1.0000
16: delta[5] -21.712 154.67 -3.280e+02 270.00 4102 0.9999
17: delta[6] 1476.541 147.84 1.181e+03 1755.92 5320 1.0010
18: delta[7] -1603.056 145.40 -1.891e+03 -1313.72 5601 1.0000
19: delta[8] -820.386 145.03 -1.102e+03 -538.55 5335 1.0000
20: delta[9] 56.727 146.65 -2.301e+02 345.23 5396 1.0000
21: delta[10] -1931.169 143.26 -2.212e+03 -1658.16 5660 1.0009
22: delta[11] 429.876 168.20 9.977e+01 752.36 3441 1.0018
23: delta[12] 2682.222 155.78 2.381e+03 2990.30 4191 1.0004
24: delta[13] 1547.696 157.58 1.240e+03 1852.64 3458 1.0004
25: delta[14] 1188.212 146.55 9.058e+02 1478.69 4743 0.9999
26: delta[15] 213.078 146.76 -7.575e+01 495.96 4750 0.9999
27: delta[16] 542.632 145.82 2.585e+02 829.31 5435 1.0004
28: delta[17] -60.923 156.00 -3.637e+02 241.27 4233 1.0003
29: delta[18] -408.313 163.28 -7.449e+02 -105.42 3649 1.0002
30: delta[19] -2172.886 151.46 -2.468e+03 -1880.16 4538 1.0019
31: delta[20] 7.641 151.29 -2.878e+02 301.10 4954 1.0000
32: lambda[1] 5106.531 477.92 4.121e+03 6013.13 3157 1.0016
33: lambda[2] 1339.303 806.12 5.931e-02 2705.39 639 1.0065
parameter mean sd lower upper ess rhat
<char> <num> <num> <num> <num> <num> <num>
Scores of the bilinear terms: x$summary (blocks alpha, gamma); biplot coordinates: x$scores
Sources: Crossa et al. (2011); Spiegelhalter et al. (2002); Watanabe (2010).
Full references: bam_cite(x).The print shows the number of empty cells. bam_predict() gives the posterior of every cell mean \(\eta_{ij}\), including the empty cells:
pu <-
bam_predict(
fit = fu
, .level = 0.95
)
pu$cells[n == 0] Gen Env n observed mean sd lower upper
<fctr> <fctr> <int> <num> <num> <num> <num> <num>
1: 2 5 0 NA 4369.859 279.9944 3794.720 4891.308
2: 7 11 0 NA 4903.347 322.4456 4269.447 5523.364
3: 9 18 0 NA 4761.894 311.5227 4165.312 5390.920Maiz_cells <- data.table::as.data.table(x = Maiz)[, list(full = mean(y)), by = list(
Gen = factor(entry)
, Env = factor(site)
)]
merge(x = pu$cells[n == 0], y = Maiz_cells, by = c("Gen", "Env"))[, list(
Gen
, Env
, mean
, lower
, upper
, full
)]Key: <Gen, Env>
Gen Env mean lower upper full
<fctr> <fctr> <num> <num> <num> <num>
1: 2 5 4369.859 3794.720 4891.308 5201.475
2: 7 11 4903.347 4269.447 5523.364 5051.575
3: 9 18 4761.894 4165.312 5390.920 5244.400The column full holds the cell means of the complete data, which the fit never saw. 2 of the 3 HPD intervals contain them. The miss, genotype 2 in environment 5, is a cell whose interaction is not carried by the two bilinear terms: the model predicts an empty cell only from the pattern of the rest of the trial. The posterior standard deviations of the empty cells are on average 1.1 times those of the cells with all four plots.
plot(x = pu, which = "cells")
With .het = TRUE the plots of environment \(j\) have their own variance, \(e_{ijl} \sim N(0, \sigma^2_j)\), and every \(\sigma^2_j\) has the scaled inverse chi-squared prior of the common variance. Write \(w_j = 1/\sigma^2_j\). The conditional distributions change as follows.
Perez-Elizalde et al. (2012) give the general linear-bilinear model with these conditionals.
fh <-
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 = 4
, .align = TRUE
, .model = "AMMI"
, .het = TRUE
, .waic = TRUE
)
fh$summary[block == "sigma2", list(parameter, mean, lower, upper, rhat)] parameter mean lower upper rhat
<char> <num> <num> <num> <num>
1: sigma2[1] 899051.19 511891.64 1349890.6 1.0006236
2: sigma2[2] 359098.16 192026.52 555745.0 0.9998866
3: sigma2[3] 1358450.15 751176.80 2064839.8 1.0065478
4: sigma2[4] 265204.01 143568.41 406218.9 0.9998684
5: sigma2[5] 486396.32 270327.39 743027.4 1.0003240
6: sigma2[6] 542883.85 302578.54 865847.9 1.0000071
7: sigma2[7] 485314.39 269353.19 740659.9 1.0036182
8: sigma2[8] 1083672.16 514349.66 1779086.2 1.0011059
9: sigma2[9] 1735925.45 1033048.68 2624506.7 0.9998791
10: sigma2[10] 90216.12 47092.66 139392.2 0.9998561
11: sigma2[11] 987914.43 558046.74 1522479.9 1.0007315
12: sigma2[12] 639387.39 368112.12 997579.5 1.0002132
13: sigma2[13] 912822.78 513857.36 1405516.9 0.9999362
14: sigma2[14] 1048576.95 599100.73 1565103.0 1.0005215
15: sigma2[15] 582948.13 304895.26 882299.0 1.0003107
16: sigma2[16] 301506.87 170387.93 455563.0 1.0000115
17: sigma2[17] 589627.10 309397.79 904723.1 1.0035253
18: sigma2[18] 1369988.68 742355.05 2056858.1 1.0005439
19: sigma2[19] 2031831.15 1162647.09 3044867.7 0.9998868
20: sigma2[20] 392824.32 208244.34 594441.1 1.0014998
parameter mean lower upper rhat
<char> <num> <num> <num> <num>The posterior means range from 90,216 to 2,031,831, a ratio of 23. One common variance cannot describe both extremes.
ggplot2::ggplot(
data = fh$summary[block == "sigma2"]
, mapping = ggplot2::aes(x = reorder(label, mean), y = mean, ymin = lower, ymax = upper)
) +
ggplot2::geom_pointrange(colour = "#2c7fb8") +
ggplot2::scale_y_continuous(breaks = scales::pretty_breaks(n = 6), labels = scales::label_comma()) +
ggplot2::labs(x = "Environment", y = "Error variance (posterior mean, 95% HPD)") +
ggplot2::theme_bw()
fc <-
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 = 4
, .align = TRUE
, .model = "AMMI"
, .het = FALSE
, .waic = TRUE
)
cp <-
bam_compare(
common = fc
, environment_var = fh
)
cpComparison of Bayesian models (smaller is better)
model terms variances DIC pD WAIC p_waic
<char> <int> <char> <num> <num> <num> <num>
1: environment_var 2 per environment 11735.63 85.6869 11741.45 82.0777
2: common 2 common 11862.02 67.9290 11865.68 65.9309
se lambda_last_lower lambda_last_mean dWAIC dWAIC_se
<num> <num> <num> <num> <num>
1: 41.9641 910.1783 2093.077 0.0000 0.0000
2: 44.2329 63.7490 1686.287 124.2365 22.4796
Sources: Spiegelhalter et al. (2002); Watanabe (2010); Vehtari et al. (2017).
Full references: bam_cite(x).The environment-specific variances lower WAIC by 124, 5.5 times the standard error of the difference. They also sharpen the interaction: the lower HPD limit of \(\lambda_2\) rises from 64 to 910, because the plots of the noisy environments no longer blur the second term. The posterior predictive check of the common-variance model tells the same story through the ratio of the largest to the smallest environment variance:
pp <-
bam_ppc(
fit = fc
, .draws = 500
, .seed = 1
)
ppPosterior predictive checks (500 replicated trials)
statistic observed mean_rep p_value
<char> <num> <num> <num>
1: variance of cell means 1.842e+06 1.901e+06 0.682
2: largest cell mean 8.385e+03 8.608e+03 0.728
3: smallest cell mean 1.632e+03 1.743e+03 0.672
4: variance of genotype means 1.247e+05 1.429e+05 0.662
5: first singular value 5.923e+03 5.902e+03 0.482
6: within-cell variance 7.559e+05 7.660e+05 0.526
7: largest / smallest environment variance 2.582e+01 2.924e+00 0.000
Sources: Gelman et al. (1996). Full references: bam_cite(x).The observed ratio is 26, against 2.9 on average in the replicated trials of the common-variance model (posterior predictive p-value 0.000). The other statistics are reproduced well. Vignette 9 returns to these tools.
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
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
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.
Tanner, M. A. and Wong, W. H. (1987). The calculation of posterior distributions by data augmentation. Journal of the American Statistical Association, 82(398), 528–540.
Little, R. J. A. and Rubin, D. B. (2019). Statistical Analysis with Missing Data, 3rd edn. Wiley, Hoboken.