MultiEnvironmentTrial
MultiEnvironmentTrial.RmdIntroduction
This tutorial introduces use to the main functions of the
MegaLMM R package.
We will use MegaLMM to do Genomic Prediction for a set
of maize lines in a large multi-environmental trial. Multi-environment
trials are used to evaluate candidate varieties under different
environments to learn which varieties might be useful for particular
locations. This is important because Gene-Environment
Interactions are very common in plants, which means that the
relative performances of varieties may change across different
environments, so the same line won’t necessarily be best everywhere.
Gene-environment interactions are often thought of as reaction norms, where we plot the change in a line’s performance as a function of the environment. However, an equivalent model for gene-environment interactions is to think of the trait value in each environment as a separate trait, and model the correlation in trait values across environments. As reaction norms, gene-environment interactions are represented by lines with different slopes. As correlated traits, gene-environment interactions are represented by correlations in traits that are less than one.
In MegaLMM we model gene-environment interactions as
correlated traits, because this takes advantage of
MegaLMM’s ability to model the covariances among a large
number of traits. We will use MegaLMM to estimate the
additive genetic and non-additive genetic covariances among all trials,
and then use these covariances to predict the genetic values of every
line in every trial. This is particularly useful when
multi-environmental trials are incomplete meaning that not
every line is evaluated in every trial. Specifically, we will leverage
the relative line performances in some trials and the covariances among
trials to predict the line performances in trials where they were not
observed. We will use cross-validation to evaluate the
accuracy of these predictions, and compare them to predictions in each
trial that we would have made treating each trial independently.
The data are based on data from the Genomes To Fields Initiative which is a large consortium growing maize hybrids across a large number of trials across North America, but have been anonymized and subsetted to a smaller set for demonstration.
Preliminaries
We will use the MegaLMM, rrBLUP, and
ggplot2 packages.
rrBLUP and ggplot2 can be installed from
CRAN if you do not have them already:
if(!require(rrBLUP)) { install.packages("rrBLUP"); library(rrBLUP) }
#> Loading required package: rrBLUP
if(!require(ggplot2)) { install.packages("ggplot2"); library(ggplot2) }
#> Loading required package: ggplot2MegaLMM is installed from GitHub:
if(!require(devtools)) { install.packages("devtools"); library(devtools) }
#> Loading required package: devtools
#> Loading required package: usethis
if(!require(MegaLMM)) {
devtools::install_github('deruncie/MegaLMM')
library(MegaLMM)
}
#> Loading required package: MegaLMMLoad the data and format for MegaLMM
The data files for this tutorial are included with the
MegaLMM package and can be accessed with the
data() function:
Yield data are in the file Yield_trial_BLUPs and include
3,318 yield measurements from 502 lines and 19 environments.
data('yield_data',package='MegaLMM')
yield_data
#> Line Population Env Yield
#> 1 Line001 1 Env01 7.663660e-02
#> 2 Line001 1 Env02 5.736228e-01
#> 3 Line001 1 Env03 -1.057396e-01
#> 4 Line001 1 Env04 4.053487e-01
#> 5 Line001 1 Env05 -9.638836e-02
#> 6 Line001 1 Env12 7.502145e-02
#> 7 Line002 1 Env01 -3.429877e-02
#> 8 Line002 1 Env02 -4.004347e-02
#> 9 Line002 1 Env03 -2.653557e-01
#> 10 Line002 1 Env04 1.136820e-01
#> 11 Line002 1 Env05 -3.726350e-03
#> 12 Line002 1 Env12 -3.714474e-01
#> 13 Line003 1 Env01 2.105977e-01
#> 14 Line003 1 Env02 -6.395920e-01
#> 15 Line003 1 Env03 3.995948e-01
#> 16 Line003 1 Env04 2.926501e-01
#> 17 Line003 1 Env05 1.146499e-01
#> 18 Line003 1 Env12 -2.370533e-01
#> 19 Line004 1 Env01 -7.522661e-01
#> 20 Line004 1 Env02 -1.431944e-01
#> 21 Line004 1 Env03 -1.200977e+00
#> 22 Line004 1 Env04 -4.363149e-01
#> 23 Line004 1 Env05 -2.494169e-01
#> 24 Line004 1 Env12 -7.852241e-01
#> 25 Line005 1 Env01 5.040685e-01
#> 26 Line005 1 Env02 1.487476e-01
#> 27 Line005 1 Env03 6.330452e-01
#> 28 Line005 1 Env04 6.561490e-01
#> 29 Line005 1 Env05 1.237169e-01
#> 30 Line005 1 Env12 -2.390269e-01
#> 31 Line006 1 Env01 -2.619325e-02
#> 32 Line006 1 Env02 1.682000e-01
#> 33 Line006 1 Env03 1.799732e-01
#> 34 Line006 1 Env04 -2.938856e-01
#> 35 Line006 1 Env05 2.111995e-02
#> 36 Line006 1 Env12 2.513783e-01
#> 37 Line007 1 Env01 6.059105e-01
#> 38 Line007 1 Env02 8.658490e-02
#> 39 Line007 1 Env03 3.821189e-01
#> 40 Line007 1 Env04 4.749081e-01
#> 41 Line007 1 Env05 1.062514e-01
#> 42 Line007 1 Env12 -2.105132e-01
#> 43 Line008 1 Env01 7.674815e-02
#> 44 Line008 1 Env02 -1.872101e-01
#> 45 Line008 1 Env03 -3.152836e-01
#> 46 Line008 1 Env04 -4.100128e-01
#> 47 Line008 1 Env05 9.233675e-02
#> 48 Line008 1 Env12 -2.581535e-01
#> 49 Line009 1 Env01 1.487860e-01
#> 50 Line009 1 Env02 -5.105435e-01
#> 51 Line009 1 Env03 3.582082e-01
#> 52 Line009 1 Env04 -7.643776e-02
#> 53 Line009 1 Env05 1.296953e-01
#> 54 Line009 1 Env12 1.624819e-01
#> 55 Line010 1 Env01 2.427229e-01
#> 56 Line010 1 Env02 6.540042e-02
#> 57 Line010 1 Env03 6.162406e-01
#> 58 Line010 1 Env04 -2.463628e-01
#> 59 Line010 1 Env05 9.485659e-02
#> 60 Line010 1 Env12 -7.857771e-01
#> 61 Line011 1 Env01 -5.387960e-02
#> 62 Line011 1 Env02 -1.654603e-01
#> 63 Line011 1 Env03 -5.442302e-01
#> 64 Line011 1 Env04 -2.303881e-01
#> 65 Line011 1 Env05 -2.921937e-02
#> 66 Line011 1 Env12 1.048671e-01
#> 67 Line012 1 Env01 -2.315558e-01
#> 68 Line012 1 Env02 3.777948e-02
#> 69 Line012 1 Env03 -2.529660e-02
#> 70 Line012 1 Env04 -5.408300e-01
#> 71 Line012 1 Env05 3.335723e-02
#> 72 Line012 1 Env12 -2.931048e-01
#> 73 Line013 1 Env01 -6.446561e-01
#> 74 Line013 1 Env02 -6.717251e-01
#> 75 Line013 1 Env03 -4.344309e-01
#> 76 Line013 1 Env04 2.185775e-01
#> 77 Line013 1 Env05 -1.013940e-01
#> 78 Line013 1 Env12 -6.828011e-01
#> 79 Line014 1 Env01 -3.126808e-01
#> 80 Line014 1 Env02 1.714737e-02
#> 81 Line014 1 Env03 -1.775868e-02
#> 82 Line014 1 Env04 1.652616e-01
#> 83 Line014 1 Env05 2.601497e-02
#> 84 Line014 1 Env12 -1.526309e-01
#> 85 Line015 1 Env01 4.100197e-01
#> 86 Line015 1 Env02 -2.147732e-01
#> 87 Line015 1 Env03 1.789289e-01
#> 88 Line015 1 Env04 -3.884937e-02
#> 89 Line015 1 Env05 7.092664e-02
#> 90 Line015 1 Env12 2.762909e-01
#> 91 Line016 1 Env01 -5.932312e-02
#> 92 Line016 1 Env02 5.224478e-01
#> 93 Line016 1 Env03 2.155236e-01
#> 94 Line016 1 Env04 4.689738e-01
#> 95 Line016 1 Env05 6.285470e-02
#> 96 Line016 1 Env12 2.774458e-01
#> 97 Line017 1 Env01 -2.878808e-01
#> 98 Line017 1 Env02 -7.946321e-01
#> 99 Line017 1 Env03 -2.054899e-01
#> 100 Line017 1 Env04 1.537212e-01
#> 101 Line017 1 Env05 -2.266715e-02
#> 102 Line017 1 Env12 6.754159e-01
#> 103 Line018 1 Env01 -3.223057e-02
#> 104 Line018 1 Env02 8.191515e-02
#> 105 Line018 1 Env03 -9.293057e-01
#> 106 Line018 1 Env04 -5.352318e-02
#> 107 Line018 1 Env05 -1.890778e-01
#> 108 Line018 1 Env12 2.315757e-01
#> 109 Line019 1 Env01 1.208312e-01
#> 110 Line019 1 Env02 -5.974399e-01
#> 111 Line019 1 Env03 -6.146869e-01
#> 112 Line019 1 Env04 -1.371508e-01
#> 113 Line019 1 Env05 -9.664683e-02
#> 114 Line019 1 Env12 5.763020e-01
#> 115 Line020 1 Env01 -6.742635e-01
#> 116 Line020 1 Env02 3.595038e-01
#> 117 Line020 1 Env03 2.996193e-01
#> 118 Line020 1 Env04 -4.208846e-02
#> 119 Line020 1 Env05 -4.244346e-02
#> 120 Line020 1 Env12 2.418972e-01
#> 121 Line021 1 Env01 9.942649e-02
#> 122 Line021 1 Env02 2.386349e-01
#> 123 Line021 1 Env03 -1.664182e-01
#> 124 Line021 1 Env04 8.595582e-02
#> 125 Line021 1 Env05 -1.387228e-01
#> 126 Line021 1 Env12 5.283409e-01
#> 127 Line022 1 Env01 2.403872e-01
#> 128 Line022 1 Env02 -2.705558e-01
#> 129 Line022 1 Env03 2.787914e-01
#> 130 Line022 1 Env04 -2.859170e-01
#> 131 Line022 1 Env05 1.834085e-02
#> 132 Line022 1 Env12 -7.379959e-01
#> 133 Line023 1 Env01 -3.542020e-02
#> 134 Line023 1 Env02 -4.687080e-01
#> 135 Line023 1 Env03 2.074113e-01
#> 136 Line023 1 Env04 -4.540796e-01
#> 137 Line023 1 Env05 3.697371e-02
#> 138 Line023 1 Env12 -6.936249e-02
#> 139 Line024 1 Env01 -9.153470e-02
#> 140 Line024 1 Env02 1.475635e-01
#> 141 Line024 1 Env03 3.102428e-01
#> 142 Line024 1 Env04 -1.042002e-01
#> 143 Line024 1 Env05 7.933363e-02
#> 144 Line024 1 Env12 -1.555026e-02
#> 145 Line025 1 Env01 4.431633e-01
#> 146 Line025 1 Env02 9.331619e-01
#> 147 Line025 1 Env03 7.725887e-01
#> 148 Line025 1 Env04 2.904366e-01
#> 149 Line025 1 Env05 -1.341809e-01
#> 150 Line025 1 Env12 -8.723935e-02
#> 151 Line026 1 Env01 2.471259e-01
#> 152 Line026 1 Env02 1.619154e-01
#> 153 Line026 1 Env03 8.276160e-01
#> 154 Line026 1 Env04 -7.328924e-02
#> 155 Line026 1 Env05 1.090861e-01
#> 156 Line026 1 Env12 4.949538e-01
#> 157 Line027 1 Env01 -1.251651e-02
#> 158 Line027 1 Env02 1.688593e-01
#> 159 Line027 1 Env03 -8.178109e-01
#> 160 Line027 1 Env04 -4.061314e-02
#> 161 Line027 1 Env05 -3.665523e-02
#> 162 Line027 1 Env12 -8.397086e-02
#> 163 Line028 1 Env01 9.571128e-02
#> 164 Line028 1 Env02 -4.433641e-01
#> 165 Line028 1 Env03 -3.501224e-02
#> 166 Line028 1 Env04 7.969534e-02
#> 167 Line028 1 Env05 6.996588e-02
#> 168 Line028 1 Env12 -1.205333e-01
#> 169 Line029 1 Env01 -8.560861e-02
#> 170 Line029 1 Env02 -2.906025e-01
#> 171 Line029 1 Env03 3.588538e-01
#> 172 Line029 1 Env04 -6.047660e-01
#> 173 Line029 1 Env05 1.793650e-01
#> 174 Line029 1 Env12 2.988244e-02
#> 175 Line030 1 Env01 -2.450239e-01
#> 176 Line030 1 Env02 -3.768555e-01
#> 177 Line030 1 Env03 -3.359933e-01
#> 178 Line030 1 Env04 1.122401e-01
#> 179 Line030 1 Env05 -6.430218e-02
#> 180 Line030 1 Env12 3.176897e-01
#> 181 Line031 1 Env01 4.327484e-01
#> 182 Line031 1 Env02 3.169358e-01
#> 183 Line031 1 Env03 -3.953674e-01
#> 184 Line031 1 Env04 7.013470e-02
#> 185 Line031 1 Env05 -2.842641e-01
#> 186 Line031 1 Env12 3.005638e-01
#> 187 Line032 1 Env01 2.022828e-02
#> 188 Line032 1 Env02 6.938523e-01
#> 189 Line032 1 Env03 1.962519e-01
#> 190 Line032 1 Env04 6.652834e-01
#> 191 Line032 1 Env05 2.513995e-01
#> 192 Line032 1 Env12 4.023299e-01
#> 193 Line033 1 Env01 -3.672013e-02
#> 194 Line033 1 Env02 2.193808e-01
#> 195 Line033 1 Env03 4.384160e-01
#> 196 Line033 1 Env04 1.855260e-01
#> 197 Line033 1 Env05 -3.754366e-02
#> 198 Line033 1 Env12 4.405533e-01
#> 199 Line034 1 Env01 -2.630628e-01
#> 200 Line034 1 Env02 1.068427e-01
#> 201 Line034 1 Env03 2.448363e-01
#> 202 Line034 1 Env04 2.300229e-01
#> 203 Line034 1 Env05 -1.819871e-01
#> 204 Line034 1 Env12 3.934037e-01
#> 205 Line035 1 Env01 4.568233e-01
#> 206 Line035 1 Env02 9.331069e-01
#> 207 Line035 1 Env03 7.616089e-01
#> 208 Line035 1 Env04 9.123068e-01
#> 209 Line035 1 Env05 2.698053e-01
#> 210 Line035 1 Env12 -2.611717e-01
#> 211 Line036 1 Env01 -3.122496e-02
#> 212 Line036 1 Env02 -1.471982e-02
#> 213 Line036 1 Env03 -6.590165e-01
#> 214 Line036 1 Env04 1.323698e-01
#> 215 Line036 1 Env05 1.406086e-01
#> 216 Line036 1 Env12 -5.751703e-01
#> 217 Line037 1 Env01 1.992879e-01
#> 218 Line037 1 Env02 -6.110940e-02
#> 219 Line037 1 Env03 3.360152e-02
#> 220 Line037 1 Env04 5.278882e-02
#> 221 Line037 1 Env05 -2.321549e-01
#> 222 Line037 1 Env12 -3.036594e-01
#> 223 Line038 1 Env01 2.596149e-03
#> 224 Line038 1 Env02 -3.585840e-01
#> 225 Line038 1 Env03 -6.083689e-01
#> 226 Line038 1 Env04 -4.153578e-01
#> 227 Line038 1 Env05 9.797793e-02
#> 228 Line038 1 Env12 -6.350388e-01
#> 229 Line039 1 Env01 -5.147031e-02
#> 230 Line039 1 Env02 1.711045e-01
#> 231 Line039 1 Env03 -2.609646e-01
#> 232 Line039 1 Env04 5.303502e-01
#> 233 Line039 1 Env05 1.869182e-01
#> 234 Line039 1 Env12 2.622916e-01
#> 235 Line040 1 Env01 4.230289e-01
#> 236 Line040 1 Env02 -9.530414e-02
#> 237 Line040 1 Env03 1.089190e+00
#> 238 Line040 1 Env04 3.964316e-01
#> 239 Line040 1 Env05 1.417472e-01
#> 240 Line040 1 Env12 8.100431e-01
#> 241 Line041 1 Env01 -1.191161e-02
#> 242 Line041 1 Env02 -6.758489e-02
#> 243 Line041 1 Env03 4.133780e-01
#> 244 Line041 1 Env04 -2.104227e-01
#> 245 Line041 1 Env05 3.342105e-02
#> 246 Line041 1 Env12 -4.969144e-01
#> 247 Line042 1 Env01 -5.194868e-01
#> 248 Line042 1 Env02 -5.224029e-01
#> 249 Line042 1 Env03 -1.198224e+00
#> 250 Line042 1 Env04 -5.853565e-01
#> 251 Line042 1 Env05 -3.110742e-01
#> 252 Line042 1 Env12 1.521808e-01
#> 253 Line043 1 Env01 8.223788e-01
#> 254 Line043 1 Env02 4.890592e-02
#> 255 Line043 1 Env03 1.063901e+00
#> 256 Line043 1 Env04 2.600288e-01
#> 257 Line043 1 Env05 1.920498e-01
#> 258 Line043 1 Env12 1.923032e-01
#> 259 Line044 1 Env01 -3.940269e-02
#> 260 Line044 1 Env02 3.604565e-01
#> 261 Line044 1 Env03 2.237813e-01
#> 262 Line044 1 Env04 4.369172e-01
#> 263 Line044 1 Env05 -1.426801e-01
#> 264 Line044 1 Env12 2.598595e-02
#> 265 Line045 1 Env01 -4.065801e-01
#> 266 Line045 1 Env02 1.870945e-01
#> 267 Line045 1 Env03 -6.283372e-02
#> 268 Line045 1 Env04 -9.191246e-02
#> 269 Line045 1 Env05 -6.948758e-02
#> 270 Line045 1 Env12 1.529945e-01
#> 271 Line046 1 Env01 -9.779920e-02
#> 272 Line046 1 Env02 7.071879e-03
#> 273 Line046 1 Env03 -1.360393e-01
#> 274 Line046 1 Env04 6.041856e-02
#> 275 Line046 1 Env05 -1.590995e-01
#> 276 Line046 1 Env12 3.241246e-01
#> 277 Line047 1 Env01 1.130650e-02
#> 278 Line047 1 Env02 3.298617e-01
#> 279 Line047 1 Env03 -8.371703e-01
#> 280 Line047 1 Env04 3.593239e-01
#> 281 Line047 1 Env05 -1.160482e-01
#> 282 Line047 1 Env12 3.455347e-01
#> 283 Line048 1 Env01 -8.341143e-02
#> 284 Line048 1 Env02 1.522246e-01
#> 285 Line048 1 Env03 4.302926e-01
#> 286 Line048 1 Env04 -6.841019e-01
#> 287 Line048 1 Env05 6.639779e-02
#> 288 Line048 1 Env12 4.091434e-01
#> 289 Line049 1 Env01 -1.690703e-01
#> 290 Line049 1 Env02 -6.894051e-01
#> 291 Line049 1 Env03 -6.326231e-01
#> 292 Line049 1 Env04 -5.262413e-01
#> 293 Line049 1 Env05 8.013351e-02
#> 294 Line049 1 Env12 -3.289724e-01
#> 295 Line050 1 Env01 -9.826602e-02
#> 296 Line050 1 Env02 -1.025849e-01
#> 297 Line050 1 Env03 -1.658634e-01
#> 298 Line050 1 Env04 3.945151e-01
#> 299 Line050 1 Env05 2.780061e-02
#> 300 Line050 1 Env12 -3.349608e-01
#> 301 Line051 1 Env01 6.295461e-02
#> 302 Line051 1 Env02 -6.491773e-01
#> 303 Line051 1 Env03 1.855970e-01
#> 304 Line051 1 Env04 -3.538478e-01
#> 305 Line051 1 Env05 1.955815e-01
#> 306 Line051 1 Env12 -1.784969e-01
#> 307 Line052 1 Env01 2.300118e-01
#> 308 Line052 1 Env02 1.092563e+00
#> 309 Line052 1 Env03 4.116734e-01
#> 310 Line052 1 Env04 3.119957e-01
#> 311 Line052 1 Env05 -1.271968e-01
#> 312 Line052 1 Env12 2.818212e-01
#> 313 Line053 1 Env01 -8.038103e-01
#> 314 Line053 1 Env02 6.456338e-01
#> 315 Line053 1 Env03 -4.408843e-01
#> 316 Line053 1 Env04 7.154657e-02
#> 317 Line053 1 Env05 8.454802e-02
#> 318 Line053 1 Env12 -2.140067e-01
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#> 320 Line054 1 Env02 3.199930e-01
#> 321 Line054 1 Env03 1.219829e+00
#> 322 Line054 1 Env04 4.887317e-01
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#> 324 Line054 1 Env12 3.193326e-01
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#> 326 Line055 1 Env02 4.712043e-02
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#> 333 Line056 1 Env03 7.668241e-01
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#> 2986 Line447 2 Env10 1.654151e-01
#> 2987 Line447 2 Env11 4.740068e-01
#> 2988 Line447 2 Env18 7.056005e-01
#> 2989 Line448 2 Env07 -2.075655e-01
#> 2990 Line448 2 Env08 3.025091e-01
#> 2991 Line448 2 Env09 -3.399877e-01
#> 2992 Line448 2 Env10 4.692409e-01
#> 2993 Line448 2 Env11 -1.079045e-01
#> 2994 Line448 2 Env18 -5.219622e-01
#> 2995 Line449 2 Env07 2.521464e-01
#> 2996 Line449 2 Env08 -4.260738e-01
#> 2997 Line449 2 Env09 1.136695e+00
#> 2998 Line449 2 Env10 4.440225e-01
#> 2999 Line449 2 Env11 1.239444e-01
#> 3000 Line449 2 Env18 -5.342003e-02
#> 3001 Line450 2 Env07 -1.773016e-01
#> 3002 Line450 2 Env08 -5.864637e-01
#> 3003 Line450 2 Env09 -8.960906e-01
#> 3004 Line450 2 Env10 9.295341e-01
#> 3005 Line450 2 Env11 -3.496573e-01
#> 3006 Line450 2 Env18 -1.103313e+00
#> 3007 Line451 2 Env07 -5.368574e-01
#> 3008 Line451 2 Env08 -2.162229e-01
#> 3009 Line451 2 Env09 2.273259e-02
#> 3010 Line451 2 Env10 -1.839032e-01
#> 3011 Line451 2 Env11 -2.734482e-01
#> 3012 Line451 2 Env18 -1.246375e-01
#> 3013 Line452 2 Env07 3.857974e-01
#> 3014 Line452 2 Env08 3.895974e-01
#> 3015 Line452 2 Env09 7.241085e-01
#> 3016 Line452 2 Env10 4.296917e-01
#> 3017 Line452 2 Env11 4.835300e-01
#> 3018 Line452 2 Env18 9.976889e-01
#> 3019 Line453 2 Env07 -7.875247e-01
#> 3020 Line453 2 Env08 -1.422300e-01
#> 3021 Line453 2 Env09 -5.047220e-02
#> 3022 Line453 2 Env10 1.286886e-01
#> 3023 Line453 2 Env11 -4.585328e-02
#> 3024 Line453 2 Env18 2.593400e-01
#> 3025 Line454 2 Env07 1.897565e-01
#> 3026 Line454 2 Env08 -2.170448e-02
#> 3027 Line454 2 Env09 -3.619273e-01
#> 3028 Line454 2 Env10 1.710258e-01
#> 3029 Line454 2 Env11 1.820602e-01
#> 3030 Line454 2 Env18 -6.295293e-02
#> 3031 Line455 2 Env07 1.257541e-01
#> 3032 Line455 2 Env08 6.028350e-01
#> 3033 Line455 2 Env09 3.758814e-01
#> 3034 Line455 2 Env10 4.666427e-01
#> 3035 Line455 2 Env11 3.932886e-01
#> 3036 Line455 2 Env18 -3.855363e-01
#> 3037 Line456 2 Env07 2.380393e-01
#> 3038 Line456 2 Env08 6.454923e-02
#> 3039 Line456 2 Env09 -8.852665e-02
#> 3040 Line456 2 Env10 4.936016e-01
#> 3041 Line456 2 Env11 5.840086e-02
#> 3042 Line456 2 Env18 6.345307e-01
#> 3043 Line457 2 Env07 -1.484142e-01
#> 3044 Line457 2 Env08 7.509217e-01
#> 3045 Line457 2 Env09 -2.443983e-01
#> 3046 Line457 2 Env10 1.886972e-01
#> 3047 Line457 2 Env11 2.169015e-01
#> 3048 Line457 2 Env18 1.208942e-01
#> 3049 Line458 2 Env07 1.658718e-01
#> 3050 Line458 2 Env08 2.946521e-01
#> 3051 Line458 2 Env09 9.327704e-01
#> 3052 Line458 2 Env10 8.310815e-01
#> 3053 Line458 2 Env11 -5.911329e-02
#> 3054 Line458 2 Env18 7.618187e-01
#> 3055 Line459 2 Env07 -1.602071e-01
#> 3056 Line459 2 Env08 -5.636245e-01
#> 3057 Line459 2 Env09 -4.430454e-01
#> 3058 Line459 2 Env10 -4.323722e-01
#> 3059 Line459 2 Env11 -4.496299e-01
#> 3060 Line459 2 Env18 -1.805055e-01
#> 3061 Line460 2 Env07 -5.128661e-01
#> 3062 Line460 2 Env08 5.613691e-01
#> 3063 Line460 2 Env09 4.471507e-01
#> 3064 Line460 2 Env10 -8.085988e-01
#> 3065 Line460 2 Env11 -3.800844e-01
#> 3066 Line460 2 Env18 5.801176e-01
#> 3067 Line461 2 Env07 -3.774906e-02
#> 3068 Line461 2 Env08 8.918928e-01
#> 3069 Line461 2 Env09 2.046373e-01
#> 3070 Line461 2 Env10 2.050786e-01
#> 3071 Line461 2 Env11 -3.821651e-02
#> 3072 Line461 2 Env18 4.516897e-01
#> 3073 Line462 2 Env07 -2.101380e-01
#> 3074 Line462 2 Env08 3.528009e-01
#> 3075 Line462 2 Env09 1.059709e+00
#> 3076 Line462 2 Env10 6.487733e-01
#> 3077 Line462 2 Env11 5.611301e-01
#> 3078 Line462 2 Env18 8.983984e-01
#> 3079 Line463 2 Env07 -5.580144e-02
#> 3080 Line463 2 Env08 1.136258e-01
#> 3081 Line463 2 Env09 -3.582224e-01
#> 3082 Line463 2 Env10 -2.230686e-01
#> 3083 Line463 2 Env11 -5.300265e-01
#> 3084 Line463 2 Env18 -3.421853e-01
#> 3085 Line464 2 Env07 -5.976535e-02
#> 3086 Line464 2 Env08 2.980213e-01
#> 3087 Line464 2 Env09 1.925046e-01
#> 3088 Line464 2 Env10 -3.311046e-01
#> 3089 Line464 2 Env11 -4.655951e-01
#> 3090 Line464 2 Env18 7.367072e-02
#> 3091 Line465 2 Env07 -3.359528e-02
#> 3092 Line465 2 Env08 3.700589e-01
#> 3093 Line465 2 Env09 3.196115e-02
#> 3094 Line465 2 Env10 -5.867339e-03
#> 3095 Line465 2 Env11 4.607880e-02
#> 3096 Line465 2 Env18 -2.813679e-01
#> 3097 Line466 2 Env07 8.109439e-02
#> 3098 Line466 2 Env08 2.710562e-02
#> 3099 Line466 2 Env09 2.868791e-01
#> 3100 Line466 2 Env10 3.649212e-01
#> 3101 Line466 2 Env11 3.286566e-01
#> 3102 Line466 2 Env18 1.696620e-01
#> 3103 Line467 2 Env07 -2.049983e-01
#> 3104 Line467 2 Env08 -2.067242e-01
#> 3105 Line467 2 Env09 -4.809010e-01
#> 3106 Line467 2 Env10 4.059622e-01
#> 3107 Line467 2 Env11 5.938833e-01
#> 3108 Line467 2 Env18 -1.662090e-02
#> 3109 Line468 2 Env07 -2.196019e-03
#> 3110 Line468 2 Env08 -1.652654e-01
#> 3111 Line468 2 Env09 8.438293e-02
#> 3112 Line468 2 Env10 1.861029e-01
#> 3113 Line468 2 Env11 4.379570e-01
#> 3114 Line468 2 Env18 2.866262e-01
#> 3115 Line469 2 Env07 3.256827e-01
#> 3116 Line469 2 Env08 2.682046e-01
#> 3117 Line469 2 Env09 1.509331e-01
#> 3118 Line469 2 Env10 1.163935e-01
#> 3119 Line469 2 Env11 1.950116e-01
#> 3120 Line469 2 Env18 7.982267e-02
#> 3121 Line470 2 Env07 -7.660226e-02
#> 3122 Line470 2 Env08 -2.184190e-02
#> 3123 Line470 2 Env09 -3.583300e-01
#> 3124 Line470 2 Env10 -5.917390e-01
#> 3125 Line470 2 Env11 -2.694019e-02
#> 3126 Line470 2 Env18 -2.676855e-01
#> 3127 Line471 2 Env07 9.886897e-02
#> 3128 Line471 2 Env08 5.148411e-02
#> 3129 Line471 2 Env09 1.041769e+00
#> 3130 Line471 2 Env10 4.991517e-01
#> 3131 Line471 2 Env11 -2.759165e-01
#> 3132 Line471 2 Env18 -4.006020e-02
#> 3133 Line472 2 Env07 -3.968355e-02
#> 3134 Line472 2 Env08 -5.892998e-01
#> 3135 Line472 2 Env09 -8.433605e-01
#> 3136 Line472 2 Env10 2.247841e-01
#> 3137 Line472 2 Env11 -8.860557e-02
#> 3138 Line472 2 Env18 -1.575884e-01
#> 3139 Line473 2 Env07 2.449058e-01
#> 3140 Line473 2 Env08 -2.679446e-01
#> 3141 Line473 2 Env09 4.910809e-01
#> 3142 Line473 2 Env10 -8.247019e-01
#> 3143 Line473 2 Env11 -1.341941e-01
#> 3144 Line473 2 Env18 -1.772371e-01
#> 3145 Line474 2 Env07 -1.631212e-01
#> 3146 Line474 2 Env08 -5.457100e-01
#> 3147 Line474 2 Env09 -7.936695e-01
#> 3148 Line474 2 Env10 1.800190e-01
#> 3149 Line474 2 Env11 -3.492748e-01
#> 3150 Line474 2 Env18 -4.816460e-01
#> 3151 Line475 2 Env07 3.858371e-01
#> 3152 Line475 2 Env08 1.351180e-02
#> 3153 Line475 2 Env09 -2.184656e-01
#> 3154 Line475 2 Env10 3.992395e-01
#> 3155 Line475 2 Env11 -1.765522e-01
#> 3156 Line475 2 Env18 -4.230817e-01
#> 3157 Line476 2 Env07 -6.354329e-02
#> 3158 Line476 2 Env08 -5.974567e-02
#> 3159 Line476 2 Env09 8.854801e-01
#> 3160 Line476 2 Env10 4.938334e-01
#> 3161 Line476 2 Env11 6.847668e-01
#> 3162 Line476 2 Env18 4.796469e-01
#> 3163 Line477 2 Env07 -2.480675e-01
#> 3164 Line477 2 Env08 -1.722255e-01
#> 3165 Line477 2 Env09 -6.256433e-01
#> 3166 Line477 2 Env10 -5.548491e-01
#> 3167 Line477 2 Env11 -1.382821e-01
#> 3168 Line477 2 Env18 3.172788e-01
#> 3169 Line478 2 Env07 -5.626158e-01
#> 3170 Line478 2 Env08 1.938583e-01
#> 3171 Line478 2 Env09 3.964626e-03
#> 3172 Line478 2 Env10 -5.849700e-01
#> 3173 Line478 2 Env11 -1.274783e-01
#> 3174 Line478 2 Env18 -5.038894e-01
#> 3175 Line479 2 Env07 6.964489e-01
#> 3176 Line479 2 Env08 7.010937e-02
#> 3177 Line479 2 Env09 -8.699923e-02
#> 3178 Line479 2 Env10 3.969676e-01
#> 3179 Line479 2 Env11 -5.048246e-02
#> 3180 Line479 2 Env18 2.397630e-01
#> 3181 Line480 2 Env07 4.771511e-01
#> 3182 Line480 2 Env08 -5.631871e-02
#> 3183 Line480 2 Env09 3.124875e-01
#> 3184 Line480 2 Env10 1.050427e+00
#> 3185 Line480 2 Env11 2.121208e-01
#> 3186 Line480 2 Env18 3.931429e-02
#> 3187 Line481 2 Env07 -2.360902e-01
#> 3188 Line481 2 Env08 2.474122e-01
#> 3189 Line481 2 Env09 4.789110e-01
#> 3190 Line481 2 Env10 1.073238e-01
#> 3191 Line481 2 Env11 3.659585e-01
#> 3192 Line481 2 Env18 -2.208459e-01
#> 3193 Line482 2 Env07 -6.746652e-01
#> 3194 Line482 2 Env08 -1.423498e-01
#> 3195 Line482 2 Env09 -5.491483e-01
#> 3196 Line482 2 Env10 -1.611212e-01
#> 3197 Line482 2 Env11 1.238833e-01
#> 3198 Line482 2 Env18 -3.415031e-01
#> 3199 Line483 2 Env07 -6.510996e-01
#> 3200 Line483 2 Env08 2.305794e-01
#> 3201 Line483 2 Env09 -4.535972e-01
#> 3202 Line483 2 Env10 1.430234e-01
#> 3203 Line483 2 Env11 1.920677e-01
#> 3204 Line483 2 Env18 1.908374e-01
#> 3205 Line484 2 Env07 1.750004e-01
#> 3206 Line484 2 Env08 1.281273e-01
#> 3207 Line484 2 Env09 1.541320e-01
#> 3208 Line484 2 Env10 3.580099e-01
#> 3209 Line484 2 Env11 1.230135e-02
#> 3210 Line484 2 Env18 -4.327763e-03
#> 3211 Line485 2 Env07 8.591130e-01
#> 3212 Line485 2 Env08 3.984696e-01
#> 3213 Line485 2 Env09 2.724308e-01
#> 3214 Line485 2 Env10 4.959955e-01
#> 3215 Line485 2 Env11 1.647880e-01
#> 3216 Line485 2 Env18 1.878780e-01
#> 3217 Line486 2 Env07 1.406309e-01
#> 3218 Line486 2 Env08 -3.299947e-01
#> 3219 Line486 2 Env09 -1.746237e-01
#> 3220 Line486 2 Env10 4.978671e-02
#> 3221 Line486 2 Env11 -1.524626e-01
#> 3222 Line486 2 Env18 2.150668e-01
#> 3223 Line487 2 Env07 -3.691694e-02
#> 3224 Line487 2 Env08 -2.587806e-01
#> 3225 Line487 2 Env09 -2.456653e-01
#> 3226 Line487 2 Env10 -4.860604e-01
#> 3227 Line487 2 Env11 6.688308e-02
#> 3228 Line487 2 Env18 4.277078e-01
#> 3229 Line488 2 Env07 4.389565e-01
#> 3230 Line488 2 Env08 -9.803447e-02
#> 3231 Line488 2 Env09 -3.120294e-01
#> 3232 Line488 2 Env10 4.592851e-01
#> 3233 Line488 2 Env11 6.431404e-01
#> 3234 Line488 2 Env18 -9.491138e-02
#> 3235 Line489 2 Env07 -1.522128e-01
#> 3236 Line489 2 Env08 3.822185e-01
#> 3237 Line489 2 Env09 -5.523805e-02
#> 3238 Line489 2 Env10 -1.571521e-01
#> 3239 Line489 2 Env11 -2.835575e-01
#> 3240 Line489 2 Env18 -6.877879e-03
#> 3241 Line490 2 Env07 -2.498936e-01
#> 3242 Line490 2 Env08 5.544483e-01
#> 3243 Line490 2 Env09 -2.316629e-01
#> 3244 Line490 2 Env10 5.589257e-01
#> 3245 Line490 2 Env11 1.454931e-01
#> 3246 Line490 2 Env18 8.089218e-01
#> 3247 Line491 2 Env07 3.095620e-01
#> 3248 Line491 2 Env08 -6.489997e-01
#> 3249 Line491 2 Env09 3.180641e-01
#> 3250 Line491 2 Env10 3.248508e-01
#> 3251 Line491 2 Env11 -4.742402e-01
#> 3252 Line491 2 Env18 -1.483494e-01
#> 3253 Line492 2 Env07 6.465929e-02
#> 3254 Line492 2 Env08 8.085793e-03
#> 3255 Line492 2 Env09 5.320146e-02
#> 3256 Line492 2 Env10 -6.958482e-01
#> 3257 Line492 2 Env11 3.284597e-01
#> 3258 Line492 2 Env18 -2.947430e-01
#> 3259 Line493 2 Env07 7.230805e-03
#> 3260 Line493 2 Env08 -8.563154e-02
#> 3261 Line493 2 Env09 -5.886486e-02
#> 3262 Line493 2 Env10 -1.122066e+00
#> 3263 Line493 2 Env11 -1.118962e-01
#> 3264 Line493 2 Env18 -3.695091e-01
#> 3265 Line494 2 Env07 2.353324e-01
#> 3266 Line494 2 Env08 1.755170e-01
#> 3267 Line494 2 Env09 -4.601296e-01
#> 3268 Line494 2 Env10 5.684251e-01
#> 3269 Line494 2 Env11 -2.017712e-01
#> 3270 Line494 2 Env18 4.211957e-01
#> 3271 Line495 2 Env07 -6.067863e-02
#> 3272 Line495 2 Env08 -5.220268e-01
#> 3273 Line495 2 Env09 1.253751e-01
#> 3274 Line495 2 Env10 3.137329e-01
#> 3275 Line495 2 Env11 6.684349e-02
#> 3276 Line495 2 Env18 -5.873161e-01
#> 3277 Line496 2 Env07 5.285133e-01
#> 3278 Line496 2 Env08 3.439581e-01
#> 3279 Line496 2 Env09 1.830050e-01
#> 3280 Line496 2 Env10 1.743509e-01
#> 3281 Line496 2 Env11 2.887337e-01
#> 3282 Line496 2 Env18 4.261468e-01
#> 3283 Line497 2 Env07 -3.236221e-01
#> 3284 Line497 2 Env08 -7.411147e-02
#> 3285 Line497 2 Env09 -9.793533e-02
#> 3286 Line497 2 Env10 -6.400513e-02
#> 3287 Line497 2 Env11 -2.123664e-01
#> 3288 Line497 2 Env18 1.393497e-01
#> 3289 Line498 2 Env07 -3.750728e-01
#> 3290 Line498 2 Env08 2.455763e-01
#> 3291 Line498 2 Env09 7.922759e-01
#> 3292 Line498 2 Env10 6.522879e-01
#> 3293 Line498 2 Env11 -4.426905e-02
#> 3294 Line498 2 Env18 4.403647e-02
#> 3295 Line499 2 Env07 -1.369626e-01
#> 3296 Line499 2 Env08 -3.156295e-02
#> 3297 Line499 2 Env09 -2.025579e-01
#> 3298 Line499 2 Env10 9.563605e-02
#> 3299 Line499 2 Env11 -1.196079e-01
#> 3300 Line499 2 Env18 -5.601190e-01
#> 3301 Line500 2 Env07 -1.382644e-01
#> 3302 Line500 2 Env08 -2.600126e-01
#> 3303 Line500 2 Env09 -1.408302e-01
#> 3304 Line500 2 Env10 -3.051666e-01
#> 3305 Line500 2 Env11 -2.826251e-01
#> 3306 Line500 2 Env18 -7.882763e-01
#> 3307 Line501 2 Env07 -1.180153e-01
#> 3308 Line501 2 Env08 1.766371e-01
#> 3309 Line501 2 Env09 -1.508217e-01
#> 3310 Line501 2 Env10 -4.457630e-01
#> 3311 Line501 2 Env11 -5.420383e-01
#> 3312 Line501 2 Env18 3.649028e-01
#> 3313 Line502 2 Env07 6.399184e-01
#> 3314 Line502 2 Env08 5.382367e-01
#> 3315 Line502 2 Env09 -1.978510e-01
#> 3316 Line502 2 Env10 1.147028e+00
#> 3317 Line502 2 Env11 -1.925140e-02
#> 3318 Line502 2 Env18 4.235418e-01We can see the incidence matrix of lines by environments using the
Image function in MegaLMM
Image(as.matrix(table(yield_data$Line,yield_data$Env))) + theme(legend.position = 'none') + xlab('Environment') + ylab('Line')
As you can see, no line is grown in every environment, and no environment includes every line. In fact, there seems to be largely 2 sets of lines, one grown in ~1/4 the environments and the other grown in a portion of the remaining environments. These two sets of lines are designated as different “populations” in the input data.
We can look at the number of observations by line and by environment:
The genetic data is available as an additive genomic relationship matrix calculated from GBS SNPs.
data('K',package='MegaLMM')We can view the matrix also using Image
Image(K)
This shows we also have two groups of fairly related lines, with low relationships between groups.
Formatting data from MegaLMM
The yield data was provided in the tall format,
meaning a single observation per row. In this format we would say we
have 1 trait (Yield) with values measured in many
environments.
But MegaLMM isn’t good for modeling GxE like this.
Instead, we want to consider the yield in each environment as a separate
trait, and each line is measured for 6-9 of these traits. So we need to
construct a 502x19 trait matrix. The MegaLMM
package includes a helper function to do this called
create_data_matrices. This uses tidyr’s
pivot_wider function to create the matrix, and the
arguments are the same.
data_matrices = create_data_matrices(
tall_data = yield_data, # your input tall data.frame,
id_cols = c('Line','Population'), # vector giving the set of columns of tall_data used to identify each individual, and any covariates you'll want to use to model the trait data across individuals.
names_from = 'Env', # vector giving the set of columns of tall_data used to identify each trait
values_from = 'Yield' # name of the trait data column
)The output of create_data_matrices is a list with 3
elements. We’ll only use the first two.
The first is a new data.frame with one row per individual, and a single column giving the Line identifier. If you have covariates among lines (e.g. sex, population, etc), those variables should be included here too.
sample_data = data_matrices$data
sample_data
#> Line Population
#> 1 Line001 1
#> 2 Line002 1
#> 3 Line003 1
#> 4 Line004 1
#> 5 Line005 1
#> 6 Line006 1
#> 7 Line007 1
#> 8 Line008 1
#> 9 Line009 1
#> 10 Line010 1
#> 11 Line011 1
#> 12 Line012 1
#> 13 Line013 1
#> 14 Line014 1
#> 15 Line015 1
#> 16 Line016 1
#> 17 Line017 1
#> 18 Line018 1
#> 19 Line019 1
#> 20 Line020 1
#> 21 Line021 1
#> 22 Line022 1
#> 23 Line023 1
#> 24 Line024 1
#> 25 Line025 1
#> 26 Line026 1
#> 27 Line027 1
#> 28 Line028 1
#> 29 Line029 1
#> 30 Line030 1
#> 31 Line031 1
#> 32 Line032 1
#> 33 Line033 1
#> 34 Line034 1
#> 35 Line035 1
#> 36 Line036 1
#> 37 Line037 1
#> 38 Line038 1
#> 39 Line039 1
#> 40 Line040 1
#> 41 Line041 1
#> 42 Line042 1
#> 43 Line043 1
#> 44 Line044 1
#> 45 Line045 1
#> 46 Line046 1
#> 47 Line047 1
#> 48 Line048 1
#> 49 Line049 1
#> 50 Line050 1
#> 51 Line051 1
#> 52 Line052 1
#> 53 Line053 1
#> 54 Line054 1
#> 55 Line055 1
#> 56 Line056 1
#> 57 Line057 1
#> 58 Line058 1
#> 59 Line059 1
#> 60 Line060 1
#> 61 Line061 1
#> 62 Line062 1
#> 63 Line063 1
#> 64 Line064 1
#> 65 Line065 1
#> 66 Line066 1
#> 67 Line067 1
#> 68 Line068 1
#> 69 Line069 1
#> 70 Line070 1
#> 71 Line071 1
#> 72 Line072 1
#> 73 Line073 1
#> 74 Line074 1
#> 75 Line075 1
#> 76 Line076 1
#> 77 Line077 1
#> 78 Line078 1
#> 79 Line079 1
#> 80 Line080 1
#> 81 Line081 1
#> 82 Line082 1
#> 83 Line083 1
#> 84 Line084 1
#> 85 Line085 1
#> 86 Line086 1
#> 87 Line087 1
#> 88 Line088 1
#> 89 Line089 1
#> 90 Line090 1
#> 91 Line091 1
#> 92 Line092 1
#> 93 Line093 1
#> 94 Line094 1
#> 95 Line095 1
#> 96 Line096 1
#> 97 Line097 1
#> 98 Line098 1
#> 99 Line099 1
#> 100 Line100 1
#> 101 Line101 1
#> 102 Line102 1
#> 103 Line103 1
#> 104 Line104 1
#> 105 Line105 1
#> 106 Line106 1
#> 107 Line107 1
#> 108 Line108 1
#> 109 Line109 1
#> 110 Line110 1
#> 111 Line111 1
#> 112 Line112 1
#> 113 Line113 1
#> 114 Line114 1
#> 115 Line115 1
#> 116 Line116 1
#> 117 Line117 1
#> 118 Line118 1
#> 119 Line119 1
#> 120 Line120 1
#> 121 Line121 1
#> 122 Line122 1
#> 123 Line123 1
#> 124 Line124 1
#> 125 Line125 1
#> 126 Line126 1
#> 127 Line127 1
#> 128 Line128 1
#> 129 Line129 1
#> 130 Line130 1
#> 131 Line131 1
#> 132 Line132 1
#> 133 Line133 1
#> 134 Line134 1
#> 135 Line135 1
#> 136 Line136 1
#> 137 Line137 1
#> 138 Line138 1
#> 139 Line139 1
#> 140 Line140 1
#> 141 Line141 1
#> 142 Line142 1
#> 143 Line143 1
#> 144 Line144 1
#> 145 Line145 1
#> 146 Line146 1
#> 147 Line147 1
#> 148 Line148 1
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#> 502 Line502 2The second is the nxp trait matrix. The rows of the
trait matrix are aligned with the rows of the individual identifier
data.frame. We can extract these for use in MegaLMM:
Y = data_matrices$Y
head(Y)[,1:5]
#> Env01 Env02 Env03 Env04 Env05
#> Line001 0.07663660 0.57362277 -0.1057396 0.4053487 -0.09638836
#> Line002 -0.03429877 -0.04004347 -0.2653557 0.1136820 -0.00372635
#> Line003 0.21059771 -0.63959202 0.3995948 0.2926501 0.11464992
#> Line004 -0.75226609 -0.14319441 -1.2009773 -0.4363149 -0.24941693
#> Line005 0.50406848 0.14874755 0.6330452 0.6561490 0.12371692
#> Line006 -0.02619325 0.16820004 0.1799732 -0.2938856 0.02111995One check we need to do is ensure all our individuals in our data are represented in the genomic relationship matrix:
Set up Cross Validation
The goal of genomic prediction is to accurately predict the genetic values of individuals that are not observed in a particular environment. The standard way to estimate this accuracy is to mask a portion of the lines in the input data, use a model to predict these masked values, and then measure the correlation between the predicted values and the original data. In this tutorial we will do only 1 round of a k-fold cross-validation. Generally you would repeat this with other training / testing partition.
Because we are evaluating the accuracy for incomplete multi-environment trial prediction, we will mask different set of individuals in each environment, so each individual maintains input data in at least some individuals.
Because the individuals are stratified between two populations, we will ensure all testing individuals come from the same population. The masking algorithm will be:
- For each environment, decide which population is best represented
- For the observed individuals in that population, divide them into k-folds
- Create a matrix with the testing sets (fold_ID) for each observation
Not not worry about understanding this code! The call to
set.seed() at the beginning makes it repeatable.
set.seed(1)
k_fold = 5 # we will hold out 1/5 = 20% of the observations from each environment
fold_ID_matrix = matrix(NA,nrow = nrow(Y),ncol = ncol(Y),dimnames = dimnames(Y))
for(i in 1:ncol(fold_ID_matrix)) {
observed_lines = sample_data[!is.na(Y[,i]),]
pop = names(sort(table(observed_lines$Population),decreasing=T))[1]
observed_lines = subset(observed_lines,Population == pop)
n_lines = nrow(observed_lines)
observed_lines$fold = sample(rep(1:k_fold,(n_lines/k_fold)+1))[1:n_lines]
fold_ID_matrix[match(observed_lines$Line,rownames(fold_ID_matrix)),i] = observed_lines$fold
}Now that we have divided the observed data into folds, we can chose to mask fold==1 to create our training data, and extract the corresponding values as our testing data
fold_ID = 1
Y_train = Y_testing = Y
Y_train[fold_ID_matrix == fold_ID] = NA
Y_testing[fold_ID_matrix != fold_ID | is.na(fold_ID_matrix)] = NARun univariate GBLUP as reference
To evaluate whether the multi-trait prediction from
MegaLMM is useful, we’ll run normal univariate genomic
prediction using the GBLUP model using the rrBLUP
package.
library(rrBLUP)
rrBLUP_predictions = matrix(NA,nrow(Y),ncol(Y),dimnames = dimnames(Y))
for(i in 1:ncol(Y)) {
X = model.matrix(~Population,sample_data) # we will include Population as a covariate if it is variable among the individuals for this environment
if(var(X[!is.na(Y_train[,i]),2]) == 0) X = X[,-2,drop=FALSE]
res = mixed.solve(y = Y_train[,i],
X = X,
K = K)
rrBLUP_predictions[,i] = c(X %*% res$beta) + res$u
}Here are the correlations between the predictions and the testing data:
diag(cor(Y_testing,rrBLUP_predictions,use='p'))
#> Env01 Env02 Env03 Env04 Env05 Env12
#> 0.278425958 0.187881694 0.625000316 0.389125876 -0.033478375 0.029312408
#> Env14 Env16 Env17 Env15 Env06 Env13
#> -0.109251581 0.339391947 0.334737933 0.373848343 -0.272951670 0.111330746
#> Env18 Env19 Env07 Env08 Env09 Env10
#> 0.426321030 0.002766824 0.309384501 0.349614409 -0.097410915 0.104172216
#> Env11
#> -0.071876574Run MegaLMM
Now, we’ll move to MegaLMM and fit a multivariate GBLUP
model to all trials at once.
First, I’ll review the MegaLMM model, and then describe
the implementation and usage of the R package.
Background
The MegaLMM model
MegaLMM implements multivariate linear mixed models of the form:
Y = X*B + Z*U + Ewhere Y is a n x t matrix of observations
for n individuals and t traits, X
is a design matrix for b fixed effects (including an
intercept), Z is a design matrix for the random effects,
and E is a n x t matrix of residual errors.
The random effects are U are independent of the residuals,
but columns of U matrix can be correlated, and each column
vector marginally follows a multivariate normal distribution with a
known covariance matrix K.
MvLMMs like this are notoriously difficult to fit. We address this by re-paramterizing the MvLMM as a mixed effect factor model:
Y = F*Lambda + Y_R
Y_R = X*B_R + Z*U_R + E_R
F = X*B_F + Z*U_F + E_Fwhere F is a n x k matrix of latent factor
traits and Lambda is a k x t matrix of factor
loadings. This is the model actually fit by MegaLMM.
Basically, we break Y which is a set of t
correlated traits into two sets of uncorrelated traits: Y_R
and F. These are sets of t and K
traits all of which are independently related to the fixed and random
effects. All covariances within and among these sets of traits are
captured by Lambda. Because of this, we can treat each of
the columns of Y_R or F independently and
specific a LMM for each of them. Generally, we use the same
X, Z and K for all these traits.
However in MegaLMM we allow some additional
flexibility:
- The fixed effect design matrix
Xis split into two parts:X_1andX_2.X_1are true fixed effects meaning the corresponding coefficients (B_1) are given flat priors. Because of this, we can’t allowFto depend onX_1, so this is only part of the model forY_r.X_2are regularized effects, so the corresponding coefficients are given an informative prior (e.g. BayesC). We allow bothY_RandFto depend onX_2, and potentially on different subsets ofX_2:X_2FandX_2R, with corresponding coefficient matricesB_2FandB_2R. We do not make use of these matrices in this tutorial. The full models thus are:Y_R = X_1*B_1 + X_2R*B_2R + Z*U_R + E_RandF = X_2F*B_2F + Z*U_F + E_F. - While the same model structure applies to all columns of
Y_R, because of missing values not all columns ofX_1may be variable for a particular trait. Therefore we drop columns ofX_1as needed and assign the coefficients to 0. - While I only wrote a single random effect above,
MegaLMMdoes allow you to specific multiple independent random effects with different covariance matrices. However the memory and time complexities increase exponentially with more random effects. And, we cannot account for correlations amongU_iandU_j.
Taking a single column of Y_R, the LMM is:
$$ y_r = X\times b_r + Z\times u_r + e_r \\ u_r \sim N(0,\sigma^2*h^2*K) \\ e_r \sim N(0,\sigma^2*(1-h^2)*I) $$
The model for each column of F is similar. This differs
from most Bayesian LMMs in the parameterization of the variance
components, but has some conceptual and algorithmic advantages. For
priors, we use an inverse gamma prior for
and a discrete prior on
.
Specification of the priors is described below.
The unique aspects of MegaLMM relative to other factor models are:
- The residuals of
Yafter accounting for the factors are not assumed to be iid, but are modeled with independent (across traits) LMMs accounting for both fixed and random effects. - The factors themselves are also not assumed to be iid, but are modeled with the same LMMs. This highlights the parallel belief that these latent factors represent traits that we just didn’t measure directly.
- Each factor is shared by all modeled sources of variation (fixed effects, random effects and residuals), rather than being unique to a particular source.
- The factor loadings are strongly regularized so ensure that estimation is efficient. We accomplish this by ordering the factors from most-to-least important using a prior similar to that proposed by Bhattarchya and Dunson (2011)
- Optionally, we also allow regression models for the factor loadings. This feature was described in Hu et al (2024). Multiple covariate matrices can be passed to MegaLMM as predictors of the variation within each row of factor loadings.
Model formula
We use R’s formula syntax to construct the design
matrices X and Z. In default usage, we
specific a single formula and assume it applies to all columns of both
Y_R and F, except the fixed effects do not
apply to F.
Random effects
The random effect syntax in a formula is (a|X). This
specifies a variance for each level of a (e.g.
environment) for the location effects for each level of X
(e.g. genotype, i.e. variance among genotypes in each
environment). In lme4 syntax, there would additionally be
covariances between the levels of a within each level of
X. However we cannot model these covariances in
MegaLMM, so this syntax makes independent variances for
each level of a. Note, however, that each level of
a introduces a new variance into the model, which
exponentially increases the memory requirements! It is much better, if
possible, to introduce each level of a as a separate
trait!
Random effects have two parts: location effects
which are the values for each level (e.g. breeding values for
each individual) and variances which are the population
variances of the location effects. We model the location effects as
following a multivariate normal distribution with covariance equal to a
known covariance matrix (K) times a variance
proportion
()
times a phenotypic variance
().
This differs slightly from typical parameterizations of random effects,
which uses a separate variance for each random effect. In
MegaLMM we instead model the proportion that each random
effect contributes to the total, so all
values sum to 1, and use a discrete prior over the interval [0,1] for
this parameter. This gives you a lot of flexibility for specifying prior
distributions.
Factors
In most factor models the number of factors K is a
critical parameter, and models with different numbers of parameters
(either larger or smaller than optimal) may give very different answers.
This is generally not the case in MegaLMM. In
MegaLMM we use a prior to order and regularize the
importance of the factors, enforcing that high-order factors explain
less and less of the overall variation. Therefore the highest-order
factors are generally extremely unimportant, and adding a few more or
fewer of these unimportant factors won’t change the influence of the
first factors. It is important to set K large enough to
capture most of the covariation in Y, but once it’s large
enough, additional values will not likely affect the model much.
A related note, though, is that the precise ordering of the factors
is not well learned by the MCMC algorithm, and so inferences that rely
on this should be treated with extreme caution. The rate of decay of
factor importance is highly sensitive to the prior, and factor ordering
does not mix well. Routines are described below to help the convergence
of factor ordering to a useful value, but that is all we can do. This
also means that the precise values of individual factor loadings may
drift during the MCMC as factor orderings change slowly. This would
greatly impact the inference on factor identities, but is not very
important if the goal is prediction of U or
Y.
Missing data
In a Bayesian model, we can treat missing data as additional parameters that need to be learned, so imputation of missing data happens naturally. However, if we construct the model correctly, some missing data points are not needed for the inference of any other parameters, and so can be simply predicted from the posterior values of other parameters. The more missing values we can treat this way the better, because conditioning on imputed values in MCMC greatly reduces the mixing rate of the chain.
In MegaLMM, if we can identify groups of traits that
share missing values across a group of individuals (rows), we can
declare that this block of values to be only predicted, not imputed.
There is a tradeoff here in that the more groups of traits are
specified, the greater the memory overhead of MegaLMM. But
the improvement in MCMC mixing can be great.
MCMC and Posterior samples
MegaLMM uses a Gibbs sampler to draw samples from
posterior of all unknown parameters. There are a lot of parameters in
the MegaLMM model, and not all of them may be of interest
to a user. You can choose which specific parameters should be tracked as
described below. Additionally, you may be interested in a function of
several parameters, and there is a function to calculate these values on
each iteration as well. Finally, the sets of posterior samples can
themselves be very large. If you’re tracking large matrices of predicted
values for thousands of traits these posterior samples can take of Gbs
of memory. Therefore MegaLMM has a way to store the
posterior samples as a database on the disk, only holding small chunks
of a chain in memory at a time.
Running MegaLMM
Set control parameters
The first function for MegaLMM sets several parameters
of the model. Only a few are noted here. See the help page for more
control parameters. The output is a list that will be passed to the main
model construction function below.
run_parameters = MegaLMM_control(
h2_divisions = 20,
# Each variance component is allowed to explain between 0% and 100% of the
# total variation. How many segments should the range [0,100) be divided
# into for each random effect?
burn = 0,
# number of burn in samples before saving posterior samples. I set this to
# zero and instead run the chain in small chunks, doing the burning manually, a
# s described below.
thin = 2,
# during sampling, we'll save every 2nd sample to the posterior database.
K = 15 # number of factors. With 19 traits, this is likely way higher than needed.
)Create the model object
The function setup_model_MegaLMM parses the model
formulas, links the GRM to the random effects, and creates an object to
store all components of the model.
MegaLMM_state = setup_model_MegaLMM(
Y = Y_train,
# The n x p trait matrix
formula = ~ Population + (1|Line),
# This is syntax like lme4 for mixed effect models.
# We specify a fixed effect of population and a random effect for genotype (Line)
data = sample_data,
# the data.frame with information for constructing the model matrices
relmat = list(Line = K),
# A list of covariance matrices to link to the random effects in formula.
# each grouping variable in formula can be linked to a covariance matrix.
# If so, every level of the grouping variable must be in the rownames of K.
# additional rows of K not present in data will still be predicted
# (and therefore will use memory and computational time!)
run_parameters=run_parameters,
# This list of control parameters created above
run_ID = sprintf('MegaLMM_fold_%02d',fold_ID)
# A run identifier. The function will create a folder with this name
# and store lots of useful data inside it
)Note: There is an important optional extra argument
extra_regressions that can be used to pass covariates
specifically for the factors. By default, X_F, the design
matrix for the factors, is empty. Qu
et al (2022) used this argument to pass genetic marker data as
priors for the variation in each column of F. Hu
et al (2024) used this to allow an intercept for each factor which
is useful for multi-environment trials. The syntax is:
# extra_regressions = list(X=X_F_mat,factors=T), #This specifices the variable X_F in MegaLMM is assigned to X_F_mat, and applies to the factors FThe output is the variable MegaLMM_state which is an
object of class MegaLMM_state including the following
slots:
-
current_state: a list with elements holding the current values for all model parameters. Each parameter is stored as a 2d matrix. Variable names correspond as closely as possible to those described in the manuscript: Runcie et al 2020. -
Posterior: a list with elements 3d or 2d arrays holding posterior samples (or posterior means) of specified model parameters. By default, samples of all parameters are stored. However these matrices can be large if data is large, so parameters can be dropped from this list by removing their names from the listsMegaLMM_state$Posterior$posteriorSample_paramsandMegaLMM_state$Posterior$posteriorMean_params. -
run_ID: The current state of the chain plus Posterior samples and any diagnostic plots are automatically saved in a folder with this name during the run.
Before we can run the model, we have to do a few more steps
Set priors
We need to set priors for the variance components
(
and
for Y_R and F, and for the parameters of the
factor loadings Lambda.
For Lambda, we have several types of priors as described
in the MegaLMM papers. In this tutorial we will use the horseshoe prior
from the Genome Biology paper:
Lambda_prior = list(
sampler = sample_Lambda_prec_ARD,
# function that implements the ARD Lambda prior
# described in Runcie et al 2013 paper.
#See code to see requirements for this function.
# other options are:
# ?sample_Lambda_prec_horseshoe
# ?sample_Lambda_prec_BayesC
Lambda_df = 3,
delta_1 = list(shape = 2, rate = 1),
delta_2 = list(shape = 3, rate = 1), # parameters of the gamma distribution giving the expected change in proportion of non-zero loadings in each consecutive factor
# parameters of the gamma distribution giving the expected change
# in proportion of non-zero loadings in each consecutive factor
delta_iterations_factor = 100
# parameter that affects mixing of the MCMC sampler. This value is generally fine.
)Environmental covariates
We can specify matrices of environmental covariates as priors for the
factor loadings. The covariate matrices should be matrices with the
number of rows equal to the number of traits. These should be pasted
column-wise into a big matrix X_Env. Then, make a vector
X_Env_groups that ‘labels’ each column of
X_Env based on which set of covariates it belongs to. For
example, if the first column of X_Env is a vector of 1’s as
an intercept, then the next 5 columns are temperature covariates, and
the final 3 columns are soil covariates, we might specify:
X_Env_groups = c(1,2,2,2,2,2,3,3,3). We pass
X_Env, X_Env_groups, as well as
hyperparameters of the inverse gamma prior on the variance of each set
of covariate’s coefficients: Lambda_beta_var_shape and
Lambda_beta_var_rate
# Lambda_prior = list(
# sampler = sample_Lambda_prec_ARD,
# # function that implements the ARD Lambda prior
# # described in Runcie et al 2013 paper.
# #See code to see requirements for this function.
# # other options are:
# # ?sample_Lambda_prec_horseshoe
# # ?sample_Lambda_prec_BayesC
# prop_0 = 0.1,
# # prior guess at the number of non-zero loadings in the first and most important factor
# delta = list(shape = 3, scale = 1),
# # parameters of the gamma distribution giving the expected change
# # in proportion of non-zero loadings in each consecutive factor
# delta_iterations_factor = 100,
# # parameter that affects mixing of the MCMC sampler. This value is generally fine.
# X = X_Env,
# X_group = X_Env_groups,
# fit_X = F, # we start by letting Lambda converge without X, but then turn it on during burnins.
# Lambda_beta_var_shape = 3,
# Lambda_beta_var_rate = 1
# )For the remaining priors we use MegaLMM_priors
priors = MegaLMM_priors(
tot_Y_var = list(V = 0.5, nu = 5),
# Prior variance of trait residuals after accounting for fixed effects and factors
# See MCMCglmm for meaning of V and nu
tot_F_var = list(V = 18/20, nu = 20),
# Prior variance of factor traits. This is included to improve MCMC mixing,
# but can be turned off by setting nu very large
h2_priors_resids_fun = function(h2s,n) 1,
# Function that returns the prior density for any value of the h2s vector
# (ie the vector of random effect proportional variances across all random effects.
# 1 means constant prior.
# n is the number of h2 divisions above (here=20)
# 1-n*sum(h2s)/n linearly interpolates between 1 and 0,
# giving more weight to lower values
h2_priors_factors_fun = function(h2s,n) 1,
# See above.
# sum(h2s) linearly interpolates between 0 and 1,
# giving more weight to higher values
# Another choice is one that gives 50% weight to h2==0: ifelse(h2s == 0,n,n/(n-1))
Lambda_prior = Lambda_prior
# from above
)We then assign them to the MegaLMM_state object:
MegaLMM_state = set_priors_MegaLMM(MegaLMM_state,priors)Deal with the missing data
As described above, if missing values can be grouped into line:environment sets that are 100% missing, these sets can be dropped from the model and only predicted from the posterior of other parameters. The following code attempts to find an optimal partitioning of values to maximize the number of dropped NA values in the smallest number of groups
maps = make_Missing_data_map(MegaLMM_state,max_NA_groups = ncol(Y)+1,verbose=F)
maps$map_results
#> map N_groups max_group_size total_kept_NAs
#> 1 1 1 502 6797
#> 2 2 3 327 3115
#> 3 3 4 312 2560
#> 4 4 5 296 2018
#> 5 5 6 296 1437
#> 6 6 7 296 1143
#> 7 7 8 296 869
#> 8 8 9 296 677
#> 9 9 10 296 611
#> 10 10 11 296 561
#> 11 11 12 296 525
#> 12 12 13 287 382
#> 13 13 14 287 348
#> 14 14 15 287 281
#> 15 15 16 287 215
#> 16 16 17 287 155
#> 17 17 18 287 101
#> 18 18 19 287 48
#> 19 19 20 287 0Using the 15th map above might be a good option:
MegaLMM_state = set_Missing_data_map(MegaLMM_state,maps$Missing_data_map_list[[15]])Initialize the model object
Next, we create random starting values for all parameters:
MegaLMM_state = initialize_variables_MegaLMM(MegaLMM_state)
#> [1] "initializing B_prec horseshoe"
MegaLMM_state$run_parameters$burn = run_parameters$burnInitialize parameters and posterior database
Now, we need to calculate some matrices that MegaLMM
will use repeatedly during the Gibbs sampler. These calculations can
take quite a bit of time for large models, particular when there are a
lot of individuals, more than 1 random effect, and many groups of traits
from the missing data map.
Memory usage
The stored matrices can also use a lot of RAM. It is a good idea to first get an estimate of how much RAM the model will need, before jumping in to the calculations. We can estimate the memory usage using the following function:
estimate_memory_initialization_MegaLMM(MegaLMM_state)
#> Loading required package: pryr
#> [1] "Random effects: Line"
#> [1] "16 groups of traits and 20 h2 grid cells"
#> [1] "Estimated initialized size: 0.01 B Gb"Because this dataset is small and there is only 1 random effect, the memory requirements are low.
Pre-calculate useful matrices
Now we can run these preliminary calculations:
MegaLMM_state = initialize_MegaLMM(MegaLMM_state,verbose = T)
#> [1] "Pre-calculating random effect inverse matrices for 16 groups of traits and 20 sets of random effect weights"
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As described above, the MegaLMM has many parameters, and
we could store posterior samples of all parameters. But there’s not much
use in storing large parameter arrays if we’re not actually interested
in the values of those parameters.
Also, sometimes our interest is not really in any of the
parameters, but instead in some function we can calculate from a set of
parameters. For example, we’re interested in U, the
additive genetic values, or Y the genetic values, but those
are not parameters of our MegaLMM model. We could store all
parameters and then calculated these predicted values at the end, but
that would be a waste of space.
Instead, we can control which specific parameters are stored by the program.
By default, MegaLMM stores individual posterior samples
of some parameters, and posterior means of others. You can see the list
of defaults here:
These parameters have individual samples stores:
MegaLMM_state$Posterior$posteriorSample_params
#> [1] "Lambda" "U_F" "F" "delta" "tot_F_prec"
#> [6] "F_h2" "tot_Eta_prec" "resid_h2" "B1" "B2_F"
#> [11] "B2_R" "U_R" "cis_effects" "Eta"These parameters have only posterior means stores:
MegaLMM_state$Posterior$posteriorMean_params
#> [1] "Eta_mean"Eta_mean is the internal parameter for the predicted
phenotypic value Y.
In our case, many of these values are not useful, so I’ll re-specify these lists:
MegaLMM_state$Posterior$posteriorSample_params = c('Lambda','F_h2','resid_h2','tot_Eta_prec','B1')
MegaLMM_state$Posterior$posteriorMean_params = 'Eta_mean'But we also want to calculate the predicted genetic values. From the
MegaLMM model, the predicted genetic values are the
combination of the genetic component of Y_R
(U_R), and the genetic component of F
(U_F) rotated by the factor loadings:
U = U_F * Lambda + U_R
We can ask MegaLMM to calculate this value for us and
save the posterior samples. I’ve also included code to calculate the
genetic (G) and residual (R)
covariances among environments, and the additive heritability of each
environment because they might be interesting.
MegaLMM_state$Posterior$posteriorFunctions = list(
U = 'U_F %*% Lambda + U_R + X1 %*% B1',
G = 't(Lambda) %*% diag(F_h2[1,]) %*% Lambda + diag(resid_h2[1,]/tot_Eta_prec[1,])',
R = 't(Lambda) %*% diag(1-F_h2[1,]) %*% Lambda + diag((1-resid_h2[1,])/tot_Eta_prec[1,])',
h2 = '(colSums(F_h2[1,]*Lambda^2)+resid_h2[1,]/tot_Eta_prec[1,])/(colSums(Lambda^2)+1/tot_Eta_prec[1,])'
)Now that we’ve decided which values to save, we initialize the posterior database:
MegaLMM_state = clear_Posterior(MegaLMM_state) As a final check, we should also assess how much memory the posterior samples will require.
We can estimate with the estimate_memory_posterior()
function, giving it a number of iterations we plan to run in a
single chunk (see below).
estimate_memory_posterior(MegaLMM_state,100)
#> [1] "Estimated posterior size for n_samples: 0.00404036 Gb"Since we’re not saving any large matrices, the memory requirements will be low.
Run the MCMC and diagnose convergence
We’re finally ready to fit the model! Fitting the means running the
Gibbs sampler. This is accomplished with the
sample_MegaLMM() function, which takes a
MegaLMM_state object and the number of iterations to run as
arguments. We do run the chain in two stages: burnin and
sampling.
Burnin
The burnin period is a period we wait until the chain comes to the stationary distribution. We can either wait a defined number of steps, or we can monitor convergence diagnostics, such as trace plots.
I prefer to use trace plots of the parameters that I am interested in. Yes it is not completely safe to declare stationarity until all parameters are stationary, but especially when we are only interested in the posterior mean of something like breeding values or genetic covariance, this seems to work well, and correlations across replicate runs is generally high.
While Gibbs samplers will eventually reach the stationary distribution, it is OK during the burnin phase to use some deliberate artificial jumps to push the chain into a location that likely has higher posterior mass. The one place that I’ve found this useful is in the order of the factors. Factor order is very sticky in the chain - it can take hundreds of iterations for any factor to switch. This means that factor order will never achieve a high effective sample size from this Gibbs sampler. However, I find that if I periodically check the observed importance of each factor during the burnin phase and then re-sort the factors, I achieve convergence of other parameters much more readily.
Therefore, my recommended manual burnin goes through a few rounds of:
re-order the factors
draw a set of new samples from the chain
look at some trace plots.
If they look good, clear the samples and start collecting real posterior samples
If not, repeat again.
#> Warning in cor(F): the standard deviation is zero
#> Warning in cor(F): the standard deviation is zero
#> Warning in cor(F): the standard deviation is zero
#> Warning in cor(F): the standard deviation is zeroThe function traceplot_array() saves a pdf booklet in
the MegaLMM_state$run_ID directory. To see them, navigate
to this directory in finder and look for Lambda.pdf and
U.pdf
Collect posterior samples
If we think the model is reasonable converged to stationary, we can now collect posterior samples.
Since we’re not going to collect a lot of samples, and we’re not
storing large matrices, we could do this in one run. But I’m still going
to do it in a few chunks to demonstrate the
save_posterior_chunk() function which saves the posterior
samples to the database on disk, clears the posterior samples in memory,
and continues sampling. We then load the samples we want back at the
end.
n_iter = 250
for(i in 1:4) {
print(sprintf('Sampling run %d',i))
MegaLMM_state = sample_MegaLMM(MegaLMM_state,n_iter)
MegaLMM_state = save_posterior_chunk(MegaLMM_state)
print(MegaLMM_state)
}
#> [1] "Sampling run 1"
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#> Time difference of 10.19775 secs
#>
#> Current iteration: 750, Posterior_samples: 125
#> Total time: 30.96193 secs
#>
#> [1] "Sampling run 2"
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#> Time difference of 9.970276 secs
#>
#> Current iteration: 1000, Posterior_samples: 250
#> Total time: 40.93221 secs
#>
#> [1] "Sampling run 3"
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#> Time difference of 9.727893 secs
#>
#> Current iteration: 1250, Posterior_samples: 375
#> Total time: 50.6601 secs
#>
#> [1] "Sampling run 4"
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#> Time difference of 9.599143 secs
#>
#> Current iteration: 1500, Posterior_samples: 500
#> Total time: 1.004321 minsSince our thinning rate is 2, and we run a total of 1000 sampling iterations, we end up with 500 posterior samples.
Calculate predicted values and other posterior statistics
While we’ve collected 500 posterior samples, if we actually look at
the Posterior slot of MegaLMM_state, we’ll find the
posterior is empty:
This is because the save_posterior_chunk function saves
the samples to the disk. The posterior database is in the folder:
MegaLMM_fold_01/Posterior/*. To reload samples of a
particular parameter, use:
Lambda_samples = load_posterior_param(MegaLMM_state,'Lambda')
U_samples = load_posterior_param(MegaLMM_state,'U')
dim(U_samples)
#> [1] 500 502 19We can get posterior means with the get_posterior_mean()
function:
U_hat = get_posterior_mean(U_samples)U_hat is our predicted additive genetic value for every
line in every trial.
We can also access the predicted total genetic value
Eta_mean, which we stored as a posterior mean instead of as
individual samples during the chain:
Eta_mean = load_posterior_param(MegaLMM_state,'Eta_mean')Estimate Genomic Prediction accuracy
Let’s compare the accuracy of MegaLMM's predictions
(U_hat or Eta_mean) to those of
rrBLUP:
rrBLUP_accuracy = diag(cor(Y_testing,rrBLUP_predictions,use='p'))
MegaLMM_Uhat_accuracy = diag(cor(Y_testing,U_hat,use='p'))
MegaLMM_Eta_mean_accuracy = diag(cor(Y_testing,Eta_mean,use='p'))
plot(rrBLUP_accuracy,MegaLMM_Uhat_accuracy);abline(0,1)
We see that in most trials we gained considerable accuracy through the multi-trait modeling.
We also see that because MegaLMM can also look at
non-additive-genetic covariances among lines (i.e. residual
correlations that are not explained by K but still must be
genetic), we generally gained a bit of additional accuracy.
Additional package details
You can get a summary of the MCMC chain with the print
and summary methods:
print(MegaLMM_state)
#>
#> Current iteration: 1500, Posterior_samples: 500
#> Total time: 1.004321 mins
summary(MegaLMM_state)
#> Model dimensions: factors = 15, fixed = 2, regression_R = 0, regression_F = 0, random = 502
#> Current iteration: 1500, Posterior_samples: 500
#> Total time: 1.004321 minsWorking with the posterior samples
As I mentioned above, posterior samples can be saved to the disk like this:
MegaLMM_state = save_posterior_chunk(MegaLMM_state)When you do this, you no longer have direct access to the samples
you’ve collected inside the MegaLMM_state object. Instead,
they are stored in the folder: [run_ID]/Posterior/ where
[run_ID] is the name you gave to this model run above.
dim(MegaLMM_state$Posterior$Lambda)
#> [1] 0 15 19To load all posterior samples of a particular parameter back into
MegaLMM_state so that you can work with them, you can
either call:
U = load_posterior_param(MegaLMM_state,'U')
dim(U)
#> [1] 500 502 19or you can reload all samples of all stored parameers with:
MegaLMM_state$Posterior = reload_Posterior(MegaLMM_state)
dim(MegaLMM_state$Posterior$Lambda)
#> [1] 500 15 19
dim(MegaLMM_state$Posterior$F_h2)
#> [1] 500 1 15As you can see above, we have collected 500 posterior samples. The
samples for each parameter are stored as a 3-dimensional array. All
parameters of the MegaLMM model are stored as 2-dimensional
matrices. So MegaLMM_state$Posterior\$Lambda[1,,] will
return the 1st posterior sample of the parameter Lambda, which has
dimension
in this model because K=15 and t=19. The
parameter F_h2 stores the variance component proportions
for the random effect Line for the 15 latent factors. There
is only 1 random effect, so the dimension of this matrix is
.
To assess convergence of a parameter, it’s helpful to look at traceplots. You make a traceplot of a single parameter by extracting its chain and plotting it:
plot(U[,1,2],type='l')
But it’d take a lot to make this plot for every element of every
matrix. As a shortcut, the function traceplot_array can
make lots of traceplots for a matrix parameter:
traceplot_array(MegaLMM_state$Posterior$Lambda,facet_dim = 2,name = 'Lambda')This will create a pdf booklet stored in the [run_ID]
folder (note in the next update, this will be changed to directly
use the file name provided). This will take the rows
(facet_dim=2) or columns (facet_dim=3) of the
provided parameter array and make a faceted plot, where within each
facet a sampling of the values in that row/column will be selected
(those with the largest posterior means) and traceplots will be made.
Generally it is the largest values that are the most interesting, and
will be most diagnostic of sampling issues.
Often we want to calculate summaries of the posterior samples. Two functions are provided:
U_hat = get_posterior_mean(U)
dim(U_hat)
#> [1] 502 19
U_HPD = get_posterior_HPDinterval(U,prob = 0.95)
dim(U_HPD)
#> [1] 2 502 19The latter function will calculate lower and upper 0.95 Highest
Posterior Density bounds for each element of the matrix
U.
Finally, we can calculate functions of the parameters, as long as all are stored in the Posterior database. For example, we can calculate the phenotypic covariance matrix at each posterior sample like this:
P_samples = get_posterior_FUN(MegaLMM_state,t(Lambda) %*% Lambda + diag(1/tot_Eta_prec[1,]))
dim(P_samples)
#> [1] 500 19 19Extracting random effect covariances
We’ve focused on predicting location effects of the random effects of line for each trait (yield in each environment). But we can also extract the estimates and posterior distributions on the key variance-covariance parameters and . The model for the genetic covariance in MegaLMM is:
We can calculate this using the same syntax:
G_samples = get_posterior_FUN(MegaLMM_state,
t(Lambda) %*% diag(F_h2[1,]) %*% Lambda + diag(resid_h2[,1]/tot_Eta_prec[1,])
)
dim(G_samples)
#> [1] 500 19 19But, if you look back, we actually defined this as one of the
posterior_functions we specifed in the beginning, so it’s
actually already calculated for us, and we can just load the samples of
this matrix directly:
G_samples = load_posterior_param(MegaLMM_state,'G')
R_samples = load_posterior_param(MegaLMM_state,'R')
dim(G_samples)
#> [1] 500 19 19
dim(R_samples)
#> [1] 500 19 19Extending the chain
If you look at your posterior samples and decide that the model hasn’t really converged, you can treat the current chain as an extended burnin and re-start the collection of posterior samples. I won’t run the code here so we don’t lose our current samples!
#MegaLMM_state2 = clear_Posterior(MegaLMM_state)
#print(MegaLMM_state2)You’ll see the Posterior_samples value has been set to
0.
Saving the model object
Our current model object is not that big, and so you can save it directly:
saveRDS(MegaLMM_state,file = 'MegaLMM_state_run_01.rds')And then come back and reload it and go:
MegaLMM_state = readRDS('MegaLMM_state_run_01.rds')
print(MegaLMM_state)
#>
#> Current iteration: 1500, Posterior_samples: 500
#> Total time: 1.004321 minsHowever, to re-start the sampling, all you actually need is the
initialized MegaLMM_state object with a stored
current_state slot and the posterior database in the
Posterior slot. The first slot is a list with the current
state of all parameters as well as the random number generator. This
gets automatically saved in the [run_ID] directory, so you
can re-run the setup_model_MegaLMM() function, add priors,
initialize, etc, and then reload the current_state and
resume the chain where you left off. The second is a list with the
posterior matrices and associated information that gets saved as
Posterior/Posterior_base.rds. You can load base like
this:
Convergence issues
You may have noticed in some diagnostics that parameters of
Lambda do seem to be drifting a lot, suggesting that the
chain has not converged. This suggests you should probably run this
model much longer. That is probably true. However, in my experience,
posterior distributions of location effects like U do not
change much with much longer chains - what’s happening is that the
magnitude of values of Lambda and the magnitude of
corresponding columns of F are not identified in the
likelihood, because the true term in the model is
F * Lambda. So there’s a lot of drift (poor mixing) in the
magnitudes of those parameters, ever if their product is mixing well.
Also, the order of columns of F (and rows of Lambda) is not identified
in the likelihood. The prior does provide a fairly clear ordering, but
it’s still not unusual to have factors switch order. When this happens,
Lambda[1,3] may take on the previous identity of
Lambda[2,3] and so traceplots of Lambda[1,3]
will not look good (or posterior means of this parameter. Therefore, I
advise caution interpreting Lambda.