The sommer package was developed to provide R users a powerful and reliable multivariate mixed model solver. The package is focused in problems of the type p > n (more effects to estimate than observations) and its core algorithm is coded in C++ using the Armadillo library. This package allows the user to fit mixed models with the advantage of specifying the variance-covariance structure for the random effects, and specify heterogeneous variances, and obtain other parameters such as BLUPs, BLUEs, residuals, fitted values, variances for fixed and random effects, etc.
The purpose of this vignette is to show how to fit different genotype by environment (GxE) models using the sommer package:
When the breeder decides to run a trial and apply selection in a single environment wheter because the amount of seed is a limitation or there’s no availability for any location the breeder takes the risk of selecting material for a target population of environments (TPEs) and this environment tested not being representative of the larger TPE. Therefore, many breeding programs try to based their selection decision using multi-environment trial (MET) data. Although, models could be adjusted by adding additional information like spatial information, experimental design information, etc., in this tutorial we will focus mainly on the covariance structures for GxE and the incorporation of relationship matrices for the genotype effect.
A single environment model is the one that is fitted when the breeding program can only afford one location leaving out the possible information available from other environments. This will be used to further expand to GxE models.
library(sommer)
data(DT_example)
DT <- DT_example
A <- A_example
ansSingle <- mmer(Yield~1,
random= ~ vs(Name, Gu=A),
rcov= ~ units,
data=DT)## iteration LogLik wall cpu(sec) restrained
## 1 -80.9858 21:27:32 0 0
## 2 -79.2137 21:27:32 0 0
## 3 -78.8346 21:27:32 0 0
## 4 -78.8088 21:27:32 0 0
## 5 -78.8087 21:27:32 0 0## ============================================================
## Multivariate Linear Mixed Model fit by REML
## ********************** sommer 4.1 **********************
## ============================================================
## logLik AIC BIC Method Converge
## Value -78.80875 159.6175 162.8378 NR TRUE
## ============================================================
## Variance-Covariance components:
## VarComp VarCompSE Zratio Constraint
## u:Name.Yield-Yield 6.529 2.202 2.965 Positive
## units.Yield-Yield 13.868 1.633 8.494 Positive
## ============================================================
## Fixed effects:
## Trait Effect Estimate Std.Error t.value
## 1 Yield (Intercept) 11.74 0.4876 24.07
## ============================================================
## Groups and observations:
## Yield
## u:Name 41
## ============================================================
## Use the '$' sign to access results and parametersIn this model the only term to be estimated is the one for the germplasm (here called Name). For the sake of example we have added a relationship matrices among the levels of the random effect Name. This is just a diagonal matrix with as many rows and columns as levels present in the random effect Name, but any other non-diagonal relationship matrix could be used.
A multi environment model is the one that is fitted when the breeding program can afford more than one location. This assumes that GxE doesn’t exist and that the main genotype effect plus the fixed effect for environment is enough to predict the genotype effect in all locations of interest.
## iteration LogLik wall cpu(sec) restrained
## 1 -36.8096 21:27:32 0 0
## 2 -33.211 21:27:32 0 0
## 3 -32.6234 21:27:32 0 0
## 4 -32.5942 21:27:32 0 0
## 5 -32.5942 21:27:32 0 0## ============================================================
## Multivariate Linear Mixed Model fit by REML
## ********************** sommer 4.1 **********************
## ============================================================
## logLik AIC BIC Method Converge
## Value -32.59421 71.18842 80.84949 NR TRUE
## ============================================================
## Variance-Covariance components:
## VarComp VarCompSE Zratio Constraint
## u:Name.Yield-Yield 4.856 1.5233 3.188 Positive
## units.Yield-Yield 8.109 0.9615 8.434 Positive
## ============================================================
## Fixed effects:
## Trait Effect Estimate Std.Error t.value
## 1 Yield (Intercept) 16.385 0.5849 28.012
## 2 Yield EnvCA.2012 -5.688 0.5741 -9.908
## 3 Yield EnvCA.2013 -6.218 0.6107 -10.182
## ============================================================
## Groups and observations:
## Yield
## u:Name 41
## ============================================================
## Use the '$' sign to access results and parametersA multi environment model is the one that is fitted when the breeding program can afford more than one location. This assumes that GxE doesn’t exist and that the main genotype effect plus the fixed effect for environment is enough to predict the genotype effect in all locations of interest.
## iteration LogLik wall cpu(sec) restrained
## 1 -42.26 21:27:32 0 0
## 2 -26.3735 21:27:32 0 0
## 3 -21.5756 21:27:32 0 0
## 4 -21.05 21:27:32 0 0
## 5 -21.0417 21:27:32 0 0
## 6 -21.0416 21:27:32 0 0## ============================================================
## Multivariate Linear Mixed Model fit by REML
## ********************** sommer 4.1 **********************
## ============================================================
## logLik AIC BIC Method Converge
## Value -21.04157 48.08315 57.74421 NR TRUE
## ============================================================
## Variance-Covariance components:
## VarComp VarCompSE Zratio Constraint
## CA.2011:Name.Yield-Yield 17.493 6.1099 2.863 Positive
## CA.2012:Name.Yield-Yield 5.337 1.7662 3.022 Positive
## CA.2013:Name.Yield-Yield 7.884 2.5526 3.089 Positive
## units.Yield-Yield 4.381 0.6493 6.747 Positive
## ============================================================
## Fixed effects:
## Trait Effect Estimate Std.Error t.value
## 1 Yield (Intercept) 16.621 0.948 17.532
## 2 Yield EnvCA.2012 -5.958 1.045 -5.699
## 3 Yield EnvCA.2013 -6.662 1.098 -6.067
## ============================================================
## Groups and observations:
## Yield
## CA.2011:Name 41
## CA.2012:Name 41
## CA.2013:Name 41
## ============================================================
## Use the '$' sign to access results and parametersA multi environment model is the one that is fitted when the breeding program can afford more than one location. This assumes that GxE doesn’t exist and that the main genotype effect plus the fixed effect for environment is enough to predict the genotype effect in all locations of interest.
E <- diag(length(unique(DT$Env)))
rownames(E) <- colnames(E) <- unique(DT$Env)
EA <- kronecker(E,A, make.dimnames = TRUE)
ansCS <- mmer(Yield~Env,
random= ~ vs(Name, Gu=A) + vs(Env:Name, Gu=EA),
rcov= ~ units,
data=DT)## iteration LogLik wall cpu(sec) restrained
## 1 -31.2668 21:27:32 0 0
## 2 -23.2804 21:27:32 0 0
## 3 -20.4746 21:27:32 0 0
## 4 -20.1501 21:27:32 0 0
## 5 -20.1454 21:27:32 0 0
## 6 -20.1454 21:27:32 0 0## ============================================================
## Multivariate Linear Mixed Model fit by REML
## ********************** sommer 4.1 **********************
## ============================================================
## logLik AIC BIC Method Converge
## Value -20.14538 46.29075 55.95182 NR TRUE
## ============================================================
## Variance-Covariance components:
## VarComp VarCompSE Zratio Constraint
## u:Name.Yield-Yield 3.682 1.691 2.177 Positive
## u:Env:Name.Yield-Yield 5.173 1.495 3.460 Positive
## units.Yield-Yield 4.366 0.647 6.748 Positive
## ============================================================
## Fixed effects:
## Trait Effect Estimate Std.Error t.value
## 1 Yield (Intercept) 16.496 0.6855 24.065
## 2 Yield EnvCA.2012 -5.777 0.7558 -7.643
## 3 Yield EnvCA.2013 -6.380 0.7960 -8.015
## ============================================================
## Groups and observations:
## Yield
## u:Name 41
## u:Env:Name 123
## ============================================================
## Use the '$' sign to access results and parametersA multi environment model is the one that is fitted when the breeding program can afford more than one location. This assumes that GxE doesn’t exist and that the main genotype effect plus the fixed effect for environment is enough to predict the genotype effect in all locations of interest.
ansMain <- mmer(Yield~Env,
random= ~ vs(Name, Gu=A) + vs(ds(Env),Name, Gu=A),
rcov= ~ units,
data=DT)## iteration LogLik wall cpu(sec) restrained
## 1 -31.2668 21:27:32 0 0
## 2 -21.0887 21:27:32 0 0
## 3 -18.4752 21:27:32 0 0
## 4 -18.1673 21:27:32 0 0
## 5 -18.1618 21:27:32 0 0
## 6 -18.1616 21:27:32 0 0## ============================================================
## Multivariate Linear Mixed Model fit by REML
## ********************** sommer 4.1 **********************
## ============================================================
## logLik AIC BIC Method Converge
## Value -18.16164 42.32327 51.98434 NR TRUE
## ============================================================
## Variance-Covariance components:
## VarComp VarCompSE Zratio Constraint
## u:Name.Yield-Yield 2.965 1.5055 1.969 Positive
## CA.2011:Name.Yield-Yield 10.424 4.4544 2.340 Positive
## CA.2012:Name.Yield-Yield 2.658 1.8032 1.474 Positive
## CA.2013:Name.Yield-Yield 5.702 2.5113 2.271 Positive
## units.Yield-Yield 4.398 0.6517 6.748 Positive
## ============================================================
## Fixed effects:
## Trait Effect Estimate Std.Error t.value
## 1 Yield (Intercept) 16.511 0.8269 19.967
## 2 Yield EnvCA.2012 -5.809 0.8593 -6.760
## 3 Yield EnvCA.2013 -6.423 0.9358 -6.864
## ============================================================
## Groups and observations:
## Yield
## u:Name 41
## CA.2011:Name 41
## CA.2012:Name 41
## CA.2013:Name 41
## ============================================================
## Use the '$' sign to access results and parametersA multi environment model is the one that is fitted when the breeding program can afford more than one location. This assumes that GxE doesn’t exist and that the main genotype effect plus the fixed effect for environment is enough to predict the genotype effect in all locations of interest.
## iteration LogLik wall cpu(sec) restrained
## 1 -37.9059 21:27:32 0 0
## 2 -19.0506 21:27:32 0 0
## 3 -14.6786 21:27:32 0 0
## 4 -14.2203 21:27:32 0 0
## 5 -14.2098 21:27:32 0 0
## 6 -14.2095 21:27:32 0 0## ==================================================================
## Multivariate Linear Mixed Model fit by REML
## ************************* sommer 4.1 *************************
## ==================================================================
## logLik AIC BIC Method Converge
## Value -14.20951 34.41901 44.08008 NR TRUE
## ==================================================================
## Variance-Covariance components:
## VarComp VarCompSE Zratio Constraint
## CA.2011:Name.Yield-Yield 15.994 5.381 2.972 Positive
## CA.2012:CA.2011:Name.Yield-Yield 6.172 2.503 2.465 Unconstr
## CA.2012:Name.Yield-Yield 5.273 1.750 3.013 Positive
## CA.2013:CA.2011:Name.Yield-Yield 6.366 3.069 2.074 Unconstr
## CA.2013:CA.2012:Name.Yield-Yield 0.376 1.535 0.245 Unconstr
## CA.2013:Name.Yield-Yield 7.689 2.490 3.088 Positive
## units.Yield-Yield 4.386 0.650 6.748 Positive
## ==================================================================
## Fixed effects:
## Trait Effect Estimate Std.Error t.value
## 1 Yield (Intercept) 16.341 0.8141 20.072
## 2 Yield EnvCA.2012 -5.696 0.7406 -7.692
## 3 Yield EnvCA.2013 -6.286 0.8202 -7.664
## ==================================================================
## Groups and observations:
## Yield
## CA.2011:Name 41
## CA.2012:CA.2011:Name 82
## CA.2012:Name 41
## CA.2013:CA.2011:Name 82
## CA.2013:CA.2012:Name 82
## CA.2013:Name 41
## ==================================================================
## Use the '$' sign to access results and parametersA multi environment model is the one that is fitted when the breeding program can afford more than one location. This assumes that GxE doesn’t exist and that the main genotype effect plus the fixed effect for environment is enough to predict the genotype effect in all locations of interest.
## Loading required package: polynomDT$EnvN <- as.numeric(as.factor(DT$Env))
ansRR <- mmer(Yield~Env,
random= ~ vs(leg(EnvN,1),Name),
rcov= ~ units,
data=DT)## iteration LogLik wall cpu(sec) restrained
## 1 -40.232 21:27:32 0 0
## 2 -29.2803 21:27:32 0 0
## 3 -27.8646 21:27:32 0 0
## 4 -27.7107 21:27:32 0 0
## 5 -27.7036 21:27:32 0 0
## 6 -27.7032 21:27:32 0 0## ============================================================
## Multivariate Linear Mixed Model fit by REML
## ********************** sommer 4.1 **********************
## ============================================================
## logLik AIC BIC Method Converge
## Value -27.70318 61.40636 71.06743 NR TRUE
## ============================================================
## Variance-Covariance components:
## VarComp VarCompSE Zratio Constraint
## leg0:Name.Yield-Yield 10.392 3.1473 3.302 Positive
## leg1:Name.Yield-Yield 2.079 0.9792 2.123 Positive
## units.Yield-Yield 6.297 0.8442 7.459 Positive
## ============================================================
## Fixed effects:
## Trait Effect Estimate Std.Error t.value
## 1 Yield (Intercept) 16.541 0.6770 24.432
## 2 Yield EnvCA.2012 -5.832 0.6425 -9.078
## 3 Yield EnvCA.2013 -6.472 0.8239 -7.854
## ============================================================
## Groups and observations:
## Yield
## leg0:Name 41
## leg1:Name 41
## ============================================================
## Use the '$' sign to access results and parametersA multi environment model is the one that is fitted when the breeding program can afford more than one location. This assumes that GxE doesn’t exist and that the main genotype effect plus the fixed effect for environment is enough to predict the genotype effect in all locations of interest.
E <- CS(DT$Env)
rownames(E) <- colnames(E) <- unique(DT$Env)
EA <- kronecker(E,A, make.dimnames = TRUE)
ansCS <- mmer(Yield~Env,
random= ~ vs(Name, Gu=A) + vs(Env:Name, Gu=EA),
rcov= ~ units,
data=DT)## iteration LogLik wall cpu(sec) restrained
## 1 -31.1056 21:27:33 0 0
## 2 -23.39 21:27:33 0 0
## 3 -20.4917 21:27:33 0 0
## 4 -20.1504 21:27:33 0 0
## 5 -20.1454 21:27:33 0 0
## 6 -20.1454 21:27:33 0 0## ============================================================
## Multivariate Linear Mixed Model fit by REML
## ********************** sommer 4.1 **********************
## ============================================================
## logLik AIC BIC Method Converge
## Value -20.14538 46.29075 55.95182 NR TRUE
## ============================================================
## Variance-Covariance components:
## VarComp VarCompSE Zratio Constraint
## u:Name.Yield-Yield 1.958 1.932 1.013 Positive
## u:Env:Name.Yield-Yield 6.897 1.994 3.460 Positive
## units.Yield-Yield 4.366 0.647 6.748 Positive
## ============================================================
## Fixed effects:
## Trait Effect Estimate Std.Error t.value
## 1 Yield (Intercept) 16.496 0.6855 24.065
## 2 Yield EnvCA.2012 -5.777 0.7558 -7.643
## 3 Yield EnvCA.2013 -6.380 0.7960 -8.015
## ============================================================
## Groups and observations:
## Yield
## u:Name 41
## u:Env:Name 123
## ============================================================
## Use the '$' sign to access results and parametersKeep in mind that sommer uses direct inversion (DI) algorithm which can be very slow for large datasets. The package is focused in problems of the type p > n (more random effect levels than observations) and models with dense covariance structures. For example, for experiment with dense covariance structures with low-replication (i.e. 2000 records from 1000 individuals replicated twice with a covariance structure of 1000x1000) sommer will be faster than MME-based software. Also for genomic problems with large number of random effect levels, i.e. 300 individuals (n) with 100,000 genetic markers (p). For highly replicated trials with small covariance structures or n > p (i.e. 2000 records from 200 individuals replicated 10 times with covariance structure of 200x200) asreml or other MME-based algorithms will be much faster and we recommend you to opt for those software.
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