Quantitative genetics using the sommer package
Giovanny Covarrubias-Pazaran
2017-07-08
The sommer package was developed to provide R users with a powerful multivariate mixed model solver for different genetic and non-genetic analysis in diploid and polyploid organisms. This package allows the user to estimate variance components for a mixed model with the advantage of specifying the variance-covariance structure of the random effects and obtain other parameters such as BLUPs, BLUEs, residuals, fitted values, variances for fixed and random effects, etc.
The package is focused on genomic prediction (hybrid prediction & genomic selection) and GWAS analysis, although general mixed models can be fitted as well. The package provides kernels to estimate additive (A.mat), dominance (D.mat), and epistatic (E.mat) relationship matrices that have been shown to increase prediction accuracy under certain scenarios. The package provides flexibility to fit other genetic models such as full and half diallel models as well.
Vignettes aim to provide several examples in how to use the sommer package under different scenarios in breeding and genetics. We will spend the rest of the space providing examples for:
- Heritability (\(h^2\)) calculation
- Specifying heterogeneous variances in univariate mixed models
- Using the pin calculator
- Half and full diallel designs
- Genome wide association analysis (GWAS) in diploids and tetraploids
- Genomic selection
- Single cross prediction
- Multivariate genetic models and genetic correlations
- Multivariate GWAS
Background
The core of the package are the mmer (matrix-based) and mmer2 (formula-based) functions which solve the mixed model equations. The functions are an interface to call one of the 4 ML/REML methods supported in the package; EMMA efficient mixed model association (Kang et al. 2008), AI Direct-Inversion Average Information (Gilmour et al. 1995; Lee et al. 2015), EM expectation maximization (Searle 1993; Bernardo 2010), and the default NR Direct-Inversion Newton-Raphson (Tunnicliffe 1989). All methods can handle multiple random effects and covariance structures and multivariate (multiple responses) mixed models.
The univariate mixed models that sommer can solve have the form:
\(y = X\beta+Zu+\epsilon\)
or
\(y = X\beta+Zu_1+...+Zu_i+\epsilon\)
where:
X is an incidence matrix for fixed effects
Z is an incidence matrix for random effects
\(\beta\) is the vector for BLUEs of fixed effects
u is the vector for BLUPs of random effects
\(\epsilon\) are the residuals
The variance of the response is known to be the random part of the model:
Var(y) = Var(\(Zu+\epsilon\)) = ZGZ + R = V
and with
u ~ MVN(u, G)
\(\epsilon\) ~ MVN(u, R)
When multiple random effects are present the Z matrix becomes the column binding of each of the \(Z_i\) matrices for the i random effects. And the G matrix becomes the diagonal binding of each of the variance covariance structures (K matrices) for the random effects:
\[\mathbf{Z} = \left[\begin{array}
{rrr}
Z_1 & ... & Z_i \\
\end{array}\right]
\]
\[\mathbf{G} = \left[\begin{array}
{rrr}
K_1 \sigma^2_u1 & 0 & 0 \\
0 & ... & 0 \\
0 & 0 & K_i \sigma^2_ui
\end{array}\right]
\]
Since version 2.0 sommer can handle multivariate models. These have the form:
\(Y = X\beta + Zu + \epsilon\)
with:
\[\mathbf{Y} = \left[\begin{array}
{r}
y_1 \\
y_2 \\
... \\
y_t \\
\end{array}\right]
\]
\[\mathbf{X} = \left[\begin{array}
{rrr}
X & ... & ... \\
... & ... & ...\\
... & ... & X \\
\end{array}\right]
\]
\[\mathbf{V} = \left[\begin{array}
{rrr}
Z_1 G_1 Z_1' + ... + Z_1 R_1 Z_1' & ... & Z_1 H_1 Z_t' + ... + Z_1 S_1 Z_t' \\
... & ... & ...\\
Z_t H_1 Z_1' + ... + Z_t S_1 Z_1' & ... & Z_t G_1 Z_t' + ... + Z_t R_1 Z_t' \\
\end{array}\right]
\]
for ‘t’ traits, where G are H are variance and covariance matrices among random effects for the “t” trait, and R and S are variance and covariance matrices among residuals. Here R=S=I\(\sigma_\epsilon\) , where I is an identity matrix. We can specify the covariance matrices. BLUPs will also be corrected for such covariances usually leading to more accurate predictions.
The function mmer takes the Zs and Ks for each random effect and construct the neccesary structure inside and estimates the variance components by ML/REML using any of the 4 methods available in sommer. The mmer2 function is enabled to work in a model-based fashion so user don’t have to build the Z’s and K matrices. Please refer to the canonical papers listed in the Literature section to check how the algorithms work. We have tested widely the methods to make sure they provide the same solution when the likelihood behaves well but for complex problems they might lead to slightly different answers. If you have any concern please contact me at cova_ruber@live.com.mx.
In the following section we will go in detail over several examples on how to use mixed models in univariate and multivariate case and their use in quantitative genetics.
1) Marker and non-marker based heritability calculation
The heritability is one of the most popular parameters in the breeding and genetics community. The heritability is usually estimated as narrow sense (\(h^2\); only additive variance in the numerator \(\sigma^2_A\)), and broad sense (\(H^2\); all genetic variance in the numerator \(\sigma^2_G\)).
In a classical experiment with no molecular markers, special designs are performed to estimate and disect the additive (\(\sigma^2_A\)) and dominance (\(\sigma^2_D\)) variance along with environmental variability. Designs such as generation analysis, North Carolina designs are used to disect \(\sigma^2_A\) and \(\sigma^2_D\) to estimate the narrow sense heritability (\(h^2\)). When no special design is available we can still disect the genetic variance (\(\sigma^2_G\)) and estimate the broad sense heritability. In this example we will show the broad sense estimation which doesn’t use covariance structures for random effects. For big models with no covariance structures, sommer’s direct inversion is a bad idea to use but we will show anyways how to do it, for very sparse models we recommend using the lmer function from the lme4 package from Douglas Bates or change to the EM algorithm which uses MME-based algorithms.
The dataset has 41 potato lines evaluated in 5 locations across 3 years in an RCBD design. We show how to fit the model and extract the variance components to calculate the \(h^2\).
## Loading required package: Matrix
## Warning: package 'Matrix' was built under R version 3.2.5
## Loading required package: MASS
## Warning: package 'MASS' was built under R version 3.2.2
## Loading required package: parallel
## Name Env Loc Year Block y
## 1 W8822-3 FL.2012 FL 2012 FL.2012.1 2
## 2 W8867-7 FL.2012 FL 2012 FL.2012.2 2
## 3 MSL007-B MO.2011 MO 2011 MO.2011.1 3
## 4 CO00270-7W FL.2012 FL 2012 FL.2012.2 3
## 5 Manistee(MSL292-A) FL.2013 FL 2013 FL.2013.2 3
## 6 MSM246-B FL.2012 FL 2012 FL.2012.2 3
ans1 <- mmer2(y~1, random=~Name + Env + Name:Env + Block,data=h2, silent = TRUE)
vc <- ans1$var.comp
V_E <- vc[2,1];V_GE <- vc[3,1];V_G <- vc[1,1];Ve <- vc[5,1]
n.env <- length(levels(h2$Env))
h2 <- V_G/(V_G + V_GE/n.env + Ve/(2*n.env)) #the 2 is a reference for block
h2
## [1] 0.8594795
The same model can be fitted with the mmer function that is actually used by the mmer2 function in the background. This is just to show that you can create your customized matrices and use the mixed model solver. This is how you would do it:
library(sommer)
data(h2)
head(h2)
## Name Env Loc Year Block y
## 1 W8822-3 FL.2012 FL 2012 FL.2012.1 2
## 2 W8867-7 FL.2012 FL 2012 FL.2012.2 2
## 3 MSL007-B MO.2011 MO 2011 MO.2011.1 3
## 4 CO00270-7W FL.2012 FL 2012 FL.2012.2 3
## 5 Manistee(MSL292-A) FL.2013 FL 2013 FL.2013.2 3
## 6 MSM246-B FL.2012 FL 2012 FL.2012.2 3
Z1 <- model.matrix(~Name-1, h2)
Z2 <- model.matrix(~Env-1, h2)
Z3 <- model.matrix(~Env:Name-1, h2)
Z4 <- model.matrix(~Block-1, h2)
ETA <- list(name=list(Z=Z1),env=list(Z=Z2),name.env=list(Z=Z3),block=list(Z=Z4))
y <- h2$y
ans1 <- mmer(Y=y, Z=ETA, silent = TRUE)
vc <- ans1$var.comp
Recently with markers becoming cheaper, thousand of markers can be run in the breeding materials. When markers are available, an special design is not neccesary to disect the additive genetic variance. The availability of the additive, dominance and epistatic relationship matrices allow us to estimate \(\sigma^2_A\), \(\sigma^2_D\) and \(\sigma^2_I\).
Assume you have a population, and a similar model like the one displayed previously has been fitted. Now we have BLUPs for the genotypes but in addition we have genetic markers.
data(CPdata)
CPpheno <- CPdata$pheno; CPpheno$idd <-CPpheno$id; CPpheno$ide <-CPpheno$id
CPgeno <- CPdata$geno
### look at the data
head(CPpheno)
## id Row Col Year color Yield FruitAver Firmness idd ide
## P003 P003 3 1 2014 0.10075269 154.67 41.93 588.917 P003 P003
## P004 P004 4 1 2014 0.13891940 186.77 58.79 640.031 P004 P004
## P005 P005 5 1 2014 0.08681502 80.21 48.16 671.523 P005 P005
## P006 P006 6 1 2014 0.13408561 202.96 48.24 687.172 P006 P006
## P007 P007 7 1 2014 0.13519278 174.74 45.83 601.322 P007 P007
## P008 P008 8 1 2014 0.17406685 194.16 44.63 656.379 P008 P008
## scaffold_50439_2381 scaffold_39344_153 uneak_3436043 uneak_2632033
## P003 0 0 0 1
## P004 0 0 0 1
## P005 0 -1 0 1
## P006 -1 -1 -1 0
## P007 0 0 0 1
## fit a model including additive and dominance effects
A <- A.mat(CPgeno) # additive relationship matrix
D <- D.mat(CPgeno) # dominance relationship matrix
E <- E.mat(CPgeno) # epistatic relationship matrix
ans.ADE <- mmer2(color~1, random=~g(id) + g(idd) + g(ide),
G=list(id=A,idd=D,ide=E), silent = TRUE, data=CPpheno)
(H2 <- sum(ans.ADE$var.comp[1:3,1])/sum(ans.ADE$var.comp[,1]))
## [1] 0.7512293
(h2 <- sum(ans.ADE$var.comp[1,1])/sum(ans.ADE$var.comp[,1]))
## [1] 0.6215569
In the previous example we showed how to estimate the additive (\(\sigma^2_A\)), dominance (\(\sigma^2_D\)), and epistatic (\(\sigma^2_I\)) variance components based on markers and estimate broad (\(H^2\)) and narrow sense heritability (\(h^2\)). Notice that we used the g() function which indicates that the random effect inside the parenthesis (i.e. id, idd or ide) has a covariance matrix (A, D, or E), that will be specified in the G argument in the form of a list and using the name of the random efect to allow the program to recognize which variance covariance matrix belongs to each random effect. Please DO NOT provide the inverse but the original covariance matrix. This is why we have called the function g() and no giv() as the popular software asreml.
Just to show one more time that you can use your own matrices we will repeat the same calculation using the mmer function:
data(CPdata)
CPgeno <- CPdata$geno
### look at the data
head(CPpheno)
## id Row Col Year color Yield FruitAver Firmness idd ide
## P003 P003 3 1 2014 0.10075269 154.67 41.93 588.917 P003 P003
## P004 P004 4 1 2014 0.13891940 186.77 58.79 640.031 P004 P004
## P005 P005 5 1 2014 0.08681502 80.21 48.16 671.523 P005 P005
## P006 P006 6 1 2014 0.13408561 202.96 48.24 687.172 P006 P006
## P007 P007 7 1 2014 0.13519278 174.74 45.83 601.322 P007 P007
## P008 P008 8 1 2014 0.17406685 194.16 44.63 656.379 P008 P008
## scaffold_50439_2381 scaffold_39344_153 uneak_3436043 uneak_2632033
## P003 0 0 0 1
## P004 0 0 0 1
## P005 0 -1 0 1
## P006 -1 -1 -1 0
## P007 0 0 0 1
## fit a model including additive and dominance effects
Z1 <- model.matrix(~id-1, CPpheno); colnames(Z1) <- gsub("id","",colnames(Z1))
A <- A.mat(CPgeno) # additive relationship matrix
D <- D.mat(CPgeno) # dominance relationship matrix
E <- E.mat(CPgeno) # epistatic relationship matrix
y <- CPpheno$color
ETA <- list(id=list(Z=Z1,K=A),idd=list(Z=Z1,K=D),ide=list(Z=Z1,K=E))
ans.ADE <- mmer(Y=y, Z=ETA, silent = TRUE)
ans.ADE$var.comp
## component constraint
## Var(id) 0.005200536 Positive
## Var(idd) 0.001084963 Positive
## Var(ide) 0.000000000 Boundary
## Var(units) 0.002081452 Positive
2) Specifying heterogeneous variances in univariate models
Very often in multi-environment trials, the assumption that genetic variance is the same across locations may be too naive. Because of that, specifying a general genetic component and a location specific genetic variance is the way to go. Although the function ‘mmer’ implemented in sommer can be used to do that, can be quite cumbersome and messy to create the incidence and variance covariance matrices for fitting those models. For that reason the function ‘mmer2’ was added to the package to make such models easier to fit.
We estimate variance components for \(GCA_2\) and \(SCA\) specifying the variance structure.
data(cornHybrid)
hybrid2 <- cornHybrid$hybrid # extract cross data
head(hybrid2)
## Location GCA1 GCA2 SCA Yield PlantHeight
## 1 1 A258 AS5707 A258:AS5707 NA NA
## 2 1 A258 B2 A258:B2 NA NA
## 3 1 A258 B99 A258:B99 NA NA
## 4 1 A258 LH51 A258:LH51 NA NA
## 5 1 A258 Mo44 A258:Mo44 NA NA
## 6 1 A258 NC320 A258:NC320 NA NA
### fit the model
modFD <- mmer2(Yield~1, random=~ at(Location,c("3","4")):GCA2,
rcov= ~ at(Location):units,
data=hybrid2, silent = TRUE)
summary(modFD)
##
## Information contained in this fitted model:
## * Variance components, Residuals, Fitted values
## * BLUEs and BLUPs, Inverse phenotypic variance(V)
## * Variance-covariance matrix for fixed & random effects
## * Predicted error variance (PEV), LogLikelihood
## Use the '$' symbol to access such information
## ===========================================================
## Linear mixed model fit by restricted maximum likelihood
## ********************** sommer 2.9 ***********************
## ===========================================================
## logLik AIC BIC Method Conv DI
## Params -1382.801 2767.602 2771.594 NR TRUE TRUE
## ===========================================================
## Random effects:
## VarComp VarCompSE Zratio constraint
## Var(at(Location,c("3","4"))3:GCA2) 62.41 53.39 1.169 Positive
## Var(at(Location,c("3","4"))4:GCA2) 98.02 79.59 1.232 Positive
## Var(1:units) 216.82 30.76 7.048 Positive
## Var(2:units) 216.82 30.76 7.048 Positive
## Var(3:units) 493.08 77.29 6.380 Positive
## Var(4:units) 711.98 111.64 6.378 Positive
## Number of obs: 400 Groups: 20 20
## ===========================================================
## Fixed effects:
## Value Std.Error t.value
## Intercept 138.11278 0.94417 146.28
## ===========================================================
## Use the '$' symbol to access all information
In the previous example we showed how the at function is used in the mmer2 solver. By using the at function you can specify that i.e. the GCA2 has a different variance in different Locations, in this case locations 3 and 4, but also a main GCA variance. This is considered a CS + DIAG (compound symmetry + diagonal) model.
In addition, other functions can be added on top to fit models with covariance structures, i.e. the g() function which indicates that the random effect inside the parenthesis (i.e. GCA2) has a covariance matrix (A, pedigree or genomic relationship matrix) that will be specified in the G argument in the form of a list:
data(cornHybrid)
hybrid2 <- cornHybrid$hybrid # extract cross data
## get the covariance structure for GCA2
A <- cornHybrid$K
## fit the model
modFD <- mmer2(Yield~1, random=~ g(GCA2) + at(Location):g(GCA2),
rcov= ~ at(Location):units,
data=hybrid2, G=list(GCA2=A),
silent = TRUE, draw=FALSE)
summary(modFD)
##
## Information contained in this fitted model:
## * Variance components, Residuals, Fitted values
## * BLUEs and BLUPs, Inverse phenotypic variance(V)
## * Variance-covariance matrix for fixed & random effects
## * Predicted error variance (PEV), LogLikelihood
## Use the '$' symbol to access such information
## ===========================================================
## Linear mixed model fit by restricted maximum likelihood
## ********************** sommer 2.9 ***********************
## ===========================================================
## logLik AIC BIC Method Conv DI
## Params -1375.837 2753.673 2757.665 NR TRUE TRUE
## ===========================================================
## Random effects:
## VarComp VarCompSE Zratio constraint
## Var(g(GCA2)) 28.287 13.85 2.0424 Positive
## Var(at(Location)1:g(GCA2)) 0.000 8.80 0.0000 Boundary
## Var(at(Location)2:g(GCA2)) 0.000 8.80 0.0000 Boundary
## Var(at(Location)3:g(GCA2)) 3.936 19.72 0.1996 Positive
## Var(at(Location)4:g(GCA2)) 10.039 28.35 0.3541 Positive
## Var(1:units) 187.902 30.26 6.2089 Positive
## Var(2:units) 187.902 30.26 6.2089 Positive
## Var(3:units) 497.055 72.36 6.8690 Positive
## Var(4:units) 727.276 105.90 6.8677 Positive
## Number of obs: 400 Groups: 20 20 20 20 20
## ===========================================================
## Fixed effects:
## Value Std.Error t.value
## Intercept 138.3389 1.3246 104.44
## ===========================================================
## Use the '$' symbol to access all information
The draw argument allows you to see the progress of the likelihood and the change of the variance components, we just mention it in case you like to do that inspection but this will make the fitting process more time consuming.
3) Using the pin calculator
Sometimes the user needs to calculate ratios or functions of specific variance-covariance components and obtain the standard error for such parameters. Examples of these are the genetic correlations, heritabilities, etc. Using the CPdata we will show how to estimate the heritability and the standard error.
data(CPdata)
CPpheno <- CPdata$pheno
CPgeno <- CPdata$geno
#### create the variance-covariance matrix
A <- A.mat(CPgeno)
#### look at the data and fit the model
head(CPpheno)
## id Row Col Year color Yield FruitAver Firmness
## P003 P003 3 1 2014 0.10075269 154.67 41.93 588.917
## P004 P004 4 1 2014 0.13891940 186.77 58.79 640.031
## P005 P005 5 1 2014 0.08681502 80.21 48.16 671.523
## P006 P006 6 1 2014 0.13408561 202.96 48.24 687.172
## P007 P007 7 1 2014 0.13519278 174.74 45.83 601.322
## P008 P008 8 1 2014 0.17406685 194.16 44.63 656.379
mix1 <- mmer2(color~1,random=~g(id), G=list(id=A), data=CPpheno, silent=TRUE)
summary(mix1)
##
## Information contained in this fitted model:
## * Variance components, Residuals, Fitted values
## * BLUEs and BLUPs, Inverse phenotypic variance(V)
## * Variance-covariance matrix for fixed & random effects
## * Predicted error variance (PEV), LogLikelihood
## Use the '$' symbol to access such information
## ===========================================================
## Linear mixed model fit by restricted maximum likelihood
## ********************** sommer 2.9 ***********************
## ===========================================================
## logLik AIC BIC Method Conv DI
## Params 790.5537 -1579.107 -1575.216 NR TRUE TRUE
## ===========================================================
## Random effects:
## VarComp VarCompSE Zratio constraint
## Var(g(id)) 0.005123 0.0010394 4.929 Positive
## Var(units) 0.002743 0.0003002 9.137 Positive
## Number of obs: 362 Groups: 363
## ===========================================================
## Fixed effects:
## Value Std.Error t.value
## Intercept 0.182565 0.002755 66.267
## ===========================================================
## Use the '$' symbol to access all information
#### run the pin function
pin(mix1, h2 ~ V1 / ( V1 + V2 ) )
## Estimate SE
## h2 0.6512661 0.06109021
The same can be used for multivariate models. Please check the documentation of the pin function to see more examples.
4) Half and full diallel designs
When breeders are looking for the best single cross combinations, diallel designs have been by far the most used design in crops like maize. There are 4 types of diallel designs depending if reciprocate and self cross (omission of parents) are performed (full diallel with parents n^2; full diallel without parents n(n-1); half diallel with parents 1/2 * n(n+1); half diallel without parents 1/2 * n(n-1) ). In this example we will show a full dialle design (reciprocate crosses are performed) and half diallel designs (only one of the directions is performed).
In the first data set we show a full diallel among 40 lines from 2 heterotic groups, 20 in each. Therefore 400 possible hybrids are possible. We have pehnotypic data for 100 of them across 4 locations. We use the data available to fit a model of the form:
\(y = X\beta + Zu_1 + Zu_2 + Zu_S + \epsilon\)
We estimate variance components for \(GCA_1\), \(GCA_2\) and \(SCA\) and use them to estimate heritability. Additionally BLUPs for GCA and SCA effects can be used to predict crosses.
data(cornHybrid)
hybrid2 <- cornHybrid$hybrid # extract cross data
head(hybrid2)
## Location GCA1 GCA2 SCA Yield PlantHeight
## 1 1 A258 AS5707 A258:AS5707 NA NA
## 2 1 A258 B2 A258:B2 NA NA
## 3 1 A258 B99 A258:B99 NA NA
## 4 1 A258 LH51 A258:LH51 NA NA
## 5 1 A258 Mo44 A258:Mo44 NA NA
## 6 1 A258 NC320 A258:NC320 NA NA
modFD <- mmer2(Yield~Location, random=~GCA1+GCA2+SCA, data=hybrid2,silent = TRUE, draw=FALSE)
summary(modFD)
##
## Information contained in this fitted model:
## * Variance components, Residuals, Fitted values
## * BLUEs and BLUPs, Inverse phenotypic variance(V)
## * Variance-covariance matrix for fixed & random effects
## * Predicted error variance (PEV), LogLikelihood
## Use the '$' symbol to access such information
## ===========================================================
## Linear mixed model fit by restricted maximum likelihood
## ********************** sommer 2.9 ***********************
## ===========================================================
## logLik AIC BIC Method Conv DI
## Params -1341.685 2691.37 2707.336 NR TRUE TRUE
## ===========================================================
## Random effects:
## VarComp VarCompSE Zratio constraint
## Var(GCA1) 0.00 17.66 0.0000 Boundary
## Var(GCA2) 7.19 19.30 0.3725 Positive
## Var(SCA) 187.75 44.68 4.2016 Positive
## Var(units) 221.14 17.78 12.4386 Positive
## Number of obs: 400 Groups: 20 20 400
## ===========================================================
## Fixed effects:
## Value Std.Error t.value
## (Intercept) 1.3793e+02 2.1189e+00 65.0960
## Location2 3.1974e-13 2.1030e+00 0.0000
## Location3 7.8353e+00 2.1030e+00 3.7257
## Location4 -9.0975e+00 2.1030e+00 -4.3259
## ===========================================================
## Use the '$' symbol to access all information
Vgca <- sum(modFD$var.comp[1:2,1])
Vsca <- modFD$var.comp[3,1]
Ve <- modFD$var.comp[4,1]
Va = 4*Vgca
Vd = 4*Vsca
Vg <- Va + Vd
(H2 <- Vg / (Vg + (Ve)) )
## [1] 0.7790562
(h2 <- Va / (Vg + (Ve)) )
## [1] 0.02873332
Don’t worry too much about the small h2 value, the data was simulated to be mainly dominance variance, therefore the Va was simulated extremely small leading to such value of narrow sense h2.
In this second data set we show a small half diallel with 7 parents crossed in one direction. n(n-1)/2 crosses are possible 7(6)/2 = 21 unique crosses. Parents appear as males or females indistictly. Each with two replications in a CRD. For a half diallel design a single GCA variance component for both males and females can be estimated and an SCA as well (\(\sigma^2_GCA\) and \(\sigma^2_SCA\) respectively), and BLUPs for GCA and SCA of the parents can be extracted. We would show first how to use it with the mmer2 function using the and() function and later we will show how to do it creating customized matrices using the overlay and model.matrix functions for the GCA and SCA matrices respectively. The specific model here is:
\(y = X\beta + Zu_g + Zu_s + \epsilon\)
data(HDdata)
head(HDdata)
## rep geno male female sugar
## 1 1 12 1 2 13.950509
## 2 2 12 1 2 9.756918
## 3 1 13 1 3 13.906355
## 4 2 13 1 3 9.119455
## 5 1 14 1 4 5.174483
## 6 2 14 1 4 8.452221
HDdata$geno <- as.factor(HDdata$geno)
HDdata$male <- as.factor(HDdata$male)
HDdata$female <- as.factor(HDdata$female)
# Fit the model
modHD <- mmer2(sugar~1, random=~male + and(female) + geno,
data=HDdata, silent = TRUE)
summary(modHD)
##
## Information contained in this fitted model:
## * Variance components, Residuals, Fitted values
## * BLUEs and BLUPs, Inverse phenotypic variance(V)
## * Variance-covariance matrix for fixed & random effects
## * Predicted error variance (PEV), LogLikelihood
## Use the '$' symbol to access such information
## ===========================================================
## Linear mixed model fit by restricted maximum likelihood
## ********************** sommer 2.9 ***********************
## ===========================================================
## logLik AIC BIC Method Conv DI
## Params -58.1779 118.3558 120.0935 NR TRUE TRUE
## ===========================================================
## Random effects:
## VarComp VarCompSE Zratio constraint
## Var(and(female)) 5.509 3.579 1.539 Positive
## Var(geno) 1.816 1.363 1.332 Positive
## Var(units) 3.117 0.962 3.240 Positive
## Number of obs: 42 Groups: 7 21
## ===========================================================
## Fixed effects:
## Value Std.Error t.value
## Intercept 10.3332 1.8189 5.6811
## ===========================================================
## Use the '$' symbol to access all information
Vgca <- modHD$var.comp[1,1]
Vsca <- modHD$var.comp[2,1]
Ve <- modHD$var.comp[3,1]
Va = 4*Vgca
Vd = 4*Vsca
Vg <- Va + Vd
(H2 <- Vg / (Vg + (Ve/2)) ) # 2 technical reps
## [1] 0.9494881
(h2 <- Va / (Vg + (Ve/2)) )
## [1] 0.7140855
Notice how the and() argument makes the overlay possible making sure that male and female are joint into a single random effect. The same can be done using the mmer argument by creating the incidence and covariance matrices in case you want to see what is doing mmer2 in the background.
data(HDdata)
head(HDdata)
## rep geno male female sugar
## 1 1 12 1 2 13.950509
## 2 2 12 1 2 9.756918
## 3 1 13 1 3 13.906355
## 4 2 13 1 3 9.119455
## 5 1 14 1 4 5.174483
## 6 2 14 1 4 8.452221
#### GCA matrix for half diallel using male and female columns
#### use the 'overlay' function to create the half diallel matrix
Z1 <- overlay(HDdata[,c(3:4)])
#### Obtain the SCA matrix
Z2 <- model.matrix(~as.factor(geno)-1, data=HDdata)
#### Define the response variable and run
y <- HDdata$sugar
ETA <- list(list(Z=Z1), list(Z=Z2)) # Zu component
modHD <- mmer(Y=y, Z=ETA, draw=FALSE, silent=TRUE)
summary(modHD)
##
## Information contained in this fitted model:
## * Variance components, Residuals, Fitted values
## * BLUEs and BLUPs, Inverse phenotypic variance(V)
## * Variance-covariance matrix for fixed & random effects
## * Predicted error variance (PEV), LogLikelihood
## Use the '$' symbol to access such information
## ===========================================================
## Linear mixed model fit by restricted maximum likelihood
## ********************** sommer 2.9 ***********************
## ===========================================================
## logLik AIC BIC Method Conv DI
## Params -58.1779 118.3558 120.0935 NR TRUE TRUE
## ===========================================================
## Random effects:
## VarComp VarCompSE Zratio constraint
## Var(u.1) 5.509 3.579 1.539 Positive
## Var(u.2) 1.816 1.363 1.332 Positive
## Var(units) 3.117 0.962 3.240 Positive
## Number of obs: 42 Groups: 7 21
## ===========================================================
## Fixed effects:
## Value Std.Error t.value
## Intercept 10.3332 1.8189 5.6811
## ===========================================================
## Use the '$' symbol to access all information
5) Genome wide association analysis (GWAS) in diploids and tetraploids
With the development of modern statistical machinery the detection of markers associated to phenotypic traits have become quite straight forward. The days of QTL mapping using biparental populations exclusively are in the past. In this section we will show how to perform QTL mapping for diploid and polyploid organisms with complex genetic relationships. In addition we will show QTL mapping in biparental populations to clarify that the fact that is not required anymore doesn’t limit the capabilities of modern mixed model machinery.
First we will start doing the GWAS in a biparental population with 363 individuals genotyped with 2889 SNP markers. This is easily done by creating the variance covariance among individuals and using it in the random effect for genotypes. The markers are added in the W argument to fit the model of the form:
\(y = X\beta + Zu + Wg + \epsilon\)
In this case \(X\beta\) is the fixed part only for the intercept, \(Zu\) is the random effect for genotypes with the additive relationship matrix (A) as the variance-covariance of the random effect, \(Wg\) is the marker matrix and the effects of each marker. This is done in this way:
data(CPdata)
CPpheno <- CPdata$pheno
CPgeno <- CPdata$geno
### look at the data
head(CPpheno); CPgeno[1:5,1:4]
## id Row Col Year color Yield FruitAver Firmness
## P003 P003 3 1 2014 0.10075269 154.67 41.93 588.917
## P004 P004 4 1 2014 0.13891940 186.77 58.79 640.031
## P005 P005 5 1 2014 0.08681502 80.21 48.16 671.523
## P006 P006 6 1 2014 0.13408561 202.96 48.24 687.172
## P007 P007 7 1 2014 0.13519278 174.74 45.83 601.322
## P008 P008 8 1 2014 0.17406685 194.16 44.63 656.379
## scaffold_50439_2381 scaffold_39344_153 uneak_3436043 uneak_2632033
## P003 0 0 0 1
## P004 0 0 0 1
## P005 0 -1 0 1
## P006 -1 -1 -1 0
## P007 0 0 0 1
A <- A.mat(CPgeno) # additive relationship matrix
### fit the model
ans.A <- mmer2(color~1,random=~g(id), G=list(id=A),
W=CPgeno, data=CPpheno, silent=TRUE) # fit the model
## Response is imputed for estimation of variance components in GWAS models.
### if you have a genetic map you can use it
my.map <- CPdata$map
ans.A <- mmer2(color~1,random=~g(id), G=list(id=A),
W=CPgeno, data=CPpheno, silent=TRUE, map=my.map) # fit the model
## Response is imputed for estimation of variance components in GWAS models.
Notice that we used the random effect g(id) to adjust for the population structure whereas the markers are added in the W argument. Internally there’s 2 option for this 2-step GWAS models, in the default P3D=TRUE the model estimate variance components once and test each marker by a generalized least square process adjusting by the inverse of the V matrix (ZKZ’+R). If the user sets the argument P3D=FALSE the model will estimate variance components for each marker tested by adding the marker being tested as a fixed effect in the model so the variance components are adjusted. This can be quite time consuming and the first option presented usually yields very good results. For more details go to the help page of the mmer or mmer2 functions.
Now we will show how to do GWAS in a tetraploid using potato data. Is not very different from diploids. We only need to pay attention to the ploidy argument in the atcg1234 and A.mat functions. In addition, when running the mmer model there is more models that can be implemented according to Rosyara et al. (2016).
data(PolyData)
genotypes <- PolyData$PGeno
phenotypes <- PolyData$PPheno
## convert markers to numeric format
numo <- atcg1234(data=genotypes, ploidy=4, silent = TRUE); numo[1:5,1:4]; dim(numo)
## Obtaining reference alleles
## Checking for markers with more than 2 alleles. If found will be removed.
## Converting to numeric format
## Calculating minor allele frequency (MAF)
## Imputing missing data with mode
## c2_41437 c2_24258 c2_21332 c2_21320
## A96104-2 1 2 2 4
## A97066-42 2 3 2 4
## ACBrador 2 4 2 4
## ACLPI175395 0 4 0 4
## ADGPI195204 0 4 0 4
## [1] 221 3521
# get only plants with both genotypes and phenotypes
common <- intersect(phenotypes$Name,rownames(numo))
marks <- numo[common,]; marks[1:5,1:4]
## c2_41437 c2_24258 c2_21332 c2_21320
## A97066-42 2 3 2 4
## ACBrador 2 4 2 4
## AdirondackBlue 2 2 2 4
## AF2291-10 0 4 2 4
## AF2376-5 1 3 2 4
phenotypes2 <- phenotypes[match(common,phenotypes$Name),];
phenotypes2[1:5,1:4]
## Name total_yield chip_color tuber_eye_depth
## 1 A97066-42 13.10 2.35 3.03
## 2 ACBrador 15.56 2.63 4.37
## 3 AdirondackBlue 11.77 2.82 3.76
## 4 AF2291-10 13.43 1.50 4.50
## 5 AF2376-5 12.58 1.83 4.50
# Additive relationship matrix, specify ploidy
K1 <- A.mat(marks, ploidy=4)
# run the model you want
models <- c("additive","1-dom-alt","1-dom-ref","2-dom-alt","2-dom-ref")
ans2 <- mmer2(tuber_shape~1, random=~g(Name), G=list(Name=K1), W=marks,
method="EMMA", data=phenotypes2, silent = TRUE, models="additive")
## Response is imputed for estimation of variance components in GWAS models.
##
## Information contained in this fitted model:
## * Variance components, Residuals, Fitted values
## * BLUEs and BLUPs, Inverse phenotypic variance(V)
## * Variance-covariance matrix for fixed & random effects
## * Predicted error variance (PEV), LogLikelihood
## Use the '$' symbol to access such information
## ===========================================================
## Linear mixed model fit by restricted maximum likelihood
## ********************** sommer 2.9 ***********************
## ===========================================================
## logLik AIC BIC Method Conv DI
## Params -192.5206 387.0411 390.2723 EMMA TRUE TRUE
## ===========================================================
## Random effects:
## VarComp constraint
## Var(g(Name)) 0.60778 Positive
## Var(Error) 0.03807 Positive
## Number of obs: 187 Groups: 187
## ===========================================================
## Fixed effects:
## Value Std.Error t.value
## Intercept 3.307861 0.018302 180.74
## ===========================================================
## Use the '$' symbol to access all information
Notice the use of the models argument. The default is the additive model which applies for diploids and polyploids but for polyploids the model can be a vector with all possible models, i.e. “additive”,“1-dom-alt”,“1-dom-ref”,“2-dom-alt”,“2-dom-ref” models are supported for polyploids based on Rosyara (2016)
6) Genomic selection
In this section we will use wheat data from CIMMYT to show how is genomic selection performed. This is the case of prediction of specific individuals within a population. It basically uses a similar model of the form:
\(y = X\beta + Zu + \epsilon\)
and takes advantage of the variance covariance matrix for the genotype effect known as the additive relationship matrix (A) and calculated using the A.mat function to establish connections among all individuals and predict the BLUPs for individuals that were not measured. The prediction accuracy depends on several factors such as the heritability (\(h^2\)), training population used (TP), size of TP, etc.
data(wheatLines);
X <- wheatLines$wheatGeno; X[1:5,1:4]; dim(X)
## wPt.0538 wPt.8463 wPt.6348 wPt.9992
## [1,] -1 1 1 1
## [2,] 1 1 1 1
## [3,] 1 1 1 1
## [4,] -1 1 1 1
## [5,] -1 1 1 1
## [1] 599 1279
Y <- data.frame(wheatLines$wheatPheno); Y$id <- rownames(Y); head(Y);
## X1 X2 X4 X5 id
## 775 1.6716295 -1.72746986 -1.89028479 0.0509159 775
## 2166 -0.2527028 0.40952243 0.30938553 -1.7387588 2166
## 2167 0.3418151 -0.64862633 -0.79955921 -1.0535691 2167
## 2465 0.7854395 0.09394919 0.57046773 0.5517574 2465
## 3881 0.9983176 -0.28248062 1.61868192 -0.1142848 3881
## 3889 2.3360969 0.62647587 0.07353311 0.7195856 3889
rownames(X) <- rownames(Y)
# select environment 1
K <- A.mat(X) # additive relationship matrix
# GBLUP pedigree-based approach
set.seed(12345)
y.trn <- Y
vv <- sample(rownames(Y),round(dim(Y)[1]/5))
y.trn[vv,"X1"] <- NA
ans <- mmer2(X1~1,random=~g(id), G=list(id=K), method="EMMA",
data=y.trn, silent = TRUE) # kinship based
cor(ans$u.hat$`g(id)`[vv,],Y[vv,"X1"])
## [1] 0.4885687
## maximum prediction value that can be achieved
sqrt(ans$var.comp[1,1]/sum(ans$var.comp[,1]))
## [1] 0.5771923
7) Single cross prediction
When doing prediction of single cross performance the phenotype can be dissected in three main components, the general combining abilities (GCA) and specific combining abilities (SCA). This can be expressed with the same model analyzed in the diallel experiment mentioned before:
\(y = X\beta + Zu_1 + Zu_2 + Zu_S + \epsilon\)
with:
\(u_1\) ~ N(0, \(K_1\)\(\sigma^2_u1\))
\(u_2\) ~ N(0, \(K_2\)\(\sigma^2_u2\))
\(u_s\) ~ N(0, \(K_3\)\(\sigma^2_us\))
And we can specify the K matrices. The main difference between this model and the full and half diallel designs is the fact that this model will include variance covariance structures in each of the three random effects (GCA1, GCA2 and SCA) to be able to predict the crosses that have not ocurred yet. We will use the data published by Technow et al. (2015) to show how to do prediction of single crosses.
data(Technow_data)
A.flint <- Technow_data$AF # Additive relationship matrix Flint
A.dent <- Technow_data$AD # Additive relationship matrix Dent
pheno <- Technow_data$pheno # phenotypes for 1254 single cross hybrids
head(pheno);dim(pheno)
## hybrid dent flint GY GM hy
## 1 518.298 518 298 -8.04 -0.85 518:298
## 2 518.305 518 305 -11.10 1.70 518:305
## 3 518.306 518 306 -16.85 2.24 518:306
## 4 518.316 518 316 2.08 -1.33 518:316
## 5 518.323 518 323 5.65 -2.71 518:323
## 6 518.327 518 327 -16.95 -0.52 518:327
## [1] 1254 6
# CREATE A DATA FRAME WITH ALL POSSIBLE HYBRIDS
DD <- kronecker(A.dent,A.flint,make.dimnames=TRUE)
hybs <- data.frame(sca=rownames(DD),yield=NA,matter=NA,gcad=NA, gcaf=NA)
hybs$yield[match(pheno$hy, hybs$sca)] <- pheno$GY
hybs$matter[match(pheno$hy, hybs$sca)] <- pheno$GM
hybs$gcad <- as.factor(gsub(":.*","",hybs$sca))
hybs$gcaf <- as.factor(gsub(".*:","",hybs$sca))
head(hybs)
## sca yield matter gcad gcaf
## 1 513:316 10.02 -2.05 513 316
## 2 513:323 6.97 -3.78 513 323
## 3 513:330 NA NA 513 330
## 4 513:336 NA NA 513 336
## 5 513:340 NA NA 513 340
## 6 513:341 NA NA 513 341
# RUN THE PREDICTION MODEL
y.trn <- hybs
vv1 <- which(!is.na(hybs$yield))
vv2 <- sample(vv1, 100)
y.trn[vv2,"yield"] <- NA
anss2 <- mmer2(yield~1, random=~g(gcad) + g(gcaf), G=list(gcad=A.dent, gcaf=A.flint),
method="EM", silent=TRUE, data=y.trn)
## With dense var-cov structures (G) present you may want to try the AI or NR algorithm.
##
## Information contained in this fitted model:
## * Variance components, Residuals, Fitted values
## * BLUEs and BLUPs, Inverse phenotypic variance(V)
## * Variance-covariance matrix for fixed & random effects
## * Predicted error variance (PEV), LogLikelihood
## Use the '$' symbol to access such information
## ===========================================================
## Linear mixed model fit by restricted maximum likelihood
## ********************** sommer 2.9 ***********************
## ===========================================================
## logLik AIC BIC Method Conv DI
## Params -53925.37 107852.7 107857.8 EM FALSE TRUE
## ===========================================================
## Random effects:
## VarComp VarCompSE Zratio constraint
## g(gcad) 16.18 0.7041 22.99 Positive
## g(gcaf) 11.27 0.7119 15.83 Positive
## units 17.65 1.2314 14.33 Positive
## Number of obs: 1154 Groups: 123 86
## ===========================================================
## Fixed effects:
## Value Std.Error t.value
## Intercept 0.124518 0.048448 2.5701
## ===========================================================
## Use the '$' symbol to access all information
cor(anss2$fitted.y[vv2], hybs$yield[vv2])
## [1] 0.8797645
In the previous model we only used the GCA effects (GCA1 and GCA2) for practicity, altough it’s been shown that the SCA effect doesn’t actually help that much in increasing prediction accuracy and increase a lot the computation intensity required since the variance covariance matrix for SCA is the kronecker product of the variance covariance matrices for the GCA effects, resulting in a 10578x10578 matrix that increases in a very intensive manner the computation required.
A model without covariance structures would show that the SCA variance component is insignificant compared to the GCA effects. This is why including the third random effect doesn’t increase the prediction accuracy.
8) Multivariate genetic models and genetic correlations
Sometimes is important to estimate genetic variance-covariance among traits, multi-reponse models are very useful for such task. Let see an example with 3 traits (color, Yield, and Firmness) and a single random effect (genotype; id) although multiple effects can be modeled as well. We need to use a variance covariance structure for the random effect to be able to obtain the genetic covariance among traits.
data(CPdata)
CPpheno <- CPdata$pheno
CPgeno <- CPdata$geno
### look at the data
head(CPpheno);CPgeno[1:5,1:4]
## id Row Col Year color Yield FruitAver Firmness
## P003 P003 3 1 2014 0.10075269 154.67 41.93 588.917
## P004 P004 4 1 2014 0.13891940 186.77 58.79 640.031
## P005 P005 5 1 2014 0.08681502 80.21 48.16 671.523
## P006 P006 6 1 2014 0.13408561 202.96 48.24 687.172
## P007 P007 7 1 2014 0.13519278 174.74 45.83 601.322
## P008 P008 8 1 2014 0.17406685 194.16 44.63 656.379
## scaffold_50439_2381 scaffold_39344_153 uneak_3436043 uneak_2632033
## P003 0 0 0 1
## P004 0 0 0 1
## P005 0 -1 0 1
## P006 -1 -1 -1 0
## P007 0 0 0 1
## fit a model including additive effects
A <- A.mat(CPgeno) # additive relationship matrix
####================####
#### ADDITIVE MODEL ####
####================####
ans.A <- mmer2(cbind(color,Yield,Firmness)~1, random=~g(id),G=list(id=A),
MVM=TRUE, data=CPpheno, silent = TRUE)
summary(ans.A)
## Information contained in this structure:
## * Results for a multi response model
## Displayed:
## * Variance-covariance component summaries
## Use the '$' sign to access parameters
## =======================================================
## Multivariate Linear Mixed Model fit by REML
## ******************** sommer 2.9 *********************
## =======================================================
## Method:[1] "MNR"
## logLik AIC BIC
## MVM -435.9851 877.9703 892.9493
## =======================================================
## Variance-Covariance components:
##
## Var-Covar(g(id))
## color Yield Firmness
## color 0.005219 0.2984 0.5462
## Yield 0.298362 663.1160 -122.6967
## Firmness 0.546208 -122.6967 1264.8873
##
## Var-Covar(Residual)
## color Yield Firmness
## color 0.002713 0.2224 -0.06052
## Yield 0.222351 4011.1719 189.58719
## Firmness -0.060518 189.5872 1200.92942
## =======================================================
## Standard errors for variance components:
## VarComp VarCompSE Zratio
## g(id).color-color 5.219e-03 1.047e-03 4.9844
## g(id).color-Yield 2.984e-01 4.334e-01 0.6885
## g(id).color-Firmness 5.462e-01 3.977e-01 1.3734
## g(id).Yield-Yield 6.631e+02 3.260e+02 2.0341
## g(id).Yield-Firmness -1.227e+02 2.260e+02 -0.5430
## g(id).Firmness-Firmness 1.265e+03 2.969e+02 4.2602
## Residual.color-color 2.713e-03 2.979e-04 9.1068
## Residual.color-Yield 2.224e-01 2.261e-01 0.9833
## Residual.color-Firmness -6.052e-02 1.320e-01 -0.4586
## Residual.Yield-Yield 4.011e+03 3.424e+02 11.7162
## Residual.Yield-Firmness 1.896e+02 1.433e+02 1.3230
## Residual.Firmness-Firmness 1.201e+03 1.194e+02 10.0548
## =======================================================
## Fixed effects:
## color Yield Firmness
## (Intercept) 1.417 144.6 638
## =======================================================
## Groups and observations:
## Observ Groups
## g(id) 363 363
## =======================================================
## Use the '$' sign to access parameters
Now you can extract the BLUPs using the ‘randef’ function or simple accesing with the ‘$’ sign and pick ‘u.hat’. Also, genetic correlations and heritabilities can be calculated easily.
## genetic variance covariance
gvc <- ans.A$var.comp$`g(id)`
## extract variances (diagonals) and get standard deviations
sd.gvc <- as.matrix(sqrt(diag(gvc)))
## get possible products sd(Vgi) * sd(Vgi')
prod.sd <- sd.gvc %*% t(sd.gvc)
## genetic correlations cov(gi,gi')/[sd(Vgi) * sd(Vgi')]
(gen.cor <- gvc/prod.sd)
## color Yield Firmness
## color 1.0000000 0.1603880 0.2125959
## Yield 0.1603880 1.0000000 -0.1339714
## Firmness 0.2125959 -0.1339714 1.0000000
## heritabilities
(h2 <- diag(gvc) / diag(cov(CPpheno[,names(diag(gvc))], use = "complete.obs")))
## color Yield Firmness
## 0.8389640 0.1457021 0.4936389
9) Multivariate GWAS
Following the same theory of multivariate methods, theorically the marker effects can take advantage of the information contained in the correlation among traits besides exploiting the correlations between individuals. We have extended the GWAS framework to multivariate GWAS. Here, we will show a multivariate GWAS in a biparental population using the information of 2 traits for simplicity although any number of traits can be used.
data(CPdata)
CPpheno <- CPdata$pheno
CPgeno <- CPdata$geno
### look at the data
head(CPpheno);CPgeno[1:5,1:4]
## id Row Col Year color Yield FruitAver Firmness
## P003 P003 3 1 2014 0.10075269 154.67 41.93 588.917
## P004 P004 4 1 2014 0.13891940 186.77 58.79 640.031
## P005 P005 5 1 2014 0.08681502 80.21 48.16 671.523
## P006 P006 6 1 2014 0.13408561 202.96 48.24 687.172
## P007 P007 7 1 2014 0.13519278 174.74 45.83 601.322
## P008 P008 8 1 2014 0.17406685 194.16 44.63 656.379
## scaffold_50439_2381 scaffold_39344_153 uneak_3436043 uneak_2632033
## P003 0 0 0 1
## P004 0 0 0 1
## P005 0 -1 0 1
## P006 -1 -1 -1 0
## P007 0 0 0 1
## fit a model including additive effects
A <- A.mat(CPgeno) # additive relationship matrix
####================####
#### ADDITIVE MODEL ####
####================####
ans.A <- mmer2(cbind(color,Firmness)~1, random=~g(id),G=list(id=A),
MVM=TRUE, data=CPpheno, silent = TRUE, W=CPgeno, IMP=TRUE,
map=CPdata$map)
## Response is imputed for estimation of variance components in GWAS models.
## Information contained in this structure:
## * Results for a multi response model
## Displayed:
## * Variance-covariance component summaries
## Use the '$' sign to access parameters
## =======================================================
## Multivariate Linear Mixed Model fit by REML
## ******************** sommer 2.9 *********************
## =======================================================
## Method:[1] "MNR"
## logLik AIC BIC
## MVM -261.3855 526.7709 535.946
## =======================================================
## Variance-Covariance components:
##
## Var-Covar(g(id))
## color Firmness
## color 0.7695 0.1323
## Firmness 0.1323 0.4954
##
## Var-Covar(Residual)
## color Firmness
## color 0.39991 -0.01485
## Firmness -0.01485 0.46815
## =======================================================
## Standard errors for variance components:
## VarComp VarCompSE Zratio
## g(id).color-color 0.76953 0.15437 4.9848
## g(id).color-Firmness 0.13228 0.09560 1.3838
## g(id).Firmness-Firmness 0.49536 0.11634 4.2578
## Residual.color-color 0.39991 0.04392 9.1064
## Residual.color-Firmness -0.01485 0.03164 -0.4693
## Residual.Firmness-Firmness 0.46815 0.04664 10.0385
## =======================================================
## Fixed effects:
## color Firmness
## (Intercept) 0.023 -0.02618
## =======================================================
## Groups and observations:
## Observ Groups
## g(id) 363 363
## =======================================================
## Use the '$' sign to access parameters
Notice that here we made use of several arguments that were not neccesarily. Good luck with your analysis.
Literature
Covarrubias-Pazaran G. 2016. Genome assisted prediction of quantitative traits using the R package sommer. PLoS ONE 11(6):1-15.
Bernardo Rex. 2010. Breeding for quantitative traits in plants. Second edition. Stemma Press. 390 pp.
Gilmour et al. 1995. Average Information REML: An efficient algorithm for variance parameter estimation in linear mixed models. Biometrics 51(4):1440-1450.
Henderson C.R. 1975. Best Linear Unbiased Estimation and Prediction under a Selection Model. Biometrics vol. 31(2):423-447.
Kang et al. 2008. Efficient control of population structure in model organism association mapping. Genetics 178:1709-1723.
Lee et al. 2015. MTG2: An efficient algorithm for multivariate linear mixed model analysis based on genomic information. Cold Spring Harbor. doi: http://dx.doi.org/10.1101/027201.
Searle. 1993. Applying the EM algorithm to calculating ML and REML estimates of variance components. Paper invited for the 1993 American Statistical Association Meeting, San Francisco.
Yu et al. 2006. A unified mixed-model method for association mapping that accounts for multiple levels of relatedness. Genetics 38:203-208.
Abdollahi Arpanahi R, Morota G, Valente BD, Kranis A, Rosa GJM, Gianola D. 2015. Assessment of bagging GBLUP for whole genome prediction of broiler chicken traits. Journal of Animal Breeding and Genetics 132:218-228.
Tunnicliffe W. 1989. On the use of marginal likelihood in time series model estimation. JRSS 51(1):15-27.