The sommer package was developed to provide R users with flexible univariate and multivariate linear mixed-model tools for genetic and non-genetic analyses in diploid and polyploid organisms. The package can estimate variance-covariance parameters while allowing structured covariance models for random and residual effects, heterogeneous variances, and extraction of quantities such as BLUPs, BLUEs, residuals, fitted values, and prediction-error information. The current mmes() implementation uses Henderson’s mixed model equations with a sparse Average Information REML algorithm. Its numerical core uses sparse LDLT factorization, reuses the factorization for mixed-model solutions and variance-parameter differentiation, and can compute selected inverse elements with Takahashi sparse-inverse recursions when required by the REML calculations. Covariance factors supplied through vsm() are handled through a generic descriptor architecture and can be combined in arbitrary Kronecker products. The package also retains the complementary marginal-covariance MNR formulation through mmer(). The relative computational advantage of the two formulations depends on model dimensions, sparsity, and covariance structure rather than on a single p versus n rule.
The package supports a broad range of quantitative-genetic mixed models, including genomic prediction, hybrid prediction, GWAS-related analyses, multi-environment and multivariate models, and general mixed-model applications. It provides functions to construct additive (A.mat), dominance (D.mat), and epistatic (E.mat) relationship matrices, which can be incorporated as covariance or precision structures as appropriate for the solver being used. The package also provides flexibility for genetic models such as full and half diallel designs.
The vignettes aim to provide several examples in how to use the sommer package under different scenarios. We will spend the rest of the space providing examples for:
**SECTION 1: Introduction **
- Background in linear algebra
**SECTION 2: Topics in quantitative genetics **
- Heritability (\(h^2\)) calculation
- Specifying heterogeneous variances in mixed models
- Using the
vpredict()calculator - Half and full diallel designs (using the overlay)
- Genomic selection (predicting mendelian sampling)
- GBLUP
- rrBLUP
- Indirect genetic effects
- Single cross prediction (hybrid prediction)
- Multivariate genetic models and genetic correlations
- Bivariate model with repeated records
SECTION 3: Special topics in quantitative genetics
- Partitioned model
- UDU’ decomposition
- Mating designs
- GWAS by GBLUP
- Reduced models
SECTION 1: Introduction
Backgrounds in linear algebra
The package provides two complementary REML computational formulations. The current mmes() function solves Henderson’s mixed model equations and estimates covariance parameters with a sparse Average Information REML algorithm. The mixed-model coefficient matrix is factorized using sparse LDLT methods, and the same factorization is reused to solve the equations and to obtain the exact forward derivatives needed by the AI calculations. The mmer() function provides the marginal-covariance MNR formulation, in which computations are organized around the observation covariance matrix and the REML projection matrix. Since version 2.0, sommer can handle multivariate models. Following the usual multivariate mixed-model notation, the multivariate (and, by extension, univariate) model has the form:
\(y_1 = X_1\beta_1 + Z_1u_1 + \epsilon_1\)
\(y_2 = X_2\beta_2 + Z_2u_2 + \epsilon_2\)
…
\(y_i = X_i\beta_i + Z_iu_i + \epsilon_i\)
where \(y_i\) is a vector of trait phenotypes, \(\beta_i\) is a vector of fixed effects, \(u_i\) is a vector of random effects for individuals and \(e_i\) are residuals for trait i (i = 1, …, t). The random effects (\(u_1\) … \(u_i\) and \(e_i\)) are assumed to be normally distributed with mean zero. X and Z are incidence matrices for fixed and random effects respectively. The distributions of the multivariate response and the phenotypic variance covariance (V) are:
\(Y = X\beta + ZU + \epsilon_i\)
Y ~ MVN(\(X\beta\), V)
`aW$$\mathbf{Y} = \left[\begin{array} {r} y_1 \ y_2 \ … \ y_t \ \end{array}\right] $$
`aW$$\mathbf{X} = \left[\begin{array} {rrr} X_1 & … & … \ \vdots & \ddots & \vdots\ … & … & X_t \ \end{array}\right] $$
`aW$$\mathbf{V} = \left[\begin{array} {rrr} Z_1 K{\sigma^2_{g_{1}}} Z_1’ + H{\sigma^2_{\epsilon_{1}}} & … & Z_1 K{\sigma_{g_{1,t}}} Z_t’ + H{\sigma_{\epsilon_{1,t}}}\ \vdots & \ddots & \vdots\ Z_1 K{\sigma_{g_{1,t}}} Z_t’ + H{\sigma_{\epsilon_{1,t}}} & … & Z_t K{\sigma^2_{g_{t}}} Z_t’ + H{\sigma^2_{\epsilon_{t}}} \ \end{array}\right] $$
where (K) denotes a relationship or covariance structure associated with a random effect. More generally, sommer permits several random terms and structured residual covariance, so the marginal covariance can be written as (V=\sum_k Z_kG_kZ_k^\prime+R). The quantities (\sigma^2_{g_i}) and (\sigma^2_{\epsilon_i}) denote trait-specific random-effect and residual variances, while the corresponding off-diagonal terms denote covariances between traits. In the current vsm() architecture, each covariance term can itself be constructed from one or more normalized covariance factors and a single product-level variance scale. REML estimation is based on the restricted log likelihood; ignoring constants, the usual marginal form is:
\(-2\ell_R = \log|V| + \log|X^\prime V^{-1}X| + Y^\prime P Y\)
where (P=V^{-1}-V^{-1}X(X^\prime V^{-1}X)^{-1}X^\prime V^{-1}), and (|\cdot|) denotes a determinant. This expression describes the marginal REML criterion. In mmes(), the same REML objective is evaluated through identities associated with Henderson’s mixed model equations rather than by repeatedly forming and directly inverting the full observation covariance matrix. Variance parameters are updated using an Average Information step of the general form:
\(\theta^{k+1} = \theta^{k} + AI(\thetak){-1}s(\theta^k)\)
Here (\theta) denotes the working covariance-parameter vector, (s(\theta)) is the REML score, and (AI(\theta)) is the Average Information matrix. For structured covariance factors, mmes() differentiates the factorized mixed-model solve analytically with respect to the covariance parameters and assembles the AI system from those derivatives. Parameters are represented in working coordinates chosen to respect covariance constraints, and the proposed update is protected by trust scaling and a global exact-likelihood backtracking procedure. For nonlinear covariance parameterizations the AI matrix uses first covariance derivatives, rather than adding second derivatives of the covariance model that belong to an observed-Hessian formulation. The vpredict() function can subsequently use estimated variance-covariance information and the delta method to obtain standard errors for functions such as heritabilities and genetic correlations.
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.
SECTION 2: Topics in quantitative genetics
1) Marker and non-marker based heritability calculation
Heritability is one of the most popular parameters among the breeding and genetics communities because of the insight it provides in the inheritance of the trait and potential selection response. 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 breeding experiment without molecular markers, mating and experimental designs can be used to partition additive ((\sigma^2_A)) and non-additive components such as dominance ((\sigma^2_D)) and epistasis, together with environmental variation. Designs such as generation analysis and the North Carolina designs can identify additive and dominance components under their corresponding assumptions. When such information is unavailable, replicated genotype effects can still be used to estimate total genetic variance and broad-sense heritability. The first example illustrates this latter case without a genomic relationship matrix. Because the current mmes() solver is based on sparse mixed model equations, models composed of sparse incidence matrices and simple covariance structures are now directly within its intended computational setting; the older warning about a dense direct-inversion implementation no longer applies.
The following 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 \(h^2\).
library(sommer)
data(DT_example, package="enhancer")
DT <- DT_example
A <- A_example
ans1 <- mmes(Yield~1,
random= ~ Name + Env + Env:Name + Env:Block,
rcov= ~ units,
data=DT, verbose = FALSE)
## Solver selected: cholmod
summary(ans1)$varcomp
## term parameter estimate StdError Zratio
## 1 vsm(ism(Name)) sigma2 3.718394e+00 1.6329073 2.2771619823
## 2 vsm(ism(Env)) sigma2 1.200875e+01 12.3113554 0.9754204469
## 3 vsm(ism(Env:Name)) sigma2 5.152650e+00 1.4734085 3.4970954882
## 4 vsm(ism(Env:Block)) sigma2 7.839943e-05 0.1703363 0.0004602626
## 5 vsm(ism(units)) sigma2 4.366401e+00 0.6532380 6.6842431496
(n.env <- length(levels(DT$Env)))
## [1] 3
vpredict(ans1, h2 ~ V1 / ( V1 + (V3/n.env) + (V5/(2*n.env)) ) )
## Estimate SE
## h2 0.6032752 0.1160635
That is an estimate of broad-sense heritability.
Recently with markers becoming cheaper, thousand of markers can be run in the breeding materials. When markers are available, a special design is not neccesary to dissect 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 \(\sigma^2_I\), although given that A, D and E are not orthogonal the interpretation of models that fit more than the A matrix at the same time becomes cumbersome.
Assume you have a population (even unreplicated) in the field but in addition we have genetic markers. Now we can fit the model and estimate the genomic heritability that explains a portion of the additive genetic variance (with high marker density \(\sigma^2_A\) = \(\sigma^2_{markers}\))
data(DT_cpdata, package="enhancer")
DT <- DT_cpdata
GT <- GT_cpdata
MP <- MP_cpdata
DT$idd <-DT$id; DT$ide <-DT$id
### look at the data
A <- A.mat(GT) # additive relationship matrix
D <- D.mat(GT) # dominance relationship matrix
E <- E.mat(GT) # epistatic relationship matrix
Ai <- solve(A + diag(1e-5, nrow(A), nrow(A)))
Ai[lower.tri(Ai)] <- t(Ai)[lower.tri(Ai)] # fill the lower triangular
Ai <- as(as(as( Ai, "dMatrix"), "generalMatrix"), "CsparseMatrix")
attr(Ai, "inverse")=TRUE
Di <- solve(D+ diag(1e-5, nrow(A), nrow(A)))
Di <- as(as(as( Di, "dMatrix"), "generalMatrix"), "CsparseMatrix")
attr(Di, "inverse")=TRUE
# ans.ADE <- mmes(Yield~1,
# random=~vsm(ism(id),Gu=Ai) + vsm(ism(idd),Gu=Di),
# rcov=~units, nIters=10,
# data=DT,verbose = FALSE)
# (summary(ans.ADE)$varcomp)
# vpredict(ans.ADE, h2 ~ (V1) / ( V1+V3) ) # narrow sense
# vpredict(ans.ADE, h2 ~ (V1+V2) / ( V1+V2+V3) ) # broad-sense
This example illustrates how marker-derived additive and dominance relationship structures can be associated with random effects and used to estimate additive and dominance variance components, from which narrow- and broad-sense heritability functions can be formed. In the current Henderson-based mmes() interface, vsm() defines the covariance structure of a random term. When a matrix supplied through Gu is already a precision matrix, as Ai and Di are in the code above, it is converted to sparse form and marked with attr(..., "inverse")=TRUE. Thus the solver can use the supplied precision directly rather than inverting it internally. This replaces the older henderson=FALSE direct-inversion distinction.
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.
We estimate variance components for \(GCA_2\) and \(SCA\) specifying the variance structure.
data(DT_cornhybrids, package="enhancer")
DT <- DT_cornhybrids
DTi <- DTi_cornhybrids
GT <- GT_cornhybrids
### fit the model
modFD <- mmes(Yield~1,
random=~ vsm(atm(Location,c("3","4")),ism(GCA2)),
rcov= ~ vsm(dsm(Location),ism(units)),
data=DT, verbose = FALSE)
## Solver selected: ldlt
summary(modFD)
## ============================================================
## Multivariate Linear Mixed Model fit by REML
## ********************** sommer 4.4 **********************
## ============================================================
## logLik AIC BIC Method Converge
## Value -167.6796 337.3592 341.3506 AI TRUE
## ============================================================
## Variance-Covariance components:
## term parameter estimate StdError
## 1 vsm(atm(Location, c("3", "4")), ism(GCA2)) variance[3] 62.39 56.53
## 2 vsm(atm(Location, c("3", "4")), ism(GCA2)) variance[4] 98.01 82.45
## 3 vsm(dsm(Location), ism(units)) variance[1] 216.75 30.71
## 4 vsm(dsm(Location), ism(units)) variance[2] 216.83 30.73
## 5 vsm(dsm(Location), ism(units)) variance[3] 493.11 77.91
## 6 vsm(dsm(Location), ism(units)) variance[4] 712.02 112.13
## Zratio
## 1 1.104
## 2 1.189
## 3 7.057
## 4 7.056
## 5 6.329
## 6 6.350
## ============================================================
## Fixed effects:
## Estimate Std.Error t.value
## Intercept 138.1 NA NA
## ============================================================
## Use the '$' sign to access results and parameters
In the previous example atm() is used as a covariance-shaping factor inside vsm(). It permits selected levels of the factor—in this case Locations 3 and 4—to have their own relative diagonal variance parameters within the random-effect covariance. In the current parameterization the covariance factor is normalized and vsm() supplies one product-level variance scale, so the selected-level parameters describe variance ratios rather than redundant absolute scales. The residual term similarly uses dsm(Location) to allow location-specific residual variances.
3) Using the vpredict calculator
Sometimes the user needs to calculate ratios or functions of specific variance-covariance components and obtain the standard errors for such parameters. Examples of these are the genetic correlations, heritabilities, etc. Using the CPdata we will show how to estimate the heritability and the standard error using the vpredict() function that uses the delta method to come up with these parameters. This can be extended for any linear combination of the variance components.
3.1) Standar error for heritability
data(DT_cpdata, package="enhancer")
DT <- DT_cpdata
GT <- GT_cpdata
MP <- MP_cpdata
### look at the data
A <- A.mat(GT) # additive relationship matrix
Ai <- solve(A+ diag(1e-4, nrow(A),nrow(A)))
Ai[lower.tri(Ai)] <- t(Ai)[lower.tri(Ai)] # fill the lower triangular
Ai <- as(as(as( Ai, "dMatrix"), "generalMatrix"), "CsparseMatrix")
attr(Ai, "inverse")=TRUE
ans <- mmes(color~1,
random=~vsm(ism(id),Gu=Ai),
rcov=~units, nIters=10,
data=DT, verbose = FALSE)
## Solver selected: cholmod
summary(ans)$varcomp
## term parameter estimate StdError Zratio
## 1 vsm(ism(id), Gu = Ai) sigma2 0.005121546 0.0010577993 4.841699
## 2 vsm(ism(units)) sigma2 0.002743183 0.0003036588 9.033765
vpredict(ans, h2 ~ (V1) / ( V1+V2) )
## Estimate SE
## h2 0.6512044 0.05322574
The same can be used for multivariate models. Please check the documentation of the vpredict function to see more examples.
4) Half and full diallel designs (use of the overlay)
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 on whether reciprocal and self-crosses (omission of parents) are performed (full diallel with parents n^2; full diallel without parents n(n-1); half diallel with parents ½ * n(n+1); half diallel without parents ½ * n(n-1) ). In this example we will show a full diallel design (reciprocal 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(DT_cornhybrids, package="enhancer")
DT <- DT_cornhybrids
DTi <- DTi_cornhybrids
GT <- GT_cornhybrids
modFD <- mmes(Yield~Location,
random=~GCA1+GCA2+SCA,
rcov=~units,
data=DT, verbose = FALSE)
## Solver selected: ldlt
(suma <- summary(modFD)$varcomp)
## term parameter estimate StdError Zratio
## 1 vsm(ism(GCA1)) sigma2 0.01719791 24.08397 0.000714081
## 2 vsm(ism(GCA2)) sigma2 11.73477181 23.13633 0.507201032
## 3 vsm(ism(SCA)) sigma2 186.38302058 40.62892 4.587447188
## 4 vsm(ism(units)) sigma2 220.37495898 18.05278 12.207259909
Vgca <- sum(suma[1:2,"estimate"])
Vsca <- suma[3,"estimate"]
Ve <- suma[4,"estimate"]
Va = 4*Vgca
Vd = 4*Vsca
Vg <- Va + Vd
(H2 <- Vg / (Vg + (Ve)) )
## [1] 0.7824349
(h2 <- Va / (Vg + (Ve)) )
## [1] 0.04640852
Don’t worry too much about the 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 the second data set we show a small half diallel with 7 parents crossed in one direction. There are n(n-1)/2 possible crosses; 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 will show first how to do so with the mmes() function using the overlay() function. The specific model here is:
\(y = X\beta + Zu_g + Zu_s + \epsilon\)
data("DT_halfdiallel", package="enhancer")
DT <- DT_halfdiallel
head(DT)
## 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
DT$femalef <- as.factor(DT$female)
DT$malef <- as.factor(DT$male)
DT$genof <- as.factor(DT$geno)
#### model using overlay
modh <- mmes(sugar~1,
random=~vsm(ism(overlay(femalef,malef)) )
+ genof, data=DT, verbose = FALSE)
## Solver selected: ldlt
summary(modh)$varcomp
## term parameter estimate StdError Zratio
## 1 vsm(ism(overlay(femalef, malef))) sigma2 5.512698 3.5821868 1.538920
## 2 vsm(ism(genof)) sigma2 1.811638 1.3620943 1.330038
## 3 vsm(ism(units)) sigma2 3.120834 0.9636482 3.238562
Notice how the overlay() argument makes the overlap of incidence matrices possible making sure that male and female are joint into a single random effect.
5) Genomic selection: predicting mendelian sampling
In this section we will use wheat data from CIMMYT to show how genomic selection is 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(DT_wheat, package="enhancer")
DT <- DT_wheat
GT <- apply(GT_wheat,2,as.numeric)
rownames(GT) <- rownames(GT_wheat)
colnames(DT) <- paste0("X",1:ncol(DT))
DT <- as.data.frame(DT);DT$id <- as.factor(rownames(DT))
# select environment 1
K <- A.mat(GT) # additive relationship matrix
colnames(K) <- rownames(K) <- rownames(DT)
Ki <- solve(K+ diag(1e-4, nrow(K),nrow(K)))
Ki[lower.tri(Ki)] <- t(Ki)[lower.tri(Ki)] # fill the lower triangular
Ki <- as(as(as( Ki, "dMatrix"), "generalMatrix"), "CsparseMatrix")
attr(Ki, "inverse")=TRUE
# GBLUP pedigree-based approach
set.seed(12345)
y.trn <- DT
vv <- sample(rownames(DT),round(nrow(DT)/5))
y.trn[vv,"X1"] <- NA
head(y.trn)
## X1 X2 X3 X4 id
## 775 1.6716295 -1.72746986 -1.89028479 0.0509159 775
## 2166 -0.2527028 0.40952243 0.30938553 -1.7387588 2166
## 2167 NA -0.64862633 -0.79955921 -1.0535691 2167
## 2465 0.7854395 0.09394919 0.57046773 0.5517574 2465
## 3881 0.9983176 -0.28248062 1.61868192 -0.1142848 3881
## 3889 2.3360969 0.62647587 0.07353311 0.7195856 3889
## GBLUP with mmes
ans <- mmes(X1~1,
random=~vsm(ism(id),Gu=Ki),
rcov=~units,
data=y.trn, verbose = FALSE) # kinship based
## Solver selected: cholmod
cor(ans$u[vv,] ,DT[vv,"X1"], use="complete")
## [1] 0.507606
## rrBLUP with mmer
ans2 <- mmer(X1~1,
random=~vsr(list(GT)),
rcov=~units, getPEV = TRUE,
data=y.trn, verbose = FALSE) # kinship based
## This function has been deprecated. Please start using 'mmes' and its auxiliary functions. This function will be no longer maintained.
u <- GT %*% ans2$U$`u:GT`$X1 # BLUPs for individuals
rownames(u) <- rownames(GT)
cor(u[vv,],DT[vv,"X1"]) # same correlation
## [1] 0.5737681
# the same can be applied in multi-response models in GBLUP or rrBLUP
The two prediction formulations shown above use different parameterizations. The GBLUP example fits genotype effects with the sparse precision matrix Ki through mmes() and vsm(). The rrBLUP example instead fits marker effects through mmer() and vsr(list(GT)), so a separate random effect is associated with each marker column. The older buildGu=FALSE explanation does not apply to the code shown here and is not part of the current mmes() covariance-factor interface.
Marker effects can also be recovered from a GBLUP fit without refitting, with meffects_mmes(). Passing the marker matrix used in A.mat() reproduces how the relationship matrix was built; blend accounts for a matrix of the form \((1-\lambda)G+\lambda I\), and se = TRUE adds standard errors and back-solved GWAS tests (Gualdron-Duarte et al. 2014):
G <- A.mat(GT)
Ki <- solve(0.99 * G + 0.01 * diag(nrow(G))); attr(Ki, "inverse") <- TRUE
ans <- mmes(X1 ~ 1, random = ~ vsm(ism(id), Gu = Ki), data = y.trn, verbose = FALSE)
markers <- meffects_mmes(ans, 1, GT, blend = 0.01, se = TRUE)
head(markers[order(markers$p.value), ])
6) Indirect genetic effects
General variance structures can be used to fit indirect genetic effects. Here, we use an example dataset to show how we can fit the variance and covariance components between two or more different random effects.
We first fit a direct genetic effects model:
data(DT_ige, package="enhancer")
DT <- DT_ige
Af <- A_ige
An <- A_ige
# Direct genetic effects model
modDGE <- mmes(trait ~ block,
random = ~ focal,
rcov = ~ units,
data = DT, verbose=FALSE)
## Solver selected: ldlt
summary(modDGE)$varcomp
## term parameter estimate StdError Zratio
## 1 vsm(ism(focal)) sigma2 19895.48 3108.2608 6.400841
## 2 vsm(ism(units)) sigma2 10131.31 477.5705 21.214281
We now fit the indirect genetic effects model without covariance between DGE and IGE:
data(DT_ige, package="enhancer")
DT <- DT_ige
A <- A_ige
## Indirect genetic effects model
modIGE <- mmes(trait ~ block, dateWarning = FALSE,
random = ~ focal + neighbour, verbose = FALSE,
rcov = ~ units,
data = DT)
## Solver selected: ldlt
summary(modIGE)$varcomp
## term parameter estimate StdError Zratio
## 1 vsm(ism(focal)) sigma2 20551.426 3140.4852 6.544029
## 2 vsm(ism(neighbour)) sigma2 2927.065 604.1962 4.844560
## 3 vsm(ism(units)) sigma2 7300.613 363.8944 20.062451
We now fit the indirect genetic effects model with covariance between DGE and IGE for which we will use the strm() function (covm() is its two-term special case):
modIGE <- mmes(trait ~ block, dateWarning = FALSE,
random = ~ strm(dge = vsm(ism(focal)), ige = vsm(ism(neighbour))),
rcov = ~ units, verbose = FALSE,
data = DT)
## Solver selected: ldlt
summary(modIGE)$varcomp
## term parameter
## 1 strm(dge = vsm(ism(focal)), ige = vsm(ism(neighbour))) variance[dge]
## 2 strm(dge = vsm(ism(focal)), ige = vsm(ism(neighbour))) covariance[ige,dge]
## 3 strm(dge = vsm(ism(focal)), ige = vsm(ism(neighbour))) variance[ige]
## 4 vsm(ism(units)) sigma2
## estimate StdError Zratio
## 1 21010.852 3203.8570 6.557987
## 2 -7468.348 1242.0027 -6.013150
## 3 2964.661 579.4974 5.115918
## 4 7297.476 358.2924 20.367378
On top of that we can include a relationship matrix for the two random effects that are being forced to co-vary
Ai <- solve(A_ige + diag(1e-5, nrow(A_ige),nrow(A_ige) ))
attr(Ai, "inverse") <- TRUE
modIGE <- mmes(trait ~ block, dateWarning = FALSE,
random = ~ strm(dge = vsm(ism(focal)), ige = vsm(ism(neighbour)), Gu = Ai),
rcov = ~ units, verbose = FALSE,
data = DT)
## Solver selected: cholmod
summary(modIGE)$varcomp
## term
## 1 strm(dge = vsm(ism(focal)), ige = vsm(ism(neighbour)), Gu = Ai)
## 2 strm(dge = vsm(ism(focal)), ige = vsm(ism(neighbour)), Gu = Ai)
## 3 strm(dge = vsm(ism(focal)), ige = vsm(ism(neighbour)), Gu = Ai)
## 4 vsm(ism(units))
## parameter estimate StdError Zratio
## 1 variance[dge] 7805.570 1247.3599 6.257673
## 2 covariance[ige,dge] -2800.551 493.6032 -5.673689
## 3 variance[ige] 1139.775 242.9189 4.691999
## 4 sigma2 7315.442 356.3860 20.526741
7) Genomic selection: 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$)
And we can specify the K matrices. The main difference between this model and the full and half diallel designs is the fact that this model will include variance covariance structures in each of the three random effects (GCA1, GCA2 and SCA) to be able to predict the crosses that have not ocurred yet. We will use the data published by Technow et al. (2015) to show how to do prediction of single crosses.
data(DT_technow, package="enhancer")
DT <- DT_technow
Md <- apply(Md_technow,2,as.numeric)
rownames(Md) <- rownames(Md_technow)
Mf <- apply(Mf_technow,2,as.numeric)
rownames(Mf) <- rownames(Mf_technow)
Md <- (Md*2) - 1
Mf <- (Mf*2) - 1
Ad <- A.mat(Md)
Af <- A.mat(Mf)
Adi <- solve(Ad + diag(1e-4,ncol(Ad),ncol(Ad)))
Adi[lower.tri(Adi)] <- t(Adi)[lower.tri(Adi)] # fill the lower triangular
Adi <- as(as(as( Adi, "dMatrix"), "generalMatrix"), "CsparseMatrix")
attr(Adi, 'inverse')=TRUE
Afi <- solve(Af + diag(1e-4,ncol(Af),ncol(Af)))
Afi[lower.tri(Afi)] <- t(Afi)[lower.tri(Afi)] # fill the lower triangular
Afi <- as(as(as( Afi, "dMatrix"), "generalMatrix"), "CsparseMatrix")
attr(Afi, 'inverse')=TRUE
# RUN THE PREDICTION MODEL
y.trn <- DT
vv1 <- which(!is.na(DT$GY))
vv2 <- sample(vv1, 100)
y.trn[vv2,"GY"] <- NA
anss2 <- mmes(GY~1, henderson=TRUE,
random=~vsm(ism(dent),Gu=Adi) + vsm(ism(flint),Gu=Afi),
rcov=~units, nIters=15,
data=y.trn, verbose = FALSE)
## Solver selected: cholmod
summary(anss2)$varcomp
## term parameter estimate StdError Zratio
## 1 vsm(ism(dent), Gu = Adi) sigma2 16.06059 2.6426609 6.077430
## 2 vsm(ism(flint), Gu = Afi) sigma2 11.41708 2.2419014 5.092589
## 3 vsm(ism(units)) sigma2 16.81820 0.7742311 21.722458
# zu1 <- model.matrix(~dent-1,y.trn) %*% anss2$uList$`vsm(ism(dent), Gu = Adi)`
# zu2 <- model.matrix(~flint-1,y.trn) %*% anss2$uList$`vsm(ism(flint), Gu = Afi)`
# u <- zu1+zu2+as.vector(anss2$b)
# cor(u[vv2,], DT$GY[vv2])
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, but does 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 10578 x 10578 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
Multivariate mixed models are useful for estimating genetic variances and covariances among traits. The example below represents two traits (color and Yield) in long format, with a single response column and a factor identifying the trait. This makes trait a covariance dimension that can be included explicitly in vsm(). The random term vsm(usm(trait), ism(id), Gu=Ai) represents the genotype covariance as a Kronecker product of an unstructured trait covariance factor and the genotype relationship structure, while vsm(dsm(trait), ism(units)) allows trait-specific residual variances with zero residual covariance between traits.
# data(DT_cpdata, package="enhancer")
# DT <- DT_cpdata
# GT <- GT_cpdata
# MP <- MP_cpdata
# traits <- c("color","Yield")
# DT[,traits] <- apply(DT[,traits],2,scale)
# DTL <- reshape(DT[,c("id", traits)],
# idvar = c("id"),
# varying = traits,
# v.names = "value", direction = "long",
# timevar = "trait", times = traits )
# DTL <- DTL[with(DTL, order(trait)), ]
# head(DTL)
#
# A <- A.mat(GT) # additive relationship matrix
# # if using mmes=TRUE you need to provide the inverse
# Ai <- solve(A + diag(1e-4,ncol(A),ncol(A)))
# Ai[lower.tri(Ai)] <- t(Ai)[lower.tri(Ai)] # fill the lower triangular
# Ai <- as(as(as( Ai, "dMatrix"), "generalMatrix"), "CsparseMatrix")
# attr(Ai, 'inverse')=TRUE
# #### be patient this model is heavier
# ansm <- mmes( value ~ trait, # henderson=TRUE,
# random=~ vsm(usm(trait), ism(id), Gu=Ai), # Ai if henderson
# rcov=~ vsm(dsm(trait), ism(units)),
# data=DTL)
# cov2cor(ansm$theta[[1]])
For wide data, stackTraits() creates the long response and an explicit record key. Use that key as the final residual incidence when traits measured on the same experimental unit should have correlated errors; covmatrix_mmes() returns the fitted covariance, correlation and their standard errors.
# DTL <- stackTraits(DT, traits = c("color", "Yield"), keep = "id")
# ansm <- mmes(value ~ trait,
# random = ~ vsm(usm(trait), ism(id), Gu = Ai),
# rcov = ~ vsm(usm(trait), ism(record)), data = DTL)
# covmatrix_mmes(ansm, 1)$correlation # genetic correlation
# covmatrix_mmes(ansm, 2)$correlation # residual correlation
Mixed-family traits are also supported through familym(). For example, a binary disease score and a Gaussian yield can be fitted jointly with an unstructured genetic covariance. The residual covariance uses corgm(trait, variance="heterogeneous") so that the fixed binomial residual variance and the estimated Gaussian residual variance can coexist; see the familym() and corgm() examples for a complete reproducible fit.
Other examples of multi-trait models can be found in the documentation of the rice dataset (DT_rice) and in the example dataset (DT_example).
9) Bivariate model with repeated records
We first mimic the availability of 2 traits (TRAIT) with repeated records across several days (X) for a response variable (Y).
data(DT_legendre)
DT <- DT_legendre
head(DT)
## SUBJECT X Y Xf
## 1.1 1 1 -0.7432795 1
## 2.1 2 1 -0.6669945 1
## 3.1 3 1 -4.2802751 1
## 4.1 4 1 4.1092149 1
## 5.1 5 1 -3.0317213 1
## 6.1 6 1 1.3506577 1
DT$SUBJECT <- paste("s",DT$SUBJECT,sep="_")
DT1 <- DT2 <- DT
DT1$TRAIT <- "T1"
DT2$TRAIT <- "T2"
DT2$Y <- sample(DT2$Y)
DTC <- rbind(DT1,DT2)
Now we fit an unstructured model for a legendre polynomial with two traits and a diagonal model for the residual.
#
# library(orthopolynom)
#
# Z <- with(DTC, dsm(leg(X,1)) )$Z
# for(i in 1:ncol(Z)){DTC[,colnames(Z)[i]] <- Z[,i]}
#
# X <- with(DTC, dsm(TRAIT) )$Z
# for(i in 1:ncol(X)){DTC[,colnames(X)[i]] <- X[,i]}
#
# A <- diag(length(unique(DTC$SUBJECT)))
# rownames(A) <- colnames(A) <- unique(DTC$SUBJECT)
# Ai <- solve(A + diag(1e-4,ncol(A),ncol(A)))
# Ai[lower.tri(Ai)] <- t(Ai)[lower.tri(Ai)] # fill the lower triangular
# Ai <- as(as(as( Ai, "dMatrix"), "generalMatrix"), "CsparseMatrix")
# attr(Ai, 'inverse')=TRUE
# ##
# M <- model.matrix(~ T1:leg0 + T1:leg1 + T2:leg0 + T2:leg1 - 1 , data=DTC)
# mRR2b<-mmes(Y ~ Xf,
# random=~ vsm( usm( M ) , ism(SUBJECT) , Gu = Ai),
# rcov = ~ vsm( dsm(TRAIT), ism(units) ),
# nIters = 10, verbose = FALSE,
# data=DTC)
# summary(mRR2b)$varcomp
SECTION 3: Special topics in Quantitative genetics
1) Partitioned model
Marker-effect BLUPs can be recovered from an equivalent GBLUP parameterization by exploiting the relationship between the marker matrix (M), the genomic covariance (MM^\prime), and the corresponding genotype BLUPs. The example below fits the marker model directly and then fits a partitioned GBLUP model based on (MM^\prime). Marker effects are recovered from genotype effects with the linear transformation (M^\prime(MM^\prime)^{-1}), using the regularized inverse constructed in the example. This illustrates the algebraic equivalence of the two parameterizations under the covariance scaling used here.
library(sommer)
data("DT_cpdata", package="enhancer")
DT <- DT_cpdata
M <- GT_cpdata
################
# MARKER MODEL
################
mix.marker <- mmer(Yield~1,
random=~Rowf+vsr(list(M)),
rcov=~units,data=DT,
verbose = FALSE)
## This function has been deprecated. Please start using 'mmes' and its auxiliary functions. This function will be no longer maintained.
me.marker <- mix.marker$U$`u:M`$Yield
################
# PARTITIONED GBLUP MODEL
################
MMT <-tcrossprod(M) ## MM' = additive relationship matrix
MMTinv<-solve(MMT + diag(1e-4, nrow(MMT), nrow(MMT))) ## inverse
MTMMTinv<-t(M)%*%MMTinv # M' %*% (M'M)-
MMTinv <- as(as(as( MMTinv, "dMatrix"), "generalMatrix"), "CsparseMatrix")
attr(MMTinv, 'inverse')=TRUE
mix.part <- mmes(Yield~1, nIters = 20,
random=~Rowf+vsm(ism(id), Gu=MMTinv),
rcov=~units,data=DT,
verbose = FALSE)
## Solver selected: cholmod
#convert BLUPs to marker effects me=M'(M'M)- u
me.part<-MTMMTinv%*%matrix(mix.part$uList$`vsm(ism(id), Gu = MMTinv`,ncol=1)
# compare marker effects between both models
plot(me.marker,me.part)
Under the covariance construction and regularization used in this example, the two parameterizations produce the corresponding marker-effect predictions through the stated linear transformation. Their relative computational cost depends on the numbers of records, genotypes and markers and on the solver formulation; therefore the partitioned form should not be assumed to be universally faster.
2) UDU’ decomposition
An eigendecomposition of a symmetric relationship matrix can be written (A=UDU^\prime), with orthogonal eigenvectors (U) and diagonal eigenvalue matrix (D). Premultiplying the model by (U^\prime) rotates the observations and design matrices into this eigenbasis, so the relationship component becomes diagonal. The mmes() interface performs this transformation internally when rotation=TRUE is supplied to the relationship vsm() term; users should provide the named relationship precision matrix through Gu, just as in an ordinary mmes() fit.
data("DT_wheat", package="enhancer")
rownames(GT_wheat) <- rownames(DT_wheat)
GT <- apply(GT_wheat,2,as.numeric)
rownames(GT) <- rownames(GT_wheat)
A <- A.mat(GT)
Ai <- solve(A + diag(1e-5, nrow(A)))
Ai[lower.tri(Ai)] <- t(Ai)[lower.tri(Ai)] # fill the lower triangular
Ai <- as(as(as(Ai, "dMatrix"), "generalMatrix"), "CsparseMatrix")
attr(Ai, "inverse") <- TRUE
isSymmetric(Ai)
## [1] TRUE
# One complete observation per relationship level gives a balanced example.
DTn <- data.frame(
id=rownames(A),
y=as.numeric(DT_wheat[,1])
)
model_regular <- mmes(
y~1,
random=~vsm(ism(id), Gu=Ai),
rcov=~units, data=DTn, verbose=FALSE
)
## Solver selected: cholmod
# Henderson MME formulation with eigenbasis random coefficients.
model_rotation_h <- mmes(
y~1,
random=~vsm(ism(id), Gu=Ai, rotation=TRUE),
rcov=~units, data=DTn, henderson=TRUE, verbose=FALSE
)
## Solver selected: ldlt
# Lee--van der Werf direct observation-covariance formulation.
model_rotation_d <- mmes(
y~1,
random=~vsm(ism(id), Gu=Ai, rotation=TRUE),
rcov=~units, data=DTn, henderson=FALSE, verbose=FALSE
)
## Engine selected: direct inversion (henderson=FALSE)
model_regular$covParNative
## term factor section parameter estimate StdError Zratio
## 1 vsm(ism(id), Gu = Ai) sigma2 <NA> sigma2 0.2090522 0.04287764 4.875552
## 2 vsm(ism(units)) sigma2 <NA> sigma2 0.6376369 0.04619642 13.802736
model_rotation_h$covParNative
## term factor section parameter estimate
## 1 vsm(ism(id), Gu = Ai, rotation = TRUE) sigma2 <NA> sigma2 0.2090522
## 2 vsm(ism(units)) sigma2 <NA> sigma2 0.6376369
## StdError Zratio
## 1 0.04287764 4.875552
## 2 0.04619642 13.802736
model_rotation_d$covParNative
## term factor section parameter estimate
## 1 vsm(ism(id), Gu = Ai, rotation = TRUE) sigma2 <NA> sigma2 0.2090522
## 2 vsm(ism(units)) sigma2 <NA> sigma2 0.6376369
## StdError Zratio
## 1 0.04287764 4.875552
## 2 0.04619642 13.802736
plot(model_rotation_h$bu[,1], model_regular$bu[,1])
plot(model_rotation_d$bu[,1], model_regular$bu[,1])
The transformed and untransformed formulations are algebraically equivalent under the rotation assumptions. The current implementation accepts one rotated relationship term in a complete balanced Gaussian model and computeCi=0 during fitting. Residual structures are supported when they are rotation invariant, i.e. when residuals may covary across rotation blocks such as traits or environments but are independent and identically structured across the levels of the rotated term: (R=\Sigma_e\otimes I). Examples are rcov=~units, rcov=~vsm(dsm(env), ism(units)), and rcov=~vsm(usm(trait), ism(units)). Residual structures correlating observations of different levels (e.g., ar1m() or maternm() over plots), residual variances differing among levels, and W matrices that are not diagonal with a constant weight within each rotation block are not invariant and are rejected. With henderson=TRUE, mmes() uses diagonal relationship precision in the eigen-coefficient basis. With henderson=FALSE, it uses the Lee–van der Werf transformed observation-covariance formulation. The relative computational benefit depends on model dimensions and sparsity.
Another example of how usefult this can be is in the multi-trait models:
data("DT_wheat", package="enhancer")
rownames(GT_wheat) <- rownames(DT_wheat)
GT <- apply(GT_wheat,2,as.numeric)
rownames(GT) <- rownames(GT_wheat)
A <- A.mat(GT)
Ai <- solve(A + diag(1e-5, nrow(A)))
Ai[lower.tri(Ai)] <- t(Ai)[lower.tri(Ai)] # fill the lower triangular
Ai <- as(as(as(Ai, "dMatrix"), "generalMatrix"), "CsparseMatrix")
attr(Ai, "inverse") <- TRUE
isSymmetric(Ai)
## [1] TRUE
# One complete observation per relationship level gives a balanced example.
data(DT_wheat)
DT <- DT_wheat
GT <- apply(GT_wheat,2,as.numeric)
rownames(GT) <- rownames(GT_wheat)
DT <- data.frame(pheno=as.vector(DT),
env=as.factor(paste0("e", sort(rep(1:4,nrow(DT))))),
id=rep(rownames(DT),4))
# Henderson MME formulation with eigenbasis random coefficients.
model_rotation_h <- mmes(
pheno~1,
random=~vsm(usm(env),ism(id), Gu=Ai, rotation=TRUE),
rcov=~vsm(dsm(env),ism(units)), data=DT, henderson=TRUE, verbose=FALSE
)
## Solver selected: ldlt
# genetic correlation
cov2cor(model_rotation_h$theta[[1]])
## [,1] [,2] [,3] [,4]
## [1,] 1.0000000 -0.2041513 -0.2901046 -0.3899389
## [2,] -0.2041513 1.0000000 0.9931139 0.7908472
## [3,] -0.2901046 0.9931139 1.0000000 0.8491582
## [4,] -0.3899389 0.7908472 0.8491582 1.0000000
3) Mating designs
Estimating variance components has been a topic of interest for the breeding community for a long time. Here we show how to calculate additive and dominance variance using the North Carolina Design I (Nested design) and North Carolina Design II (Factorial design) using the classical Expected Mean Squares method and the REML methods from sommer and how these two are equivalent.
North Carolina Design I (Nested design)
data(DT_expdesigns, package="enhancer")
DT <- DT_expdesigns$car1
DT <- aggregate(yield~set+male+female+rep, data=DT, FUN = mean)
DT$setf <- as.factor(DT$set)
DT$repf <- as.factor(DT$rep)
DT$malef <- as.factor(DT$male)
DT$femalef <- as.factor(DT$female)
# lattice::levelplot(yield~male * malef:femalef|set, data=DT, main="NC design I")
##############################
## Expected Mean Square method
##############################
mix1 <- lm(yield~ setf + setf:repf + femalef:malef:setf + malef:setf, data=DT)
MS <- anova(mix1); MS
## Analysis of Variance Table
##
## Response: yield
## Df Sum Sq Mean Sq F value Pr(>F)
## setf 1 0.1780 0.17796 1.6646 0.226012
## setf:repf 2 0.9965 0.49824 4.6605 0.037141 *
## setf:malef 4 7.3904 1.84759 17.2822 0.000173 ***
## setf:femalef:malef 6 1.6083 0.26806 2.5074 0.095575 .
## Residuals 10 1.0691 0.10691
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
ms1 <- MS["setf:malef","Mean Sq"]
ms2 <- MS["setf:femalef:malef","Mean Sq"]
mse <- MS["Residuals","Mean Sq"]
nrep=2
nfem=2
Vfm <- (ms2-mse)/nrep
Vm <- (ms1-ms2)/(nrep*nfem)
## Calculate Va and Vd
Va=4*Vm # assuming no inbreeding (4/(1+F))
Vd=4*(Vfm-Vm) # assuming no inbreeding(4/(1+F)^2)
Vg=c(Va,Vd); names(Vg) <- c("Va","Vd"); Vg
## Va Vd
## 1.579537 -1.257241
##############################
## REML method
##############################
mix2 <- mmes(yield~ setf + setf:repf,
random=~femalef:malef:setf + malef:setf,
data=DT, verbose = FALSE)
## Solver selected: ldlt
vc <- summary(mix2)$varcomp; vc
## term parameter estimate StdError Zratio
## 1 vsm(ism(femalef:malef:setf)) sigma2 0.08066013 0.08103778 0.9953397
## 2 vsm(ism(malef:setf)) sigma2 0.39473881 0.32877865 1.2006218
## 3 vsm(ism(units)) sigma2 0.10686297 0.04778064 2.2365328
Vfm <- vc[1,"estimate"]
Vm <- vc[2,"estimate"]
## Calculate Va and Vd
Va=4*Vm # assuming no inbreeding (4/(1+F))
Vd=4*(Vfm-Vm) # assuming no inbreeding(4/(1+F)^2)
Vg=c(Va,Vd); names(Vg) <- c("Va","Vd"); Vg
## Va Vd
## 1.578955 -1.256315
As can be seen the REML method is easier than manipulating the MS and we arrive to the same results.
North Carolina Design II (Factorial design)
DT <- DT_expdesigns$car2
DT <- aggregate(yield~set+male+female+rep, data=DT, FUN = mean)
DT$setf <- as.factor(DT$set)
DT$repf <- as.factor(DT$rep)
DT$malef <- as.factor(DT$male)
DT$femalef <- as.factor(DT$female)
#levelplot(yield~male*female|set, data=DT, main="NC desing II")
head(DT)
## set male female rep yield setf repf malef femalef
## 1 1 1 1 1 831.03 1 1 1 1
## 2 1 2 1 1 1046.55 1 1 2 1
## 3 1 3 1 1 853.33 1 1 3 1
## 4 1 4 1 1 940.00 1 1 4 1
## 5 1 5 1 1 802.00 1 1 5 1
## 6 1 1 2 1 625.93 1 1 1 2
N=with(DT,table(female, male, set))
nmale=length(which(N[1,,1] > 0))
nfemale=length(which(N[,1,1] > 0))
nrep=table(N[,,1])
nrep=as.numeric(names(nrep[which(names(nrep) !=0)]))
##############################
## Expected Mean Square method
##############################
mix1 <- lm(yield~ setf + setf:repf +
femalef:malef:setf + malef:setf + femalef:setf, data=DT)
MS <- anova(mix1); MS
## Analysis of Variance Table
##
## Response: yield
## Df Sum Sq Mean Sq F value Pr(>F)
## setf 1 847836 847836 45.6296 1.097e-09 ***
## setf:repf 4 144345 36086 1.9421 0.109652
## setf:malef 8 861053 107632 5.7926 5.032e-06 ***
## setf:femalef 8 527023 65878 3.5455 0.001227 **
## setf:femalef:malef 32 807267 25227 1.3577 0.129527
## Residuals 96 1783762 18581
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
ms1 <- MS["setf:malef","Mean Sq"]
ms2 <- MS["setf:femalef","Mean Sq"]
ms3 <- MS["setf:femalef:malef","Mean Sq"]
mse <- MS["Residuals","Mean Sq"]
nrep=length(unique(DT$rep))
nfem=length(unique(DT$female))
nmal=length(unique(DT$male))
Vfm <- (ms3-mse)/nrep;
Vf <- (ms2-ms3)/(nrep*nmale);
Vm <- (ms1-ms3)/(nrep*nfemale);
Va=4*Vm; # assuming no inbreeding (4/(1+F))
Va=4*Vf; # assuming no inbreeding (4/(1+F))
Vd=4*(Vfm); # assuming no inbreeding(4/(1+F)^2)
Vg=c(Va,Vd); names(Vg) <- c("Va","Vd"); Vg
## Va Vd
## 10840.192 8861.659
##############################
## REML method
##############################
mix2 <- mmes(yield~ setf + setf:repf ,
random=~femalef:malef:setf + malef:setf + femalef:setf,
data=DT, verbose = FALSE)
## Solver selected: ldlt
vc <- summary(mix2)$varcomp; vc
## term parameter estimate StdError Zratio
## 1 vsm(ism(femalef:malef:setf)) sigma2 2218.157 2284.985 0.9707533
## 2 vsm(ism(malef:setf)) sigma2 5492.783 3611.924 1.5207358
## 3 vsm(ism(femalef:setf)) sigma2 2709.204 2235.487 1.2119081
## 4 vsm(ism(units)) sigma2 18578.042 2681.306 6.9287284
Vfm <- vc[1,"estimate"]
Vm <- vc[2,"estimate"]
Vf <- vc[3,"estimate"]
Va=4*Vm; # assuming no inbreeding (4/(1+F))
Va=4*Vf; # assuming no inbreeding (4/(1+F))
Vd=4*(Vfm); # assuming no inbreeding(4/(1+F)^2)
Vg=c(Va,Vd); names(Vg) <- c("Va","Vd"); Vg
## Va Vd
## 10836.817 8872.629
As can be seen, the REML method is easier than manipulating the MS and we arrive to the same results.
4) GWAS by GBLUP
Gualdron-Duarte et al. (2014) and Bernal-Rubio et al. (2016) proved that in (SingleStep)GBLUP or RRBLUP/SNP-BLUP, dividing the estimate of the marker effect by its standard error is mathematically equivalent to fixed regression EMMAX GWAS, even if markers are estimated as random effects in GBLUP and as fixed effects in EMMAX. That way fitting a GBLUP model is enough to perform GWAS for additive and on-additive effects.
Let us use the DT_cpdata dataset to explore the GWAS by GBLUP method
data(DT_cpdata, package="enhancer")
DT <- DT_cpdata
GT <- GT_cpdata[,1:200]
MP <- MP_cpdata
#### create the variance-covariance matrix
A <- A.mat(GT) # additive relationship matrix
n <- nrow(DT) # to be used for degrees of freedom
k <- 1 # to be used for degrees of freedom (number of levels in fixed effects)
First we fit a regular GWAS/EMMAX using the GWAS function available in sommer that first calculates variance components and then fits a regression marker by marker as a fixed effect.
###########################
#### Regular GWAS/EMMAX approach
###########################
# mix2 <- GWAS(color~1,
# random=~vsm(ism(id), Gu=A) + Rowf + Colf,
# rcov=~units, M=GT, gTerm = "u:id",
# verbose = FALSE,
# data=DT)
To compare EMMAX to the approach proposed by Gualdron-Duarte et al. (2014) and Bernal-Rubio et al. (2016) we will start fitting an RRBLUP/SNP-BLUP model to show that the estimate of the marker effect by its standard error is mathematically equivalent to fixed regression EMMAX GWAS.
# ###########################
# #### GWAS by RRBLUP approach
# ###########################
# Z <- GT[as.character(DT$id),]
# mixRRBLUP <- mmer(Yield~1,
# random=~vsr(list(Z)) + Rowf + Colf,
# rcov=~units, nIters=10,
# verbose = FALSE,
# data=DT)
#
# a <- mixRRBLUP$U$`u:Z`$Yield
# se.a <- sqrt( diag(kronecker(diag(ncol(Z)),mixRRBLUP$sigma$`u:Z`) - mixRRBLUP$PevU$`u:Z`$Yield ) ) # SE of marker effects
# t.stat <- a/se.a # t-statistic
# pvalRRBLUP <- dt(t.stat,df=n-k-1) # -log10(pval)
Instead of fitting the RRBLUP/SNP-BLUP model we can fit a GBLUP model which is less computationally demanding and recover marker effects and their standard errors from the genotype effects.
# ###########################
# #### GWAS by GBLUP approach
# ###########################
# M<- GT
# MMT <-tcrossprod(M) ## MM' = additive relationship matrix
# MMTinv<-solve(MMT + diag(1e-4, ncol(MMT), ncol(MMT))) ## inverse of MM'
# MTMMTinv<-t(M)%*%MMTinv # M' %*% (M'M)-
# MMTinv <- as(as(as( MMTinv, "dMatrix"), "generalMatrix"), "CsparseMatrix")
# attr(MMTinv, 'inverse')=TRUE
#
# mixGBLUP <- mmes(Yield~1,
# random=~vsm(ism(id), Gu=MMTinv) + Rowf + Colf,
# rcov=~units, nIters=25,
# verbose = T, computeCi = 2,
# data=DT)
# a.from.g <-MTMMTinv%*%matrix(mixGBLUP$uList$`vsm(ism(id), Gu = MMTinv`,ncol=1)
# start=mixGBLUP$partitions[[1]][1]
# end=mixGBLUP$partitions[[1]][2]
# var.g <- kronecker(MMT,mixGBLUP$theta[[1]]) - mixGBLUP$Ci[start:end,start:end]
# var.a.from.g <- t(M)%*%MMTinv%*% (var.g) %*% t(MMTinv)%*%M
# se.a.from.g <- sqrt(diag(var.a.from.g))
# t.stat.from.g <- a.from.g/se.a.from.g # t-statistic
# pvalGBLUP <- dt(t.stat.from.g,df=n-k-1) # -log10(pval)
Now we can look at the p-values coming from the 3 approaches to indeed show that results are equivalent.
###########################
#### Compare results
###########################
# plot(mix2$scores[,1], main="GWAS")
# plot(-log(pvalRRBLUP), main="GWAS by RRBLUP/SNP-BLUP")
# plot(-log(pvalGBLUP), main="GWAS by GBLUP")
Final remarks
Keep in mind that the current mmes() solver uses Henderson’s mixed model equations with sparse Average Information REML; it is not the former dense direct-inversion implementation. Its performance depends strongly on sparsity, the number and structure of random effects, fill-in in the sparse factorization, and the covariance models being fitted. The complementary mmer() marginal-covariance formulation remains useful for models whose dimensions or covariance representation favor observation-space calculations. Thus solver choice should be based on the structure of the particular problem rather than only on whether the number of records exceeds the number of coefficients.
Literature
Covarrubias-Pazaran G. 2016. Genome assisted prediction of quantitative traits using the R package sommer. PLoS ONE 11(6):1-15.
Covarrubias-Pazaran G. 2018. Software update: Moving the R package sommer to multivariate mixed models for genome-assisted prediction. doi: https://doi.org/10.1101/354639
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, D.-J., Durban, M., and Eilers, P.H.C. (2013). Efficient two-dimensional smoothing with P-spline ANOVA mixed models and nested bases. Computational Statistics and Data Analysis, 61, 22 - 37.
Lee, S.H. and van der Werf, J.H.J. 2016. MTG2: An efficient algorithm for multivariate linear mixed model analysis based on genomic information. Bioinformatics 32(9):1420-1422. doi:10.1093/bioinformatics/btw012.
Maier et al. 2015. Joint analysis of psychiatric disorders increases accuracy of risk prediction for schizophrenia, bipolar disorder, and major depressive disorder. Am J Hum Genet; 96(2):283-294.
Rodriguez-Alvarez, Maria Xose, et al. Correcting for spatial heterogeneity in plant breeding experiments with P-splines. Spatial Statistics 23 (2018): 52-71.
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.
Tunnicliffe W. 1989. On the use of marginal likelihood in time series model estimation. JRSS 51(1):15-27.