Single-Step GBLUP

Introduction

  • Genotypes are available only on a few thousand (non-random) individuals

  • Phenotype and pedigree information available on millions

  • Often, phenotypes are not available on genotyped individuals (sires)

  • Training (estimation of marker effects) based on deregressed EBV

  • Marker-based EBV combined with pedigree-based EBV using selection index theory

  • An alternative, single-step approach proposed by Legarra et al. @Legarra:2009:J-Dairy-Sci:19700729 [@Misztal:2009:J-Dairy-Sci:19700728; @Aguilar:2010:J-Dairy-Sci:20105546]

Theory

Marker Effect Model

$$\mathbf{y}=\mathbf{X}\boldsymbol{\beta}+\mathbf{M}\boldsymbol{\alpha}+\mathbf{e},$$

  • $\boldsymbol{\beta}$ are fixed effects

  • $\mathbf{X}$ incidence matrix for fixed effects

  • $\mathbf{M}$ marker covariates

  • $\boldsymbol{\alpha}\sim N(\mathbf{0},\mathbf{I}\sigma_{\alpha}^{2})$

  • $\mathbf{e}\sim N(\mathbf{0},\mathbf{I\sigma_{e}^{2})}$

Breeding Value Models

  • Two mixed linear models are linearly equivalent and will lead to the same inferences if the vector $\mathbf{y}$ of observations has the same first and second moments in both models @Henderson.CR:1984.

  • In this sense, a model that is equivalent to ([eq:2-1]) can be written as

    $$\mathbf{y}=\mathbf{X}\boldsymbol{\beta}+\mathbf{g}+\mathbf{e},$$

  • $\mathbf{g}=\mathbf{M}\boldsymbol{\alpha}$ has

    • null means and

    • covariance matrix

      $$\begin{aligned} Var(\mathbf{g}|\mathbf{M}) &=Var(\mathbf{M}\boldsymbol{\alpha})\\ &=\mathbf{M}Var(\boldsymbol{\alpha})\mathbf{M}'. \end{aligned}$$

  • Then, in both models ([eq:2-1]) and ([eq:GM]), the mean of $\mathbf{y}$ is $\mathbf{X}\boldsymbol{\beta}$ and

  • the covariance matrix is $$Var(\mathbf{y}|\mathbf{M})=\mathbf{M}Var(\boldsymbol{\alpha})\mathbf{M}'+\mathbf{I}\sigma_{e}^{2}.$$ Thus, these two models are linearly equivalent and will lead to the same inferences.

  • When the number of markers is large relative to the size of $\mathbf{g}$, BLUP of $\mathbf{g}$ can be obtained efficiently @VanRaden:2008:J-Dairy-Sci:18946147 [@Stranden:2009:J-Dairy-Sci:19448030] by solving the MME that correspond to ([eq:GM]).

  • Under some assumptions,

    $$\sigma_{\alpha}^{2}=\frac{\sigma_{g}^{2}}{\sum_{j}2p_{j}(1-p_{j})}$$

  • So,

    $$\begin{aligned} Var(\mathbf{g}|\mathbf{M}) & = \frac{\mathbf{MM}'}{\sum_{j}2p_{j}(1-p_{j})}\sigma_{g}^{2}\nonumber \\ & = \mathbf{G}\sigma_{g}^{2}. \end{aligned}$$

BLUP combining genotype and pedigree relationships

  • Suppose $\mathbf{g}$ is partitioned as $$\mathbf{g}=\left[\begin{array}{c} \mathbf{g}_{1}\\ \mathbf{g}_{2} \end{array}\right]=\left[\begin{array}{l} \mathbf{g}_{1}\\ \mathbf{M}_{2}\boldsymbol{\alpha} \end{array}\right],$$
  • $\mathbf{g_{1}}$ are the genomic BVs of the animals with missing genotypes $\mathbf{M}_{1}$

  • $\mathbf{g}_{2}$ are the BVs of those with observed genotypes $\mathbf{M}_{2}$.

  • Following Legarra et al. @Legarra:2009:J-Dairy-Sci:19700729, the vector $\mathbf{g}_{1}$ can be written as

    $$\begin{aligned} \mathbf{g}_{1} & = \mathbf{A}_{12}\mathbf{A}_{22}^{-1}\mathbf{g}_{2}+(\mathbf{g}_{1}-\mathbf{A}_{12}\mathbf{A}_{22}^{-1}\mathbf{g}_{2})\nonumber \\ & = \hat{\mathbf{g}}_{1}+\boldsymbol{\epsilon}, \end{aligned}$$

  • $\mathbf{A}_{ij}$ are partitions of $\mathbf{A}$ that correspond to $\mathbf{g}_{1}$ and $\mathbf{g}_{2}$.

  • The first term in ([eq:g1.2]) is the best linear predictor (BLP) of $\mathbf{g}_{1}$ given $\mathbf{g}_{2}$,

  • the second is a residual.

  • It is easy to see that $\boldsymbol{\epsilon}$ in ([eq:g1.2]) is uncorrelated to $\mathbf{g}_{2}$,

  • therefore if $\mathbf{g}_{1}$ and $\mathbf{g}_{2}$ are multivariate normal, $\mathbf{\epsilon}$ and $\mathbf{g}_{2}$ are independent.

  • Consider first the conditional distribution of $\mathbf{g}_{1}$ given $\mathbf{P}$. Then, as expected, the

    • variance of $\mathbf{g}_{1}$ is

      $$\begin{aligned} Var(\mathbf{g_{1}|}\mathbf{P}) & = [\mathbf{A}_{12}\mathbf{A}_{22}^{-1}\mathbf{A}_{21}+(\mathbf{A}_{11}-\mathbf{A}_{12}\mathbf{A}_{22}^{-1}\mathbf{A}_{21})]\sigma_{g}^{2}\\ & = \mathbf{A}_{11}\sigma_{g}^{2},\end{aligned}$$

      where the first term of ([eq:varg1]) is the variance of the $\hat{{\bf{g}}}_{1}$ and the second term is the variance of $\boldsymbol{\epsilon}$.

    • Similarly, $Var(\mathbf{g}_{2}|\mathbf{P})=\mathbf{A}_{22}\sigma_{g}^{2}$.

  • Consider now the conditional distribution of $\mathbf{g}_{1}$ given $\mathbf{M}_{2}$.

  • Note that, given the observed genotypes $\mathbf{M}_{2}$, the distribution of $\mathbf{g}_{2}$ changes to a multivariate normal with

    • mean $\mathbf{0}$ and

    • covariance matrix $\mathbf{M}_{2}\mathbf{M}_{2}'\sigma_{\alpha}^{2}$.

  • As explained below, this change in the distribution of $\mathbf{g}_{2}$, produces an associated change in the distribution of $\mathbf{g}_{1}$ to a normal with

    • mean $\mathbf{0}$ and
    • covariance matrix: $$ Var(\mathbf{g_{1}|M}_{2})=\mathbf{A}_{12}\mathbf{A}_{22}^{-1}\mathbf{M}_{2}\mathbf{M}'_{2}\mathbf{A}_{22}^{-1}\mathbf{A}_{21}\sigma_{\alpha}^{2}+(\mathbf{A}_{11}-\mathbf{A}_{12}\mathbf{A}_{22}^{-1}\mathbf{A}_{21})\sigma_{g}^{2}, $$
    • where now the vector $\hat{\mbox{g}}_{1}=\mathbf{A}_{12}\mathbf{A}_{22}^{-1}\mathbf{g}_{2}$ has covariance matrix given by the first term of ([eq:varg|z]),
    • because $\boldsymbol{\epsilon}$ is independent of $\mathbf{g}_{2}$, the second term of ([eq:varg|z]) remains identical to that of ([eq:varg1]).
  • Similarly, the covariance between $\mathbf{g}_{1}$ and $\mathbf{g}_{2}$ conditional on $\mathbf{M}_{2}$ is $$Cov(\mathbf{g}_{1},\mathbf{g}_{2})=\mathbf{A}_{12}\mathbf{A}_{22}^{-1}\mathbf{M}_{2}\mathbf{M}'_{2}\sigma_{\alpha}^{2}.$$
  • Further, assuming ([eq:MarkerGenVar]), the above results can be combined to show that conditional on $\mathbf{M}_{2}$,

    • $\mathbf{g}$ has a multivariate normal distribution with null mean and covariance matrix:

      $$Var(\mathbf{g}|\mathbf{M}_{2})=\mathbf{H}=\left[\begin{array}{cc} \mathbf{A}_{12}\mathbf{A}_{22}^{-1}\mathbf{G}\mathbf{A}_{22}^{-1}\mathbf{A}_{21}+(\mathbf{A}_{11}-\mathbf{A}_{12}\mathbf{A}_{22}^{-1}\mathbf{A}_{21}) & \mathbf{A}_{12}\mathbf{A}_{22}^{-1}\mathbf{G}\\ \mathbf{G}\mathbf{A}_{22}^{-1}\mbox{A}_{21} & \mathbf{G} \end{array}\right]\sigma_{g}^{2},$$

      where $\mathbf{G=}\mathbf{M}_{2}\mathbf{M}_{2}'/[\sum2p_{i}(1-p_{i})]$.

    • The inverse of this matrix is needed to setup the MME, and this is computed as $$\mathbf{H}^{-1}=\mathbf{A}^{-1}+\left[\begin{array}{ll} \mathbf{0} & \mathbf{0}\\ \mathbf{0} & \mathbf{G}^{-1}-\mathbf{A}_{22}^{-1} \end{array}\right].$$

  • Note that this requires computing both $\mathbf{G}^{-1}$ and $\mathbf{A}_{22}^{-1}$, which are dense and not easy to compute.

  • Due to the increased adoption of SNP genotyping in livestock, $\mathbf{G}^{-1}$ and $\mathbf{A}_{22}^{-1}$ are becoming too large for SS-GBLUP to remain computationally feasible (e.g. >100,000 animals).

  • A second problem in SS-GBLUP is related to the scaling that is done using the SNP frequencies.

  • As mentioned earlier, when all data that were used for selection are available for computing the conditional mean, it can be computed as if selection had not taken place @Gianola.D.ea:1986 [@Im.S.ea:1989; @Sorensen.DA.ea:2001].

  • If selection has taken place, this requires using SNP frequencies from the founders, as these frequencies are not changed by selection.

  • In most situations, however, SNP genotypes are not available in the founders and frequencies observed in the genotyped animals are used, which can lead to biased evaluations, particularly in a multi-breed context.

  • An approach very similar to that of using ([eq:g1.2]) to model missing genotypes was proposed by Fernando (see equation (43) [email protected]:_genom) in the context of genomic prediction using kernel regression, where missing genotypes were replaced by their conditional expectation computed using all available information, in contrast to using BLP as in SS-GBLUP.

  • Also, a residual that is similar to $\boldsymbol{\epsilon}$ was included in the model.

  • When these residuals are modeled exactly, the inverse of their covariance matrix is not sparse and SS-GBLUP would not be computationally feasible.