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Training set determination for genomic selection

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Abstract

Key message

A new optimality criterion is proposed to determine a training set for genomic selection, which is derived from Pearson’s correlation between GEBVs and phenotypic values of a test set. R functions are provided to generate the optimal training set.

Abstract

For a specified test set, we develop a highly efficient algorithm to determine an optimal subset from a large candidate set in which the individuals have been genotyped but not phenotyped yet. The chosen subset serves as a training set to be phenotyped, and then a genomic selection (GS) model is built based on its phenotype and genotype data. In this study, we consider the additive effects whole-genome regression model and adopt ridge regression estimation for marker effects in the GS model. The resulting GS model is then employed to predict genomic estimated breeding values (GEBVs) for the individuals of the test set, which have been genotyped only. We propose a new optimality criterion to determine the required training set, which is derived directly from Pearson’s correlation between GEBVs and phenotypic values of the test set. Pearson’s correlation is the standard measure for prediction accuracy of a GS model. Our proposed methods can be applied to data with the varying degree of population structure. All the R functions for implementing our training set determination algorithms are available from the R package TSDFGS. The algorithms are illustrated with two datasets which have strong (rice genome dataset) and mild (wheat genome dataset) population structures. Our methods are shown to be advantageous over existing ones, mainly because they fully use the genomic relationship between the test set and the training set by taking into account both the variance and bias for predicting GEBVs.

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Acknowledgements

This research was supported by the Ministry of Science and Technology Taiwan (Grant No.: MOST 106-2118-M-002-002-MY2).

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Correspondence to Chen-Tuo Liao.

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Communicated by Hiroyoshi Iwata.

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Supplementary material 1 (DOCX 396 kb)

Appendix: The derivation of r-score

Appendix: The derivation of r-score

Let \( \varvec{w}_{1} = \varvec{y}_{0} - \bar{y}_{0} {\mathbf{1}}_{{n_{0} }} \) and \( \varvec{w}_{2} = \varvec{g}_{0} - \bar{g}_{0} {\mathbf{1}}_{{n_{0} }} \), then Pearson’s correlation between \( \varvec{y}_{0} \) and \( \varvec{g}_{0} \) can be expressed in the matrix form

$$ r = \frac{{\varvec{w}_{1}^{{ \top }} \varvec{w}_{2} }}{{\sqrt {\left( {\varvec{w}_{1}^{{ \top }} \varvec{w}_{1} } \right)\left( {\varvec{w}_{2}^{{ \top }} \varvec{w}_{2} } \right)} }}. $$

We now consider a natural surrogate to the expectation of \( r \) by replacing \( \varvec{w}_{1}^{{ \top }} \varvec{w}_{2} \), \( \varvec{w}_{1}^{{ \top }} \varvec{w}_{1} \) and \( \varvec{w}_{2}^{{ \top }} \varvec{w}_{2} \) with their respective expectations, yielding

$$ E\left( {\tilde{r}} \right) = \frac{{E\left( {\varvec{w}_{1}^{{ \top }} \varvec{w}_{2} } \right)}}{{\sqrt {E\left( {\varvec{w}_{1}^{{ \top }} \varvec{w}_{1} } \right)E\left( {\varvec{w}_{2}^{{ \top }} \varvec{w}_{2} } \right)} }}. $$

According to the identity (Searle 1982, p. 355),

$$ E\left( {\varvec{v}_{1}^{{ \top }} \varvec{v}_{2} } \right) = {\text{Tr}}\left[ {{\text{Cov}}\left( {\varvec{v}_{1} ,\varvec{v}_{2} } \right)} \right] + E\left( {\varvec{v}_{1}^{{ \top }} } \right)E\left( {\varvec{v}_{2} } \right) $$

for two random variables vectors \( \varvec{v}_{1} \) and \( \varvec{v}_{2} \) of same order, where \( {\text{Cov}}\left( {\varvec{v}_{1} , \varvec{v}_{2} } \right) \) denotes the covariance matrix between \( \varvec{v}_{1} \) and \( \varvec{v}_{2} \), we have that

$$ E\left( {\varvec{w}_{1}^{{ \top }} \varvec{w}_{1} } \right) = {\text{Tr}}\left[ {\left( {\varvec{I}_{{n_{0} }} - \bar{\varvec{J}}_{{n_{0} }} } \right)} \right]\sigma_{e}^{2} +\varvec{\beta}^{{ \top }} \varvec{X}_{0}^{{ \top }} \left( {\varvec{I}_{{n_{0} }} - \bar{\varvec{J}}_{{n_{0} }} } \right)\varvec{X}_{0}\varvec{\beta}; $$
(8)
$$ E\left( {\varvec{w}_{2}^{{ \top }} \varvec{w}_{2} } \right) = {\text{Tr}}\left[ {\varvec{A}^{{ \top }} \varvec{X}_{0}^{{ \top }} \left( {\varvec{I}_{{n_{0} }} - \bar{\varvec{J}}_{{n_{0} }} } \right)\varvec{X}_{0} \varvec{A}} \right]\sigma_{e}^{2} +\varvec{\beta}^{{ \top }} \varvec{X}^{{ \top }} \varvec{A}^{{ \top }} \varvec{X}_{0}^{{ \top }} \left( {\varvec{I}_{{n_{0} }} - \bar{\varvec{J}}_{{n_{0} }} } \right)\varvec{X}_{0} \varvec{AX\beta }; $$
(9)
$$ E\left( {\varvec{w}_{1}^{{ \top }} \varvec{w}_{2} } \right) = \varvec{\beta}^{{ \top }} \varvec{X}_{0}^{{ \top }} \left( {\varvec{I}_{{n_{0} }} - \bar{\varvec{J}}_{{n_{0} }} } \right)\varvec{X}_{0} \varvec{AX\beta }. $$
(10)

If matrix \( \varvec{H} \) can be factorized as \( \varvec{H} = \varvec{RR}^{{ \top }} \) for some matrix \( \varvec{R} \), then \( \varvec{H} \) is nonnegative definite (Searle 1982, p. 206). Furthermore, \( h_{ii} \ge 0 \) for all \( i \), where \( h_{ii} \) denotes the ith diagonal element of \( \varvec{H} \). Thus, the quadratic form \( \varvec{\beta}^{{ \top }} \varvec{H\beta } \) is proportion to \( \sum h_{ii} \beta_{i}^{2} \), hence proportional to \( {\text{Tr}}\left( \varvec{H} \right) = \sum h_{ii} \) for any value of \( \beta_{i} \). Based on the above result, we have that \( E\left( {\varvec{w}_{1}^{{ \top }} \varvec{w}_{1} } \right) \) of Eq. (8) is proportional to

$$ q_{1} = \left( {n_{0} - 1} \right) + {\text{Tr}}\left[ {\varvec{X}_{0}^{{ \top }} \left( {\varvec{I}_{{n_{0} }} - \bar{\varvec{J}}_{{n_{0} }} } \right)\varvec{X}_{0} } \right]. $$

Also, \( E\left( {\varvec{w}_{2}^{{ \top }} \varvec{w}_{2} } \right) \) of Eq. (9) is an increasing function of

$$ q_{2} = {\text{Tr}}\left[ {\varvec{A}^{{ \top }} \varvec{X}_{0}^{{ \top }} \left( {\varvec{I}_{{n_{0} }} - \bar{\varvec{J}}_{{n_{0} }} } \right)\varvec{X}_{0} \varvec{A}} \right] + {\text{Tr}}\left[ {\varvec{X}^{{ \top }} \varvec{A}^{{ \top }} \varvec{X}_{0}^{{ \top }} \left( {\varvec{I}_{{n_{0} }} - \bar{\varvec{J}}_{{n_{0} }} } \right)\varvec{X}_{0} \varvec{AX}} \right]. $$

From Eq. (10), we consider

$$ \begin{aligned} & {\text{Tr}}\left[ {\varvec{X}_{0}^{{ \top }} \left( {\varvec{I}_{{n_{0} }} - \bar{\varvec{J}}_{{n_{0} }} } \right)\varvec{X}_{0} \varvec{AX}} \right] \\ & = {\text{Tr}}\left[ {\varvec{X}_{0}^{{ \top }} \left( {\varvec{I}_{{n_{0} }} - \bar{\varvec{J}}_{{n_{0} }} } \right)\varvec{X}_{0} \varvec{X}^{{ \top }} \left( {\varvec{XX}^{{ \top }} + \lambda \varvec{I}_{n} } \right)^{ - 1} \varvec{X}} \right] \\ & = {\text{Tr}}\left[ {\varvec{RR}^{{ \top }} } \right] \\ \end{aligned} $$

where \( \varvec{R} = \left( {\varvec{XX}^{{ \top }} + \lambda \varvec{I}_{n} } \right)^{{ - \frac{1}{2}}} \varvec{XX}_{0}^{{ \top }} \left( {\varvec{I}_{{n_{0} }} - \bar{\varvec{J}}_{{n_{0} }} } \right) \). Here \( \varvec{B}^{{ - \frac{1}{2}}} \varvec{B}^{{ - \frac{1}{2}}} = \varvec{B}^{ - 1} \). So \( E\left( {\varvec{w}_{1}^{{ \top }} \varvec{w}_{2} } \right) \) is proportional to

$$ q_{12} = {\text{Tr}}\left[ {\varvec{X}_{0}^{{ \top }} \left( {\varvec{I}_{{n_{0} }} - \bar{\varvec{J}}_{{n_{0} }} } \right)\varvec{X}_{0} \varvec{AX}} \right]. $$

Consequently, it is reasonable to define r-score as \( \frac{{q_{12} }}{{\sqrt {q_{1} q_{2} } }} \).

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Ou, JH., Liao, CT. Training set determination for genomic selection. Theor Appl Genet 132, 2781–2792 (2019). https://doi.org/10.1007/s00122-019-03387-0

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