# Correction: Theoretical accuracy for indirect predictions based on SNP effects from single-step GBLUP

André Garcia, Ignacio Aguilar, Andres Legarra, Shogo Tsuruta, Ignacy

Misztal, Daniela Lourenco

► To cite this version:

André Garcia, Ignacio Aguilar, Andres Legarra, Shogo Tsuruta, Ignacy Misztal, et al. Correction: Theoretical accuracy for indirect predictions based on SNP effects from single-step GBLUP. Genetics Selection Evolution, 2023, 55 (1), pp.26. 10.1186/s12711-023-00799-x. hal-04095912

HAL Id: hal-04095912

https://hal.inrae.fr/hal-04095912

Submitted on 25 Jul 2023

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CORRECTION

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# Correction: Theoretical accuracy for indirect predictions based on SNP effects from single-step GBLUP

Andre Garcia<fnref n="1" />*, Ignacio Aguilar<fnref n="2" />, Andres Legarra<fnref n="3" />, Shogo Tsuruta<fnref n="1" />, Ignacy Misztal<fnref n="1" /> and Daniela Lourenco<fnref n="1" />

**Correction: Genetics Selection Evolution (2022) 54:66**
https://doi.org/10.1186/s12711-022-00752-4

After publication of this work [1], we noticed that there was an error in Eqs. (14) to (19) presented on page 4 of the paper, where we refer to the term var($\hat{\mathbf{a}}$) as the

prediction error covariance (PEC) of SNP effects which is not correct.

The correct equations should be:

$$ \mathbf{var}(\hat{\mathbf{a}}) = \mathrm{var}\left( (1-\alpha)b \frac{1}{2\sum p_i(1-p_i)} \mathbf{Z}'\mathbf{G}^{-1}\hat{\mathbf{u}} \right) \quad (14) $$

$$ = (1 - \alpha)b \frac{1}{2\sum p_i(1 - p_i)} \mathbf{Z}'\mathbf{G}^{-1}\mathrm{var}(\hat{\mathbf{u}}) \left( (1 - \alpha)b \frac{1}{2\sum p_i(1 - p_i)} \mathbf{Z}'\mathbf{G}^{-1} \right)' \quad (15) $$

$$ = (1 - \alpha)b \frac{1}{2\sum p_i(1 - p_i)} \mathbf{Z}'\mathbf{G}^{-1}(\mathrm{var}(\mathbf{u}) - \mathrm{var}(\hat{\mathbf{u}} - \mathbf{u}))\mathbf{G}^{-1}\mathbf{Z} \frac{1}{2\sum p_i(1 - p_i)} b(1 - \alpha). \quad (16) $$

The original article can be found online at https://doi.org/10.1186/s12711-022-00752-4.

<footnote>*Correspondence:</footnote>

<footnote>Andre Garcia
andreluizseccatto@gmail.com</footnote>

<footnote n="1">¹ Department of Animal and Dairy Science, University of Georgia, Athens, GA 30602, USA</footnote>

<footnote n="2">² Instituto Nacional de Investigación Agropecuaria (INIA), 11500 Montevideo, Uruguay</footnote>

<footnote n="3">³ UMR GenPhySE, INRA Toulouse, BP52626, 31326 Castanet Tolosan, France</footnote>
Then,

$$ \mathrm{var}(\hat{\mathbf{a}}) = (1 - \alpha)b \frac{1}{2 \sum p_i (1 - p_i)} \mathbf{Z}' \mathbf{G}^{-1} (\mathbf{G} \sigma_u^2 - \mathbf{C}^{u_2 u_2}) \mathbf{G}^{-1} \mathbf{Z} \frac{1}{2 \sum p_i (1 - p_i)} b(1 - \alpha). \quad (17) $$

Therefore,

$$ \mathrm{var}(\hat{\mathbf{a}}) = (1 - \alpha)b \frac{1}{2 \sum p_i (1 - p_i)} (\mathbf{Z}' \mathbf{G}^{-1} \mathbf{Z} \sigma_u^2 - \mathbf{Z}' \mathbf{G}^{-1} \mathbf{C}^{u_2 u_2} \mathbf{G}^{-1} \mathbf{Z}) \frac{1}{2 \sum p_i (1 - p_i)} b(1 - \alpha). \quad (18) $$

$$ \mathrm{ACC}_{\mathrm{IP}_j} = \sqrt{\frac{\mathbf{z}_j \mathrm{var}(\hat{\mathbf{a}}) \mathbf{z}_j'}{\sigma_u^2}}. \quad (19) $$

Please note that the equations are correctly programmed in the software used for analyses, and the corrections above do not affect the end results presented in the manuscript.

We thank Dr. Matias Berman from The University of Georgia and Dr. Jeremie Vandenplas from Wageningen University for their interest and suggestions to correct the manuscript.

Published online: 17 April 2023

### Reference

1. Garcia A, Aguilar I, Legarra A, Tsuruta S, Misztal I, Lourenco D. Theoretical accuracy for indirect predictions based on SNP effects from single-step GBLUP. Genet Sel Evol. 2022;54:66. https://doi.org/10.1186/s12711-022-00752-4.

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