Correction: Theoretical accuracy for indirect predictions based on SNP effects from single-step GBLUP,Genetics Selection Evolution

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Correction: Theoretical accuracy for indirect predictions based on SNP effects from single-step GBLUP
Genetics Selection Evolution ( IF 3.6 ) Pub Date : 2023-04-17 , DOI: 10.1186/s12711-023-00799-x
Andre Garcia ₁ , Ignacio Aguilar ₂ , Andres Legarra ₃ , Shogo Tsuruta ₁ , Ignacy Misztal ₁ , Daniela Lourenco ₁

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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 $\mathrm{var}(\widehat{\mathbf{a}})$ as the prediction error covariance (PEC) of SNP effects which is not correct.

The correct equations should be:

$${\mathbf{var}}\left(\widehat{\mathbf{a}}\right)\text{ = var}\left(\left({1}-{\alpha }\right){{b}}{\frac{1}{2\sum {\mathrm{p}}_{i}\left(1-{\mathrm{p}}_{i}\right)}\mathbf{Z}}{\boldsymbol{^{\prime}}}{\mathbf{G}}^{-{1}}\widehat{\mathbf{u}}\right)$$(14)$$=\left({1}-\alpha \right){{b}}{\frac{1}{2\sum {\mathrm{p}}_{i}\left(1-{\mathrm{p}}_{i}\right)}\mathbf{Z}}{\mathbf{^{\prime}}}{\mathbf{G}}^{-{\boldsymbol{1}}}\mathrm{var}(\widehat{\mathbf{u}}){\left(\left(1-\alpha \right){{b}}\frac{1}{2\sum {\mathrm{p}}_{i}\left(1-{\mathrm{p}}_{i}\right)}{\mathbf{Z}}{\boldsymbol{^{\prime}}}{\mathbf{G}}^{-{\boldsymbol1}}\right)}^{^{\prime}}$$(15)$$=\left({1}-\mathrm{\alpha }\right){{b}}{\frac{1}{2\sum {\mathrm{p}}_{i}\left(1-{\mathrm{p}}_{i}\right)}\mathbf{Z}}{\mathbf{^{\prime}}}{\mathbf{G}}^{-{\boldsymbol{1}}}\left(\mathrm{var}\left(\mathbf{u}\right)-\mathrm{var}\left(\widehat{\mathbf{u}}-\mathbf{u}\right)\right){\mathbf{G}}^{-{\boldsymbol{1}}}{\mathbf{Z}}\frac{1}{2\sum {\mathrm{p}}_{i}\left(1-{\mathrm{p}}_{i}\right)}{{b}}\left({1}-{\alpha }\right).$$(16)

Then,

$${\text{var}}\left(\widehat{\mathbf{a}}\right)= \text{ } \left({1}-{\alpha }\right){b}{\frac{1}{2\sum {\mathrm{p}}_{i}\left(1-{\mathrm{p}}_{i}\right)}\mathbf{Z}}{\boldsymbol{^{\prime}}}{\mathbf{G}}^{-{\boldsymbol{1}}}\left({\mathbf{G}}{\upsigma}_{\text{u}}^{2}-{\mathbf{C}}^{{\mathrm{u}}_{2}{\mathrm{u}}_{2}}\right){\mathbf{G}}^{-{\boldsymbol{1}}}{\mathbf{Z}}\frac{1}{2\sum {\mathrm{p}}_{i}\left(1-{\mathrm{p}}_{i}\right)}b\left({1}-{\alpha }\right).$$(17)

Therefore,

$${\text{var}}\left(\widehat{\mathbf{a}}\right)= \text{ } \left({1}-\alpha \right){{b}}{\frac{1}{2\sum {\mathrm{p}}_{i}\left(1-{\mathrm{p}}_{i}\right)}(\mathbf{Z}}{\boldsymbol{^{\prime}}}{\mathbf{G}}^{-{\boldsymbol{1}}}\mathbf{Z}{\upsigma}_{\text{u}}^{2} \, -{\mathbf{Z}}{\boldsymbol{^{\prime}}{\mathbf{G}}^{-{\boldsymbol{1}}}{\mathbf{C}}^{{\mathrm{u}}_{2}{\mathrm{u}}_{2}}{\mathbf{G}}^{-{\boldsymbol1}}}\mathbf{Z)}\frac{1}{2\sum {\mathrm{p}}_{i}\left(1-{\mathrm{p}}_{i}\right)}b\left({1}-\alpha \right).$$(18)$${\mathrm{ACC}}_{{\mathrm{IP}}_{j}} \, \text{=}\sqrt{\frac{{\mathbf{z}}_{{j}}{\text{var}}\left(\widehat{\mathbf{a}}\right){\mathbf{z}}_{{j}}^{\prime}}{{\upsigma}_{\text{u}}^{2}}}.$$(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.

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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Authors and Affiliations

Department of Animal and Dairy Science, University of Georgia, Athens, GA, 30602, USA
Andre Garcia, Shogo Tsuruta, Ignacy Misztal & Daniela Lourenco
Instituto Nacional de Investigación Agropecuaria (INIA), 11500, Montevideo, Uruguay
Ignacio Aguilar
UMR GenPhySE, INRA Toulouse, BP52626, 31326, Castanet Tolosan, France
Andres Legarra

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Andre GarciaView author publications
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Ignacio AguilarView author publications
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Andres LegarraView author publications
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Ignacy MisztalView author publications
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Correspondence to Andre Garcia.

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Garcia, A., Aguilar, I., Legarra, A. et al. Correction: Theoretical accuracy for indirect predictions based on SNP effects from single-step GBLUP. Genet Sel Evol 55, 26 (2023). https://doi.org/10.1186/s12711-023-00799-x

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Published: 17 April 2023
DOI: https://doi.org/10.1186/s12711-023-00799-x

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中文翻译：

更正：基于单步 GBLUP 的 SNP 效应的间接预测的理论准确性

更正：遗传选择进化（2022）54:66 https://doi.org/10.1186/s12711-022-00752-4

这项工作 [1] 发表后，我们注意到方程式中存在错误。(14) 到 (19) 出现在论文的第 4 页，其中我们将术语$\mathrm{var}(\widehat{\mathbf{a}})$称为预测误差协方差 (PEC)不正确的 SNP 效应。

正确的方程应该是：

$${\mathbf{var}}\left(\widehat{\mathbf{a}}\right)\text{ = var}\left(\left({1}-{\alpha }\right){{b }}{\frac{1}{2\sum {\mathrm{p}}_{i}\left(1-{\mathrm{p}}_{i}\right)}\mathbf{Z}}{ \boldsymbol{^{\prime}}}{\mathbf{G}}^{-{1}}\widehat{\mathbf{u}}\right)$$ (14) $$=\left ( {1} -\alpha \right){{b}}{\frac{1}{2\sum {\mathrm{p}}_{i}\left(1-{\mathrm{p}}_{i}\right )}\mathbf{Z}}{\mathbf{^{\prime}}}{\mathbf{G}}^{-{\boldsymbol{1}}}\mathrm{var}(\widehat{\mathbf{u }}){\left(\left(1-\alpha \right){{b}}\frac{1}{2\sum {\mathrm{p}}_{i}\left(1-{\mathrm {p}}_{i}\right)}{\mathbf{Z}}{\boldsymbol{^{\prime}}}{\mathbf{G}}^{-{\boldsymbol1}}\right)}^ {^{\prime}}$$ (15)$$=\left({1}-\mathrm{\alpha }\right){{b}}{\frac{1}{2\sum {\mathrm{p}}_{i}\left(1- {\mathrm{p}}_{i}\right)}\mathbf{Z}}{\mathbf{^{\prime}}}{\mathbf{G}}^{-{\boldsymbol{1}}} \left(\mathrm{var}\left(\mathbf{u}\right)-\mathrm{var}\left(\widehat{\mathbf{u}}-\mathbf{u}\right)\right){ \mathbf{G}}^{-{\boldsymbol{1}}}{\mathbf{Z}}\frac{1}{2\sum {\mathrm{p}}_{i}\left(1-{ \mathrm{p}}_{i}\right)}{{b}}\left({1}-{\alpha }\right).$$ (16)

然后，

$${\text{var}}\left(\widehat{\mathbf{a}}\right)= \text{ } \left({1}-{\alpha }\right){b}{\frac{ 1}{2\sum {\mathrm{p}}_{i}\left(1-{\mathrm{p}}_{i}\right)}\mathbf{Z}}{\boldsymbol{^{\ prime}}}{\mathbf{G}}^{-{\boldsymbol{1}}}\left({\mathbf{G}}{\upsigma}_{\text{u}}^{2}-{ \mathbf{C}}^{{\mathrm{u}}_{2}{\mathrm{u}}_{2}}\right){\mathbf{G}}^{-{\boldsymbol{1} }}{\mathbf{Z}}\frac{1}{2\sum {\mathrm{p}}_{i}\left(1-{\mathrm{p}}_{i}\right)}b \left({1}-{\alpha }\right).$$ (17)

所以，

$${\text{var}}\left(\widehat{\mathbf{a}}\right)= \text{ } \left({1}-\alpha \right){{b}}{\frac{ 1}{2\sum {\mathrm{p}}_{i}\left(1-{\mathrm{p}}_{i}\right)}(\mathbf{Z}}{\boldsymbol{^{ \prime}}}{\mathbf{G}}^{-{\boldsymbol{1}}}\mathbf{Z}{\upsigma}_{\text{u}}^{2} \, -{\mathbf {Z}}{\boldsymbol{^{\prime}}{\mathbf{G}}^{-{\boldsymbol{1}}}{\mathbf{C}}^{{\mathrm{u}}_{ 2}{\mathrm{u}}_{2}}{\mathbf{G}}^{-{\boldsymbol1}}}\mathbf{Z)}\frac{1}{2\sum {\mathrm{p }}_{i}\left(1-{\mathrm{p}}_{i}\right)}b\left({1}-\alpha \right).$$ (18) $${ \ mathrm {ACC}}_{{\mathrm{IP}}_{j}} \, \text{=}\sqrt{\frac{{\mathbf{z}}_{{j}}{\text{var} }\left(\widehat{\mathbf{a}}\right){\mathbf{z}}_{{j}}^{\prime}}{{\upsigma}_{\text{u}}^{ 2}}}.$$ (19)

请注意，方程式在用于分析的软件中正确编程，上述更正不会影响手稿中显示的最终结果。

我们感谢佐治亚大学的 Matias Berman 博士和瓦赫宁根大学的 Jeremie Vandenplas 博士对手稿的修改和建议。

Garcia A、Aguilar I、Legarra A、Tsuruta S、Misztal I、Lourenco D. 基于来自单步 GBLUP 的 SNP 效应的间接预测的理论准确性。Genet Sel Evol。2022；54：66。https://doi.org/10.1186/s12711-022-00752-4。
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佐治亚大学动物与乳制品科学系，雅典，乔治亚州，30602，美国
Andre Garcia、Shogo Tsuruta、Ignacy Misztal 和 Daniela Lourenco
Instituto Nacional de Investigación Agropecuaria (INIA), 11500, 蒙得维的亚, 乌拉圭
伊格纳西奥·阿吉拉尔
UMR GenPhySE, INRA Toulouse, BP52626, 31326, Castanet Tolosan, 法国
安德烈斯·莱加拉

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