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

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.