Me for the reference population

Jack C. M. Dekkers; Hailin Su; Jian Cheng

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Me for the reference population

JD Jack C. M. Dekkers

HS Hailin Su

JC Jian Cheng

This method is extracted from research article: Genet Sel Evol, Jun 2021

Predicting the accuracy of genomic predictions

DOI: 10.1186/s12711-021-00647-w

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If an estimate of $M_{e}$ for the reference population is available, the accuracy of GEBV in the reference population can be estimated using Eqs. (1), (2), and (3) to predict $r_{D_{r}} .$ Cross-validation or pseudoBLUP methodology can be used to predict $r_{A_{r}} .$ These two accuracies can then be combined to predict $r_{G_{r}},$ using either the Fisher approach (Eqs. (6) and (7)) or the Index approach (Eq. (9)).

If $M_{e}$ for the reference population is not known, it can be derived using different approaches:

Based on theoretical functions of effective population size ( $N_{e}$ ), reference size ( $N$ ), and genome size in terms of number of chromosomes ( $k$ ) and the individual or average ( $L$ in Morgans) size of chromosomes [5, 7, 9, 14]. Here, two such theoretical predictions of $M_{e}$ will be used: $M_{e} = 2 N_{e} L k$ based on [8], and $M_{e} = 2 N_{e} Lk / l n (N_{e} L)$ based on [7].

Based on the inverse of the variance of relationships [8]. Because $M_{e}$ is used to estimate the accuracy of DEBV, the variance of genomic minus pedigree relationships among all pairs of individuals in the reference population was used.

Based on observed accuracies of GEBV and of PEBV in the reference population, $r_{G_{r}}$ and $r_{A_{r}}$ , using the relationships among the accuracies derived above based on either the Fisher or the Index approach:

Using the Fisher approach, $θ_{G}$ and $θ_{A}$ can be computed from the observed $r_{G_{r}}$ and $r_{A_{r}}$ using Eq. (4), with $q_{G}^{2} = q_{A}^{2} = 1$ . Fisher information statistic $θ_{D}$ can then be computed as $θ_{D} = θ_{G} - θ_{A}$ based on Eq. (5). Substituting Eq. (3) into Eq. (2) results in the following quadratic form in $M_{e}$ : $θ_{D} M_{e}^{2} + θ_{D} M M_{e} - N M h^{2} = 0$ , which can be solved for $M_{e}$ as:

Using the Index approach, $r_{D_{r}}^{2}$ can be computed from the observed $r_{G_{r}}$ and $r_{A_{r}}$ using Eq. (9), which can then be used to compute $θ_{D}$ for a given value of $q_{D}^{2}$ using Eq. (6). $M_{e}$ can then be derived from $θ_{D}$ using Eq. (2), resulting in:

Because $q_{D}^{2} = M / (M + M_{e})$ based on Eq. (3), the solution for $M_{e}$ must be obtained in an iterative manner by substituting the new value of $q_{D}^{2}$ based on Eq. (3) back into Eq. (11) until a stable value of $M_{e}$ is obtained (see Appendix 1).

Open AccessThis article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. The Creative Commons Public Domain Dedication waiver (http://creativecommons.org/publicdomain/zero/1.0/) applies to the data made available in this article, unless otherwise stated in a credit line to the data.

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