Orthogonal Estimates of Variances for Additive, Dominance, and Epistatic Effects in Populations
Authors: Andrés Legarra
Luis Varona
Miguel A Toro
Zulma G Vitezica
Journal: Genetics (2017)
Discipline: MOLECULAR BIOLOGY & GENETICS
Keywords: GenPred; NOIA approach; dominance; epistasis; genetic variance components
Affiliations: Institut National Polytechnique, École Nationale Supérieure Agronomique de Toulouse, Université de Toulouse, UMR 1388 Génétique, Physiologie et Systèmes d'Elevage, F-31326 Castanet-Tolosan, France ., Institut National de la Recherche Agronomique, UMR 1388 Génétique, Physiologie et Systèmes d'Elevage, F-31326 Castanet-Tolosan, France., Escuela Técnica Superior de Ingenieros Agrónomos, Universidad Politécnica de Madrid, 28040, Spain., Departamento de Anatomía, Embriología y Genética, Universidad de Zaragoza, 50013, Spain., Instituto Agroalimentario de Aragón, 50013 Zaragoza, Spain.

Abstract:  Genomic prediction methods based on multiple markers have potential to include nonadditive effects in prediction and analysis of complex traits. However, most developments assume a Hardy-Weinberg equilibrium (HWE). Statistical approaches for genomic selection that account for dominance and epistasis in a general context, without assuming HWE (e.g., crosses or homozygous lines), are therefore needed. Our method expands the natural and orthogonal interactions (NOIA) approach, which builds incidence matrices based on genotypic (not allelic) frequencies, to include genome-wide epistasis for an arbitrary number of interacting loci in a genomic evaluation context. This results in an orthogonal partition of the variances, which is not warranted otherwise. We also present the partition of variance as a function of genotypic values and frequencies following Cockerham's orthogonal contrast approach. Then we prove for the first time that, even not in HWE, the multiple-loci NOIA method is equivalent to construct epistatic genomic relationship matrices for higher-order interactions using Hadamard products of additive and dominant genomic orthogonal relationships. A standardization based on the trace of the relationship matrices is, however, needed. We illustrate these results with two simulated F1 (not in HWE) populations, either in linkage equilibrium (LE), or in linkage disequilibrium (LD) and divergent selection, and pure biological dominant pairwise epistasis. In the LE case, correct and orthogonal estimates of variances were obtained using NOIA genomic relationships but not if relationships were constructed assuming HWE. For the LD simulation, differences were smaller, due to the smaller deviation of the F1 from HWE. Wrongly assuming HWE to build genomic relationships and estimate variance components yields biased estimates, inflates the total genetic variance, and the estimates are not empirically orthogonal. The NOIA method to build genomic relationships, coupled with the use of Hadamard products for epistatic terms, allows the obtaining of correct estimates in populations either in HWE or not in HWE, and extends to any order of epistatic interactions. 

Retracted: No

PubPeer comments:
 The authors state in the abstract *  However, most developments assume a Hardyu2013Weinberg equilibrium (HWE). Statistical approaches for genomic selection that account for dominance and epistasis in a general context, without assuming HWE (e.g., crosses or homozygous lines), are therefore needed. [...] Then we prove for the first time that, even not in HWE, the multiple-loci NOIA method is equivalent to construct epistatic genomic relationship matrices for higher-order interactions using Hadamard products of additive and dominant genomic orthogonal relationships. This is not true. Earlier proofs did not assume HWE, but were generally true for any coding. *  A standardization based on the trace of the relationship matrices is, however, needed. This is not true in the context of genomic prediction. The factor does not matter at all. *  This results in an orthogonal partition of the variances, which is not warranted otherwise. This is not true in the n p context. Having orthogonality between a marker and its dominance effect or between two markers and their interaction is not sufficient to obtain an orthogonal partition of the variances. You would need a joint orthogonality between the groups of predictors (which is not possible if n p). The results of two markers cannot simply be generalized to p markers, without considering the dependence between them. The partition of the variance will not be orthogonal for n p. There are many additional mathematical. I recommend that Genetics reconsiders the article with experts having a solid mathematical background. ,