The use of a genetic relationship matrix biases the best linear unbiased prediction
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Ó Indian Academy of Sciences (0123456789().,-volV) (0123456789().,-volV)
RESEARCH ARTICLE
The use of a genetic relationship matrix biases the best linear unbiased prediction BONGSONG KIM RiceTec, Inc, Alvin, TX 77511, USA E-mail: [email protected]. Received 12 December 2019; revised 20 April 2020; accepted 22 April 2020 Abstract. The best linear unbiased prediction (BLUP), derived from the linear mixed model (LMM), has been popularly used to estimate animal and plant breeding values (BVs) for a few decades. Conventional BLUP has a constraint that BVs are estimated from the assumed covariance among unknown BVs, namely conventional BLUP assumes that its covariance matrix is a kK, in which k is a coefficient that leads to the minimum mean square error of the LMM, and K is a genetic relationship matrix. The uncertainty regarding the use of kK in conventional BLUP was recognized by past studies, but it has not been sufficiently investigated. This study was motivated to answer the following question: is it indeed reasonable to use a kK in conventional BLUP? The mathematical investigation concluded: (i) the use of a kK in conventional BLUP biases the estimated BVs, and (ii) the objective BLUP, mathematically derived from the LMM, has the same representation as the least squares. Keywords.
best linear unbiased prediction; breeding values; genetic relationship matrix; numerator relationship matrix; least squares.
Introduction A breeding value (BV) refers to the combining ability of an entity as a parent. Estimating the BVs is purposeful in seed and livestock industries, aiming to save time and resources in increasing genetic gains by maximizing selection accuracies and efficiencies. The best linear unbiased prediction (BLUP), derived from the linear mixed model (LMM), has been popularly used to estimate BVs for a few decades (Piepho 1994; Panter and Allen 1995a, b; Meuwissen et al. 2001; Choi et al. 2017). Conventional BLUP has a constraint that the estimated breeding values (EBVs) are derived from the assumed covariance among unknown BVs. Conventional BLUP uses a matrix representing pairwise genetic variations for an assumed covariance matrix (Henderson 1975). Pedigrees or genomic fingerprints are typically required for calculating the assumed covariance matrix (Emik and Terrill 1949; VanRaden 2008; Kim et al. 2016; Kim and Beavis 2017). Previous studies reported that EBVs obtained by conventional BLUP have a high correlation with empirically
observed combining abilities, based on field tests and computer simulations (Belonsky and Kennedy 1988; Piepho 1994; Panter and Allen 1995a, b; Bauer et al. 2006; Nielsen et al. 2011; Choi et al. 2017; ManzanillaPech et al. 2017). However, no studies clarified two uncertainties regarding conventional BLUP. First, there are no clues that the assumed covariance matrix is objective. If the assumed and objective covariance matrices are not equal, conventional BLUP is biased. Second, BVs are not physically measurable but conceptual. In fact, the combining ability for an entity can vary accor
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