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 λK, in which λ 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 λK 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 λK in conventional BLUP? The mathematical investigation concluded: (i) the use of a λK 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.
Highlights
A breeding value means a combining ability of an entity as a parent
The linear mixed model (LMM) is a statistical method that is widely used for estimating breeding values (Piepho, 1994; Panter and Allen, 1995a; Panter and Allen, 1995b; Choi et al, 2017)
Every value of for Equation 9 represents the arithmetic mean of multiple phenotypic observations per each entity, given all phenotypic observations adjusted to the mean of zero
Summary
The linear mixed model (LMM) is a statistical method that is widely used for estimating breeding values (Piepho, 1994; Panter and Allen, 1995a; Panter and Allen, 1995b; Choi et al, 2017). The random-effect variable has its own variance-covariance, which characterizes the LMM differently from the linear model (linear regression) that is restricted to estimating the fixed-effect variable. A number of previous studies validated the effectiveness of BLUP taking advantage of the kinship matrix by empirically measuring the combining abilities through field tests and computer simulations (Belonsky and Kennedy, 1988; Piepho, 1994; Panter and Allen, 1995a; Panter and Allen, 1995b; Bauer et al, 2006; Nielsen et al, 2011; Choi et al, 2017; Manzanila-Pech et al, 2017). This study investigated the integrity of BLUP from the mathematical perspective
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