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An efficient exact method to obtain GBLUP and singlestep GBLUP when the genomic relationship matrix is singular
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https://doi.org/10.1186/s1271101602607Abstract
Background
The mixed linear model employed for genomic best linear unbiased prediction (GBLUP) includes the breeding value for each animal as a random effect that has a mean of zero and a covariance matrix proportional to the genomic relationship matrix ([Formula: see text]), where the inverse of [Formula: see text] is required to set up the usual mixed model equations (MME). When only some animals have genomic information, genomic predictions can be obtained by an extension known as singlestep GBLUP, where the covariance matrix of breeding values is constructed by combining the pedigreebased additive relationship matrix with [Formula: see text]. The inverse of the combined relationship matrix can be obtained efficiently, provided [Formula: see text] can be inverted. In some livestock species, however, the number [Formula: see text] of animals with genomic information exceeds the number of marker covariates used to compute [Formula: see text], and this results in a singular [Formula: see text]. For such a case, an efficient and exact method to obtain GBLUP and singlestep GBLUP is presented here.Results
Exact methods are already available to obtain GBLUP when [Formula: see text] is singular, but these require working with large dense matrices. Another approach is to modify [Formula: see text] to make it nonsingular by adding a small value to all its diagonals or regressing it towards the pedigreebased relationship matrix. This, however, results in the inverse of [Formula: see text] being dense and difficult to compute as [Formula: see text] grows. The approach presented here recognizes that the number r of linearly independent genomic breeding values cannot exceed the number of marker covariates, and the mixed linear model used here for genomic prediction only fits these r linearly independent breeding values as random effects.Conclusions
The exact method presented here was compared to ApyGBLUP and to Apy singlestep GBLUP, both of which are approximate methods that use a modified [Formula: see text] that has a sparse inverse which can be computed efficiently. In a small numerical example, predictions from the exact approach and Apy were almost identical, but the MME from Apy had a condition number about 1000 times larger than that from the exact approach, indicating illconditioning of the MME from Apy. The practical application of exact SSGBLUP is not more difficult than implementation of Apy.Many UCauthored scholarly publications are freely available on this site because of the UC's open access policies. Let us know how this access is important for you.
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