Academic journal article Genetics

Increasing the Efficiency of Variance Component Quantitative Trait Loci Analysis by Using Reduced-Rank Identity-by-Descent Matrices

Academic journal article Genetics

Increasing the Efficiency of Variance Component Quantitative Trait Loci Analysis by Using Reduced-Rank Identity-by-Descent Matrices

Article excerpt

ABSTRACT

Recent technological development in genetics has made large-scale marker genotyping fast and practicable, facilitating studies for detection of QTL in large general pedigrees. We developed a method that speeds up restricted maximum-likelihood (REML) algorithms for QTL analysis by simplifying the inversion of the variance-covariance matrix of the trait vector. The method was tested in an experimental chicken pedigree including 767 phenotyped individuals and 14 genotyped markers on chicken chromosome 1. The computation time in a chromosome scan covering 475 cM was reduced by 43% when the analysis was based on linkage only and by 72% when linkage disequilibrium information was included. The relative advantage of using our method increases with pedigree size, marker density, and linkage disequilibrium, indicating even greater improvements in the future.

(ProQuest-CSA LLC: ... denotes formulae omitted.)

THE use of variance component models is rapidly increasing in the field of QTL analysis (Lynch andWalsh 1998). As the cost for genotyping decreases, the sizes of the analyzed pedigrees are likely to increase, making full genome scans computationally slow or even infeasible. However, current algorithms commonly used for variance component estimation were not specifically developed for QTL analysis and there is a need to reevaluate the computational efficiency and robustness of these algorithms.

Variance component estimation has been included in general statistical software, such as Proc Mixed in SAS (Littell et al. 1996),where anarbitrary covariance structure of the random effect can be given by the user. These programs have in common that they use iterative procedures, Fisher's scoring or Newton-Raphson, to maximize the likelihood or the restricted likelihood (Pawitan 2001). More specific programs for applications in animal breeding have also been developed over the last two decades such as ASReml, DMU, and VCE (see Druet and Ducrocq 2006 and references therein). These programs use a mixture of Fisher's scoring and Newton-Raphson to maximize the restricted likelihood, called the "average information restricted maximum-likelihood (AI-REML) algorithm." The most computationally demanding part of AI-REML is the inversion of the variance-covariance matrix (V) of the response vector (y). This inversion has to be performed on each iteration. We study a model with a random QTL effect and a residual effect, with variance- covariance matrix of the form ..., where Π is the symmetric identity-by-descent (IBD) matrix, ... is the QTL variance, I is the identity matrix, and ... is the residual variance. If Π is positive definite, then the inversion of V can be simplified by inverting V in parts. This has been implemented in the software package ASReml by setting up the mixed-model equations (MME) (Gilmour et al. 1995; Johnson and Thompson 1995; Jensen et al. 1997).

The AI-REML algorithm can be implemented by combining the MME with sparse matrix techniques (as done in ASReml), which gives fast solutions when the covariance structure of the random effect is sparse and positive definite. In traditional animal breeding applications, the covariance structure is given by the average relationship between individuals (Lynch and Walsh 1998). This covariance structure is usually sparse and always positive definite. The IBD matrix is not necessarily sparse or positive definite, however, and the advantage of using sparse matrix techniques together with inversion of V by means of MME in AI-REML may be questioned. Lee and van der Werf (2006) found that the AI-REML algorithm could be faster and more robust in QTL analysis if direct inversion of V is used, especially in linkage-disequilibriumlinkage (LDL)mapping (Meuwissen and Goddard 2000) since the IBD matrices used in LDL are usually dense and positive semidefinite. The reason for this is that a covariance structure is added to the base generation alleles (Meuwissen and Goddard 2000; Hernandez-Sanchez et al. …

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