Highly Efficient Multivariate GREML

Genomic-relatedness-matrix restricted maximum-likelihood (GREML) estimation of a bivariate linear mixed model (LMM) is often used for estimating the genetic correlation and covariance between traits. An extension of such... [ view full abstract ]

Genomic-relatedness-matrix restricted maximum-likelihood (GREML) estimation of a bivariate linear mixed model (LMM) is often used for estimating the genetic correlation and covariance between traits. An extension of such bivariate GREML methods is a multivariate approach. Instead of estimating multivariate LMMs, researchers often estimate pairwise bivariate LMMs for each combination of two phenotypes within a set of phenotypes. This practice for inferring genetic covariance between multiple traits can pose a problem, since the resulting (combined) covariance matrix is not necessarily positive (semi)-definite. To overcome this problem, we present a multivariate GREML (MGREML) estimation method. This method consists of an iterative procedure, based on a Newton-Raphson algorithm, to obtain consistent estimates of the parameters of a multivariate SNP-based LMM for balanced data on P phenotypes, hence, observed in the same set of N individuals. We provide a parametrization of the LMM, such that resulting estimates of the P x P genetic and environment covariance matrices are positive (semi)-definite, irrespective of starting values and updates of the estimates throughout the iterations. Provided one has access to the eigendecomposition of the genomic-relatedness matrix, we are able, from there on, to reduce the computational complexity of MGREML estimation from the order (NP)³ to an order of N(P)³.  Therefore, the MGREML estimation method we propose can be applied to a large set of phenotypes and individuals, provided the data are balanced.