Better Estimates of Genetic Covariance Matrices by “Bending” Using Penalized Maximum Likelihood

Abstract

Obtaining accurate estimates of the genetic covariance matrix S G for multivariate data is a fundamental task in quantitative genetics and important for both evolutionary biologists and plant or animal breeders. Classical methods for estimating S G are well known to suffer from substantial sampling errors; importantly, its leading eigenvalues are systematically overestimated. This article proposes a framework that exploits information in the phenotypic covariance matrix S P in a new way to obtain more accurate estimates of S G . The approach focuses on the ''canonical heritabilities'' (the eigenvalues of S À1 P S G ), which may be estimated with more precision than those of S G because S P is estimated more accurately. Our method uses penalized maximum likelihood and shrinkage to reduce bias in estimates of the canonical heritabilities. This in turn can be exploited to get substantial reductions in bias for estimates of the eigenvalues of S G and a reduction in sampling errors for estimates of S G . Simulations show that improvements are greatest when sample sizes are small and the canonical heritabilities are closely spaced. An application to data from beef cattle demonstrates the efficacy this approach and the effect on estimates of heritabilities and correlations. Penalized estimation is recommended for multivariate analyses involving more than a few traits or problems with limited data.

Social and genetic interactions drive fitness variation in a free-living dolphin population

“…Relatedness estimates measured in this way result in approximate marginal probabilities in the relatedness matrix instead of a joint distribution. The occurrence of negative-definite matrices is common in genetics (48)(49)(50)(51). Genetic covariance matrices are well known for being non-negative-definite, as are identity-by-descent matrices (50,51).…”

“…The occurrence of negative-definite matrices is common in genetics (48)(49)(50)(51). Genetic covariance matrices are well known for being non-negative-definite, as are identity-by-descent matrices (50,51). Bending procedures (48,49), are commonly used to correct non-negative-definiteness in genetic matrices (52), as they bend matrices until their lowest eigenvalues exceed a preset limit (e.g., zero to achieve non-negative-definiteness).…”

A Flexible Estimating Equations Approach for Mapping Function-Valued Traits

“…We evaluate this choice in a simulation study in the next section. Shrinkage estimation of the genetic covariance for function-valued data has been considered by Meyer and Kirkpatrick (2010).…”

Simple Penalties on Maximum-Likelihood Estimates of Genetic Parameters to Reduce Sampling Variation

“…For instance, shrinkage of eigenvalues toward their mean through a quadratic penalty on the likelihood is equivalent to assuming a normal distribution of the eigenvalues while the assumption of a double exponential prior distribution results in a LASSO-type penalty (Huang et al 2006). Meyer and Kirkpatrick (2010) demonstrated that a REML analog to bending is obtained by imposing a penalty proportional to the variance of the canonical eigenvalues on the likelihood and showed that this can yield substantial reductions in loss for estimates of both S G and S E ; the residual covariance matrix. Subsequent simulations Meyer 2011) examined the scope for penalties based on different functions of the parameters to be estimated and prior distributions for them and found them to be similarly effective, depending on the population values for the covariance matrices to be estimated.…”

“…A method-of-scoring algorithm together with simple derivative-free search steps was used to locate the maximum of the (penalized) likelihood function. To facilitate easy computation of derivatives of P l ; this was done using a parameterization to the elements of the canonical decomposition (see Meyer and Kirkpatrick 2010), restraining estimates of l i to the interval of ½0:0001; 0:9999: Direct estimation of n was performed by evaluating points on the profile likelihood for n [i.e., maximizing logL P ðuÞ with respect to the covariance components to be estimated for selected, fixed values of n], combined with quadratic approximation steps of the profile to locate its maximum, using Powell's (2006) Fortran subroutine NEWUOA. To avoid numerical problems, estimates of n were constrained to the interval ½2:01; 50: All calculations were carried out using custom Fortran programs (available on request).…”