Heritability estimation in high dimensional sparse linear mixed models

Abstract

Motivated by applications in genetic fields, we propose to estimate the heritability in high dimensional sparse linear mixed models. The heritability determines how the variance is shared between the different random components of a linear mixed model. The main novelty of our approach is to consider that the random effects can be sparse, that is may contain null components, but we do not know neither their proportion nor their positions. The estimator that we consider is strongly inspired by the one proposed by [16], and is based on a maximum likelihood approach. We also study the theoretical properties of our estimator, namely we establish that our estimator of the heritability is ? n-consistent when both the number of observations n and the number of random effects N tend to infinity under mild assumptions. We also prove that our estimator of the heritability satisfies a central limit theorem which gives as a byproduct a confidence interval for the heritability. SomeMonte-Carlo experiments are also conducted in order to show the finite sample performances of our estimator.

Improving Heritability Estimation by a Variable Selection Approach in Sparse High Dimensional Linear Mixed Models

“…from Bonnet et al . (): that is why we choose here to compare EstHer with HiLMM and not with GCTA. We also compare these results with the software GEMMA that is described in Zhou and Stephens ().…”

“…The results that we obtained in Bonnet et al . () suggest the following. If we knew the set of causal SNPs, then, considering only this (small) subset in the genetic information matrix, we would obtain with HiLMM an estimator having a smaller standard error than when using all SNPs in the genetic information matrix.…”

“…3 Table 3. In Table 3, the confidence intervals displayed for the rows 'Oracle' and 'Without selection' are obtained by using the asymptotic variance that was derived in Bonnet et al (2015) which corresponds to the classical inverse of the Fisher information in the case q = 1. We observe that our method without the selection step provides similar results, i.e.…”

“…We proposed in Bonnet et al . () another method based on a maximum likelihood strategy to estimate the heritability and implemented in the R package HiLMM. We proved in Bonnet et al .…”

“…We proposed in Bonnet et al (2015) another method based on a maximum likelihood strategy to estimate the heritability and implemented in the R package HiLMM. We proved in Bonnet et al (2015) the following theoretical result: though the computation of the likelihood is based on the modelling assumption (2), the estimator is consistent (unbiased) under the less restrictive modelling assumption (4). We believe that this consistency result remains true for the estimators that are produced using the restricted maximum likelihood algorithm, EMMA and GCTA.…”