Motivated by applications in neuroanatomy, we propose a novel methodology forestimating the heritability which corresponds to the proportion of phenotypicvariance which can be explained by genetic factors. Estimating this quantityfor neuroanatomical features is a fundamental challenge in psychiatric diseaseresearch. Since the phenotypic variations may only be due to a small fractionof the available genetic information, we propose an estimator of theheritability that can be used in high dimensional sparse linear mixed models.Our method consists of three steps. Firstly, a variable selection stage isperformed in order to recover the support of the genetic effects -- also calledcausal variants -- that is to find the genetic effects which really explain thephenotypic variations. Secondly, we propose a maximum likelihood strategy forestimating the heritability which only takes into account the causal geneticeffects found in the first step. Thirdly, we compute the standard error and the95% confidence interval associated to our heritability estimator thanks to anonparametric bootsrap approach. Our main contribution consists in providing anestimation of the heritability with standard errors substantially smaller thanmethods without variable selection when the genetic effects are very sparse.Since the real genetic architecture is in general unknown in practice, we alsopropose an empirical criterion which allows the user to decide whether it isrelevant to apply a variable selection based approach or not. We illustrate theperformance of our methodology on synthetic and real neuroanatomic data comingfrom the Imagen project. We also show that our approach has a very lowcomputational burden and is very efficient from a statistical point of view.
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