Abstract
Random-effects models are a popular tool for analysing total narrow-sense heritability for quantitative phenotypes, on the basis of large-scale SNP data. Recently, there have been disputes over the validity of conclusions that may be drawn from such analysis. We derive some of the fundamental statistical properties of heritability estimates arising from these models, showing that the bias will generally be small. We show that that the score function may be manipulated into a form that facilitates intelligible interpretations of the results. We go on to use this score function to explore the behavior of the model when certain key assumptions of the model are not satisfied — shared environment, measurement error, and genetic effects that are confined to a small subset of sites.The variance and bias depend crucially on the variance of certain functionals of the singular values of the genotype matrix. A useful baseline is the singular value distribution associated with genotypes that are completely independent — that is, with no linkage and no relatedness — for a given number of individuals and sites. We calculate the corresponding variance and bias for this setting.
Highlights
Genome-Wide Complex-Trait Analysis, known as GCTA, introduced by Jian Yang and collaborators in 2010 in [51], has led both to a profusion of research findings across the biomedical and social sciences and to exuberant controversy
We refer to it as “the GREML model”. It is an example of a “linear mixed model,” in which the contribution of SNPs to trait values are treated as random effects. (We do not consider fixed effects in this paper, but the same arguments would apply to reduced phenotypes after elimination of fixed effects in a mixed model.) Alternative estimation methods for the model such as LD-score Regression and HasemanElston regression have come into wide use, bringing up statistical issues paralleling those we examine here for GREML
Much of the dispute between [20] and [53] centers around the question of whether the specification of the GREML model and algorithm allows for linkage disequilibrium, and whether population stratification is adequately accounted for. [20] considers, by means of subsampling real data, the question of whether the choice of SNPs to be investigated — as a subset of the full complement of SNPs — increases the variance of heritability estimates in ways that the standard analysis fails to capture
Summary
Genome-Wide Complex-Trait Analysis, known as GCTA, introduced by Jian Yang and collaborators in 2010 in [51], has led both to a profusion of research findings across the biomedical and social sciences and to exuberant controversy. Unlike the setting in Simple GREML, in more complicated settings only a subset of all SNPs may have causal effects on the phenotype, sometimes observed SNPs, sometimes unobserved ones This sort of misspecification forms the central subject of the recent work by Jiang et al [16]. Our analysis of bias and variance for Simple GREML aims at this goal
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