Reply to Yang et al.: GCTA produces unreliable heritability estimates

Abstract

aIn our recent paper in PNAS (1), and subsequently (2), we have analyzed the mathematical model that is stated precisely by Yang et al. (3). As written, their model assumes that (i) the Genetic Relatedness Matrix (GRM) is known exactly, (ii) the phenotypic contributions of each of the P SNPs are independent identically distributed draws from the same normal distribution with mean 0 and variance σ 2 , and (iii) σ 2 is independent of P.The empirical facts are that (i) we do not know the GRM, but only have an estimate of it; (ii) the number of SNPs (P) used in the analysis depends on the technology; and (iii) there is no "empirical" evidence for the heritability estimates obtained using genome-wide complex trait analysis (GCTA) per se [notwithstanding the claims made by Yang et al. (4) in their response to our article (1)].We and others have shown that the singular value distribution is skewed, and hence it is unlikely that all of the singular values and singular vectors of the (unknown) true GRM can be reliably estimated.In a paper published elsewhere (2), we provide multiple examples of inconsistent σ 2 estimates published by the coauthors of the GCTA model (3). We show that the σ 2 estimates are unreliable when (i) N is fixed and P varies (ref. 2, p. 12), (ii) when P is fixed and N varies (ref. 2, p. 13), and (iii) when both N and P vary (ref. 2, p. 11). These authors claim that the estimate of σ 2 will decrease as P increases, but this specification is not a part of their model as we understand it.We do not understand the basis for the claim that "the GREML fits all of the SNPs jointly in a randomeffect model so that each SNP effect is fitted conditioning on the joint effects of all of the SNPs" (4). Although Yang and colleagues (4) insist on this fact, they do not provide any mathematical justification for this conclusion. We have responded in detail to all other critiques listed in this letter elsewhere (2).

Statistical properties of simple random-effects models for genetic heritability

“…Our findings about bias and variance run contrary to conclusions of Kumar et al in [ 19 ]. In particular, we see the limitations on accurate GREML estimation in the absence of stratification arising not from bias but from large standard errors, in contrast to their conclusions about the salience of bias.…”

“…Recently, Kumar et al [ 20 ] have taken a different tack, criticising GREML on statistical grounds, on what might be considered issues of inherent mathematical fallibility for estimation in models relying on high-dimensional covariates. Their paper has elicited rebuttals from Yang et al [ 53 ] and [ 54 ] and rejoinders to the rebuttal [ 21 ] and [ 19 ] from Kumar et al . Parts of that exchange are devoted to GREML and the GREML model in more complicated settings, but our results for “Simple GREML” in this paper settle some of the points in contention.…”

Best Prediction of the Additive Genomic Variance in Random-Effects Models

“…To sum up, the application of the unconditional expectation of the additive genomic variance combined with the model assumptions on the marker effects in random effect models cause, at least partially, the missing contribution of LD to the estimated additive genomic variance. This goes hand-in-hand with the critique expressed in Krishna Kumar et al (2016a,b). It is, however, less important when estimating the additive genomic variance in the base population, where the individuals are uncorrelated and less LD persists (although the marker genotypes need not be uncorrelated).…”

“…Recently, there has been a general discussion whether estimators for the genomic variance account for LD between markers, which is defined as the covariance between the marker genotypes (Bulmer 1971). Some authors argue that estimators similar to GCTA-GREML lack the contribution of LD (Krishna Kumar et al 2016a,b; Lehermeier et al 2017), whereas others (Yang et al 2016) resolutely disagree. More specifically, Krishna Kumar et al (2016a,b) state that, in GCTA-GREML, the contributions of the p markers to the phenotypic values are assumed to be independent normally distributed random variables with equal variances.…”

“…Some authors argue that estimators similar to GCTA-GREML lack the contribution of LD (Krishna Kumar et al 2016a,b; Lehermeier et al 2017), whereas others (Yang et al 2016) resolutely disagree. More specifically, Krishna Kumar et al (2016a,b) state that, in GCTA-GREML, the contributions of the p markers to the phenotypic values are assumed to be independent normally distributed random variables with equal variances. Thus, they claim that the random contribution made by each marker is not correlated with the random contributions made by any other marker, which leads to a negligence of the contribution of LD to the additive genomic variance.…”