REPLY TO YANG ET AL.:
GCTA produces unreliable heritability estimates Siddharth Krishna Kumara,1, Marcus W. Feldmana, David H. Rehkopfb, and Shripad Tuljapurkara
In 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).
1 Krishna Kumar S, Feldman MW, Rehkopf DH, Tuljapurkar S (2016) Limitations of GCTA as a solution to the missing heritability problem. Proc Natl Acad Sci USA 113(1):E61–E70. 2 Kumar SK, Feldman MW, Rehkopf DH, Tuljapurkar S (2016) Response to commentary on “Limitations of GCTA as a solution to the missing heritability problem.” bioRxiv, 10.1101/039594. 3 Yang J, et al. (2010) Common SNPs explain a large proportion of the heritability for human height. Nat Genet 42(7):565–569. 4 Yang J, Lee SH, Wray NR, Goddard ME, Visscher PM (2016) GCTA-GREML accounts for linkage disequilibrium when estimating genetic variance from genome-wide SNPs. Proc Natl Acad Sci USA 113:E4579–E4580.
a
Department of Biology, Stanford University, Stanford, CA 94305-5020; and bSchool of Medicine, Stanford University, Stanford, CA 94305-5020 Author contributions: S.K.K., M.W.F., and S.T. designed research; S.K.K., M.W.F., and S.T. performed research; S.K.K., M.W.F., and S.T. contributed new reagents/analytic tools; S.K.K. and S.T. analyzed data; and S.K.K., M.W.F., D.H.R., and S.T. wrote the paper. The authors declare no conflict of interest. 1 To whom correspondence should be addressed. Email: [email protected].
www.pnas.org/cgi/doi/10.1073/pnas.1608425113
PNAS | August 9, 2016 | vol. 113 | no. 32 | E4581
LETTER
LETTER
GCTA produces unreliable heritability estimates Siddharth Krishna Kumara,1, Marcus W. Feldmana, David H. Rehkopfb, and Shripad Tuljapurkara
In 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).
1 Krishna Kumar S, Feldman MW, Rehkopf DH, Tuljapurkar S (2016) Limitations of GCTA as a solution to the missing heritability problem. Proc Natl Acad Sci USA 113(1):E61–E70. 2 Kumar SK, Feldman MW, Rehkopf DH, Tuljapurkar S (2016) Response to commentary on “Limitations of GCTA as a solution to the missing heritability problem.” bioRxiv, 10.1101/039594. 3 Yang J, et al. (2010) Common SNPs explain a large proportion of the heritability for human height. Nat Genet 42(7):565–569. 4 Yang J, Lee SH, Wray NR, Goddard ME, Visscher PM (2016) GCTA-GREML accounts for linkage disequilibrium when estimating genetic variance from genome-wide SNPs. Proc Natl Acad Sci USA 113:E4579–E4580.
a
Department of Biology, Stanford University, Stanford, CA 94305-5020; and bSchool of Medicine, Stanford University, Stanford, CA 94305-5020 Author contributions: S.K.K., M.W.F., and S.T. designed research; S.K.K., M.W.F., and S.T. performed research; S.K.K., M.W.F., and S.T. contributed new reagents/analytic tools; S.K.K. and S.T. analyzed data; and S.K.K., M.W.F., D.H.R., and S.T. wrote the paper. The authors declare no conflict of interest. 1 To whom correspondence should be addressed. Email: [email protected].
www.pnas.org/cgi/doi/10.1073/pnas.1608425113
PNAS | August 9, 2016 | vol. 113 | no. 32 | E4581
LETTER
LETTER
