Linear mixed models (LMMs) are among the most commonly used tools for genetic association studies. However, the standard method for estimating variance components in LMMs—the restricted maximum likelihood estimation method (REML)—suffers from several important drawbacks: REML requires individual-level genotypes and phenotypes from all samples in the study, is computationally slow, and produces downward-biased estimates in case control studies. To remedy these drawbacks, we present an alternative framework for variance component estimation, which we refer to as MQS. MQS is based on the method of moments (MoM) and the minimal norm quadratic unbiased estimation (MINQUE) criterion, and brings two seemingly unrelated methods—the renowned Haseman–Elston (HE) regression and the recent LD score regression (LDSC)—into the same unified statistical framework. With this new framework, we provide an alternative but mathematically equivalent form of HE that allows for the use of summary statistics. We provide an exact estimation form of LDSC to yield unbiased and statistically more efficient estimates. A key feature of our method is its ability to pair marginal $z$-scores computed using all samples with SNP correlation information computed using a small random subset of individuals (or individuals from a proper reference panel), while capable of producing estimates that can be almost as accurate as if both quantities are computed using the full data. As a result, our method produces unbiased and statistically efficient estimates, and makes use of summary statistics, while it is computationally efficient for large data sets. Using simulations and applications to 37 phenotypes from 8 real data sets, we illustrate the benefits of our method for estimating and partitioning SNP heritability in population studies as well as for heritability estimation in family studies. Our method is implemented in the GEMMA software package, freely available at www.xzlab.org/software.html.
Ann. Appl. Stat.
11(4):
2027-2051
(December 2017).
DOI: 10.1214/17-AOAS1052
Abecasis, G. R., Cardon, L. R. and Cookson, W. O. (2000). A general test of association for quantitative traits in nuclear families. Am. J. Hum. Genet. 66 279–292.Abecasis, G. R., Cardon, L. R. and Cookson, W. O. (2000). A general test of association for quantitative traits in nuclear families. Am. J. Hum. Genet. 66 279–292.
Allen, H. L., Estrada, K., Lettre, G. Berndt, S. I., Weedon, M. N., Rivadeneira, F., Willer, C. J., Jackson, A. U., Vedantam, S. et al. (2010). Hundreds of variants clustered in genomic loci and biological pathways affect human height. Nature 467 832–838.Allen, H. L., Estrada, K., Lettre, G. Berndt, S. I., Weedon, M. N., Rivadeneira, F., Willer, C. J., Jackson, A. U., Vedantam, S. et al. (2010). Hundreds of variants clustered in genomic loci and biological pathways affect human height. Nature 467 832–838.
Browning, S. R. and Browning, B. L. (2013). Identity-by-descent-based heritability analysis in the Northern Finland Birth Cohort. Hum. Genet. 132 129–138.Browning, S. R. and Browning, B. L. (2013). Identity-by-descent-based heritability analysis in the Northern Finland Birth Cohort. Hum. Genet. 132 129–138.
Bulik-Sullivan, B. K., Loh, P.-R., Finucane, H. K., Ripke, S., Yang, J., Schizophrenia Working Group of the Psychiatric Genomics Consortium, Patterson, N., Daly, M. J., Price, A. L. and Neale, B. M. (2015a). LD score regression distinguishes confounding from polygenicity in genome-wide association studies. Nat. Genet. 47 291–295.Bulik-Sullivan, B. K., Loh, P.-R., Finucane, H. K., Ripke, S., Yang, J., Schizophrenia Working Group of the Psychiatric Genomics Consortium, Patterson, N., Daly, M. J., Price, A. L. and Neale, B. M. (2015a). LD score regression distinguishes confounding from polygenicity in genome-wide association studies. Nat. Genet. 47 291–295.
Bulik-Sullivan, B., Finucane, H. K., Anttila, V., Gusev, A., Day, F. R., Loh, P.-R., ReproGen Consortium, Psychiatric Genomics Consortium, Genetic Consortium for Anorexia Nervosa of the Wellcome Trust Case Control Consortium 3 et al. (2015b). An atlas of genetic correlations across human diseases and traits. Nat. Genet. 47 1236–1241.Bulik-Sullivan, B., Finucane, H. K., Anttila, V., Gusev, A., Day, F. R., Loh, P.-R., ReproGen Consortium, Psychiatric Genomics Consortium, Genetic Consortium for Anorexia Nervosa of the Wellcome Trust Case Control Consortium 3 et al. (2015b). An atlas of genetic correlations across human diseases and traits. Nat. Genet. 47 1236–1241.
Chen, G.-B. (2014). Estimating heritability of complex traits from genome-wide association studies using IBS-based Haseman–Elston regression. Front. Genet. 5 107.Chen, G.-B. (2014). Estimating heritability of complex traits from genome-wide association studies using IBS-based Haseman–Elston regression. Front. Genet. 5 107.
Chen, W.-M., Broman, K. W. and Liang, K.-Y. (2004). Quantitative trait linkage analysis by generalized estimating equations: Unification of variance components and Haseman–Elston regression. Genet. Epidemiol. 26 265–272.Chen, W.-M., Broman, K. W. and Liang, K.-Y. (2004). Quantitative trait linkage analysis by generalized estimating equations: Unification of variance components and Haseman–Elston regression. Genet. Epidemiol. 26 265–272.
Crawford, L., Zeng, P., Mukherjee, S. and Zhou, X. (2017). Detecting epistasis with the marginal epistasis test in genetic mapping studies of quantitative traits. BioRxiv.Crawford, L., Zeng, P., Mukherjee, S. and Zhou, X. (2017). Detecting epistasis with the marginal epistasis test in genetic mapping studies of quantitative traits. BioRxiv.
Finucane, H. K., Bulik-Sullivan, B., Gusev, A., Trynka, G., Reshef, Y., Loh, P.-R., Anttilla, V., Xu, H., Zang, C. et al. (2015). Partitioning heritability by functional category using GWAS summary statistics. Nat. Genet. 47 1228–1235.Finucane, H. K., Bulik-Sullivan, B., Gusev, A., Trynka, G., Reshef, Y., Loh, P.-R., Anttilla, V., Xu, H., Zang, C. et al. (2015). Partitioning heritability by functional category using GWAS summary statistics. Nat. Genet. 47 1228–1235.
García-Cortés, L. A., Moreno, C., Varona, L. and Altarriba, J. (1992). Variance component estimation by resampling. J. Anim. Breed. Genet. 109 358–363.García-Cortés, L. A., Moreno, C., Varona, L. and Altarriba, J. (1992). Variance component estimation by resampling. J. Anim. Breed. Genet. 109 358–363.
Gilmour, A. R., Thompson, R. and Cullis, B. R. (1995). Average information REML: An efficient algorithm for variance parameter estimation in linear mixed models. Biometrics 51 1440–1450.Gilmour, A. R., Thompson, R. and Cullis, B. R. (1995). Average information REML: An efficient algorithm for variance parameter estimation in linear mixed models. Biometrics 51 1440–1450.
Golan, D., Lander, E. S. and Rosseta, S. (2014). Measuring missing heritability: Inferring the contribution of common variants. Proc. Natl. Acad. Sci. USA 111 E5272–E5281.Golan, D., Lander, E. S. and Rosseta, S. (2014). Measuring missing heritability: Inferring the contribution of common variants. Proc. Natl. Acad. Sci. USA 111 E5272–E5281.
Gusev, A., Bhatia, G., Zaitlen, N., Vilhjalmsson, B. J., Diogo, D., Stahl, E. A., Gregersen, P. K., Worthington, J., Klareskog, L. et al. (2013). Quantifying missing heritability at known GWAS loci. PLoS Genet. 9 e1003993.Gusev, A., Bhatia, G., Zaitlen, N., Vilhjalmsson, B. J., Diogo, D., Stahl, E. A., Gregersen, P. K., Worthington, J., Klareskog, L. et al. (2013). Quantifying missing heritability at known GWAS loci. PLoS Genet. 9 e1003993.
Gusev, A., Lee, S. H., Trynka, G., Finucane, H., Vilhjálmsson, B. J., Xu, H., Zang, C., Ripke, S., Bulik-Sullivan, B. et al. (2014). Partitioning heritability of regulatory and cell-type-specific variants across 11 common diseases. Am. J. Hum. Genet. 5 535–552.Gusev, A., Lee, S. H., Trynka, G., Finucane, H., Vilhjálmsson, B. J., Xu, H., Zang, C., Ripke, S., Bulik-Sullivan, B. et al. (2014). Partitioning heritability of regulatory and cell-type-specific variants across 11 common diseases. Am. J. Hum. Genet. 5 535–552.
Hayes, B. J., Visscher, P. M. and Goddard, M. E. (2009). Increased accuracy of artificial selection by using the realized relationship matrix. Genet. Res. (Camb.) 91 47–60.Hayes, B. J., Visscher, P. M. and Goddard, M. E. (2009). Increased accuracy of artificial selection by using the realized relationship matrix. Genet. Res. (Camb.) 91 47–60.
Hayes, M. G., del Bosque-Plata, L., Tsuchiya, T., Hanis, C. L., Bell, G. I. and Cox, N. J. (2005). Patterns of linkage disequilibrium in the type 2 diabetes gene calpain-10. Diabetes 54 3573–3576.Hayes, M. G., del Bosque-Plata, L., Tsuchiya, T., Hanis, C. L., Bell, G. I. and Cox, N. J. (2005). Patterns of linkage disequilibrium in the type 2 diabetes gene calpain-10. Diabetes 54 3573–3576.
Howie, B., Fuchsberger, C., Stephens, M., Marchini, J. and Abecasis, G. R. (2012). Fast and accurate genotype imputation in genome-wide association studies through pre-phasing. Nat. Genet. 44 955–959.Howie, B., Fuchsberger, C., Stephens, M., Marchini, J. and Abecasis, G. R. (2012). Fast and accurate genotype imputation in genome-wide association studies through pre-phasing. Nat. Genet. 44 955–959.
Jostins, L., Ripke, S., Weersma, R. K., Duerr, R. H., McGovern, D. P., Hui, K. Y., Lee, J. C., Schumm, L. P., Sharma, Y. et al. (2012). Host–microbe interactions have shaped the genetic architecture of inflammatory bowel disease. Nature 491 119–124.Jostins, L., Ripke, S., Weersma, R. K., Duerr, R. H., McGovern, D. P., Hui, K. Y., Lee, J. C., Schumm, L. P., Sharma, Y. et al. (2012). Host–microbe interactions have shaped the genetic architecture of inflammatory bowel disease. Nature 491 119–124.
Kang, H. M., Zaitlen, N. A., Wade, C. M., Kirby, A., Heckerman, D., Daly, M. J. and Eskin, E. (2008). Efficient control of population structure in model organism association mapping. Genetics 178 1709–1723.Kang, H. M., Zaitlen, N. A., Wade, C. M., Kirby, A., Heckerman, D., Daly, M. J. and Eskin, E. (2008). Efficient control of population structure in model organism association mapping. Genetics 178 1709–1723.
Kang, H. M., Sul, J. H., Service, S. K., Zaitlen, N. A., Kong, S.-Y., Freimer, N. B., Sabatti, C. and Eskin, E. (2010). Variance component model to account for sample structure in genome-wide association studies. Nat. Genet. 42 348–354.Kang, H. M., Sul, J. H., Service, S. K., Zaitlen, N. A., Kong, S.-Y., Freimer, N. B., Sabatti, C. and Eskin, E. (2010). Variance component model to account for sample structure in genome-wide association studies. Nat. Genet. 42 348–354.
Kostem, E. and Eskin, E. (2013). Improving the accuracy and efficiency of partitioning heritability into the contributions of genomic regions. Am. J. Hum. Genet. 92 558–564.Kostem, E. and Eskin, E. (2013). Improving the accuracy and efficiency of partitioning heritability into the contributions of genomic regions. Am. J. Hum. Genet. 92 558–564.
Lee, S. H., Wray, N. R., Goddard, M. E. and Visscher, P. M. (2011). Estimating missing heritability for disease from genome-wide association studies. Am. J. Hum. Genet. 88 294–305.Lee, S. H., Wray, N. R., Goddard, M. E. and Visscher, P. M. (2011). Estimating missing heritability for disease from genome-wide association studies. Am. J. Hum. Genet. 88 294–305.
Lippert, C., Listgarten, J., Liu, Y., Kadie, C. M., Davidson, R. I. and Heckerman, D. (2011). FaST linear mixed models for genome-wide association studies. Nat. Methods 8 833–835.Lippert, C., Listgarten, J., Liu, Y., Kadie, C. M., Davidson, R. I. and Heckerman, D. (2011). FaST linear mixed models for genome-wide association studies. Nat. Methods 8 833–835.
Loh, P.-R., Tucker, G., Bulik-Sullivan, B. K., Vilhjalmsson, B. J., Finucane, H. K., Chasman, D. I., Ridker, P. M., Neale, B. M., Berger, B. et al. (2015a). Efficient Bayesian mixed model analysis increases association power in large cohorts. Nat. Genet. 47 284–290.Loh, P.-R., Tucker, G., Bulik-Sullivan, B. K., Vilhjalmsson, B. J., Finucane, H. K., Chasman, D. I., Ridker, P. M., Neale, B. M., Berger, B. et al. (2015a). Efficient Bayesian mixed model analysis increases association power in large cohorts. Nat. Genet. 47 284–290.
Loh, P.-R., Bhatia, G., Gusev, A., Finucane, H. K., Bulik-Sullivan, B. K., Pollack, S. J., Schizophrenia Working Group of the Psychiatric Genomics Consortium, de Candia, T. R., Lee, S. H. et al. (2015b). Contrasting genetic architectures of schizophrenia and other complex diseases using fast variance-components analysis. Nat. Genet. 47 1385–1392.Loh, P.-R., Bhatia, G., Gusev, A., Finucane, H. K., Bulik-Sullivan, B. K., Pollack, S. J., Schizophrenia Working Group of the Psychiatric Genomics Consortium, de Candia, T. R., Lee, S. H. et al. (2015b). Contrasting genetic architectures of schizophrenia and other complex diseases using fast variance-components analysis. Nat. Genet. 47 1385–1392.
Makowsky, R., Pajewski, N. M., Klimentidis, Y. C., Vazquez, A. I., Duarte, C. W., Allison, D. B. and de los Campos, G. (2011). Beyond missing heritability: Prediction of complex traits. PLoS Genet. 7 e1002051.Makowsky, R., Pajewski, N. M., Klimentidis, Y. C., Vazquez, A. I., Duarte, C. W., Allison, D. B. and de los Campos, G. (2011). Beyond missing heritability: Prediction of complex traits. PLoS Genet. 7 e1002051.
Manning, A. K., Hivert, M.-F., Scott, R. A., Grimsby, J. L., Bouatia-Naji, N., Chen, H., Rybin, D., Liu, C.-T., Bielak, L. F. et al. (2012). A genome-wide approach accounting for body mass index identifies genetic variants influencing fasting glycemic traits and insulin resistance. Nat. Genet. 44 659-669.Manning, A. K., Hivert, M.-F., Scott, R. A., Grimsby, J. L., Bouatia-Naji, N., Chen, H., Rybin, D., Liu, C.-T., Bielak, L. F. et al. (2012). A genome-wide approach accounting for body mass index identifies genetic variants influencing fasting glycemic traits and insulin resistance. Nat. Genet. 44 659-669.
Matilainen, K., Mäntysaari, E. A., Lidauer, M. H., Strandén, I. and Thompson, R. (2012). Employing a Monte Carlo algorithm in expectation maximization restricted maximum likelihood estimation of the linear mixed model. J. Anim. Breed. Genet. 129 457–468.Matilainen, K., Mäntysaari, E. A., Lidauer, M. H., Strandén, I. and Thompson, R. (2012). Employing a Monte Carlo algorithm in expectation maximization restricted maximum likelihood estimation of the linear mixed model. J. Anim. Breed. Genet. 129 457–468.
Pickrell, J. K. (2014). Joint analysis of functional genomic data and genome-wide association studies of 18 human traits. Am. J. Hum. Genet. 94 559–573.Pickrell, J. K. (2014). Joint analysis of functional genomic data and genome-wide association studies of 18 human traits. Am. J. Hum. Genet. 94 559–573.
Pirinen, M., Donnelly, P. and Spencer, C. C. A. (2013). Efficient computation with a linear mixed model on large-scale data sets with applications to genetic studies. Ann. Appl. Stat. 7 369–390.Pirinen, M., Donnelly, P. and Spencer, C. C. A. (2013). Efficient computation with a linear mixed model on large-scale data sets with applications to genetic studies. Ann. Appl. Stat. 7 369–390.
Price, A. L., Weale, M. E., Patterson, N., Myers, S. R., Need, A. C., Shianna, K. V., Ge, D., Rotter, J. I., Torres, E. et al. (2008). Long-range LD can confound genome scans in admixed populations. Am. J. Hum. Genet. 1 132–135.Price, A. L., Weale, M. E., Patterson, N., Myers, S. R., Need, A. C., Shianna, K. V., Ge, D., Rotter, J. I., Torres, E. et al. (2008). Long-range LD can confound genome scans in admixed populations. Am. J. Hum. Genet. 1 132–135.
Price, A. L., Helgason, A., Thorleifsson, G., McCarroll, S. A., Kong, A. and Stefansson, K. (2011). Single-tissue and cross-tissue heritability of gene expression via identity-by-descent in related or unrelated individuals. PLoS Genet. 7 e1001317.Price, A. L., Helgason, A., Thorleifsson, G., McCarroll, S. A., Kong, A. and Stefansson, K. (2011). Single-tissue and cross-tissue heritability of gene expression via identity-by-descent in related or unrelated individuals. PLoS Genet. 7 e1001317.
Sabatti, C., Service, S. K., Hartikainen, A.-L., Pouta, A., Ripatti, S., Brodsky, J., Jones, C. G., Zaitlen, N. A., Varilo, T. et al. (2008). Genome-wide association analysis of metabolic traits in a birth cohort from a founder population. Nat. Genet. 41 35–46.Sabatti, C., Service, S. K., Hartikainen, A.-L., Pouta, A., Ripatti, S., Brodsky, J., Jones, C. G., Zaitlen, N. A., Varilo, T. et al. (2008). Genome-wide association analysis of metabolic traits in a birth cohort from a founder population. Nat. Genet. 41 35–46.
Sham, P. C. and Purcell, S. (2001). Equivalence between Haseman–Elston and variance-components linkage analyses for sib pairs. Am. J. Hum. Genet. 68 1527–1532.Sham, P. C. and Purcell, S. (2001). Equivalence between Haseman–Elston and variance-components linkage analyses for sib pairs. Am. J. Hum. Genet. 68 1527–1532.
Sham, P. C., Purcell, S., Cherny, S. S. and Abecasis, G. R. (2002). Powerful regression-based quantitative-trait linkage analysis of general pedigrees. Am. J. Hum. Genet. 71 238–253.Sham, P. C., Purcell, S., Cherny, S. S. and Abecasis, G. R. (2002). Powerful regression-based quantitative-trait linkage analysis of general pedigrees. Am. J. Hum. Genet. 71 238–253.
Speed, D., Hemani, G., Johnson, M. R. and Balding, D. J. (2012). Improved heritability estimation from genome-wide SNPs. Am. J. Hum. Genet. 91 1011–1021.Speed, D., Hemani, G., Johnson, M. R. and Balding, D. J. (2012). Improved heritability estimation from genome-wide SNPs. Am. J. Hum. Genet. 91 1011–1021.
Speliotes, E. K., Willer, C. J., Berndt, S. I., Monda, K. L., Thorleifsson, G., Jackson, A. U., Allen, H. L., Lindgren, C. M., Luan, J. et al. (2010). Association analyses of 249,796 individuals reveal 18 new loci associated with body mass index. Nat. Genet. 42 937-948.Speliotes, E. K., Willer, C. J., Berndt, S. I., Monda, K. L., Thorleifsson, G., Jackson, A. U., Allen, H. L., Lindgren, C. M., Luan, J. et al. (2010). Association analyses of 249,796 individuals reveal 18 new loci associated with body mass index. Nat. Genet. 42 937-948.
Splansky, G. L., Corey, D., Yang, Q., Atwood, L. D., Cupples, L. A., Benjamin, E. J., D’Agostino, R. B., Fox, C. S., Larson, M. G. et al. (2007). The third generation cohort of the National Heart, Lung, and Blood Institute’s Framingham Heart Study: Design, recruitment, and initial examination. Am. J. Epidemiol. 165 1328–1335.Splansky, G. L., Corey, D., Yang, Q., Atwood, L. D., Cupples, L. A., Benjamin, E. J., D’Agostino, R. B., Fox, C. S., Larson, M. G. et al. (2007). The third generation cohort of the National Heart, Lung, and Blood Institute’s Framingham Heart Study: Design, recruitment, and initial examination. Am. J. Epidemiol. 165 1328–1335.
Teslovich, T. M., Musunuru, K., Smith, A. V., Edmondson, A. C., Stylianou, I. M., Koseki, M., Pirruccello, J. P., Ripatti, S., Chasman, D. I. et al. (2010). Biological, clinical and population relevance of 95 loci for blood lipids. Nature 466 707–713.Teslovich, T. M., Musunuru, K., Smith, A. V., Edmondson, A. C., Stylianou, I. M., Koseki, M., Pirruccello, J. P., Ripatti, S., Chasman, D. I. et al. (2010). Biological, clinical and population relevance of 95 loci for blood lipids. Nature 466 707–713.
The Wellcome Trust Case Control Consortium (2007). Genome-wide association study of 14,000 cases of seven common diseases and 3000 shared controls. Nature 447 661–678.The Wellcome Trust Case Control Consortium (2007). Genome-wide association study of 14,000 cases of seven common diseases and 3000 shared controls. Nature 447 661–678.
Thompson, E. A. and Shaw, R. G. (1990). Pedigree analysis for quantitative traits: Variance components without matrix inversion. Biometrics 46 399–413.Thompson, E. A. and Shaw, R. G. (1990). Pedigree analysis for quantitative traits: Variance components without matrix inversion. Biometrics 46 399–413.
Wen, X. and Stephens, M. (2010). Using linear predictors to impute allele frequencies from summary or pooled genotype data. Ann. Appl. Stat. 4 1158–1182.Wen, X. and Stephens, M. (2010). Using linear predictors to impute allele frequencies from summary or pooled genotype data. Ann. Appl. Stat. 4 1158–1182.
Wray, N. R., Yang, J., Hayes, B. J., Price, A. L., Goddard, M. E. and Visscher, P. M. (2013). Pitfalls of predicting complex traits from SNPs. Nat. Rev. Genet. 14 507–515.Wray, N. R., Yang, J., Hayes, B. J., Price, A. L., Goddard, M. E. and Visscher, P. M. (2013). Pitfalls of predicting complex traits from SNPs. Nat. Rev. Genet. 14 507–515.
Wu, T. T., Chen, Y. F., Hastie, T., Sobel, E. and Lange, K. (2009). Genome-wide association analysis by lasso penalized logistic regression. Bioinformatics 25 714–721.Wu, T. T., Chen, Y. F., Hastie, T., Sobel, E. and Lange, K. (2009). Genome-wide association analysis by lasso penalized logistic regression. Bioinformatics 25 714–721.
Yang, J., Benyamin, B., McEvoy, B. P., Gordon, S., Henders, A. K., Nyholt, D. R., Madden, P. A., Heath, A. C., Martin, N. G. et al. (2010). Common SNPs explain a large proportion of the heritability for human height. Nat. Genet. 42 565–569.Yang, J., Benyamin, B., McEvoy, B. P., Gordon, S., Henders, A. K., Nyholt, D. R., Madden, P. A., Heath, A. C., Martin, N. G. et al. (2010). Common SNPs explain a large proportion of the heritability for human height. Nat. Genet. 42 565–569.
Yang, J., Manolio, T. A., Pasquale, L. R., Boerwinkle, E., Caporaso, N., Cunningham, J. M., de Andrade, M., Feenstra, B., Feingold, E. et al. (2011a). Genome partitioning of genetic variation for complex traits using common SNPs. Nat. Genet. 43 519–525.Yang, J., Manolio, T. A., Pasquale, L. R., Boerwinkle, E., Caporaso, N., Cunningham, J. M., de Andrade, M., Feenstra, B., Feingold, E. et al. (2011a). Genome partitioning of genetic variation for complex traits using common SNPs. Nat. Genet. 43 519–525.
Yang, J., Weedon, M. N., Purcell, S., Lettre, G., Estrada, K., Willer, C. J., Smith, A. V., Ingelsson, E., O’Connell, J. R. et al. (2011b). Genomic inflation factors under polygenic inheritance. Eur. J. Hum. Genet. 19 807–812.Yang, J., Weedon, M. N., Purcell, S., Lettre, G., Estrada, K., Willer, C. J., Smith, A. V., Ingelsson, E., O’Connell, J. R. et al. (2011b). Genomic inflation factors under polygenic inheritance. Eur. J. Hum. Genet. 19 807–812.
Yang, J., Ferreira, T., Morris, A. P., Medland, S. E., Genetic Investigation of ANthropometric Traits (GIANT) Consortium, DIAbetes Genetics Replication And Meta-analysis (DIAGRAM) Consortium, Madden, P. A. F., Heath, A. C., Martin, N. G. et al. (2012). Conditional and joint multiple-SNP analysis of GWAS summary statistics identifies additional variants influencing complex traits. Nat. Genet. 44 369–375.Yang, J., Ferreira, T., Morris, A. P., Medland, S. E., Genetic Investigation of ANthropometric Traits (GIANT) Consortium, DIAbetes Genetics Replication And Meta-analysis (DIAGRAM) Consortium, Madden, P. A. F., Heath, A. C., Martin, N. G. et al. (2012). Conditional and joint multiple-SNP analysis of GWAS summary statistics identifies additional variants influencing complex traits. Nat. Genet. 44 369–375.
Yang, J., Zaitlen, N. A., Goddard, M. E., Visscher, P. M. and Price, A. L. (2014). Advantages and pitfalls in the application of mixed-model association methods. Nat. Genet. 46 100–106.Yang, J., Zaitlen, N. A., Goddard, M. E., Visscher, P. M. and Price, A. L. (2014). Advantages and pitfalls in the application of mixed-model association methods. Nat. Genet. 46 100–106.
Yang, J., Bakshi, A., Zhu, Z., Hemani, G., Vinkhuyzen, A. A. E., Lee, S. H., Robinson, M. R., Perry, J. R. B., Nolte, I. M. et al. (2015). Genetic variance estimation with imputed variants finds negligible missing heritability for human height and body mass index. Nat. Genet. 47 1114–1120.Yang, J., Bakshi, A., Zhu, Z., Hemani, G., Vinkhuyzen, A. A. E., Lee, S. H., Robinson, M. R., Perry, J. R. B., Nolte, I. M. et al. (2015). Genetic variance estimation with imputed variants finds negligible missing heritability for human height and body mass index. Nat. Genet. 47 1114–1120.
Yu, J., Pressoir, G., Briggs, W. H., Bi, I. V., Yamasaki, M., Doebley, J. F., McMullen, M. D., Gaut, B. S., Nielsen, D. M. et al. (2006). A unified mixed-model method for association mapping that accounts for multiple levels of relatedness. Nat. Genet. 38 203–208.Yu, J., Pressoir, G., Briggs, W. H., Bi, I. V., Yamasaki, M., Doebley, J. F., McMullen, M. D., Gaut, B. S., Nielsen, D. M. et al. (2006). A unified mixed-model method for association mapping that accounts for multiple levels of relatedness. Nat. Genet. 38 203–208.
Zaykin, D. V., Meng, Z. and Ehm, M. G. (2006). Contrasting linkage-disequilibrium patterns between cases and controls as a novel association-mapping method. Am. J. Hum. Genet. 78 737–746.Zaykin, D. V., Meng, Z. and Ehm, M. G. (2006). Contrasting linkage-disequilibrium patterns between cases and controls as a novel association-mapping method. Am. J. Hum. Genet. 78 737–746.
Zhang, Z., Ersoz, E., Lai, C.-Q., Todhunter, R. J., Tiwari, H. K., Gore, M. A., Bradbury, P. J., Yu, J., Arnett, D. K. et al. (2010). Mixed linear model approach adapted for genome-wide association studies. Nat. Genet. 42 355–360.Zhang, Z., Ersoz, E., Lai, C.-Q., Todhunter, R. J., Tiwari, H. K., Gore, M. A., Bradbury, P. J., Yu, J., Arnett, D. K. et al. (2010). Mixed linear model approach adapted for genome-wide association studies. Nat. Genet. 42 355–360.
Zhou, X. (2017). Supplement to “A unified framework for variance component estimation with summary statistics in genome-wide association studies.” DOI:10.1214/17-AOAS1052SUPP.Zhou, X. (2017). Supplement to “A unified framework for variance component estimation with summary statistics in genome-wide association studies.” DOI:10.1214/17-AOAS1052SUPP.