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Volumn 12, Issue 1, 2016, Pages

Fast and Rigorous Computation of Gene and Pathway Scores from SNP-Based Summary Statistics

Author keywords

[No Author keywords available]

Indexed keywords

COMPUTATIONAL EFFICIENCY; STATISTICAL TESTS; STATISTICS;

EID: 84956753111     PISSN: 1553734X     EISSN: 15537358     Source Type: Journal    
DOI: 10.1371/journal.pcbi.1004714     Document Type: Article
Times cited : (247)

References (49)
  • 2
    • 84892385595 scopus 로고    scopus 로고
    • A review of post-GWAS prioritization approaches
    • Hou L, Zhao H, A review of post-GWAS prioritization approaches. Front Genet. 2013;4: 280. doi: 10.3389/fgene.2013.00280 24367376
    • (2013) Front Genet , vol.4 , pp. 280
    • Hou, L.1    Zhao, H.2
  • 3
    • 77957342306 scopus 로고    scopus 로고
    • Common inherited variation in mitochondrial genes is not enriched for associations with type 2 diabetes or related glycemic traits
    • Segrè A V, Groop L, Mootha VK, Daly MJ, Altshuler D, Common inherited variation in mitochondrial genes is not enriched for associations with type 2 diabetes or related glycemic traits. PLoS Genet. 2010;6: e1001058. doi: 10.1371/journal.pgen.1001058 20714348
    • (2010) PLoS Genet , vol.6 , pp. 1001058
    • Segrè A, V.1    Groop, L.2    Mootha, V.K.3    Daly, M.J.4    Altshuler, D.5
  • 4
    • 84923096381 scopus 로고    scopus 로고
    • Biological interpretation of genome-wide association studies using predicted gene functions
    • Pers TH, Karjalainen JM, Chan Y, Westra H-J, Wood AR, Yang J, et al. Biological interpretation of genome-wide association studies using predicted gene functions. Nat Commun. Nature Publishing Group, a division of Macmillan Publishers Limited. All Rights Reserved.; 2015;6: 5890. Available: doi: http://dx.doi.org/10.1038/ncomms6890 25597830
    • (2015) Nat Commun , vol.6 , pp. 5890
    • Pers, T.H.1    Karjalainen, J.M.2    Chan, Y.3    Westra, H.-J.4    Wood, A.R.5    Yang, J.6
  • 5
    • 84908890496 scopus 로고    scopus 로고
    • Defining the role of common variation in the genomic and biological architecture of adult human height
    • Wood AR, Esko T, Yang J, Vedantam S, Pers TH, Gustafsson S, et al. Defining the role of common variation in the genomic and biological architecture of adult human height. Nat Genet. 2014;46: 1173–86. doi: 10.1038/ng.3097 25282103
    • (2014) Nat Genet , vol.46 , pp. 1173-1186
    • Wood, A.R.1    Esko, T.2    Yang, J.3    Vedantam, S.4    Pers, T.H.5    Gustafsson, S.6
  • 6
    • 0034069495 scopus 로고    scopus 로고
    • Gene ontology: tool for the unification of biology. The Gene Ontology Consortium
    • Ashburner M, Ball CA, Blake JA, Botstein D, Butler H, Cherry JM, et al. Gene ontology: tool for the unification of biology. The Gene Ontology Consortium. Nat Genet. 2000;25: 25–29. doi: 10.1038/75556 10802651
    • (2000) Nat Genet , vol.25 , pp. 25-29
    • Ashburner, M.1    Ball, C.A.2    Blake, J.A.3    Botstein, D.4    Butler, H.5    Cherry, J.M.6
  • 7
    • 84891760956 scopus 로고    scopus 로고
    • Data, information, knowledge and principle: Back to metabolism in KEGG
    • Kanehisa M, Goto S, Sato Y, Kawashima M, Furumichi M, Tanabe M, Data, information, knowledge and principle: Back to metabolism in KEGG. Nucleic Acids Res. 2014;42: D199–D205. doi: 10.1093/nar/gkt1076 24214961
    • (2014) Nucleic Acids Res , vol.42 , pp. 199-205
    • Kanehisa, M.1    Goto, S.2    Sato, Y.3    Kawashima, M.4    Furumichi, M.5    Tanabe, M.6
  • 8
    • 0141742293 scopus 로고    scopus 로고
    • PANTHER: A library of protein families and subfamilies indexed by function
    • Thomas PD, Campbell MJ, Kejariwal A, Mi H, Karlak B, Daverman R, et al. PANTHER: A library of protein families and subfamilies indexed by function. Genome Res. 2003;13: 2129–2141. doi: 10.1101/gr.772403 12952881
    • (2003) Genome Res , vol.13 , pp. 2129-2141
    • Thomas, P.D.1    Campbell, M.J.2    Kejariwal, A.3    Mi, H.4    Karlak, B.5    Daverman, R.6
  • 9
    • 78651275182 scopus 로고    scopus 로고
    • Reactome: A database of reactions, pathways and biological processes
    • Croft D, O’Kelly G, Wu G, Haw R, Gillespie M, Matthews L, et al. Reactome: A database of reactions, pathways and biological processes. Nucleic Acids Res. 2011;39: D691–D697. doi: 10.1093/nar/gkq1018 21067998
    • (2011) Nucleic Acids Res , vol.39 , pp. 691-697
    • Croft, D.1    O’Kelly, G.2    Wu, G.3    Haw, R.4    Gillespie, M.5    Matthews, L.6
  • 11
    • 77955064853 scopus 로고    scopus 로고
    • A versatile gene-based test for genome-wide association studies
    • Liu JZ, McRae AF, Nyholt DR, Medland SE, Wray NR, Brown KM, et al. A versatile gene-based test for genome-wide association studies. Am J Hum Genet. The American Society of Human Genetics; 2010;87: 139–145. doi: 10.1016/j.ajhg.2010.06.009 20598278
    • (2010) Am J Hum Genet , vol.87 , pp. 139-145
    • Liu, J.Z.1    McRae, A.F.2    Nyholt, D.R.3    Medland, S.E.4    Wray, N.R.5    Brown, K.M.6
  • 12
    • 79952467494 scopus 로고    scopus 로고
    • GATES: A rapid and powerful gene-based association test using extended Simes procedure
    • Li MX, Gui HS, Kwan JSH, Sham PC, GATES: A rapid and powerful gene-based association test using extended Simes procedure. Am J Hum Genet. 2011;88: 283–293. doi: 10.1016/j.ajhg.2011.01.019 21397060
    • (2011) Am J Hum Genet , vol.88 , pp. 283-293
    • Li, M.X.1    Gui, H.S.2    Kwan, J.S.H.3    Sham, P.C.4
  • 13
    • 79959898376 scopus 로고    scopus 로고
    • Prioritizing candidate disease genes by network-based boosting of genome-wide association data
    • Lee I, Blom UM, Wang PI, Shim JE, Marcotte EM, Prioritizing candidate disease genes by network-based boosting of genome-wide association data. Genome Res. 2011;21: 1109–21. doi: 10.1101/gr.118992.110 21536720
    • (2011) Genome Res , vol.21 , pp. 1109-1121
    • Lee, I.1    Blom, U.M.2    Wang, P.I.3    Shim, J.E.4    Marcotte, E.M.5
  • 14
    • 79951966907 scopus 로고    scopus 로고
    • An efficient hierarchical generalized linear mixed model for pathway analysis of genome-wide association studies
    • Wang L, Jia P, Wolfinger RD, Chen X, Grayson BL, Aune TM, et al. An efficient hierarchical generalized linear mixed model for pathway analysis of genome-wide association studies. Bioinformatics. 2011;27: 686–692. doi: 10.1093/bioinformatics/btq728 21266443
    • (2011) Bioinformatics , vol.27 , pp. 686-692
    • Wang, L.1    Jia, P.2    Wolfinger, R.D.3    Chen, X.4    Grayson, B.L.5    Aune, T.M.6
  • 15
    • 36249029788 scopus 로고    scopus 로고
    • Pathway-based approaches for analysis of genomewide association studies
    • Wang K, Li M, Bucan M, Pathway-based approaches for analysis of genomewide association studies. Am J Hum Genet. 2007;81: 1278–1283. doi: 10.1086/522374 17966091
    • (2007) Am J Hum Genet , vol.81 , pp. 1278-1283
    • Wang, K.1    Li, M.2    Bucan, M.3
  • 16
    • 84868369445 scopus 로고    scopus 로고
    • A multi-SNP locus-association method reveals a substantial fraction of the missing heritability
    • Ehret GB, Lamparter D, Hoggart CJ, Whittaker JC, Beckmann JS, Kutalik Z, A multi-SNP locus-association method reveals a substantial fraction of the missing heritability. Am J Hum Genet. 2012;91: 863–871. doi: 10.1016/j.ajhg.2012.09.013 23122585
    • (2012) Am J Hum Genet , vol.91 , pp. 863-871
    • Ehret, G.B.1    Lamparter, D.2    Hoggart, C.J.3    Whittaker, J.C.4    Beckmann, J.S.5    Kutalik, Z.6
  • 17
    • 84862832570 scopus 로고    scopus 로고
    • Conditional and joint multiple-SNP analysis of GWAS summary statistics identifies additional variants influencing complex traits
    • Yang J, Ferreira T, Morris AP, Medland SE, Madden PAF, Heath AC, et al. Conditional and joint multiple-SNP analysis of GWAS summary statistics identifies additional variants influencing complex traits. Nature Genetics. 2012. pp. 369–375. doi: 10.1038/ng.2213 22426310
    • (2012) Nature Genetics , pp. 369-375
    • Yang, J.1    Ferreira, T.2    Morris, A.P.3    Medland, S.E.4    Madden, P.A.F.5    Heath, A.C.6
  • 18
    • 67649584052 scopus 로고    scopus 로고
    • Gene Ontology Analysis of GWA Study Data Sets Provides Insights into the Biology of Bipolar Disorder
    • Holmans P, Green EK, Pahwa JS, Ferreira M a R, Purcell SM, Sklar P, et al. Gene Ontology Analysis of GWA Study Data Sets Provides Insights into the Biology of Bipolar Disorder. Am J Hum Genet. 2009;85: 13–24. doi: 10.1016/j.ajhg.2009.05.011 19539887
    • (2009) Am J Hum Genet , vol.85 , pp. 13-24
    • Holmans, P.1    Green, E.K.2    Pahwa, J.S.3    Ferreira M a, R.4    Purcell, S.M.5    Sklar, P.6
  • 19
    • 84910614680 scopus 로고    scopus 로고
    • A Method for Gene-Based Pathway Analysis Using Genomewide Association Study Summary Statistics Reveals Nine New Type 1 Diabetes Associations Genetic Epidemiology
    • Evangelou M, Smyth DJ, Fortune MD, Burren OS, Walker NM, Guo H, et al. A Method for Gene-Based Pathway Analysis Using Genomewide Association Study Summary Statistics Reveals Nine New Type 1 Diabetes Associations Genetic Epidemiology. Genet Epidemiol. 2014;38: 661–670. doi: 10.1002/gepi.21853 25371288
    • (2014) Genet Epidemiol , vol.38 , pp. 661-670
    • Evangelou, M.1    Smyth, D.J.2    Fortune, M.D.3    Burren, O.S.4    Walker, N.M.5    Guo, H.6
  • 20
    • 80051499915 scopus 로고    scopus 로고
    • Rare-variant association testing for sequencing data with the sequence kernel association test
    • Wu MC, Lee S, Cai T, Li Y, Boehnke M, Lin X, Rare-variant association testing for sequencing data with the sequence kernel association test. Am J Hum Genet. 2011;89: 82–93. doi: 10.1016/j.ajhg.2011.05.029 21737059
    • (2011) Am J Hum Genet , vol.89 , pp. 82-93
    • Wu, M.C.1    Lee, S.2    Cai, T.3    Li, Y.4    Boehnke, M.5    Lin, X.6
  • 21
    • 36749079336 scopus 로고    scopus 로고
    • So many correlated tests, so little time! Rapid adjustment of P values for multiple correlated tests
    • Conneely KN, Boehnke M, So many correlated tests, so little time! Rapid adjustment of P values for multiple correlated tests. Am J Hum Genet. 2007;81: 1158–1168. doi: 10.1086/522036 17966093
    • (2007) Am J Hum Genet , vol.81 , pp. 1158-1168
    • Conneely, K.N.1    Boehnke, M.2
  • 24
    • 77956331627 scopus 로고    scopus 로고
    • Integrating common and rare genetic variation in diverse human populations
    • The International HapMap 3 ConsortiumIntegrating common and rare genetic variation in diverse human populations. Nature. 2010;467: 52–8. doi: 10.1038/nature09298 20811451
    • (2010) Nature , vol.467 , pp. 52-58
  • 25
    • 84894288992 scopus 로고    scopus 로고
    • Genetics of rheumatoid arthritis contributes to biology and drug discovery
    • Okada Y, Wu D, Trynka G, Raj T, Terao C, Ikari K, et al. Genetics of rheumatoid arthritis contributes to biology and drug discovery. Nature. 2014;506: 376–81. doi: 10.1038/nature12873 24390342
    • (2014) Nature , vol.506 , pp. 376-381
    • Okada, Y.1    Wu, D.2    Trynka, G.3    Raj, T.4    Terao, C.5    Ikari, K.6
  • 26
    • 84922634287 scopus 로고    scopus 로고
    • VEGAS2 : Software for More Flexible Gene-Based Testing
    • Mishra A, Macgregor S, VEGAS2 : Software for More Flexible Gene-Based Testing. Twin Res Hum Genet. 2015;18: 86–91. doi: 10.1017/thg.2014.79 25518859
    • (2015) Twin Res Hum Genet , vol.18 , pp. 86-91
    • Mishra, A.1    Macgregor, S.2
  • 27
    • 42449133365 scopus 로고    scopus 로고
    • The CoLaus study: a population-based study to investigate the epidemiology and genetic determinants of cardiovascular risk factors and metabolic syndrome
    • Firmann M, Mayor V, Vidal P, Bochud M, Pécoud A, Hayoz D, et al. The CoLaus study: a population-based study to investigate the epidemiology and genetic determinants of cardiovascular risk factors and metabolic syndrome. BMC Cardiovascular Disorders. 2008. p. 6. doi: 10.1186/1471-2261-8-6 18366642
    • (2008) BMC Cardiovascular Disorders , pp. 6
    • Firmann, M.1    Mayor, V.2    Vidal, P.3    Bochud, M.4    Pécoud, A.5    Hayoz, D.6
  • 28
    • 77957126059 scopus 로고    scopus 로고
    • A trans-acting locus regulates an anti-viral expression network and type 1 diabetes risk
    • Heinig M, Petretto E, Wallace C, Bottolo L, Rotival M, Lu H, et al. A trans-acting locus regulates an anti-viral expression network and type 1 diabetes risk. Nature. 2010;467: 460–464. doi: 10.1038/nature09386 20827270
    • (2010) Nature , vol.467 , pp. 460-464
    • Heinig, M.1    Petretto, E.2    Wallace, C.3    Bottolo, L.4    Rotival, M.5    Lu, H.6
  • 29
    • 84930731753 scopus 로고    scopus 로고
    • Burren OS, Guo H, Wallace C. VSEAMS : A pipeline for variant set enrichment analysis using summary GWAS data identifies IKZF3, BATF and ESRRA as key transcription factors in type 1 diabetes. 2014;30: 0–26. doi: 10.1093/bioinformatics/btu571
  • 30
    • 34250727580 scopus 로고    scopus 로고
    • Davis J, Goadrich M. The Relationship Between Precision-Recall and ROC Curves. Proc 23rd Int Conf Mach Learn—ICML’06. 2006; 233–240. doi: 10.1145/1143844.1143874
  • 31
    • 78649489009 scopus 로고    scopus 로고
    • Genome-wide meta-analysis increases to 71 the number of confirmed Crohn’s disease susceptibility loci
    • Franke A, McGovern DPB, Barrett JC, Wang K, Radford-Smith GL, Ahmad T, et al. Genome-wide meta-analysis increases to 71 the number of confirmed Crohn’s disease susceptibility loci. Nat Genet. Nature Publishing Group; 2010;42: 1118–25. doi: 10.1038/ng.717 21102463
    • (2010) Nat Genet , vol.42 , pp. 1118-1125
    • Franke, A.1    McGovern, D.P.B.2    Barrett, J.C.3    Wang, K.4    Radford-Smith, G.L.5    Ahmad, T.6
  • 32
    • 70649113728 scopus 로고    scopus 로고
    • Common variants at five new loci associated with early-onset inflammatory bowel disease
    • Imielinski M, Baldassano RN, Griffiths A, Russell RK, Annese V, Dubinsky M, et al. Common variants at five new loci associated with early-onset inflammatory bowel disease. Nat Genet. 2009;41: 1335–1340. doi: 10.1038/ng.489 19915574
    • (2009) Nat Genet , vol.41 , pp. 1335-1340
    • Imielinski, M.1    Baldassano, R.N.2    Griffiths, A.3    Russell, R.K.4    Annese, V.5    Dubinsky, M.6
  • 33
    • 84969213492 scopus 로고    scopus 로고
    • Genome-wide association study of 14, 000 cases of seven common diseases and
    • Wellcome T, Case T, Consortium C, Genome-wide association study of 14, 000 cases of seven common diseases and. Nature. 2007;447: 661–78. doi: 10.1038/nature05911 17554300
    • (2007) Nature , vol.447 , pp. 661-678
    • Wellcome, T.1    Case, T.2    Consortium, C.3
  • 34
    • 75749086085 scopus 로고    scopus 로고
    • New genetic loci implicated in fasting glucose homeostasis and their impact on type 2 diabetes risk
    • Dupuis J, Langenberg C, Prokopenko I, Saxena R, Soranzo N, Jackson AU, et al. New genetic loci implicated in fasting glucose homeostasis and their impact on type 2 diabetes risk. Nat Genet. 2010;42: 105–116. doi: 10.1038/ng.520 20081858
    • (2010) Nat Genet , vol.42 , pp. 105-116
    • Dupuis, J.1    Langenberg, C.2    Prokopenko, I.3    Saxena, R.4    Soranzo, N.5    Jackson, A.U.6
  • 35
    • 84860331458 scopus 로고    scopus 로고
    • Estrada K, Styrkarsdottir U, Evangelou E, Hsu YH, Duncan EL, Ntzani EE, et al. Genome-wide meta-analysis identifies 56 bone mineral density loci and reveals 14 loci associated with risk of fracture. 2012;44: 491–501. doi: 10.1038/ng.2249
  • 36
    • 40149089555 scopus 로고    scopus 로고
    • Wnt and hedgehog signaling pathways in bone development
    • Day TF, Yang Y, Wnt and hedgehog signaling pathways in bone development. J Bone Joint Surg Am. 2008;90 Suppl 1: 19–24. doi: 10.2106/JBJS.G.01174 18292352
    • (2008) J Bone Joint Surg Am , vol.90 , pp. 19-24
    • Day, T.F.1    Yang, Y.2
  • 37
    • 77951711343 scopus 로고    scopus 로고
    • Genome-wide meta-analyses identify multiple loci associated with smoking behavior
    • Tobacco T, Consortium G, Genome-wide meta-analyses identify multiple loci associated with smoking behavior. Nat Genet. 2010;42: 441–7. doi: 10.1038/ng.571 20418890
    • (2010) Nat Genet , vol.42 , pp. 441-447
    • Tobacco, T.1    Consortium, G.2
  • 38
    • 79958727628 scopus 로고    scopus 로고
    • Complement in age-related macular degeneration: a focus on function
    • Bradley DT, Zipfel PF, Hughes AE, Complement in age-related macular degeneration: a focus on function. Eye (Lond). 2011;25: 683–693. doi: 10.1038/eye.2011.37
    • (2011) Eye (Lond) , vol.25 , pp. 683-693
    • Bradley, D.T.1    Zipfel, P.F.2    Hughes, A.E.3
  • 39
    • 84856985610 scopus 로고    scopus 로고
    • Lipids, lipoproteins, and age-related macular degeneration
    • Ebrahimi KB, Handa JT, Lipids, lipoproteins, and age-related macular degeneration. J Lipids. 2011;2011: 802059. doi: 10.1155/2011/802059 21822496
    • (2011) J Lipids , vol.2011 , pp. 802059
    • Ebrahimi, K.B.1    Handa, J.T.2
  • 40
    • 84927739557 scopus 로고    scopus 로고
    • JEPEG: a summary statistics based tool for gene-level joint testing of functional variants
    • Lee D, Williamson VS, Bigdeli TB, Riley BP, Fanous a. H, Vladimirov VI, et al. JEPEG: a summary statistics based tool for gene-level joint testing of functional variants. Bioinformatics. 2014;31: 1176–1182. doi: 10.1093/bioinformatics/btu816 25505091
    • (2014) Bioinformatics , vol.31 , pp. 1176-1182
    • Lee, D.1    Williamson, V.S.2    Bigdeli, T.B.3    Riley, B.P.4    Fanous, H.5    Vladimirov, V.I.6
  • 41
    • 0001341675 scopus 로고
    • Numerical Computation of Multivariate Normal Probabilities
    • Genz A, Numerical Computation of Multivariate Normal Probabilities. J Comput Graph Stat. 1992;1: 141–149. doi: 10.1080/10618600.1992.10477010
    • (1992) J Comput Graph Stat , vol.1 , pp. 141-149
    • Genz, A.1
  • 43
    • 27344435774 scopus 로고    scopus 로고
    • Gene set enrichment analysis: a knowledge-based approach for interpreting genome-wide expression profiles
    • Subramanian A, Tamayo P, Mootha VK, Mukherjee S, Ebert BL, Gillette M a, et al. Gene set enrichment analysis: a knowledge-based approach for interpreting genome-wide expression profiles. Proc Natl Acad Sci U S A. 2005;102: 15545–50. doi: 10.1073/pnas.0506580102 16199517
    • (2005) Proc Natl Acad Sci U S A , vol.102 , pp. 15545-15550
    • Subramanian, A.1    Tamayo, P.2    Mootha, V.K.3    Mukherjee, S.4    Ebert, B.L.5    Gillette, M.6
  • 44
    • 84943622735 scopus 로고    scopus 로고
    • DISSCO: direct imputation of summary statistics allowing covariates
    • Xu Z, Duan Q, Yan S, Chen W, Li M, Lange E, et al. DISSCO: direct imputation of summary statistics allowing covariates. Bioinformatics. 2015;31: 2434–2442. doi: 10.1093/bioinformatics/btv168 25810429
    • (2015) Bioinformatics , vol.31 , pp. 2434-2442
    • Xu, Z.1    Duan, Q.2    Yan, S.3    Chen, W.4    Li, M.5    Lange, E.6
  • 45
    • 84868369445 scopus 로고    scopus 로고
    • A multi-SNP locus-association method reveals a substantial fraction of the missing heritability
    • Ehret GB, Lamparter D, Hoggart CJ, Whittaker JC, Beckmann JS, Kutalik Z, A multi-SNP locus-association method reveals a substantial fraction of the missing heritability. Am J Hum Genet. 2012;91: 863–871. doi: 10.1016/j.ajhg.2012.09.013 23122585
    • (2012) Am J Hum Genet , vol.91 , pp. 863-871
    • Ehret, G.B.1    Lamparter, D.2    Hoggart, C.J.3    Whittaker, J.C.4    Beckmann, J.S.5    Kutalik, Z.6
  • 46
    • 43249090541 scopus 로고    scopus 로고
    • A multiple testing correction method for genetic association studies using correlated single nucleotide polymorphisms
    • Gao X, Starmer J, Martin ER, A multiple testing correction method for genetic association studies using correlated single nucleotide polymorphisms. Genet Epidemiol. 2008;32: 361–369. doi: 10.1002/gepi.20310 18271029
    • (2008) Genet Epidemiol , vol.32 , pp. 361-369
    • Gao, X.1    Starmer, J.2    Martin, E.R.3
  • 47
    • 0001008882 scopus 로고
    • The Distribution of a Linear Combination of x2 Random Variables
    • B DR, The Distribution of a Linear Combination of x2 Random Variables. J R Stat Soc Ser C. 1980;29: 323–333.
    • (1980) J R Stat Soc Ser C , vol.29 , pp. 323-333
    • B, D.R.1
  • 48
    • 0041337276 scopus 로고
    • Algorithm AS 204: the distribution of a positive linear combination of chi2 random variables
    • Farebrother R, Algorithm AS 204: the distribution of a positive linear combination of chi2 random variables. J R Stat Soc Ser C. 1984;33: 332–339. doi: 10.2307/2347721
    • (1984) J R Stat Soc Ser C , vol.33 , pp. 332-339
    • Farebrother, R.1
  • 49
    • 73149087027 scopus 로고    scopus 로고
    • Computing the distribution of quadratic forms: Further comparisons between the Liu-Tang-Zhang approximation and exact methods
    • Duchesne P, Lafaye De Micheaux P, Computing the distribution of quadratic forms: Further comparisons between the Liu-Tang-Zhang approximation and exact methods. Comput Stat Data Anal. 2010;54: 858–862. doi: 10.1016/j.csda.2009.11.025
    • (2010) Comput Stat Data Anal , vol.54 , pp. 858-862
    • Duchesne, P.1    Lafaye De Micheaux, P.2


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