Abstract
Motivated by a genome-wide association study to discover risk variants for the progression of Age-related Macular Degeneration (AMD), we develop a computationally efficient copula-based score test, in which the dependence between bivariate progression times is taken into account. Specifically, a two-step estimation approach with numerical derivatives to approximate the score function and observed information matrix is proposed. Both parametric and weakly parametric marginal distributions under the proportional hazards assumption are considered. Extensive simulation studies are conducted to evaluate the Type I error control and power performance of the proposed method. Finally, we apply our method to a large randomized trial data, the Age-related Eye Disease Study, to identify susceptible risk variants for AMD progression. The top variants identified on Chromosome 10 show significantly differential progression profiles for different genetic groups, which are critical in characterizing and predicting the risk of progression-to-late-AMD for patients with mild to moderate AMD.
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References
Akaike H (1998) Information theory and an extension of the maximum likelihood principle. In: Parzen E, Tanabe K, Kitagawa G (eds) Selected Papers of Hirotugu Akaike. Springer, New York, pp 477–485
AREDS Group (1999) The age-related eye disease study (AREDS): design implications. Control Clin Trials 20(6):573–600
Breslow NE (1972) Discussion of the paper by D. R. Cox. J R Stat Soc Ser B 34:216–217
Cantor RM, Lange K, Sinsheimer JS (2010) Prioritizing GWAS results: a review of statistical methods and recommendations for their application. Am J Hum Genet 86(1):6–22
Chen X, Fan Y, Pouzo D, Ying Z (2010) Estimation and model selection of semiparametric multivariate survival functions under general censorship. J Econom 157(2):129–142
Chen Z (2012) A flexible copula model for bivariate survival data. PhD thesis, University of Rochester
Clayton DG (1978) A model for association in bivariate life tables and its application in epidemiological studies of familial tendency in chronic disease incidence. Biometrika 65(1):141–151
Cox DR, Hinkley DV (1979) Theor Stat. Chapman & Hall/CRC, London
Ding Y, Nan B (2011) A sieve m-theorem for bundled parameters in semiparametric models, with application to the efficient estimation in a linear model for censored data. Ann Stat 39(1):2795–3443
Ding Y, Liu Y, Yan Q, Fritsche LG, Cook RJ, Clemons T, Ratnapriya R, Klein ML, Abecasis GR, Swaroop A, Chew EY, Weeks DE, Chen W, The AREDS2 Research Group (2017) Bivariate analysis of age-related macular degeneration progression using genetic risk scores. Genetics 206(1):119–133
Fritsche LG, Chen W, Schu M, Yaspan BL, Yu Y, Thorleifsson G, Zack DJ, Arakawa S, Cipriani V, Ripke S, Igo RP Jr, Buitendijk GHS, Sim X, Weeks DE, Guymer RH, Merriam JE, Francis PJ, Hannum G, Agarwal A, Armbrecht AM, Audo I, Aung T, Barile GR, Benchaboune M, Bird AC, Bishop PN, Branham KE, Brooks M, Brucker AJ, Cade WH, Cain MS, Campochiaro PA, Chan CC, Cheng CY, Chew EY, Chin KA, Chowers I, Clayton DG, Cojocaru R, Conley YP, Cornes BK, Daly MJ, Dhillon B, Edwards AO, Evangelou E, Fagerness J, Ferreyra HA, Friedman JS, Geirsdottir A, George RJ, Gieger C, Gupta N, Hagstrom SA, Harding SP, Haritoglou C, Heckenlively JR, Holz FG, Hughes G, Ioannidis JPA, Ishibashi T, Joseph P, Jun G, Kamatani Y, Katsanis N, Keilhauer C, Khan JC, Kim IK, Kiyohara Y, Klein BEK, Klein R, Kovach JL, Kozak I, Lee CJ, Lee KE, Lichtner P, Lotery AJ, Meitinger T, Mitchell P, Mohand-Sad S, Moore AT, Morgan DJ, Morrison MA, Myers CE, Naj AC, Nakamura Y, Okada Y, Orlin A, Ortube MC, Othman MI, Pappas C, Park KH, Pauer GJT, Peachey NS, Poch O, Priya RR, Reynolds R, Richardson AJ, Ripp R, Rudolph G, Ryu E, Sahel JA, Schaumberg DA, Scholl HPN, Schwartz SG, Scott WK, Shahid H, Sigurdsson H, Silvestri G, Sivakumaran TA, Smith RT, Sobrin L, Souied EH, Stambolian DE, Stefansson H, Sturgill-Short GM, Takahashi A, Tosakulwong N, Truitt BJ, Tsironi EE, Uitterlinden A, van Duijn CM, Vijaya L, Vingerling JR, Vithana EN, Webster AR, Wichmann HE, Winkler TW, Wong TY, Wright AF, Zelenika D, Zhang M, Zhao L, Zhang K, Klein ML, Hageman GS, Lathrop GM, Stefansson K, Allikmets R, Baird PN, Gorin MB, Wang JJ, Klaver CCW, Seddon JM, Pericak-Vance MA, Iyengar SK, Yates JRW, Swaroop A, Weber BHF, Kubo M, DeAngelis MM, Lveillard T, Thorsteinsdottir U, Haines JL, Farrer LA, Heid IM, Abecasis GR, AMD Gene Consortium (2013) Seven new loci associated with age-related macular degeneration. Nat Genet 45(4):433–439
Fritsche LG, Igl W, Bailey JNC, Grassmann F, Sengupta S, Bragg-Gresham JL, Burdon KP, Hebbring SJ, Wen C, Gorski M, Kim IK, Cho D, Zack D, Souied E, Scholl HPN, Bala E, Lee KE, Hunter DJ, Sardell RJ, Mitchell P, Merriam JE, Cipriani V, Hoffman JD, Schick T, Lechanteur YTE, Guymer RH, Johnson MP, Jiang Y, Stanton CM, Buitendijk GHS, Zhan X, Kwong AM, Boleda A, Brooks M, Gieser L, Ratnapriya R, Branham KE, Foerster JR, Heckenlively JR, Othman MI, Vote BJ, Liang HH, Souzeau E, McAllister IL, Isaacs T, Hall J, Lake S, Mackey DA, Constable IJ, Craig JE, Kitchner TE, Yang Z, Su Z, Luo H, Chen D, Ouyang H, Flagg K, Lin D, Mao G, Ferreyra H, Stark K, von Strachwitz CN, Wolf A, Brandl C, Rudolph G, Olden M, Morrison MA, Morgan DJ, Schu M, Ahn J, Silvestri G, Tsironi EE, Park KH, Farrer LA, Orlin A, Brucker A, Li M, Curcio CA, Mohand-Sad S, Sahel JA, Audo I, Benchaboune M, Cree AJ, Rennie CA, Goverdhan SV, Grunin M, Hagbi-Levi S, Campochiaro P, Katsanis N, Holz FG, Blond F, Blanch H, Deleuze JF, Igo RP Jr, Truitt B, Peachey NS, Meuer SM, Myers CE, Moore EL, Klein R, Hauser MA, Postel EA, Courtenay MD, Schwartz SG, Kovach JL, Scott WK, Liew G, Tan AG, Gopinath B, Merriam JC, Smith RT, Khan JC, Shahid H, Moore AT, McGrath JA, Laux R, Brantley MA Jr, Agarwal A, Ersoy L, Caramoy A, Langmann T, Saksens NTM, de Jong EK, Hoyng CB, Cain MS, Richardson AJ, Martin TM, Blangero J, Weeks DE, Dhillon B, van Duijn CM, Doheny KF, Romm J, Klaver CCW, Hayward C, Gorin MB, Klein ML, Baird PN, den Hollander AI, Fauser S, Yates JRW, Allikmets R, Wang JJ, Schaumberg DA, Klein BEK, Hagstrom SA, Chowers I, Lotery AJ, Lveillard T, Zhang K, Brilliant MH, Hewitt AW, Swaroop A, Chew EY, Pericak-Vance MA, DeAngelis M, Stambolian D, Haines JL, Iyengar SK, Weber BHF, Abecasis GR, Heid IM (2016) A large genome-wide association study of age-related macular degeneration highlights contributions of rare and common variants. Nat Genet 48(2):134–143
Goethals K, Janssen P, Duchateau L (2008) Frailty models and copulas: similarities and differences. J Appl Stat 35(9):1071–1079
Gumbel EJ (1960) Bivariate exponential distributions. J Am Stat Assoc 55(292):698–707
He W, Lawless JF (2003) Flexible maximum likelihood methods for bivariate proportional hazards models. Biometrics 59(4):837–848
Hougaard P (2000) Anal Multivar Surviv Data. Springer, New York
Joe H (1997) Multivariate models and dependence concepts. Chapman & Hall/CRC, London
Kim G, Silvapulle MJ, Silvapulle P (2007) Comparison of semiparametric and parametric methods for estimating copulas. Comput Stat Data Anal 51(6):2836–2850
Lawless JF, Yilmaz YE (2011) Semiparametric estimation in copula models for bivariate sequential survival times. Biom J 53(5):779–796
Lee EW, Wei LJ, Amato DA (1992) Cox-type regression analysis for large numbers of small groups of correlated failure time observations. In: Klein J, Goel P (eds) Surviv Anal State Art, vol 211. Springer, Dordrecht, pp 237–247
Lindfield GR, Penny JET (1989) Microcomputers in numerical analysis. Halsted Press, New York
Mei M (2016) A goodness-of-fit test for semi-parametric copula models of right-censored bivariate survival times. Master’s thesis, Simon Fraser University
Nelsen RB (2006) An introduction to Copulas. Springer, New York
Oakes D (1982) A model for association in bivariate survival data. J R Stat Soc Ser B 44(3):414–422
Sardell RJ, Persad PJ, Pan SS, Whitehead P, Adams LD, Laux R, Fortun JA, Brantley MA Jr, Kovach JL, Schwartz SG, Agarwal A, Haines JL, Scott WK, Pericak-Vance MA (2016) Progression rate from intermediate to advanced age-related macular degeneration is correlated with the number of risk alleles at the CFH locus. Investig Ophthalmol Visual Sci 57(14):6107–6115
Schwarz G (1978) Estimating the dimension of a model. Ann Stat 6(2):461–464
Seddon JM, Reynolds R, Maller J, Fagerness JA, Daly MJ, Rosner B (2009) Prediction model for prevalence and incidence of advanced age-related macular degeneration based on genetic, demographic, and environmental variables. Investig Ophthalmol Visual Sci 50(5):2044–2053
Seddon JM, Reynolds R, Yu Y, Rosner B (2014) Three new genetic loci are independently related to progression to advanced macular degeneration. PLoS ONE 9(1):1–11
Sha Q, Zhang Z, Zhang S (2011) An improved score test for genetic association studies. Genet Epidemiol 35(5):350–359
Shih JH, Louis TA (1995) Inferences on the association parameter in copula models for bivariate survival data. Biometrics 51(4):1384–1399
Sklar A (1959) Fonctions de répartition à n dimensions et leurs marges. Publications de L’Institut de Statistique de L’Université de Paris 8:229–231
Swaroop A, Chew EY, Rickman CB, Abecasis GR (2009) Unraveling a multifactorial late-onset disease: from genetic susceptibility to disease mechanisms for age-related macular degeneration. Annu Rev Genomics Human Genet 10:19–43
The Eye Diseases Prevalence Research Group (2004) Causes and prevalence of visual impairment among adults in the united states. Arch Ophthalmol 122(4):477–485
Wang W, Wells MT (2000) Model selection and semiparametric inference for bivariate failure-time data. J Am Stat Assoc 95(449):62–72
Wei LJ, Lin D, Weissfeld L (1989) Regression analysis of multivariate incomplete failure time data by modeling marginal distributions. J Am Stat Assoc 84(408):1065–1073
Yan Q, Ding Y, Liu Y, Sun T, Fritsche LG, Clemons T, Ratnapriya R, Klein ML, Cook RJ, Liu Y, Fan R, Wei L, Abecasis GR, Swaroop A, Chew EY, Group AR, Weeks DE, Chen W (2018) Genome-wide analysis of disease progression in age-related macular degeneration. Hum Mol Genet 27(5):929–940
Zhang S, Okhrin O, Zhou Q, Song P (2016) Goodness-of-fit test for specification of semiparametric copula dependence models. J Econom 193(1):215–233
Acknowledgements
This research is supported by the National Institute of Health (EY024226). We would like to thank the participants in the AREDS study, who made this research possible, and International AMD Genomics Consortium for generating the genetic data and performing quality check.
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Sun, T., Liu, Y., Cook, R.J. et al. Copula-based score test for bivariate time-to-event data, with application to a genetic study of AMD progression. Lifetime Data Anal 25, 546–568 (2019). https://doi.org/10.1007/s10985-018-09459-5
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DOI: https://doi.org/10.1007/s10985-018-09459-5