Regression Analysis of Bivariate Interval-Censored Data Under Semiparametric Partly Linear Transformation Models

Abstract

Bivariate interval-censored failure time data occur in many areas such as biomedical and epidemiological research and their regression analysis has been discussed by many authors. However, most of the existing methods have some limitations on either data structures or the assumed models. In this paper, we propose a sieve maximum likelihood estimation approach for general interval-censored data under a class of flexible copula-based semiparametric partly linear transformation models. The approach allows for potential nonlinear effects and makes use of Bernstein polynomials. The resulting estimators of regression parameters are shown to be consistent and asymptotically efficient and normal. A simulation study is conducted to assess the finite-sample performance of the proposed approach and suggests that it works well in practice. Also we apply it to a set of real data arising from dental research.

Keywords

Copula / Interval-censored / Sieve maximum likelihood estimation / Partly linear model / 62J99 / 62N02 / 62G20

References

[1] Carnicer JM, Peña JM. Shape preserving representations and optimality of the Bernstein basis. Adv. Comput. Math., 1993, 1: 173-196.

[2] Chang IS, Wen CC, Wu YJ. A profile likelihood theory for the correlated gamma-frailty model with current status family data. Stat. Sin., 2007, 17(3): 1023-1046

[3] Chen DGD, Sun J, Peace KE. Interval-Censored Time-to-Event Data: Methods and Applications, 2012. Boca Raton, CRC Press.

[4] Chen M, Tong X, Sun J. The proportional odds model for multivariate interval-censored failure time data. Stat. Med., 2007, 26(28): 5147-5161.

[5] Chen M, Tong X, Sun J. A frailty model approach for regression analysis of multivariate current status data. Stat. Med., 2009, 28(27): 3424-3436.

[6] Chen M, Tong X, Zhu L. A linear transformation model for multivariate interval-censored failure time data. Can. J. Stat., 2013, 41(2): 275-290.

[7] Cheng SC, Wei LJ, Ying Z. Analysis of transformation models with censored data. Biometrika, 1995, 82(4): 835-845.

[8] Clayton DG. A model for association in bivariate life tables and its application in epidemiological studies of familial tendency in chronic disease incidence. Biometrika, 1978, 65: 141-151.

[9] Cook RJ, Tolusso D. Second-order estimating equations for the analysis of clustered current status data. Biostatistics, 2009, 10: 756-772.

[10] Efron B. Bootstrap methods: another look at the jackknife. Ann. Stat., 1979, 7: 1-26.

[11] Efron B, Tibshirani RJ. An Introduction to the Bootstrap, 1994. London, Chapman and Hall.

[12] Finkelstein DM. A proportional hazards model for interval-censored failure time data. Biometrics, 1968, 42: 845-854.

[13] Gumbel EJ. Bivariate exponential distributions. J. Am. Stat. Assoc., 1960, 55: 698-707.

[14] Goggins WB, Finkelstein DM. A proportional hazards model for multivariate interval-censored failure time data. Biometrics, 2000, 56(3): 940-943.

[15] Huang J. Efficient estimation for the proportional hazards model with interval censoring. Ann. Stat., 1996, 24: 540-568.

[16] Huang J, Rossini A. Sieve estimation for the proportional-odds failure-time regression model with interval censoring. J. Am. Stat. Assoc., 1997, 92: 960-967.

[17] Hu T, Zhou Q, Sun J. Regression analysis of bivariate current status data under the proportional hazards model. Can. J. Stat., 2017, 45: 410-424.

[18] Joe H. Multivariate Models and Dependence Concepts, 1997. London, Chapman and Hall.

[19] Kor CT, Cheng KF, Chen YH. A method for analyzing clustered interval-censored data based on Cox model. Stat. Med., 2013, 32: 822-832.

[20] Lee CY, Wong KY, Lam KF, Xu J. Analysis of clustered interval censored data using a class of semiparametric partly linear frailty transformation models. Biometrics, 2020.

[21] Lin X, Carroll RJ. Semiparametric regression for clustered data using generalized estimating equations. J. Am. Stat. Assoc., 2001, 96: 1045-1056.

[22] Lorentz GG. Bernstein Polynomials, 1953, 1, Toronto, Univ. Toronto Press

[23] Sklar A. Fonctions de répartition à n dimensions et leurs marges. Publ. Inst. Stat. Univ. Paris, 1959, 8: 229-231

[24] Sun J. The Statistical Analysis of Interval-Censored Failure Time Data, 2006. New York, Springer.

[25] Sun T, Ding Y. Copula-based semiparametric regression method for bivariate data under general interval censoring. Biostatistics, 2021, 22(2): 315-330.

[26] Shen X. On methods of sieves and penalization. Ann. Stat., 1997, 25: 2555-2591.

[27] Shen X, Wong WH. Convergence rate of sieve estimates. Ann. Stat., 1994, 22: 580-615.

[28] Tang N, Chen X, Wang X. Consistency and asymptotic normality of profilekernel and backfitting estimators in semiparametric reproductive dispersion nonlinear models. Sci. China, Ser. A Math., 2009, 52(4): 757-770.

[29] Tong X, Chen M, Sun J. Regression analysis of multivariate interval-censored failure time data with application to tumorigenicity experiments. Biom. J., 2008, 50(3): 364-374.

[30] Van der Vaart A, Wellner J. Weak Convergence and Empirical Processes, 1996. Berlin, Springer.

[31] Vanobbergen J, Martens L, Lesaffre E, Declerck D. The Signal-Tandmobiel project a longitudinal intervention health promotion study in Flanders (Belgium): baseline and first year results. Eur. J. Paediatr. Dent., 2000, 2: 87-96

[32] Wang N, Wang L, McMahan CS. Regression analysis of bivariate current status data under the gamma-frailty proportional hazards model using the EM algorithm. Comput. Stat. Data Anal., 2015, 83: 140-150.

[33] Wen CC, Chen YH. A frailty model approach for regression analysis of bivariate interval-censored survival data. Stat. Sin., 2013, 83(1): 383-408

[34] Wu Q, Zhao H, Zhu L, Sun J. Variable selection for high-dimensional partly linear additive Cox model with application to Alzheimer’s disease. Stat. Med., 2020, 39(23): 3120-3134.

[35] Xue H, Lam KF, Li G. Sieve maximum likelihood estimator for semiparametric regression models with current status data. J. Am. Stat. Assoc., 2004, 99: 346-356.

[36] Zeng D, Gao F, Lin D. Maximum likelihood estimation for semiparametric regression models with multivariate interval-censored data. Biometrika, 2017, 104(3): 505-525.

[37] Zhou Q, Hu T, Sun J. A sieve semiparametric maximum likelihood approach for regression analysis of bivariate interval-censored failure time data. J. Am. Stat. Assoc., 2017, 112(518): 664-672.

Funding

National Natural Science Foundation of China(12171328)

RIGHTS & PERMISSIONS

School of Mathematical Sciences, University of Science and Technology of China and Springer-Verlag GmbH Germany, part of Springer Nature