Estimating Kendall's τ when both times are subject to interval censoring

Document Type : Original Scientific Paper

Authors

Department of Statistics‎, Sistan and Baluchestan University, Zahedan, Iran

Abstract

A common challenge in working with longitudinal data is dealing with incomplete data‎. ‎According to the existing studies on the dependence structure of survival times‎, ‎it is a riveting topic for researchers to estimate survival functions and dependence parameters‎, ‎especially in biology and medical research‎. ‎Some researchers have studied the aforementioned subjects with left‎- ‎or right-truncated or censored data‎. ‎When the data involves interval censoring‎, ‎the mentioned issues still need to be solved or modified‎. ‎In this article‎, ‎we propose two alternative approaches to the estimation of a dependence parameter and Kendall's τ‎, ‎given an interesting covariate and interval-censored dataset‎. ‎More precisely‎, ‎these approaches include non-parametric and semi-parametric methods to estimate the copula dependence parameter and Kendall's τ‎, ‎which are evaluated by simulation‎. ‎Finally‎, ‎we apply the mentioned approaches to a real-world dataset and copula's goodness-of-fit tests.

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Beaudoin, D., Duchesne, T. and Genest, C. (2007). Improving the estimation of Kendall's τ when censoring affects only one of the variables. Computational Statistics & Data Analysis, 51, 5743–5764.
Beran, R. (1981). Non-parametric regression with randomly censored survival data. Statistics and Probability Letters, 20, 225–234.
Betensky, R.A. and Finkelstein, D.M. (1999). An extension of Kendall's coefficient of concordance to bivariate interval-censored data. Statistics in Medicine, 18, 3101–3109.
Clayton, D. (1978). A model for association in bivariate life tables and its application in epidemiological studies of familial tendency in chronic disease incidence. Biometrika, 65, 141–151.
Dabrowska, D.M. (1988). Kaplan-Meier estimate on the plane. The Annals of Statistics, 16, 1475–1489.
Debye, P. (1912). Zur Theorie der spezifischen Wärmen. Annalen der Physik, 39(10), 789–839.
Dehghan, M.H. and Duchesne, T. (2011). A generalization of Turnbull's estimator for non-parametric estimation of the conditional survival function with interval-censored data. Lifetime Data Analysis, 17, 234–255.
Dehghan, M.H. and Duchesne, T. (2015). Generalization of Turnbull's estimator. R Package. Available from: http://127.0.0.1:15947/library/gte/html/gte.html.
Derumigny, A. and Fermanian, J.D. (2019). On kernel-based estimation of conditional Kendall's tau: finite-distance bounds and asymptotic behavior. Dependence Modeling, 7(1), 292–321.
Finkelstein, D.M. and Goggins, G. (2000). Analysis of failure time data with dependent interval censoring. Biometrics, 584, 298–304.
Frank, M.J. (1979). On the simultaneous associativity of F(x,y) and x+y-F(x,y). Aequationes Mathematicae, 19, 194–226.
Genest, C. and Rivest, L.P. (1993). Statistical inference procedures for bivariate Archimedean copulas. Journal of the American Statistical Association, 88, 1034–1043.
Genest, C., Goudi, K. and Rivest, L.P. (1995). A semi-parametric estimation procedure for dependence parameters in multivariate families of distributions. Biometrika, 82, 543–552.
Genest, C., Quessy, J.F. and Rémillard, B. (2006). Goodness of fit procedures for copula models based on the probability integral transformation. Scandinavian Journal of Statistics, 33(2), 337–366.
Gumbel, E.J. (1960). Bivariate exponential distributions. American Statistical Association Journal, 55, 698–707.
Hesieh, J.J. and Li, Z.J. (2017). Estimation and test of conditional Kendall's \tau under bivariate left-truncation data. Communications in Statistics-Theory and Methods, 46, 6635–6644.
Hoeffding, W. (1948). A class of statistics with asymptotic normal distribution. The Annals of Mathematical Statistics, 19, 293–325.
Jahanshahi, S.M.A., Habibirad, A. and Fakoor, V. (2020). Some new goodness-of-fit tests for Rayleigh distribution. Pakistan Journal of Statistics, 16(2), 305–315.
Kang, S.H. and Kim, Y.J. (2021). Association measure of doubly interval-censored data using a Kendall's τ estimator. Communications for Statistical Applications and Methods, 28, 151–159.
Kaplan, E.L. and Meier, M. (1938). Non-parametric estimation from incomplete observations. American Statistical Association Journal, 53, 457–481.
Kendall, M. (1938). A new measure of rank correlation. Biometrika, 30, 81–89.
Kim, Y.J. (2015). Estimation of conditional Kendall's τ for bivariate interval-censored data. Communications for Statistical Applications and Methods, 22, 599–604.
Koziol, A.J. (1980). Goodness-of-fit tests for randomly censored data. Biometrika, 67(3), 693–696.
Lakhal, L., Rivest, L.P. and Abdous, B. (2008). Estimating association and survival in a semi-competing risks model. Biometrics, 64, 180–188.
Lee, A.J. (1990). U-Statistics: Theory and Practice. Marcel Dekker, New York.
Lim, J. and Meier, M. (2006). Permutation procedures with censored data. Computational Statistics & Data Analysis, 50, 332–345.
Martin, E.C. and Betensky, R.A. (2005). Testing quasi-independence of failure and truncation times via conditional Kendall's \tau. Journal of the American Statistical Association, 100, 484–492.
Oakes, D. (1982). A concordance test for independence in the presence of censoring. Biometrika, 38, 451–555.
Oakes, D. (2008). On consistency of Kendall's τ under censoring. Biometrika, 95, 997–1001.
Schepsmeier, U. and Brechmann, E.C. (2013). BiCopSelect and BiCopCompare Copulas Packages. http://127.0.0.1:13916/library/VineCopula/html/VineCopula-package.html; http://127.0.0.1:13916/library/VineCopula/html/BiCopCompare.html.
 
Sklar, A. (1959). Fonctions de répartition à n dimensions et leurs marges. Institute Statistique de l'Université de Paris, 8, 229–231.
Tsai, W.Y. (1990). Testing the assumption of independence of truncation time and failure time. Biometrika, 77, 169–177.
Turnbull, B.W. (1976). The empirical distribution with arbitrarily grouped censored and truncated data. Journal of the Royal Statistical Society, 38, 290–295.
Van der Vaart, A.W. (1998). Asymptotic Statistics. Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press.
Wang, W. and Wells, M.T. (2000). Estimation of Kendall's τ under censoring. Statistica Sinica, 10, 1199–1215.
Weier, D.R. and Basu, A.P. (1980). An investigation of Kendall's τ modified for censored data with applications. Journal of Statistical Planning and Inference, 4, 381–390.