Improving the estimation of Kendall's tau when censoring affects only one of the variables

Abstract

This paper considers the estimation of Kendall's tau for bivariate data ( X , Y ) when only Y is subject to right-censoring. Although τ is estimable under weak regularity conditions, the estimators proposed by Brown et al. [1974. Nonparametric tests of independence for censored data, with applications to heart transplant studies. Reliability and Biometry, 327–354], Weier and Basu [1980. An investigation of Kendall's τ modified for censored data with applications. J. Statist. Plann. Inference 4, 381–390] and Oakes [1982. A concordance test for independence in the presence of censoring. Biometrics 38, 451–455], which are standard in this context, fail to be consistent when τ ≠ 0 because they only use information from the marginal distributions. An exception is the renormalized estimator of Oakes [2006. On consistency of Kendall's tau under censoring. Technical Report, Department of Biostatistics and Computational Biology, University of Rochester, Rochester, NY], whose consistency has been established for all possible values of τ , but only in the context of the gamma frailty model. Wang and Wells [2000. Estimation of Kendall's tau under censoring. Statist. Sinica 10, 1199–1215] were the first to propose an estimator which accounts for joint information. Four more are developed here: the first three extend the methods of Brown et al. [1974. Nonparametric tests of independence for censored data, with applications to heart transplant studies. Reliability and Biometry, 327–354], Weier and Basu [1980, An investigation of Kendall's τ modified for censored data with applications. J. Statist. Plann. Inference 4, 381–390] and Oakes [1982, A concordance test for independence in the presence of censoring. Biometrics 38, 451–455] to account for information provided by X, while the fourth estimator inverts an estimation of Pr ( Y i ⩽ y | X i = x i , Y i > c i ) to get an imputation of the value of Y i censored at C i = c i . Following Lim [2006. Permutation procedures with censored data. Comput. Statist. Data Anal. 50, 332–345], a nonparametric estimator is also considered which averages the τ ^ i obtained from a large number of possible configurations of the observed data ( X 1 , Z 1 ) , … , ( X n , Z n ) , where Z i = min ( Y i , C i ) . Simulations are presented which compare these various estimators of Kendall's tau. An illustration involving the well-known Stanford heart transplant data is also presented.