On consistency of Kendall s tau under censoring

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1 Biometria (28), 95, 4,pp C 28 Biometria Trust Printed in Great Britain doi: 1.193/biomet/asn37 Advance Access publication 17 September 28 On consistency of Kendall s tau under censoring BY DAVID OAKES Department of Biostatistics and Computational Biology, University of Rochester Medical Center, Rochester, New Yor 14642, U.S.A. oaes@bst.rochester.edu SUMMARY Necessary and sufficient conditions for consistency of a simple estimator of Kendall s tau under bivariate censoring are presented. The results are extended to data subject to bivariate left truncation as well as right censoring. Some ey words: Bivariate survival; Clayton s model; Cross-ratio function; Frailty; Left truncation. 1. INTRODUCTION Oaes (1982a) proposed a simple test for independence for bivariate data subject to censoring in both components. The test is based on a modification of Kendall s tau statistic that counts only definite concordances and definite discordances among the n(n 1)/2 pairs of bivariate observations that can be constructed from a sample of size n from the joint distribution. Many subsequent authors have followed Oaes (1982a) in pointing out that this modified tau statistic does not give a consistent estimator of the population value of tau in the absence of censoring. Some of these authors have proposed more complex estimators that do achieve consistency and improved efficiency by recovering partial information from the indefinite pairs. For testing the hypothesis of independence, it is natural to use the nonrandom denominator n(n 1)/2 proposed in Oaes (1982a). However, for estimation it would be more natural to count only the pairs giving a definite preference, leading to what has been termed the renormalized estimator. To be specific, suppose that we observe data {(T (i) 1, T (i) 2 ), (i = 1,...,n)} from a continuous bivariate survivor function S(t 1, t 2 ), but subject to right censorship in both components, so that the observations are, for = 1, 2 and all i, X (i) = min (T (i), c (i) (i) ) and V = 1(T (i) c (i) ). Here {(c(i) 1, c(i) 2 ), (i = 1,...,n)} is a sequence of nonrandom bivariate potential censoring times. Then (Cox & Oaes, 1984, p. 11), on the basis of what is observed, T (i) is nown to be less than or equal to T ( j) if and only if T (i) min (T ( j), c (i) j), c( ). Let V (ij) denote the indicator function of this event, let (ij) = V (ij) V ( ji) and set (ij) = (ij) 1 (ij) 2. Then the pair (i, j) is a definite concordance if (ij) = 1, a definite discordance if (ij) = 1 and is indeterminate if (ij) =. Let C and D denote the numbers of definite concordances and definite discordances among the n(n 1)/2 pairs. Then Oaes s (1982a) test is based on Downloaded from at Pennsylvania State University on February 2, 216 τ = whereas the renormalized estimator is ( ) n 1 (C D) = 2 ( ) n 1 (ij), (1) 2 i< j ˆτ = C D C + D. (2) The purpose of this note is to clarify the conditions under which this renormalized estimator is consistent.

2 998 DAVID OAKES 2. BIVARIATE TYPE I CENSORING Clayton (1978) proposed a model for continuous bivariate survival data, which requires that the hazard functions of T 2 given T 1 = t 1 and of T 2 given T 1 t 1 be in a fixed ratio θ, i.e. free of (t 1, t 2 ). It is easily seen that this condition is equivalent to S (t 1, t 2 )S 11 (t 1, t 2 ) = θ S 1 (t 1, t 2 )S 1 (t 1, t 2 ), t 1 >, t 2 >, (3) where, for a =, 1 and b =, 1, S ab (t 1, t 2 ) denotes ( / t 1 ) a ( / t 2 ) b S(t 1, t 2 ). Oaes (1982b, 1989)showed that the condition (1) integrates to give the representation ( S(t 1, t 2 ) = max, [ {S 1 (t 1 )} 1 θ +{S 2 (t 2 )} 1 θ 1 ] ) 1/(1 θ) (4) in terms of the marginal survivor functions S 1 (t 1 ) = S(t 1, ) and S 2 (t 2 ) = S(, t 2 ). We now show that, for bivariate Type I censoring, i.e. with c (i) = c for = 1, 2 and every i, the renormalized estimator ˆτ is consistent for the uncensored population value of τ for every possible pair of potential censoring times (c 1, c 2 ) if and only if the original uncensored data follow Clayton s model. The = (ij) are identically distributed, with pr ( = 1) = 2 pr ( = 1) = 2 The probability limit of the renormalized τ is then pr ( = 1) pr ( = 1) pr ( = 1) + pr ( = 1), S (t 1, t 2 )S 11 (t 1, t 2 )dt 1 dt 2, (5) S 1 (t 1, t 2 )S 1 (t 1, t 2 )dt 1 dt 2. (6) and this is free of (c 1, c 2 ) if and only if, for all (c 1, c 2 ) and some θ, free of (c 1, c 2 ), pr( = 1) = θ pr ( = 1). In this case τ = (θ 1)/(θ + 1). Substituting for the two probabilities from (5) and (6), and differentiating in c 1 and c 2 gives condition (3). This completes the proof. 3. GENERAL RIGHT CENSORING Consistency of ˆτ for Clayton s model easily extends to the more usual situation where the potential bivariate censoring times (c (i) 1, c(i) 2 )differoveri. The (ij) are no longer identically distributed, but for each (i, j) wemusthave pr ( (ij) = 1 ) = 2 S (t 1, t 2 )S 11 (t 1, t 2 )dt 1 dt 2, pr ( (ij) = 1 ) = 2 S 1 (t 1, t 2 )S 1 (t 1, t 2 )dt 1 dt 2, Downloaded from at Pennsylvania State University on February 2, 216 where, for = 1, 2, c = c (ij) = min (c (i) j), c( ), so that, under Clayton s model, the two probabilities have ratio θ, and ˆτ τ, provided that the sequence of potential censoring times allows C + D to become infinite as n. If the potential censoring times are random variables (C 1, C 2 ), independent of the (T 1, T 2 ), and are themselves drawn from a bivariate distribution with survivor function G(c 1, c 2 ) = pr (C 1 > c 1, C 2 > c 2 ), then the (ij) again become identically distributed with pr ( = 1) = 2 S (t 1, t 2 )S 11 (t 1, t 2 )G 2 (t 1, t 2 )dt 1 dt 2 and pr ( = 1) = 2 S 1 (t 1, t 2 )S 1 (t 1, t 2 )G 2 (t 1, t 2 )dt 1 dt 2, where both double integrals are taen over the entire positive quadrant. It is easily seen that the ratio θ = pr ( = 1)/pr ( = 1), and hence also ˆτ, is free of G(, ) if and only if condition (3) and therefore Clayton s model hold.

3 Miscellanea CENSORING OF A SINGLE VARIABLE Recently Beaudoin et al. (27) have incorporated both the original version (1) and renormalized version (2) of the concordance statistic into simulation studies in which censoring affects only one variable. Our arguments show that Clayton s model is sufficient for consistency of the renormalized tau in this case, but not that it is necessary; in fact necessity seems unliely to hold. However, we can show necessity within the class of archimedean survival copula models, which includes the models of Fran (1979), see also Genest (1987), and Hougaard (1986), considered in these simulation studies. For such models the joint survivor function taes the form S(t 1, t 2 ) = p[q{s 1 (t 1 )}+q{s 2 (t 2 )}], (7) for some twice differentiable function p( ) with p(s), p() = 1, p( ) =, p (s), p (s) and inverse function q( ). Here S 1 (t 1 ) = S(t 1, ) and S 2 (t 2 ) = S(, t 2 ) are the marginal survivor functions of T 1 and T 2 ; see for example Oaes (1989). Suppose now that T 1 is always observed, but T 2 is subject to Type I censoring at c 2. Then (5) and (6) must hold with c 1 = and all c 2. The monotone increasing substitutions u = q{s (t )} ( = 1, 2) each map (, )onto(, q()), where q() may be finite or infinite depending on the specific copula. If q() <,wemusthavep(u) = p (u) = for u > q(), so that in all cases (5) and (6) become pr ( = 1) = 2 pr ( = 1) = 2 p(u 1 + u 2 )p (u 1 + u 2 ) 2 du 1 du 2, p (u 1 + u 2 ) 2 du 1 du 2, where c = q{s 2 (c 2 )} taes all values in (, q()) as c 2 taes all values in (, ). Consistency of the renormalized estimator for all c 2 therefore requires that, for all c, p(u 1 + u 2 )p (u 1 + u 2 )du 1 du 2 = θ p (u 1 + u 2 ) 2 du 1 du 2. Differentiation in c followed by the substitution u = u 1 + c gives the equivalent condition c p(u)p (u)du = θ c p (u) 2 du. This holds for all c if and only if p(c)p (c) = θp (c) 2. This equation with the condition p() = 1integrates to give p(c) = (1 + αc) 1/(1 θ) for some constant of integration α. When substituted into (7), this yields the condition (3) for Clayton s model (4). This argument shows that the renormalized estimator will not be consistent for all values of c 2 for any continuous archimedean copula model other than Clayton s model. There remains the possibility that the renormalized estimator could be consistent for particular values of c 2. However, it is easily seen that even this possibility can be excluded if the cross-ratio function p(u)p (u)/{p (u) 2 } is strictly monotone in u, which is true, for example, for the models of Hougaard (1986) and Fran (1979). Downloaded from at Pennsylvania State University on February 2, RIGHT CENSORING COMBINED WITH LEFT TRUNCATION We can extend the concordance test statistic and the associated estimator to the case where each data point (T (i) 1, T (i) 2 ) is subject to bivariate left truncation at (l(i) 1,l(i) 2 ) as well as bivariate censorship at (c(i) 1, c(i) 2 ). It is assumed that l (i) c (i) for = 1, 2 and each i. Here, we treat the (l () 1,l() 2 ) as prespecified constants; if these are viewed as realizations of a random variable (L 1, L 2 ), extending the random censorship model of 3, then we must assume independence between (L 1, L 2, C 1, C 2 ) and (T 1, T 2 ). If we drop the superscript i for simplicity, this means that the available data are drawn from the conditional distribution of (X 1, V 1, X 2, V 2 )given(t 1 l 1, T 2 l 2 ), and that the corresponding (l 1,l 2 ) are observed. To extend Kendall s tau to this situation, we must further restrict attention to comparable pairs of observations

4 1 DAVID OAKES (Martin & Betensy, 25), that is, pairs (i, j) for which both intervals (l (i), X (i) j) ) (l(, X ( j) ), for = 1, 2, are nonempty, or equivalently l = l (ij) = max(l (i) j) (i),l( ) min (X, X ( j) ). The definition of ˆτ given in 1 applies as before except that we must now set V (ij) = 1{l (ij) T (i) min (T ( j), c (i) j), c( )}. The conditional probability that the observed pair (i, j) is comparable is { pr min ( T (i), T ( j) ) l, = 1, 2 (i) T l (i), T ( j) l ( j) }=, = 1, 2 S 2 (l 1,l 2 ) S ( l (i) 1,l(i) 2 ) S ( l ( j) 1 j) ).,l( 2 Denote this probability by R. Then, given that the observed pair (i, j) is comparable, the conditional probabilities that it is a definite concordance or definite discordance are simply 2R 1 l 1 l 2 S (t 1, t 2 )S 11 (t 1, t 2 )dt 1 dt 2, 2R 1 Again, we find that the ratio of these two probabilities is free of the c (i) model holds. 6. DISCUSSION l 1 l 2 S 1 (t 1, t 2 )S 1 (t 1, t 2 )dt 1 dt 2. and l (i) if and only if Clayton s Wang & Wells (2a) examined the performance of several estimators of τ for censored bivariate data simulated from Clayton s model. They reported that the Oaes s estimator performed well when τ was near zero but its bias became substantial when τ got larger. However, they appear to have considered only the original statistic τ, not the renormalized estimator ˆτ. The blanet statement in Wang & Wells (2b) that Oaes s estimator is not consistent if τ also appears to apply to τ. Fan et al. (2a), see also Fan et al. (2b), define a finite region censored data version of Kendall s tau that is equivalent to the population version under Type I censoring. They mention equivalence of this finite region version to the uncensored version for a particular case of Clayton s model but do not address general necessary and sufficient conditions for this equivalence. They also present simulation results for an ad hoc finite sample version of Kendall s tau calculated under a combination of the finite region restriction and random censoring, thus in effect under more stringent censoring, suggesting that this equivalence does not hold for Fran s model. The issues considered here are similar to those addressed by Oaes (25), who showed that the Clayton copula is characterized by its invariance under left truncation. The proposal in 5 to extend the tau statistic to bivariate data subject to both left truncation and right censoring may be new, although it is closely related to the wor of Martin & Betensy (25) and others on testing for quasi-independence of survival and truncation times. ACKNOWLEDGEMENT I than C. Genest for an enquiry which led me to write this note and for valuable comments. REFERENCES BEAUDOIN, D.,DUCHESNE, T.&GENEST, C. (27). Improving the estimation of Kendall s tau when censoring affects only one of the variables. Comput. Statist. Data Anal. 51, CLAYTON, D. G. (1978). A model for association in bivariate life tables and its application in epidemiologic studies of familial tendency in chronic disease incidence. Biometria 65, COX, D.R.&OAKES, D. (1984). Analysis of Survival Data. London: Chapman and Hall. FAN, J., HSU, L.& PRENTICE, R. L. (2a). Dependence estimation over a finite bivariate failure time region. Lifetime Data Anal. 6, FAN, J.,PRENTICE, R.L.&HSU, L. (2b). A class of weighted dependence measures for bivariate failure time data. J. R. Statist. Soc. B 62, FRANK, M. J. (1979). On the simultaneous associativity of F(x, y) andx + y F(x, y). Aequationes Math. 19, Downloaded from at Pennsylvania State University on February 2, 216

5 Miscellanea 11 GENEST, C. (1987). Fran s family of bivariate distributions. Biometria 74, HOUGAARD, P. (1986). A class of multivariate failure time distributions. Biometria 73, MARTIN, E.C.&BETENSKY, R. A. (25). Testing quasi-independence of failure and truncation times via conditional Kendall s tau. J. Am. Statist. Assoc. 1, OAKES, D. (1982a). A concordance test for independence in the presence of censoring. Biometrics 38, OAKES, D. (1982b). A model for association in bivariate survival data. J. R. Statist. Soc. B 44, OAKES, D. (1989). Bivariate survival models induced by frailties. J. Am. Statist. Assoc. 84, OAKES, D. (25). On the preservation of copula structure under truncation. Can. J. Statist. 33, WANG, W.&WELLS, M. T. (2a). Estimation of Kendall s tau under censoring. Statist. Sinica 1, WANG, W.& WELLS, M. T. (2b). Model selection and semiparametric inference for bivariate failure time data. J. Am. Statist. Assoc. 95, [Received September 26. RevisedMarch28] Downloaded from at Pennsylvania State University on February 2, 216