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Biometrika 2006 93(3):655-669; doi:10.1093/biomet/93.3.655
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© 2006 Biometrika Trust

Estimating survival under a dependent truncation

Lajmi Lakhal Chaieb1, Louis-Paul Rivest1 and Belkacem Abdous2

1 Département de mathématiques et de statistique, Université Laval, Ste-Foy, Québec, G1K 7P4 Canada. lakhal{at}mat.ulaval.ca, lpr{at}mat.ulaval.ca, 2 Département de médecine sociale et préventive, Université Laval, Ste-Foy, Québec, G1K 7P4 Canada. belkacem.abdous{at}msp.ulaval.ca

The product-limit estimator calculated from data subject to random left-truncation relies on the testable assumption of quasi-independence between the failure time and the truncation time. In this paper, we propose a model for a truncated sample of pairs (Xi,Yi) satisfying Yi > Xi. A possible dependency between the truncation time and the variable of interest is modelled with a parametric family of copulas. The model also features a distribution function FX(.) and a survival distribution SY(.) associated with the marginal behaviours of X and Y in the observable region Y > X. Semiparametric estimators for these two functions are proposed; they do not make any parametric assumption about either FX(.) or SY(.). We derive an estimator for the copula parameter {alpha} based on the conditional Kendall's tau. We generalise the copula-graphic estimators of Zheng & Klein (1995) to truncated variables. The asymptotic distributions of all these estimators are then investigated. The methods are illustrated with a real dataset on HIV infection by transfusion and by simulations.

Key Words: Copula; Kaplan-Meier estimator; Kendall's tau; U-statistic.

Received October 2004. Revised February 2006.


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