Conditional copula-graphic estimator for semi-competing risks data
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In semi-competing risks data, interest lies in the estimation of the survival function of a non-terminal event time, which is subject to dependent censoring by a terminal event. This problem has been extensively studied in the literature, but mostly focusing on unconditional settings. In this thesis, we propose two versions of conditional copula-graphic estimator: one allows for covariate adjustment only in the marginal survival functions of non-terminal and terminal events, and the other allows for covariate adjustment in both the marginal survival functions and the dependence structure of non-terminal and terminal event times. The proposed estimators are semiparametric. In both, the conditional copula is assumed to belong to a one-parameter Archimedean copula family, but the copula parameter is estimated parametrically in the first version and nonparametrically in the second one. Both versions employ Beran's estimator in the estimation of the conditional marginal survival functions. The performance of the conditional copula-graphic estimators is assessed using a simulation study and is compared to that of the unconditional copula-graphic estimator to investigate the cost of ignoring covariate effects. Our findings suggest that, in the presence of covariates, the conditional copula-graphic estimators are more efficient and less biased than the unconditional copula-graphic estimator. If interest centres on the estimation of the marginal survival function of the non-terminal event, both versions of the conditional copula-graphic estimator perform similarly. However, if the estimation of the conditional dependence structure is also of interest, the second version more accurately captures the underlying dependence structure. The performance of the conditional copula-graphic estimators deteriorates with the increase in the censoring rate of the non-terminal event. A real data example on breast cancer recurrence is provided to illustrate the proposed approach in comparison to the unconditional copula-graphic estimator.