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We propose a two-stage estimation procedure for a copula-based model withsemi-competing risks data, where the non-terminal event is subject to dependentcensoring by the terminal event, and both events are subject to independentcensoring. Under a copula-based model, the marginal survival functions ofindividual event times are specified by semiparametric transformation models,and the dependence between the bivariate event times is specified by aparametric copula function. For the estimation procedure, in the first stage,the parameters associated with the marginal of the terminal event are estimatedonly using the corresponding observed outcomes, and in the second stage, themarginal parameters for the non-terminal event time and the copula parameterare estimated via maximizing a pseudo-likelihood function based on the jointdistribution of the bivariate event times. We derived the asymptotic propertiesof the proposed estimator and provided an analytic variance estimator forinference. Through simulation studies, we showed that our approach leads toconsistent estimates with less computational cost and more robustness comparedto the one-stage procedure developed in Chen (2012), where all parameters wereestimated simultaneously. In addition, our approach demonstrates more desirablefinite-sample performances over another existing two-stage estimation methodproposed in Zhu et al. (2021).

Two-Stage Pseudo Maximum Likelihood Estimation of Semiparametric Copula-based Regression Models for Semi-Competing Risks Data