Combining experimental and observational follow-up datasets has received much attention lately. In a survival setting, recent work has used Medicare claims to extend the follow-up period for participants in a prostate cancer clinical trial. This allows the estimation of the long-term effect that cannot be estimated by the trial data alone. In this paper, we study the estimation of long-term effect when participants in a clinical trial are linked to an observational follow-up dataset. Such linkages are often incomplete and we formulate incomplete linkages as a missing data problem. We use the popular Cox model to define the long-term effect and we propose two approaches to deal with the missing data problem. The first approach, termed non-linked-as-censored (NLAC), is a simple approach that works when Cox model is correctly specified and linkage satisfies a conditionally independent assumption. To gain robustness against model mis-specification, we propose an inverse probability of linkage weighted approach, along with the augmented inverse probability of weighted method, based on a novel conditional linking at random (CLAR) assumption. We further extend our approach to incorporate time-dependent covariates. Simulation results confirm the validity of our method and we apply our methods to the SWOG study.

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