  Combining experimental and observational follow-up datasets has received a
lot of attention lately. In a time-to-event 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 clinical 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 with incomplete data. Such data linkages
are often incomplete for various reasons. We formulate incomplete linkages as a
missing data problem with careful considerations of the relationship between
the linkage status and the missing data mechanism. We use the popular Cox
proportional hazard model as a working model to define the long-term effect. We
propose a conditional linking at random (CLAR) assumption and an inverse
probability of linkage weighting (IPLW) partial likelihood estimator. We show
that our IPLW partial likelihood estimator is consistent and asymptotically
normal. We further extend our approach to incorporate time-dependent
covariates. Simulations results confirm the validity of our method, and we
further apply our methods to the SWOG study.
