Synthesizing secondary data into survival analysis to improve estimation efficiency

The accelerated failure time (AFT) model and Cox proportional hazards (PH) model are broadly used for survival endpoints of primary interest. However, the estimation efficiency from those models can be further enhanced by incorporating the information from secondary outcomes that are increasingly available and highly correlated with primary outcomes. Those secondary outcomes could be longitudinal laboratory measures collected from doctor visits or cross-sectional disease-relevant variables, which are believed to contain extra information related to primary survival endpoints to a certain extent. In this paper, we develop a two-stage estimation framework to combine a survival model with a secondary model that contains secondary outcomes, named as the empirical-likelihood-based weighting (ELW), which comprises two weighting schemes accommodated to the AFT model (ELW-AFT) and the Cox PH model (ELW-Cox), respectively. This innovative framework is flexibly adaptive to secondary outcomes with complex data features, and it leads to more efficient parameter estimation in the survival model even if the secondary model is misspecified. Extensive simulation studies showcase more efficiency gain from ELW compared to conventional approaches, and an application in the Atherosclerosis Risk in Communities study also demonstrates the superiority of ELW by successfully detecting risk factors at the time of hospitalization for acute myocardial infarction.