Forecasting survival risks for time-to-event data is an essential task in clinical research. Practitioners often rely on well-structured statistical models to make predictions for patient survival outcomes. The nonparametric proportional

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.

With advances in biomedical research, biomarkers are becoming increasingly important prognostic factors for predicting overall survival, while the measurement of biomarkers is often censored due to instruments' lower limits of detection. This leads to two types of censoring: random censoring in overall survival outcomes and fixed censoring in biomarker covariates, posing new challenges in statistical modeling and inference. Existing methods for analyzing such data focus primarily on linear regression ignoring censored responses or semiparametric accelerated failure time models with covariates under detection limits (DL). In this paper, we propose a quantile regression for survival data with covariates subject to DL. Comparing to existing methods, the proposed approach provides a more versatile tool for modeling the distribution of survival outcomes by allowing covariate effects to vary across conditional quantiles of the survival time and requiring no parametric distribution assumptions for outcome data. To estimate the quantile process of regression coefficients, we develop a novel multiple imputation approach based on another quantile regression for covariates under DL, avoiding stringent parametric restrictions on censored covariates as often assumed in the literature. Under regularity conditions, we show that the estimation procedure yields uniformly consistent and asymptotically normal estimators. Simulation results demonstrate the satisfactory finite‐sample performance of the method. We also apply our method to the motivating data from a study of genetic and inflammatory markers of Sepsis.

List of Figures 2.1 Estimation results for true baseline cumulative hazard function Λ 0 (t) = t 2 and survival function S 0 (t) = exp(−t 2) under Scenario I with n = 100, 200, 400. The black solid lines are the estimates, the red dot-dashed lines are the true ones, and