A Review of the Definition of Time-Dependent Covariate in Survival Analysis

Summary

Time-dependent covariates frequently arise in biomedical and epidemio-logic research when subject characteristics or exposures evolve over time. Appropriate handling of such covariates is essential for valid inference in survival analysis. This paper reviews the theoretical foundations of time-dependent covariates, emphasizing the critical distinction between external and internal processes. We define conditional cumulative hazard and survival functions, establish their relationships, and discuss key probabilistic properties. When covariates are external, the conditional cumulative hazard at time t depends only on the covariate history up to t , and the classical product-integral representation between the conditional cumulative hazard and survival functions remains valid. The framework presented provides conceptual clarity and a foundation for time-varying Cox regression, landmark analysis, and modern joint modeling techniques.