Questions: Time-Varying Covariates in Survival Models

Question 1

A study compares survival in transplant recipients vs. non-recipients by classifying patients as 'transplanted' from study entry if they ever receive a transplant. Transplant recipients show better survival. Why is this analysis biased?

• A. The transplant group is younger on average
• B. Patients must survive long enough to receive a transplant, creating immortal time bias — the pre-transplant survival is guaranteed and incorrectly attributed to the transplant
• C. The analysis should use a per-protocol approach
• D. Transplant allocation is not randomized

Answer: (B)

Immortal time bias occurs when the period between study entry and the time-varying event (transplant) is incorrectly attributed to the treated group. Patients classified as transplant recipients at baseline include their pre-transplant survival as 'treatment survival,' but they were guaranteed to survive that period (they had to be alive to receive the transplant). The correct approach treats transplant status as a time-varying covariate: each patient starts as untransplanted and switches to transplanted at the actual transplant date, so pre-transplant person-time is correctly attributed to the untransplanted group.

Question 2

In the counting process formulation, a patient observed from time 0 to time 5, with a covariate change at time 3, is represented as two intervals: (0, 3] and (3, 5]. Why is this representation necessary?

Model answer: The standard Cox model assumes each subject has fixed covariates for their entire observation period. When a covariate changes at time 3, the hazard before and after the change is different. Splitting the observation into two intervals allows the model to use the correct covariate value for each period — the old value for the (0, 3] interval and the new value for the (3, 5] interval. The partial likelihood computation then uses the appropriate covariate values when determining risk set membership and conditional event probabilities at each event time.

This is called the counting process or Andersen-Gill formulation. Each interval has a start time, stop time, event indicator (event occurs only at the final interval's end, if at all), and the covariate values in effect during that interval. The subject contributes to the risk set during each interval with the corresponding covariate values. The implementation requires restructuring the data from one-row-per-subject to multiple-rows-per-subject format.

Question 3

An external time-varying covariate (like air pollution levels) is less problematic than an internal time-varying covariate (like current blood pressure) in a Cox model because external covariates are not generated by the subject's own disease process.

• T. True
• F. False

Answer: True

External covariates are determined by processes outside the subject (weather, pollution, calendar time) and are predictable regardless of the subject's survival status. Internal covariates (biomarkers, lab values) are generated by the subject and cease to exist if the subject dies — they are often part of the disease process itself. Internal covariates can create feedback loops (worsening biomarker → higher hazard → death → no more biomarker values) and may require joint modeling of the longitudinal biomarker process and the survival process to avoid bias.