Kaplan-Meier curve, censoring and proportional hazards

Survival endpoints are awkward because at the moment of analysis most patients have not died or progressed and have been followed for different lengths of time. The Kaplan-Meier method handles this by working through time step by step: at each event it multiplies the survival probability so far by the fraction of patients still at risk who survived that moment. Patients who are alive at their last visit are censored, meaning they contribute follow-up up to that point and then drop out of the calculation without being counted as an event. The result is the stepped curve in every trial paper, with a number-at-risk table underneath showing how many patients remain at each time; when that number gets small, the tail of the curve is unreliable however dramatic it looks. The median survival is where the curve crosses 50 percent, and a landmark such as five-year survival is read off at that time.

The log-rank test asks whether two curves differ over their whole length, and the Cox model summarises the difference as a hazard ratio. Both assume proportional hazards: that the relative risk between arms is roughly constant over time. Immunotherapy trials often break that assumption. In CheckMate 067 the curves for nivolumab with ipilimumab separate from ipilimumab alone and then flatten into a plateau, with about half of patients alive at ten years when melanoma-specific survival is counted; the hazard ratio compresses a shape like that into one number and understates what the plateau means for an individual. Delayed separation, where curves overlap for months before diverging, and crossing curves, where an early excess of harm gives way to a later benefit, both make the hazard ratio misleading, and analysts turn to the restricted mean survival time or to landmark comparisons instead.

Censoring hides a trap. The method assumes patients censored at a given time are no more or less likely to have the event than those still being followed. If patients leave a trial because they are doing badly, or are censored when they start another therapy, the curve is biased upwards. Informative censoring is one reason progression-free survival read from investigator scans can flatter an open-label trial, and one reason the estimand framework asks trialists to say in advance how they will treat patients who switch, stop or are lost.