What a hazard ratio means when hazards are not proportional

A Cox model reports one hazard ratio. If the proportional-hazards assumption holds, that number is the ratio of hazards at every point in follow-up, and it means what everyone takes it to mean.

When the assumption fails — the effect is delayed, or wanes, or the curves cross — the model does not fail loudly. It returns a single number that is a weighted average of the true, time-varying log hazard ratio. The weights are not chosen by you: they are driven by how many people are still at risk at each point, which depends on the accrual, the censoring, and how long the trial ran.

That is the uncomfortable part. Two trials of the same treatment in the same population, with the same biology, will report different hazard ratios if one followed people for two years and the other for five — not because the effect differs, but because the averaging differs. A hazard ratio under non-proportionality is therefore not a property of the treatment alone, and comparing such numbers across trials, or carrying one into a model as though it were constant, quietly imports the follow-up of the trial it came from.

What to do instead depends on the question. If you want to know how the effect changes, fit it as time-dependent and report the shape. If you want a single summary that survives non-proportionality, restricted mean survival time gives one with a time scale attached. If you want absolute risk at a horizon, predict it from a model that can produce it. Testing the assumption and then reporting the single ratio anyway is the one option that is hard to defend.

MethodSurvival analysis