Checking the Proportional-Hazards Assumption: When One Hazard Ratio Can Mislead

The proportional-hazards assumption says the ratio of event rates between two groups stays constant over the whole follow-up, which is what lets a study summarize a survival difference with one hazard ratio. When it fails, for example when survival curves cross or separate only late, that single number averages very different periods and can hide or distort the real effect. Reading a trial well means checking whether the assumption was tested and what the curves actually show.

What proportional hazards means in plain terms

A hazard is the instantaneous rate of an event, the chance it happens in the next small slice of time given it has not happened yet. A hazard ratio compares that rate between two groups. The proportional-hazards assumption is the idea that this ratio holds steady across the entire follow-up: if the treatment halves the event rate early on, it halves it late as well.

That assumption is convenient because it lets one number, the hazard ratio, stand in for the whole comparison. It is also the engine of the Cox proportional-hazards model, the most common tool for analyzing time-to-event data. But convenience is not the same as truth, and the assumption is often left unchecked.

The picture: what a violation looks like

The clearest sign of a violation is survival curves that cross. If one group does better early and the other does better later, no single ratio can describe both stretches at once. A gentler sign is curves that stay together for a while and then separate, or start apart and then converge.

Real therapies produce these shapes. A treatment with early harm and late benefit, such as some surgical or immunotherapy strategies, shows curves that cross. A drug whose effect wanes shows curves that converge. When you see any of these on a Kaplan-Meier plot, a reported hazard ratio deserves a second look, because it is trying to average a story that changed over time.

How researchers test the assumption

Beyond eyeballing the curves, there are formal checks. The most common is based on Schoenfeld residuals, which examine whether the estimated effect drifts with time; a clear trend signals non-proportional hazards. Analysts also add a treatment-by-time interaction term and test whether it is needed, or plot the two groups on a log-minus-log scale, where proportional hazards should appear as roughly parallel lines.

None of these is a perfect gatekeeper. Formal tests can miss real violations when a study is small and can flag trivial ones when it is very large. That is why methodologists advise combining a test with a look at the curves rather than trusting a single p-value about the assumption.

Why one hazard ratio can mislead when it fails

When hazards are not proportional, the number a Cox model reports is a kind of weighted average of the changing effect over the follow-up, and the weights depend on when events happened and how long people were observed. Two trials of the same treatment with different follow-up lengths can then report different hazard ratios for reasons that have nothing to do with the biology.

Worse, the average can point the wrong way for the question you care about. A treatment that saves lives in the long run but carries early risk can post a hazard ratio near one, making a real long-term benefit look like nothing. The problem is not that the hazard ratio is wrong arithmetically; it is that a single summary cannot represent an effect that genuinely changes over time.

What to do instead

When the assumption fails, better summaries exist. Restricted mean survival time reports the average event-free time up to a set horizon and stays interpretable when curves cross. Reporting the survival difference at clinically meaningful landmarks, such as one year and three years, describes the pattern instead of hiding it. Sometimes the honest move is to present the whole curve and describe how the effect evolves.

None of this requires abandoning the hazard ratio entirely. Often the clearest reports show the Kaplan-Meier curves, give a hazard ratio for readers who want it, and add a time-based measure when the assumption is shaky. The goal is a summary that matches what actually happened.

What to look for as a reader

Start with the figure. Do the survival curves stay in roughly constant proportion, or do they cross, converge, or fan out late? If they are not close to parallel, treat a lone hazard ratio with caution. Then check the methods: did the authors test the assumption, and did they say how? Silence is common and is itself a small warning.

Finally, ask whether the conclusion leans entirely on one hazard ratio when the curves suggest a time-varying effect. A careful paper acknowledges non-proportional hazards and reports something that survives it. A paper that offers a single ratio, no assumption check, and curves that clearly cross is asking you to trust an average that may not mean what it seems.