Module 3 · Hazard Ratios

HR 0.70 — cuts the risk of death by 30%." So I'll live 30% longer?

A cancer drug's headline result: hazard ratio 0.70. The press release translates it: "reduces the risk of death by 30%." A patient reads that and hears: "I'll live about 30% longer."

Three reasonable-sounding statements. At most one of them means what the patient thinks — and the hazard ratio is one of the most over-read numbers in all of medicine.

What a "hazard" is

Before the ratio, the thing itself. A hazard is the instantaneous rate of the event — the chance of it happening right now, among those who have survived this far.

That last part is the subtle bit. A hazard isn't "what fraction died" (that's cumulative risk, the Kaplan-Meier curve). It's "of the people still alive at this moment, how fast are events happening right now." It's a speed, not a total — like the reading on a speedometer versus the distance on the odometer.

Because it's conditional on having survived to each moment, the hazard can change over time: high early and low later, or the reverse. Hold that — it's exactly what the hazard ratio will assume doesn't happen.

The hazard ratio

The hazard ratio (HR) compares the hazard in the treatment group to the hazard in the control group, as a single number:

• HR < 1 — the treatment lowers the event rate (benefit). HR = 0.70 means a 30% lower hazard.
• HR = 1 — no difference.
• HR > 1 — the treatment raises the event rate (harm).

So "HR 0.70" means: at any given moment, among those still at risk, the treated group is having events at 70% of the rate of the control group. That's a genuine, useful statement. The trouble is what people hear instead — and how much the HR leaves out.

What the HR is not

"HR 0.70" says nothing, by itself, about either of the things people assume it does:

• It is not "30% longer life." A 30% lower event rate translates into extra time only through the baseline risk and the shape of the curve. The same HR can mean a few extra months or a few extra years.
• It is not "30% more patients survive." It's a rate, not a head-count. The absolute difference in survivors could be large or tiny.

A hazard ratio tells you the direction and relative speed of the effect — never, on its own, the size of the benefit in time or in people. To get that, you have to go back to the curve. Let's see the same benefit told three different ways.

One benefit, three numbers

Here is one fixed pair of survival curves — the same treatment effect throughout. Switch between three honest ways of summarising it, and watch how differently the same benefit reads.

Try this: view all three. Notice that "HR 0.70" never mentions a number of months or a number of patients — but the other two do.

The assumption hiding inside the HR

A single hazard ratio carries a big, often-unstated assumption: that the treatment's advantage is constant over time — that the hazard ratio is the same at month 1 as at year 5. This is the proportional hazards assumption.

When it holds, the two curves keep a steady proportional gap, and one HR genuinely summarises the whole story. When it doesn't hold, a single HR is an average of advantages that changed over time — and averages of things that change can describe a reality that never actually happened.

Whenever you see a hazard ratio, there's a hidden claim attached: "the benefit was roughly constant throughout." Sometimes true, often not — and when it's false, that one tidy number can seriously mislead.

When it breaks

The assumption breaks most visibly when curves do something other than stay neatly parallel. Read each scenario and classify it.

From HR to real benefit

So how do you turn a hazard ratio into something a patient or a payer can feel? You go back to the baseline risk. Here's the move, worked through:

Same HR of 0.70 on a 40% baseline risk would mean roughly 12 percentage points — a completely different drug. The ratio didn't change; the baseline did all the work. (Note this is an approximation — HR and risk ratio only coincide when events are uncommon — but it's exactly the back-of-envelope translation an assessor reaches for.)

Three questions whenever you meet a hazard ratio:

• What's the absolute benefit? Translate the HR back to the curve: extra median months, or the difference in survival at a clinically relevant time point.
• Does proportional hazards hold? Parallel-ish curves → one HR is fair. Crossing, converging, or diverging late → a single HR is suspect. (When it clearly fails, newer summaries like restricted mean survival time — the average time alive up to a set point — can describe the benefit more honestly.)
• What's the baseline risk? A strong HR on a rare event is still a small absolute benefit.

The hazard ratio is a useful headline and a poor conclusion. Treat it as the opening of the survival story, then go to the curve for the size, the shape, and the timing.

Why this matters for HTA

Hazard ratios are the standard currency of survival claims in submissions — and each one needs translating before it can support a decision.

• An impressive HR ("0.65, a 35% reduction") must be converted to absolute benefit — extra months of median survival, or the gap in survival at a relevant time point — because that, not the ratio, drives cost-effectiveness.
• Always check proportional hazards. If the curves cross or the benefit is all late, a single HR misrepresents the effect, and the economic model built on it inherits the error.
• A strong HR on a low-risk population can mean a tiny absolute gain — the same relative-vs-absolute trap, now in time-to-event form.
• HRs feed survival extrapolation. The assumed hazard beyond the trial — does the benefit persist, wane, or stop? — is among the most contested and most consequential assumptions in the whole appraisal (modelling, in M8).

A hazard ratio is the most compact, and the most over-trusted, way to state a survival benefit. Your job is to unpack it: into months, into patients, into the shape of the curve it came from.

Hazard ratios, in one breath

• A hazard is the instantaneous event rate among those still at risk; the hazard ratio compares it between groups (HR < 1 = benefit).
• "HR 0.70" = a 30% lower event rate at any moment — not "30% longer life" and not "30% more survivors."
• The same benefit can be told as an HR, a median difference, or a survival gap at a time point — the HR is the most abstract and hides absolute size.
• A single HR assumes proportional hazards (a constant effect over time); crossing or late-separating curves break it, and then one HR misleads.
• Always translate an HR into absolute benefit and check the curves before trusting it.

A hazard ratio compresses an entire survival story into one number. Convenient — but always ask what months, what patients, and what shape of curve that number is standing in for.

You can now read the full language of treatment effects — whether outcomes are yes/no events or time-to-event, in relative terms or absolute, through every ratio and its traps. There's one last kind of number in this module, and it asks a different question entirely. Not "does the treatment work?" but "does this test correctly tell the sick from the well?" Diagnostic accuracy has its own measures — sensitivity, specificity, predictive values — and a famous, deeply counterintuitive trap: a near-perfect test can still be wrong most of the time. That's the finale.