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AKT Statistics Guide · Part 4 · Chapter 14

Hazard Ratios & Kaplan–Meier Curves

When the question is not just whether an event happened, but how long until it did — reading survival over time.

Where this shows up in the AKT

Interpreting a Kaplan–Meier survival curve and its censoring marks.
Reading a hazard ratio (HR) and its confidence interval.
Knowing the difference between a hazard ratio and a relative risk.
Spotting when crossing curves make the HR unreliable.

The earlier risk measures answered a yes/no question: by the end of the study, did the event happen? Survival analysis adds the dimension of time — not just whether you had the event, but how quickly. It is the natural language of cancer trials, cardiovascular studies, and any condition where timing matters.

Kaplan–Meier curves plot survival against time; the small ticks mark censored patients, and the hazard ratio is the single number that summarises the gap between the curves.

Kaplan–Meier curves: survival over time

A Kaplan–Meier (K–M) curve plots the proportion of patients still event-free as the study goes on. Read it like this:

Y-axis = proportion surviving (1.0 at the start, falling over time).
X-axis = time.
Each downward step = an event (e.g. a death).
Flatter curve = better survival; steeper curve = worse survival.

Censoring — the tick marks are not deaths

Censoring happens when we stop knowing a patient’s outcome — they left the study, moved away, or were still alive when follow-up ended.
It is shown as a small tick mark on the curve. A censored patient is not an event; treating tick marks as deaths is a classic misread.

Comparing two curves

The curve that stays higher shows better survival.
Consistent separation between the curves is strong evidence of a real difference.
Crossing curves are a red flag: the proportional-hazards assumption breaks down and the single hazard ratio becomes unreliable.

The hazard ratio

The hazard ratio is the numerical summary of what the K–M curves show visually. It compares the rate of events over time between two groups.

Reading a hazard ratio (for a survival outcome).
Hazard ratio	Meaning
HR = 1	No difference between the groups
HR < 1	Treatment reduces the event rate (good for survival)
HR > 1	Treatment increases the event rate (bad)

The crucial distinction is that the hazard ratio is about rate, not the total events at the end. An HR of 0.70 means the rate of events is 30% lower over the follow-up period — not necessarily that 30% fewer people died overall. It is an average of that rate across the whole study, which is exactly why it stops being meaningful when the curves cross.

Pearl — HR is a rate over time, RR is a yes/no at the end. A relative risk asks “by the end of the study, did you have the event?” A hazard ratio asks “how fast are events happening in one group versus the other?” Both compare risk, but only the HR carries time. And always read the HR with its confidence interval: an interval that does not cross 1 is statistically significant; its width tells you how precise the estimate is.

The traps that cost marks

Trap 1 — treating the hazard ratio as a relative risk. The HR reflects the event rate over time, not the final proportion who had the event. “HR 0.70” is not “30% fewer deaths overall”.

Trap 2 — ignoring how the curves behave. Consistent separation is strong evidence; crossing curves make a single HR unreliable, because the relative hazard is no longer constant over time.

Trap 3 — misreading censoring, or assuming HR < 1 means a big effect. Tick marks mean follow-up ended, not death. And an HR below 1 means benefit, but the confidence interval — not the point estimate alone — tells you the precision and the significance.

AKT-style question

A trial compares Drug A with placebo for survival in heart failure. The Kaplan–Meier curves show clear, consistent separation, with Drug A’s curve staying higher throughout. The hazard ratio is 0.70 (95% CI 0.55–0.90). Which statement best describes this result?

Drug A reduces the risk of death by 30% at the end of the study
Drug A reduces the rate of death over time compared with placebo
Drug A improves survival only at the final time point
The result is not statistically significant

Reveal answer

B — it reduces the rate of death over time. An HR of 0.70 means the rate of death over the follow-up is 30% lower with Drug A — not that 30% fewer patients died overall (that rules out A). Consistent K–M separation supports a real, sustained effect (not just the final time point, ruling out C). The confidence interval 0.55–0.90 does not cross 1, so the result is significant (ruling out D).

Lock it in

Kaplan–Meier: higher curve = better survival; each step = an event; tick marks = censoring, not deaths.
Hazard ratio summarises the curves: HR < 1 = lower event rate (benefit), HR > 1 = higher event rate (harm), HR = 1 = no difference.
The HR is a rate over time, averaged across follow-up — not the final headcount, and unlike a relative risk it includes time.
Crossing curves make a single HR unreliable; always read the HR with its confidence interval.

Source concepts: standard survival-analysis methodology (Kaplan–Meier estimator; Cox proportional-hazards model; e.g. Centre for Evidence-Based Medicine). A revision summary, not a clinical protocol.

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