“Thirty per cent lower hazard” and “thirty fewer events per hundred people” are different statements. So are a risk by six months and a risk by twelve months. Time-to-event research becomes much easier to interpret when the event, time horizon and effect measure stay attached to every number.
Survival analysis is not limited to death. A study might measure time to a first recurrence, a first admission or another precisely defined event. Censored participants contribute the period during which their event status is known. Cochrane Handbook chapter 6, section 6.8.
The rehabilitation study and numerical examples below are fictional. They teach interpretation, not the effectiveness of a real programme or an individual prognosis.
Define the event before reading the curve
Imagine a Canadian rehabilitation trial enrolling people free of the specified event at randomization and comparing two follow-up programmes. The event is a first activity-limiting episode meeting a prespecified study definition. The time origin is randomization, and follow-up ends at twelve months.
The event-free function, written S(t), describes the probability of remaining free of that event through time t. In this example, assume no competing events and a suitable analysis of follow-up. The estimated event risk by t is then 1 − S(t).
The fictional report supplies these event-free estimates:
| Time after randomization | Standard programme S(t) | New programme S(t) |
|---|---|---|
| 3 months | 0.90 | 0.94 |
| 6 months | 0.75 | 0.85 |
| 12 months | 0.60 | 0.72 |
At six months, the estimated event risks are 25% and 15%. The absolute reduction, standard minus new, is ten percentage points. The risk ratio, new divided by standard, is 0.15/0.25 = 0.60.
At twelve months, the estimated risks are 40% and 28%. The absolute reduction is twelve percentage points, and the risk ratio is 0.28/0.40 = 0.70.
The numbers answer different time-specific questions. Reporting “the risk ratio is 0.60” while omitting six months would obscure that distinction. Choosing the most flattering time point after inspecting the curves would also change the evidential meaning of the comparison.
A hazard ratio has a different denominator
A hazard concerns the instantaneous event rate among people who have not yet experienced the event. A hazard ratio compares those rates between groups. A proportional-hazards model assumes that their ratio is constant over the modelled follow-up, even when the hazards themselves change. Cochrane Handbook chapter 6.
A reported hazard ratio of 0.70 is therefore not automatically a six-month risk ratio of 0.70. It also cannot supply an absolute risk reduction without additional baseline event information and modelling assumptions. The fictional table above supplies risks through curves; it does not supply enough information to reconstruct a fitted hazard ratio.
A useful counterexample is two studies with very different underlying event frequencies. The same hazard ratio can accompany small or large absolute differences. The ratio alone cannot tell you how many events occur per hundred people by a stated time.
If curves cross or the relative effect changes over follow-up, a single fitted hazard ratio may also hide time variation. It can still be reported as a model-based summary, but it should not be described as a constant effect that was verified merely because software produced a coefficient.
Why censoring is more than subtracting people
Consider a separate fictional teaching dataset of ten participants, all event-free at the start:
- At month one, two experience the event. The estimated event-free probability becomes 8/10 = 0.80.
- At month two, three of the remaining eight are lost to follow-up while still known to be event-free. Five remain under observation and at risk.
- At month three, one of those five experiences the event. The conditional event-free proportion at that event time is 4/5.
The Kaplan–Meier product is 0.80 × 0.80 = 0.64. Thus the estimated event risk by month three is 36%, under the method’s assumptions. The calculation is not simply three observed events divided by ten, which would give 30%.
The lost participants supplied event-free information up to their last observation. They were not counted as definitely event-free through month three. Their censoring changed the later risk set; it did not cause an event-free probability drop at the censoring time itself.
The Kaplan–Meier method and its assumptions are explained by Clark and colleagues. For a simple group-specific curve, independent or noninformative censoring requires that those censored at a time have a comparable subsequent event experience to those remaining under observation, under the relevant assumptions. Survival Analysis Part I: Basic concepts and first analyses.
Our ten-person example assumes this condition for teaching. If worsening symptoms caused participants to leave just before an event, the observed follow-up could make event-free experience appear better than it was. Reasons for loss, timing, remaining numbers at risk and sensitivity analyses help evaluate the concern; a curve alone cannot verify the assumption.
Keep competing events separate from ordinary loss to follow-up
A competing event makes the event of interest impossible to occur afterwards. For example, death before a first nonfatal episode prevents that episode. It is not simply an unknown future episode time.
When estimating real-world cumulative incidence of a particular event, treating competing events as ordinary censoring and reporting one minus Kaplan–Meier can overestimate that event’s risk. Appropriate competing-risk methods address a different target; estimating a cause-specific hazard and estimating cumulative incidence must not be conflated. Bias by censoring for competing events in survival analysis.
The earlier table explicitly assumed no competing events so that the simple 1 − S(t) interpretation applied. In a real paper, identify what the event-free curve represents before copying that conversion.
Transfer exercise
A third fictional report gives six-month event-free estimates of 0.82 for standard care and 0.88 for a new programme. It also reports an overall hazard ratio of 0.70. A learner writes: “The new programme prevents thirty events per hundred people.”
Calculate the six-month risk difference and risk ratio from the supplied curves, assuming the same no-competing-event framework. Then identify one limitation that would remain if many participants were lost before six months.
Reasoned solution: Risks are 18% and 12%. The absolute reduction is six percentage points, corresponding to six fewer events per hundred in the estimated six-month risks. The risk ratio is 0.12/0.18, approximately 0.67. Neither number equals thirty fewer per hundred, and the overall hazard ratio does not replace this time-specific calculation.
These are point estimates. Without suitable uncertainty intervals and a credible design, they do not establish a precise causal benefit. Substantial loss related to future events would challenge the censoring assumptions; the analysis would need to address that concern. The missing proportion alone cannot determine whether bias exists.
A complete reporting sentence
For this exercise: “By six months, estimated event risks were 18% and 12%, an absolute difference of six percentage points favouring the new programme, subject to uncertainty and follow-up assumptions.” Keep the event definition beside that sentence. Report a hazard ratio separately with its model, interval and interpretation.
That habit preserves the information a reader needs: what happened, by when, how often, and under which assumptions.
Sources and further reading
- Cochrane Handbook, chapter 6, section 6.8 — time-to-event data, hazards and proportional hazards; full-text section reviewed.
- Clark et al.: Survival Analysis Part I: Basic concepts and first analyses, 2003 — Kaplan–Meier estimation and censoring assumptions; indexed publisher and article excerpts reviewed.
- Coemans et al.: Bias by censoring for competing events in survival analysis, 2022 — cumulative incidence and competing events; indexed publisher excerpts reviewed.