13 Research methods, statistics, critical review and evidence-based practice13.9 Differential and analytic statistics
13.9.6 Survival curves
7 min read3 figures
Written by Resident Doctor Committee · Published
Time-to-event data
Survival analysis takes the time until an event of interest as its outcome, such as death, relapse, readmission, self-harm or stopping treatment.
It needs time-to-event data: each person is followed from a defined starting point until the event or censoring, which only follow-up designs such as cohorts provide.
The event must be defined precisely, because readmission and a symptom threshold capture different aspects of relapse, and admission also reflects service decisions.
Length of stay is right-skewed, partly censored time-to-event data, so groups are best compared with survival methods (time to discharge), not t-tests, analysis of variance or a chi-squared test on a dichotomised outcome.
A meta-analysis that pools separately published studies is not a prospective cohort, even when it presents its result as a survival curve.
Origin, risk set and censoring
The origin is a common time at which every participant enters the analysis, such as recruitment, randomisation or discharge, not an earlier clinical milestone or the event.
The risk set is everyone still observed and event-free just before a given time.
A censored observation is a participant who has not had the event when observation stops, for example when the study closes, and censoring is not the same as drop-out.
Censoring records event-free follow-up until observation ends without assuming what happened afterwards.
Type | What is known | Bound on event time |
|---|---|---|
Right censoring | No event by the last observation | Lower bound only |
Left censoring | Event happened before the first observation, time unknown | Upper bound only |
Interval censoring | Event happened between two observations | Upper and lower bound |
Left truncation | People whose event came before entry never enter the sample | They are missing altogether |
Standard Kaplan-Meier methods need event times, so interval-censored events need interval censoring methods.
Non-informative censoring is the Kaplan-Meier assumption that censoring is independent of later event risk.
Administrative closure usually meets this assumption, but loss to follow-up may not, because disengagement can reflect deterioration.
If those lost are at higher risk, the curve overestimates event-free survival, so reasons and timing of censoring should be reported, not only totals.
Building the Kaplan-Meier estimate
At each event time, the previous estimate is multiplied by the proportion of those at risk who stay event-free: S(t) = S(previous event time) x (n - d) / n. Here n is the number at risk just before and d the number of events, and S starts at 1.
In this example, 20 patients on maintenance antipsychotics are followed for 12 months and the event is stopping treatment.
Month | At risk just before | Stopped | Censored | Survival estimate |
|---|---|---|---|---|
2 | 20 | 2 | 0 | 18/20 = 0.900 |
3 | 18 | 0 | 1 | 0.900 |
4 | 17 | 3 | 0 | 0.900 x 14/17 = 0.741 |
6 | 14 | 0 | 2 | 0.741 |
8 | 12 | 2 | 0 | 0.741 x 10/12 = 0.618 |
10 | 10 | 0 | 1 | 0.618 |
11 | 9 | 1 | 0 | 0.618 x 8/9 = 0.549 |
12 | 8 | 0 | 8 (study closes) | 0.549 |
Censoring does not lower the curve but shrinks later denominators, so the three events at month 4 are divided by 17, not 20.
One event among 9 at risk removes about 11% of the remaining survival, whereas one among 20 removes 5%.
The crude proportion stopping is 8 of 20 (40%), but the estimated cumulative proportion is 1 - 0.549, about 45%, because the estimate allows for different observation times.
Reading a Kaplan-Meier curve
The vertical axis is the survival function S(t), the estimated probability of not yet having had the event by time t; it is not the hazard, the proportion still followed up or the cumulative incidence.
The horizontal axis is elapsed time from the common origin.
The curve starts at 1.0, falls in a step at each event and stays flat between events, with one curve per group.
Tick marks show censored observations and do not lower the curve.
The number-at-risk row beneath the figure shows how many people support each part of the curve: 20, 18, 14, 10 and 8 at months 0, 3, 6, 9 and 12 in the example.
A curve that keeps stepping down means the cumulative probability of the event rises with follow-up time.
Survival and cumulative incidence
Convention | Starts at | Direction | Height at time t | A reading of 0.30 means |
|---|---|---|---|---|
Survival | 1.0 | Falls in steps | Proportion still event-free | 70% have had the event |
Cumulative incidence | 0 | Rises, never falls | Probability of having had the event | 30% have had the event |
A Kaplan-Meier curve can be drawn as cumulative incidence, the complement (1 - S(t)) of the usual survival estimate.
A cumulative incidence curve is not a curve of survival after diagnosis, of prevalence or of hazard.
With competing events such as death before relapse, a cumulative incidence curve gives absolute risk and treats death as a competing event, whereas Kaplan-Meier censors competing events.
For that reason, the plain Kaplan-Meier complement does not estimate real-world cumulative incidence when competing events occur.
Stratified cumulative incidence curves, one set for each level of a second variable, show the joint effect of two variables, which a single pooled curve cannot.
Reading values
A value is read where a vertical line from the time of interest meets the correct group's curve, on the 0 to 1 scale.
The proportion who have had the event by that time is 1 minus the Kaplan-Meier value.
Median survival time is where the curve crosses 0.5 (50%), taken as the first time the survival curve reaches or falls below 0.5, without interpolating across a step.
If the curve never reaches 0.5, as when follow-up ends first, no median can be read; the example ends at 0.549, so its median is not reached.
An unreached median refers to estimated survival, not a claim that over half the original participants are still observed.
Curves should not be extrapolated beyond follow-up without a model; restricted mean survival time summarises event-free time up to a stated horizon the data support.
Interpreting curves
Curves that meet at a time point show a similar probability in both groups at that time.
Two groups' risks are similar only when their curves run close together throughout follow-up, not when they meet only at the start, median or end.
Late plateaus or separations may rest on very few people, so they are read with confidence intervals and the risk table.
The counts printed beneath a figure show what most participants experienced, and only claims those counts support hold.
The highest-risk group is read from the position of the curves, but the reported statistics decide which differences are real.
Two groups can each differ significantly from the reference while the gap between them is not significant.
A visible gap over part of follow-up does not establish a treatment effect when the formal comparison is not significant.
Curves plus a significant between-group test support only a difference in cumulative incidence of the plotted outcome, not claims about other outcomes, unreported subgroups, early divergence or a plateau in the sparse tail.
Crossing curves complicate a single overall comparison, and visual shape alone shows neither mechanism nor constant hazards.
A better curve does not prove benefit, because prescribing decisions can confound observational depot versus oral comparisons in either direction.
Several cohorts' cumulative risk curves read at one fixed time show a wide range of risk across studies (between-study heterogeneity), and the range grows with follow-up.
Hazard ratios raised at both ends of an exposure range, with a lower middle band, show a non-linear, non-monotonic relationship, not a dose-dependent or linear one.
Comparing survival formally
Method | What it does | Adjusts for other variables |
|---|---|---|
Log-rank test | Compares whole survival curves between categorical groups, keeping censored people in the risk set; tests whether event probability is the same in both groups at every time point | No |
Cox proportional hazards regression | Relates a time-to-event outcome to several explanatory variables at once, including continuous ones | Yes |
The proportional hazards assumption of Cox regression is that the hazard ratio between groups stays constant over follow-up.
The hazard ratio is the time-to-event summary of relative risk: it uses when events happen and who remains at risk, and a Cox model averages the comparison over follow-up.
Time-to-event data are summarised with a hazard ratio rather than an odds ratio, because an odds ratio at a fixed point counts only whether the event occurred, ignoring timing and censored follow-up.