Survival Analysis & the Kaplan-Meier Curve

Overview

Many ICU outcomes are time-to-event: time to death, to extubation, to discharge, to readmission. These data share a feature that ordinary averages cannot handle - censoring - and survival analysis is the set of methods built for them.[1]

The vocabulary of survival data

Survival analysis describes the time elapsed from a defined origin (ICU admission, intubation, randomisation) to an event (death, extubation, discharge, readmission). Three functions are the mathematical core - any one can be derived from the others.[1]

• The survival function S(t) — the probability of being event-free beyond time t: S(t) = P(T > t). At t = 0, S(0) = 1 (everyone starts event-free); as events accumulate S(t) falls towards 0. The Kaplan-Meier curve is an estimate of S(t). The complement F(t) = 1 − S(t) is the cumulative incidence (the probability the event has occurred by t).[1]
• The hazard function h(t) — the instantaneous risk of the event at time t, given survival up to t: h(t) = limit (as Δt→0) of P(t ≤ T < t+Δt | T ≥ t) / Δt. It is a rate (events per unit time), not a probability, and is always ≥ 0. A constant hazard gives an exponential distribution; a hazard that rises then falls gives a bathtub curve.[1]
• The cumulative hazard H(t) — the integral of the hazard up to time t. It links the other two by the identity S(t) = exp(−H(t)), or equivalently H(t) = −ln S(t). This is why a plot of −ln(−ln S(t)) against ln(t) gives a straight line under proportional hazards (see below).[1]

These three are interchangeable: estimate S(t) and you have h(t) and H(t), and vice versa. Examiners expect you to state what each represents in one sentence, and to know that the hazard is a rate while the survival is a probability. [1]

Censoring

• A subject is censored when their event time is not observed - they are lost to follow-up, withdraw, or the study ends before the event occurs (the commonest form, right-censoring).[1]
• Censored subjects contribute information up to the point of censoring but cannot be counted as having had the event. Ignoring them (or treating them as events) biases the estimate; survival methods use the time they were observed without bias.[1]

Types of censoring

[1]

The unspoken assumption behind every Kaplan-Meier curve and Cox model is that censoring is non-informative - a censored subject has the same future risk as a subject still under observation. When this fails (sicker patients selectively lost), the curve is biased, and no amount of mathematical sophistication can fix unreported informative censoring.[1]

The Kaplan-Meier estimator

• A non-parametric estimate of the survival probability S(t) at each time point. It is the product of the conditional probabilities of surviving each interval: at each event time, multiply by (1 minus the number of events divided by the number at risk).[1]
• It produces a step function that drops at each event and is flat between events; censored subjects are marked with small ticks and removed from the at-risk count thereafter.[1]
• The median survival is the time at which the survival curve crosses 0.5 (half the population has had the event) - often a more useful summary than the mean for skewed time-to-event data.[1]

The estimator in symbols

Let $d_t$ be the number of events at time $t$ and $n_t$ the number at risk just before $t$. The Kaplan-Meier estimate is: [1]

Ŝ(t) = ∏ (1 − dᵢ/nᵢ) over all distinct event times tᵢ ≤ t [1]

The product runs only over distinct event times $t_i$ (times at which at least one event occurred). Between event times the estimate does not change — hence the step shape. At each event time the curve steps down by a factor of $(1 - d/n)$; when several events share a time (ties) the drop is correspondingly larger.[1]

A closely related quantity is the Nelson-Aalen estimator of the cumulative hazard, Ĥ(t) = Σ dᵢ/nᵢ (summed over tᵢ ≤ t), from which an alternative survival estimate S̃(t) = exp(−Ĥ(t)) can be formed. The two survival estimates agree closely in moderate-to-large samples; Nelson-Aalen is slightly better when the risk set is small or events are few.[3]

Confidence intervals

Pointwise confidence bands for Ŝ(t) are calculated on the log-log scale (to keep the bounds within 0 and 1) using Greenwood's formula for the variance of the estimator. These are the dotted lines drawn above and below the KM curve. They widen at the tail of the curve, where few subjects remain at risk — which is exactly why the tail of a KM curve is unreliable and should never be over-interpreted.[1]

Worked example — computing a Kaplan-Meier curve from ICU data

Imagine a 10-patient cohort observed for ICU mortality over 14 days. "C" denotes censoring (discharged alive / lost), "D" denotes death. The data, sorted by time: [1]

Reading the curve: S(t) sits at 0.506 at day 14, the curve steps down only on death days (2, 5, 7, 11), and the censored subjects (days 4, 7, 9, 14) would each be marked with a small tick on the step. The median survival here falls just after day 11, where the curve crosses 0.5 (0.506 → estimate ~day 11). Note how a single death at day 11, when only four patients remain at risk, produces a large step (0.675 → 0.506) — this is why the tail of a KM curve is unstable when few are at risk.[1]

Median survival

• Median survival = the smallest time t at which Ŝ(t) ≤ 0.5; read off the KM curve where it crosses the horizontal line at 0.5. Half the population has had the event by this time.[1]
• If the curve never drops to 0.5 (most patients survive the observation window), median survival is not reached and cannot be estimated — a common exam point. In that setting report the survival probability at a fixed time (e.g. "90-day survival was 0.62") instead.[1]
• It communicates intuitively ("half of patients had died by day 28") and, unlike the mean, does not require the entire survival distribution to be known — it is unaffected by censoring after the median, where most data are typically thin.[1]

Restricted mean survival time (RMST)

The mean survival time is, in principle, the area under the survival curve to infinity. But because the right tail is usually censored (and unreliable), the unrestricted mean is rarely estimable. The restricted mean survival time (RMST) solves this by integrating the survival curve only up to a pre-specified time horizon $\tau$: [1]

RMST(τ) = ∫ S(t) dt integrated from time 0 to τ [1]

Geometrically, RMST is the area under the survival curve from time 0 to $\tau$. It is measured in units of time (days, months) and represents the average event-free time enjoyed over that horizon. Between two trial arms, the difference in RMST($\tau$) is a directly interpretable measure of treatment effect ("patients survived an average of 2.4 more days alive over 90 days") — free of the proportional-hazards assumption that Cox regression imposes.[5]

Royston and Parmar argued that RMST is preferable to the hazard ratio when the proportional-hazards assumption is in doubt or when the effect varies over time, because RMST captures the whole curve difference rather than collapsing it into a single ratio. Its weaknesses: it depends on the arbitrary choice of $\tau$ (usually the minimum of the maximum follow-up in each arm), and it loses power when the curves separate late.[5]

Comparing groups

• The log-rank test compares the survival curves of two or more groups non-parametrically, testing whether they differ significantly. It does not assume a particular distribution of survival times.[1]
• Cox proportional-hazards regression models the effect of several variables on the hazard (instantaneous risk of the event) simultaneously, yielding a hazard ratio for each. Its key assumption is proportional hazards - the hazard ratio between groups is constant over time (the curves do not cross).[1]
• A hazard ratio below 1 means a lower risk (a protective factor), above 1 a higher risk (a harmful factor).[1]

The log-rank test in detail

The log-rank test (Mantel-Cox) is the non-parametric equivalent of comparing two survival curves. At each distinct event time it constructs, from the two groups' at-risk numbers, the expected number of events in each group under the null hypothesis of no difference, and compares these with the observed events. Summing across all event times gives an overall observed-minus-expected (O−E) for each group; squaring, standardising by the variance, and summing yields a chi-square statistic on (groups − 1) degrees of freedom.[1]

Key properties the examiner wants:

• It is a test of the null hypothesis that the survival distributions are the same across groups — it gives a p-value, not an effect size.
• It weights every event time equally (unlike the Gehan-Breslow generalised Wilcoxon, which weights early events more heavily by the size of the at-risk set). Use the log-rank when you believe hazards are proportional; use the Wilcoxon/Peto when the curves separate early then converge.
• It extends to more than two groups and to stratified versions (e.g. stratified by centre in a multi-centre ICU trial).
• It is most powerful when the hazard ratio is constant over time — exactly the proportional-hazards setting.[1]

Cox proportional-hazards regression

The Cox model relates the hazard for subject i with covariates xᵢ to an unspecified baseline hazard h₀(t): [1]

h(t | xᵢ) = h₀(t) · exp(βᵀxᵢ) [1]

The baseline hazard h₀(t) is left completely unspecified (the model is semi-parametric), while the covariate effects multiply it by a factor exp(βᵀx) that is constant over time. Estimation uses the partial likelihood, which avoids estimating h₀(t) at all — a clever feature that made the model practical and won it wide adoption.[2]

The hazard ratio (HR) for a one-unit increase in covariate $x_j$ is $\exp(\beta_j)$:

• HR = 1: no effect on the hazard.
• HR < 1: lower hazard (protective) — e.g. HR 0.7 means a 30% lower instantaneous risk of the event at any time.
• HR > 1: higher hazard (harmful) — e.g. HR 1.5 means a 50% higher instantaneous risk.[2]

Because the covariate effect multiplies a common baseline hazard, the ratio of hazards between any two subjects is constant over time — this is the proportional-hazards (PH) assumption, and it is the single most tested concept in this topic.[2]

Interpreting and reporting a hazard ratio

A trial reports "early goal-directed resuscitation versus usual care: HR for 90-day mortality 0.98 (95% CI 0.82–1.18), p = 0.83". Translation: at any moment during follow-up, the hazard of death in the intervention arm was 2% lower (0.98), but the confidence interval comfortably includes 1.0 (no effect), so the result is consistent with no benefit. The 95% CI is the range of HRs compatible with the data; the p-value tests HR = 1.[8]

A clinically meaningful HR is typically read alongside the absolute difference: an HR of 0.8 looks different when baseline 90-day mortality is 40% versus 5%. This is why survival data should always be accompanied by the Kaplan-Meier curves and a measure of absolute effect (risk difference, number-needed-to-treat, or RMST), not the HR alone.[2]

Checking the proportional-hazards assumption

The PH assumption must be checked, not assumed. The standard methods:[3]

Competing risks — when the event of interest can be pre-empted

In ICU data the event of interest is often competed away by another event. If the outcome is time-to-readmission, a patient who dies before readmission can never be readmitted — death is a competing risk. Treating a competing event as ordinary censoring overestimates the cumulative incidence of the event of interest, because it assumes the censored subject could still experience the event.[6]

The correct approach is the cumulative incidence function (CIF), estimated by the Fine-Gray subdistribution hazard model, which accounts for the fact that subjects who experience the competing event are no longer at risk. The subdistribution hazard ratio from Fine-Gray is interpreted like an ordinary HR but for the CIF, not the cause-specific hazard.[6]

[1]

ICU context — what the curves actually show

In critical-care trials the commonest survival endpoints are 28- or 90-day mortality (ICU and hospital), time to death, time to successful extubation, and time to ICU/hospital discharge. Two design points recur in the literature:[1]

• ICU mortality is right-censored at discharge. A patient discharged alive from ICU by day 28 but lost to the 90-day dataset is censored, not counted as a survivor. If discharge is itself prognosis-related (patients are discharged either because they are well or because they are transferred to die elsewhere), censoring can become informative — a known limitation of ICU survival datasets.
• Hospital mortality conflates two processes — the risk of death and the competing process of discharge alive. Discharge-alive is a competing risk for in-hospital death, which is why analyses of hospital mortality should account for the discharge process or use a fixed horizon (e.g. 90-day mortality) that is unaffected by discharge timing. [1]

The LUNG SAFE study of ARDS across 50 countries reported 90-day mortality with Kaplan-Meier curves stratified by severity (mild, moderate, severe ARDS) and used Cox regression to identify factors associated with mortality (lower tidal volume ventilation was associated with lower mortality; severe ARDS and higher driving pressure with higher mortality). It is a canonical example of survival methods applied to ICU epidemiology.[7]

[1] [1]

Clinical pearls — survival analysis for the viva

Worked example — interpreting an ICU trial's survival output

A fictional but representative septic-shock trial randomises 1000 patients to an early resuscitation strategy versus usual care. The investigators report: [1]

• 90-day mortality: 24.3% (intervention) vs 26.1% (control); HR 0.92 (95% CI 0.75–1.14), p = 0.46.
• Median survival: not reached in either arm (more than half survived to 90 days in both).
• Log-rank test: p = 0.44.
• RMST(90 days): intervention 72.1 days, control 70.4 days; difference 1.7 days (95% CI −1.2 to 4.6). [1]

How to read this, step by step:[2]

1. The HR of 0.92 means at any instant during follow-up the hazard of death in the intervention arm was 8% lower — but the 95% CI (0.75–1.14) comfortably spans 1.0, so the result is consistent with no effect. The p-value of 0.46 confirms this. This is a negative trial.
2. Median survival "not reached" because in both arms >50% were alive at 90 days. This is why the authors reported 90-day mortality as a binary endpoint and RMST as the time-based summary — neither the median nor the mean could be reliably estimated.
3. The log-rank p-value (0.44) tests whether the whole curves differ; it agrees with the Cox HR, reinforcing the negative result. The curves presumably did not separate (and did not cross), so the PH assumption is not in question here.
4. The RMST difference (1.7 days, 95% CI −1.2 to 4.6) gives the absolute, time-based effect: on average, intervention patients lived an estimated 1.7 more of the first 90 days alive — but the CI includes zero, consistent with the negative HR. RMST is reported here precisely because the median was not reached and because it provides an interpretable absolute measure free of the PH assumption.
5. The absolute risk difference is 26.1% − 24.3% = 1.8 percentage points — small, and not statistically significant. The number-needed-to-treat would be ~1/0.018 ≈ 55, but with a CI crossing infinity it is not meaningfully estimable. [1]

This is the pattern of a well-reported negative ICU survival trial: a Kaplan-Meier figure with at-risk tables, an adjusted Cox HR with CI, the log-rank p-value, an absolute measure (RMST or risk difference), and a statement on the PH assumption. Anything less is incomplete.[2]

Worked example — when curves cross (non-proportional hazards)

A trial of an immunomodulatory drug in severe pneumonia reports a hazard ratio of 0.85 (95% CI 0.73–0.99, p = 0.04) for 90-day mortality. The Kaplan-Meier curves, however, cross at day 12: the intervention arm has higher mortality in the first 12 days (early harm) and lower mortality thereafter (late benefit), so that the curves cross. [1]

Why the single HR is misleading:[3]

• The Cox model assumes the hazard ratio is constant, but here the HR is >1 early (harm) and <1 late (benefit). The reported 0.85 is a weighted average of two opposite effects that no single number can capture.
• The p = 0.04 reflects the average, not the biological truth. A reviewer who checks the curves immediately sees the violation.
• Correct reporting: present the crossing curves, split follow-up into early (0–12 days) and late (12–90 days) periods with separate HRs, or report RMST (which captures the net area difference without assuming proportionality). A time-varying coefficient Cox model is the formal solution. [1]

The lesson: always inspect the Kaplan-Meier curves before trusting a hazard ratio. A significant HR from a Cox model with crossing curves is an artefact of a violated assumption.[3]

Key trials and evidence

[1] [2] [3] [4] [5] [6] [7] [8] [1] [1]

Compare — choosing the method for the question

[1] [1] [1]

FlowSteps — conducting and reporting a survival analysis

Red flags

References

1. [1]Clark TG, Bradburn MJ, Love SB, Altman DG Survival analysis part I: basic concepts and first analyses Br J Cancer, 2003.PMID 12865907
2. [2]Bradburn MJ, Clark TG, Love SB, Altman DG Survival analysis part II: multivariate data analysis--an introduction to concepts and methods Br J Cancer, 2003.PMID 12888808
3. [3]Bradburn MJ, Clark TG, Love SB, Altman DG Survival analysis Part III: multivariate data analysis -- choosing a model and assessing its adequacy and fit Br J Cancer, 2003.PMID 12915864
4. [4]Clark TG, Bradburn MJ, Love SB, Altman DG Survival analysis part IV: further concepts and methods in survival analysis Br J Cancer, 2003.PMID 12942105
5. [5]Royston P, Parmar MKB Restricted mean survival time: an alternative to the hazard ratio for the design and analysis of randomized trials with a time-to-event outcome BMC Med Res Methodol, 2013.PMID 24314264
6. [6]Putter H, Fiocco M, Geskus RB Tutorial in biostatistics: competing risks and multi-state models Stat Med, 2007.PMID 17031868
7. [7]Bellani G, Laffey JG, Pham T, et al (LUNG SAFE Investigators) Epidemiology, Patterns of Care, and Mortality for Patients With Acute Respiratory Distress Syndrome in Intensive Care Units in 50 Countries JAMA, 2016.PMID 26903337
8. [8]Mouncey PR, Osborn TM, Power GS, et al (ProMISe Trial Collaboration) Trial of early, goal-directed resuscitation for septic shock N Engl J Med, 2015.PMID 25776532
9. [9]Vincent JL, Moreno R, Takala J, et al The SOFA (Sepsis-related Organ Failure Assessment) score to describe organ dysfunction/failure. On behalf of the Working Group on Sepsis-Related Problems of the European Society of Intensive Care Medicine Intensive Care Med, 1996.PMID 8844239
10. [10]Knaus WA, Draper EA, Wagner DP, Zimmerman JE APACHE II: a severity of disease classification system Crit Care Med, 1985.PMID 3928249