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Cécile Philippoteaux, for the EMEUNET Newsletter SC

How long until a patient relapses? When does a treatment stop being effective? How quickly do patients reach remission after a biologic is introduced? These questions all share a common logic: we care not only about whether an event occurs, but when it occurs. This is the territory of survival analysis, also called time-to-event analysis, and it requires its own set of tools, vocabulary, and cautions. This article provides a survival kit for survival analyses: the vocabulary you need, the methods you will encounter, and two warnings that will save you from the most common and consequential errors.

1. The vocabulary

The concept that makes survival analysis unique is censoring. Not all participants will have experienced the event by the time the study ends: some are still event-free; others are lost to follow-up. These observations are censored, we know the person survived at least until a certain point, but nothing more. Censoring is not a flaw; it is expected. What matters is that it be non-informative: the reason for censoring must not be related to the risk of experiencing the event. If sicker patients drop out more often, bias is introduced.

Table 1 – Vocabulary to survive (survival analysis essentials)

Term	Definition
Survival data / Time-to-event data	Data describing the time elapsed until a predefined event occurs.
Event of interest	The outcome being studied (e.g., death, disease onset, relapse, treatment failure).
Time origin	The starting point from which time is measured (e.g., diagnosis, treatment initiation, study entry).
Survival time	The duration between the time origin and the occurrence of the event (or end of observation).
Censoring	Situation where the exact event time is not observed.
Right censoring	Most common type: occurs when the event has not happened by end of follow-up, or the individual is lost to follow-up.
Time variable	Variable recording the observed duration for each individual.
Status indicator	Binary variable: 1 if the event occurred, 0 if censored.
Survival function S(t)	The probability of not having experienced the event by time t.
Hazard function h(t)	The instantaneous rate at which events occur at time t, given survival up to t.
Median survival time	The time point at which 50% of the population has experienced the event.
Competing event	An event that prevents the occurrence of the event of interest (e.g., death from another cause before an expected relapse).

• Statistical methods
  • Describing survival: the Kaplan–Meier estimator

The Kaplan–Meier (KM) estimator is the cornerstone of survival analysis ¹. It estimates the probability of not having experienced the event at each time point and produces the characteristic step-function curve. Three elements are non-negotiable in any KM analysis:

• Confidence intervals: the curve becomes less reliable as fewer participants remain at risk.
• The number-at-risk table shown below the x-axis, it reveals how many individuals were still under observation at each time point.
• Median survival time with 95% CI: more informative than the mean, which is often not estimable.
  • Comparing groups: the log-rank test

The log-rank test compares survival curves between two or more groups. It produces a p-value but no effect estimates, and it cannot adjust for confounders. When adjustment is needed, a regression model is required.

• Modelling survival: the Cox proportional hazards model

The Cox model is the most widely used survival regression. It estimates the effect of covariates on the hazard and produces a hazard ratio (HR): an HR > 1 indicates a higher event rate in the exposed group; HR < 1 indicates a lower rate.

Its central assumption is proportional hazards: the hazard ratio between groups must remain constant over time. This assumption needs to be verified (Schoenfeld residuals and the log-log plot). When the assumption is violated, alternatives include time-varying covariates, a stratified Cox model, or a parametric Accelerated Failure Time (AFT) model.

Table 2 — Main statistical methods used in survival analysis

Method	Type	Purpose	Key assumption / caveat
Kaplan-Meier estimator	Non-parametric	Estimate and display the survival function over time	Censoring must be non-informative. Always include the number-at-risk table.
Log-rank test	Non-parametric	Compare survival curves between groups, without covariate adjustment	Proportional hazards between groups. Cannot adjust for confounders.
Cox proportional hazards model	Semi-parametric	Estimate covariate effects on the hazard; produce adjusted hazard ratios (HR)	Proportional hazards assumption must be verified (Schoenfeld residuals, log-log plot).
Accelerated Failure Time (AFT) model	Parametric	Model survival time directly; useful when the PH assumption is violated	Requires the correct distributional assumption (e.g., Weibull, log-normal).
Fine & Gray model	Semi-parametric	Analyse time-to-event data in the presence of competing risks	Competing events must be pre-specified. Models the sub-distribution hazard.

• The warnings
  • Immortal time bias

Immortal time bias is one of the most common errors in observational survival studies². It occurs when a period during which the event cannot occur is incorrectly attributed to an exposure group, artificially inflating its apparent survival.

A classic scenario: patients are classified as “treated” if they received a drug after cohort entry. Follow-up begins at cohort entry for everyone. But treated patients must survive long enough to receive the treatment — that interval before initiation is immortal by definition (Figure 1). Counting it as treated person-time makes treatment look more protective than it is.

Figure 1 – Immortal bias (after Lévesque et al. ² )

The solutions are:

• Time-varying exposure in the Cox model, assigning exposure status at each individual time point.
• Landmark analysis: restrict to participants who were event-free at a fixed time point and define exposure status at that moment.

When reading a paper, ask: does the start of follow-up coincide exactly with the definition of exposure? If not, immortal time bias is likely present.

• Competing events

A competing event is any event that prevents or alters the evaluation of the event of interest³. The most familiar example: a patient dies of cardiovascular disease before experiencing the cancer relapse under study. Once dead, relapse is no longer possible.

The error is to treat competing events as ordinary censoring. Censoring a patient who died of another cause in a standard KM analysis assumes they had the same future risk as those still under observation, which is false. The result is a systematic overestimation of the probability of the event of interest.

The correct approach:

• Cumulative incidence function (CIF) instead of 1−KM for descriptive analyses in the presence of competing risks.
• Fine & Gray sub-distribution hazard model for covariate-adjusted analyses ⁴.

Competing risks must be pre-specified, not identified post hoc.

The overall reading path for a survival analysis paper is summarized in Figure 2.

• Take-home message

Three elements will make you a better reader and a better author of survival papers:

• Master the vocabulary, especially censoring and what it assumes.
• Match the method to the data structure and verify its assumptions.
• Actively look for the two most common traps: immortal time bias and unaddressed competing events.

Figure 2 – Flowchart reasoning in survival analyses (made with PowerPoint).

CIF: cumulative incidence function; F&G: Fine & Gray model; TVE: time-varying exposure; KM: Kaplan–Meier; HR: hazard ratio; PH: proportional hazards; CI: confidence interval.

References

1.         Bland JM, Altman DG. Survival probabilities (the Kaplan-Meier method). BMJ. 1998;317(7172):1572. doi:10.1136/bmj.317.7172.1572

2.         Lévesque LE, Hanley JA, Kezouh A, Suissa S. Problem of immortal time bias in cohort studies: example using statins for preventing progression of diabetes. BMJ. 2010;340:b5087. doi:10.1136/bmj.b5087

3.         Lau B, Cole SR, Gange SJ. Competing risk regression models for epidemiologic data. Am J Epidemiol. 2009;170(2):244-256. doi:10.1093/aje/kwp107

4.         Fine JP, Gray RJ. A Proportional Hazards Model for the Subdistribution of a Competing Risk. Journal of the American Statistical Association. 1999;94(446):496-509. doi:10.1080/01621459.1999.10474144