Survival Analysis

What survival analysis studies

It uses methods that account for incomplete follow-up and the fact that both whether and when an event occurs matter. This page explains survival and hazard functions, censoring, Kaplan-Meier estimation, regression, competing risks, extrapolation and the use of survival evidence in health-economic models.

The word survival is historical rather than restrictive. The event can be any clearly defined transition, but the time origin, event definition and censoring process must be consistent across the analysis.

Time origin, event and time scale define the estimand

Every survival analysis begins by defining when follow-up starts, which event ends the event-free period and which time scale is used. Changing the origin from diagnosis to treatment initiation or using age instead of time since treatment changes the risk set and interpretation. These choices should be specified before modelling.

Time-to-event data should retain event indicators and follow-up times rather than reducing outcomes to whether an event occurred by one arbitrary date.

Censoring preserves partial follow-up information

Right censoring occurs when a person's event time is known only to exceed the observed follow-up time. The person contributes information to the risk set until censoring. Standard methods commonly assume that, conditional on modelled information, censoring is independent of the future event process.

For person (i), observed data are:

$$ Y_i=\min(T_i,C_i) $$

and:

$$ \delta_i=I(T_i\le C_i) $$

where (T_i) is the event time, (C_i) is the censoring time and (\delta_i=1) indicates an observed event. Censoring is not the event and should not be coded as though it occurred at the censoring time.

Other forms include left censoring, where the event occurred before observation, and interval censoring, where the event is known only to lie between two assessments. Methods designed only for right censoring may be inappropriate for these data.

Delayed entry changes who is at risk

Left truncation, or delayed entry, occurs when people enter observation after the time scale has begun and are observed only if they remain event free until entry. They should join the risk set at their entry time rather than at time zero. Ignoring delayed entry can create immortal-time bias.

For entry time (E_i), person (i) contributes to a risk set at time (t) when:

$$ E_i\le t\le Y_i $$

Delayed entry is different from left censoring. In left truncation, people who experienced the event before entry are absent from the observed sample; in left censoring, they are observed but the exact earlier event time is unknown.

The survival function gives event-free probability

The survival function is the probability that event time (T) exceeds time (t). It begins at or near one and is non-increasing for a standard time-to-first-event outcome. It provides an absolute outcome scale that is directly useful for life-year and QALY calculations.

$$ S(t)=P(T>t) $$

The cumulative distribution function is:

$$ F(t)=P(T\le t)=1-S(t) $$

The median survival time is the smallest time at which estimated survival is at or below 0.5, when the curve reaches that level. If fewer than half the population experience the event during observed follow-up, median survival is not observed and should not be reported through unsupported extrapolation without qualification.

The hazard describes instantaneous event risk

The hazard function describes the instantaneous event rate among people who remain event free just before time (t). It is a rate and can exceed one; it is not the probability of an event occurring at exactly time (t). Hazard shape can reveal whether event intensity rises, falls or changes over time.

$$ h(t)= \lim_{\Delta t\to0} \frac{P(t\le T<t+\Delta t\mid T\ge t)}{\Delta t} $$

The cumulative hazard is:

$$ H(t)=\int_0^t h(u),du $$

Survival and cumulative hazard are related by:

$$ S(t)=\exp[-H(t)] $$

and, where differentiable:

$$ h(t)=-\frac{d}{dt}\log S(t) $$

This relationship allows parametric models to be expressed through survival, hazard or cumulative hazard functions consistently.

Kaplan-Meier estimation uses observed event times

The Kaplan-Meier estimator calculates survival as a product of conditional survival probabilities at observed event times. Censored people remain in the risk set until their censoring time but do not enter the event count. The estimate changes only at event times.

At ordered event times (t_j), let (d_j) be the number of events and (n_j) the number at risk immediately before (t_j). Then:

$$ \widehat S(t)= \prod_{t_j\le t} \left(1-\frac{d_j}{n_j}\right) $$

The estimator assumes that censored individuals have the same future survival prospects as comparable individuals remaining under observation, conditional on the relevant information. Heavy or differential censoring can make later portions of the curve unstable.

A worked Kaplan-Meier example

Suppose five people begin follow-up. One event occurs at time 2, one person is censored at time 3, one event occurs at time 4 and the remaining two people are event free beyond time 4. The risk set is updated immediately before each event.

At time 2, (n_1=5) and (d_1=1):

$$ \widehat S(2)=1\times\left(1-\frac{1}{5}\right)=0.80 $$

After the censoring at time 3, three people remain at risk immediately before the event at time 4. Therefore:

$$ \widehat S(4)= 0.80\times\left(1-\frac{1}{3}\right) =0.533 $$

Estimated survival just after time 4 is approximately 53.3%. Treating the censored person as an event would incorrectly reduce survival at time 3.

Uncertainty grows as the risk set shrinks

Kaplan-Meier uncertainty can be estimated using Greenwood's formula and transformed intervals that respect probability bounds. Later estimates are often less precise because fewer people remain at risk. A survival curve should therefore show numbers at risk and confidence intervals at decision-relevant times.

Greenwood's variance estimate for the log survival component leads to:

$$ \widehat{\operatorname{Var}}[\widehat S(t)] \approx \widehat S(t)^2 \sum_{t_j\le t} \frac{d_j}{n_j(n_j-d_j)} $$

Ordinary symmetric intervals can extend below zero or above one. Log-log transformed intervals are commonly used to preserve valid bounds, but the chosen method should be stated.

The log-rank test compares entire survival curves

The log-rank test compares observed and expected event counts across groups over follow-up. It is most sensitive when the hazard ratio is approximately constant and gives equal weight to event times on its standard scale. A non-significant result does not establish equivalent survival.

For two groups, a test statistic can be formed from total observed events (O_1), expected events (E_1) and variance (V_1):

$$ Z=\frac{O_1-E_1}{\sqrt{V_1}} $$

Under the null and suitable conditions:

$$ Z^2\sim\chi_1^2 $$

If curves cross or treatment effects vary over time, a single log-rank result may conceal clinically meaningful timing differences. Weighted tests answer different alternatives and should not be selected after inspecting the curves without acknowledging multiplicity.

Cox regression estimates relative hazards

The Cox proportional hazards model relates covariates to the hazard without specifying the baseline hazard shape. It estimates hazard ratios through the partial likelihood. The proportional-hazards assumption means that covariate hazard ratios remain constant over time.

$$ h_i(t)=h_0(t)\exp(x_i^{\mathsf T}\beta) $$

For a one-unit change in covariate (x_j), the hazard ratio is:

$$ HR_j=\exp(\beta_j) $$

A hazard ratio below one indicates a lower instantaneous event rate under the chosen coding, not a constant relative increase in survival probability or a direct proportion of events prevented. Absolute benefit depends on baseline hazard and time.

Proportional hazards must be assessed

When hazards are not proportional, one hazard ratio averages time-varying effects in a way that depends on event timing and censoring. It may not provide a stable effect for extrapolation or decision modelling. Assessment should combine graphical, residual and substantive evidence.

Useful checks include:

• Log-minus-log survival plots.
• Schoenfeld residual tests and plots.
• Time-by-treatment interactions.
• Piecewise hazard ratios over prespecified periods.
• Comparison of observed and predicted survival.
• Clinical assessment of delayed effect, waning or treatment switching.

A non-significant proportional-hazards test does not prove proportionality, especially with limited events. If non-proportionality matters, alternatives include time-varying coefficients, flexible parametric models, accelerated failure-time models or restricted mean survival.

Restricted mean survival captures area under the curve

Restricted mean survival time, or RMST, is the expected event-free time up to a prespecified horizon (\tau). It is the area under the survival curve and does not require proportional hazards. The horizon should be clinically meaningful and supported by follow-up for a direct empirical comparison.

$$ RMST(\tau)= \int_0^{\tau}S(t),dt $$

The treatment contrast is:

$$ \Delta RMST(\tau)= \int_0^{\tau} \left[S_1(t)-S_0(t)\right]dt $$

RMST difference is expressed in units of time. It changes with (\tau), so analyses should not choose the horizon solely because it produces a favourable result.

Parametric models support smooth estimation and extrapolation

Parametric survival models specify a probability distribution for event times. They can produce smooth hazards, individual predictions and survival beyond observed follow-up. Their extrapolations depend strongly on the selected distribution and covariate structure.

For the exponential model:

$$ S(t)=e^{-\lambda t} $$

and:

$$ h(t)=\lambda $$

For the Weibull model with scale (\lambda) and shape (p):

$$ S(t)=\exp(-\lambda t^p) $$

$$ h(t)=\lambda p t^{p-1} $$

The direction of scale parameterisation differs across software, so formulas and outputs should be reconciled before transfer into an economic model.

Flexible parametric models represent complex hazards

Spline-based and other flexible parametric models can represent changing hazards without selecting one simple distribution. They may fit observed data better, but extrapolation can become unstable or implausible if tail behaviour is weakly constrained. Knot number and placement should be justified and tested.

Flexible models may be specified on log cumulative hazard, log hazard or log odds of survival scales. Model fit within observed follow-up is not sufficient for long-term use; extrapolated hazards and survival should be inspected against clinical and external evidence.

Extrapolation is central to economic evaluation

Trials often end before the economic model horizon, requiring survival beyond observed follow-up. Extrapolation can drive incremental life-years, QALYs and costs. Statistical fit is necessary but not sufficient because several models can fit the observed period and diverge substantially later.

A robust extrapolation process should examine:

• Visual fit to observed survival and hazards.
• Information criteria and residual diagnostics.
• Proportional-hazards or accelerated-failure-time plausibility.
• External data and background mortality.
• Clinical plausibility of long-term hazards and treatment effects.
• Alternative distributions, cure structures or waning assumptions.
• The proportion of total incremental outcome arising after observed follow-up.

Model averaging or scenario analysis can represent structural uncertainty when several extrapolations remain credible.

Background mortality constrains long-term survival

Long-term all-cause mortality for a patient population should generally not be lower than credible population mortality after accounting for selection and treatment effects. Relative survival or excess-hazard models can separate disease-related mortality from expected background mortality. This can support realistic convergence as disease-specific excess risk changes.

An additive excess-hazard model is:

$$ h_{all}(t)=h_{pop}(a+t,z)+h_{excess}(t) $$

where (h_{pop}) depends on attained age and relevant population characteristics (z). Life-table matching should use appropriate sex, age, calendar period and jurisdiction when available.

Competing risks require event-specific quantities

A competing event prevents the event of interest from occurring or fundamentally changes its probability. For example, death can prevent hospital discharge or disease progression. Treating a competing event as ordinary independent censoring overestimates the cumulative incidence of the event of interest.

For cause-specific hazard (h_k(t)), cumulative incidence for cause (k) is:

$$ F_k(t)= \int_0^t S(u)h_k(u),du $$

where overall event-free survival is:

$$ S(t)= \exp\left[-\sum_j H_j(t)\right] $$

Cause-specific hazard models and subdistribution hazard models answer different questions. The method should match whether the goal is etiological association, absolute risk prediction or policy impact.

Multi-state models represent several transitions

When people can move among several clinical states, a multi-state model estimates transition-specific hazards and state occupancy over time. This can distinguish progression, treatment changes, recovery and death rather than reducing the pathway to one event. Transition structure and history dependence should reflect the clinical process.

For transition from state (r) to state (s), the transition hazard is:

$$ h_{rs}(t)= \lim_{\Delta t\to0} \frac{P[X(t+\Delta t)=s\mid X(t)=r]} {\Delta t} $$

A Markov multi-state model assumes future transitions depend on current state and modelled covariates but not on earlier history. Semi-Markov or clock-reset models allow risk to depend on time since entering the current state.

Recurrent events need methods beyond time to first event

Some outcomes can occur repeatedly, such as hospital admissions or relapses. Analysing only the first event discards later burden and may not match the economic question. Recurrent-event methods must account for within-person dependence, event order and the effect of death.

Approaches include event-rate models, gap-time or total-time models, frailty models, mean cumulative function methods and joint recurrent-event and mortality models. The estimand should specify whether it concerns event rate, cumulative event count, time to a particular recurrence or burden while alive.

Treatment switching can bias survival comparisons

Participants may switch from assigned treatment to another therapy, particularly after progression. Intention-to-treat analysis preserves randomisation but estimates the effect of treatment policy under observed switching rather than the effect of continuous receipt of the assigned treatment. Adjusted analyses rely on additional causal assumptions.

Methods can include inverse-probability weighting, rank-preserving structural failure-time models and two-stage methods. Each requires assumptions about measured confounding, treatment-effect structure or switching mechanisms. Adjusted and intention-to-treat estimates should be labelled according to their estimands rather than treated as interchangeable.

Survival curves become economic-model inputs

Survival analysis supplies probabilities, transition hazards and expected time in health states for economic models. The transformation should preserve effect scale, timing and covariance. Applying a constant hazard ratio to an unrelated baseline curve can misrepresent absolute survival, especially under non-proportional hazards.

Expected life-years over horizon (T) are:

$$ LY(T)=\int_0^T S(t),dt $$

Discounted life-years are:

$$ LY_d(T)= \int_0^T S(t)d(t),dt $$

Quality-adjusted survival is:

$$ QALY(T)= \int_0^T S(t)u(t)d_Q(t),dt $$

where (u(t)) is expected utility conditional on survival if formulated that way. The model must avoid multiplying by survival twice when utility data already include death as zero.

Common analysis and interpretation errors

Survival analysis can produce polished curves and hazard ratios while concealing incorrect risk sets, censoring assumptions or extrapolations. The following errors can materially affect clinical and economic conclusions.

• Treating censored observations as events or exclusions discards valid partial follow-up or misclassifies outcomes.
• Ignoring delayed entry can create immortal-time bias.
• Interpreting a hazard ratio as a risk ratio confuses instantaneous rates with cumulative probabilities.
• Assuming a hazard ratio below one means proportional survival gain at every time ignores baseline hazard and timing.
• Using median survival when the curve has not reached 0.5 creates an unsupported estimate.
• Treating competing events as independent censoring overstates cumulative incidence.
• Selecting an extrapolation only by best in-sample fit ignores long-term plausibility.
• Using one parametric scale convention in software and another in the economic model produces incorrect survival.
• Ignoring treatment switching or informative loss to follow-up can bias comparative effects.
• Reporting only one extrapolation without structural sensitivity analysis understates decision uncertainty.

A practical validation sequence

Validation should begin with person-level time and event records, then reconcile non-parametric estimates before fitting regression or extrapolation models. Economic-model outputs should trace back to the same survival functions and time scale. The following sequence provides an auditable minimum.

1. Define the time origin, event, time scale, competing events and censoring rules.
2. Check every record for impossible times, duplicate events and inconsistent status coding.
3. Represent delayed entry and interval censoring correctly when they occur.
4. Reproduce numbers at risk, event counts and selected Kaplan-Meier steps independently.
5. Plot survival, cumulative hazard and relevant diagnostics before choosing a regression model.
6. Assess proportional hazards and other model assumptions using statistical and clinical evidence.
7. Fit several clinically plausible parametric or flexible models when extrapolation is required.
8. Compare extrapolations with external data, background mortality and expert expectations.
9. Model competing risks, recurrent events and switching according to the target estimand.
10. Reconcile survival areas, life-years, QALYs and transition probabilities in the economic model.
11. Propagate joint parameter and structural uncertainty rather than varying one survival parameter at a time only.
12. Report how much incremental outcome arises beyond observed follow-up and whether that tail changes the decision.

Survival analysis is most useful when event timing, incomplete follow-up and long-term uncertainty remain explicit. Reliable economic conclusions require both statistically credible survival estimates and transparent assumptions about what happens after the observed data end.