Concept Architecture
Concept
Theoretically, Time-to-Event Analysis is a branch of statistical methodology that analyses the time until a specified event occurs while appropriately accounting for censored observations. Unlike conventional regression techniques, time-to-event analysis incorporates both the occurrence and timing of events, allowing valid estimation when some individuals have not yet experienced the event by the end of follow-up. The methodology forms the foundation of survival analysis and is extensively applied in medicine, epidemiology, and health economics.
Mathematically, time-to-event analysis is based on probability models describing survival functions, hazard functions, and cumulative hazard functions. Both non-parametric methods, such as the Kaplan?Meier estimator, semi-parametric methods, such as the Cox proportional hazards model, and fully parametric survival models are used depending on the objectives and assumptions of the analysis. Model parameters are typically estimated using maximum likelihood or partial likelihood methods.
In practice, time-to-event analysis is fundamental to health economic evaluation because estimates of survival directly influence lifetime costs, quality-adjusted life years, and cost-effectiveness. Applications include overall survival, progression-free survival, treatment persistence, time to disease recurrence, hospital admission, adverse events, and healthcare utilisation. Results from time-to-event analyses are frequently extrapolated beyond observed trial data to support long-term decision modelling.
Purpose
Used to estimate and compare the timing of events while accounting for censoring, supporting survival estimation, treatment comparison, risk prediction, and long-term health economic modelling.
Mathematical Formulae
Primary Formula
Survival function:
S(t) = P(T > t)
where:
- S(t) = probability of surviving beyond time t
- T = event time
Supporting Formulae
Hazard function:
h(t) = f(t) � S(t)
Cumulative hazard:
H(t) = ??? h(u) du
Relationship between survival and cumulative hazard:
S(t) = exp[?H(t)]
Hazard ratio (Cox model):
HR = exp(?)
Related Mathematical Methods
- Survival Analysis
- Kaplan?Meier Estimation
- Cox Proportional Hazards Model
- Survival Regression
- Parametric Survival Model
- Accelerated Failure Time Model
- Log-Rank Test
- Hazard Ratio
Example
A clinical trial compares two cancer treatments over five years.
At three years:
Treatment A survival probability:
S(3) = 0.72
Treatment B survival probability:
S(3) = 0.58
A Cox proportional hazards model estimates:
HR = 0.68
This indicates that patients receiving Treatment A experience a 32% lower instantaneous risk of death than those receiving Treatment B. These survival estimates are subsequently extrapolated to estimate lifetime quality-adjusted life years and incremental cost-effectiveness.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| EXP | =EXP(-C2) | Calculate survival probabilities from cumulative hazards. |
| LN | =-LN(SurvivalProbability) | Calculate cumulative hazard estimates. |
| IF | =IF(Event=1,1,0) | Code event indicators for survival analyses. |
| SORT | =SORT(TimeRange) | Order observed event times prior to analysis. |
| Solver | Maximise TotalLogLikelihood | Estimate parameters for parametric survival models using maximum likelihood. |
VBA (Optional)
Automate estimation of survival functions, hazard functions, parametric survival models, and extrapolated survival projections for health economic evaluation.
Sources
- Kaplan EL, Meier P. Nonparametric Estimation from Incomplete Observations. Journal of the American Statistical Association. 1958.
- Cox DR. Regression Models and Life-Tables. Journal of the Royal Statistical Society: Series B. 1972.
- Kalbfleisch JD, Prentice RL. The Statistical Analysis of Failure Time Data.
- Collett D. Modelling Survival Data in Medical Research.
- Klein JP, Moeschberger ML. Survival Analysis: Techniques for Censored and Truncated Data.
- Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation.
- NICE Health Technology Evaluation Manual.
Related Concepts (2)
Library
Publications
1
Statistical Analysis of Cost-Effectiveness Data — Willan & Briggs, 1st Edition ed., 2006 (John Wiley & Sons)
A synthesis of statistical methods for analysing cost-effectiveness data, including net-benefit regression, confidence intervals for the ICER, cost-effectiveness acceptability curves, and covariate adjustment. Part of the Wiley Statistics in Practice series.
BookView source →
Frequently Asked Questions (6)
What is time-to-event analysis?
Statistical methods analysing data on the time until a specific event occurs, such as death, while appropriately handling censored observations.
Source: Kalbfleisch & Prentice 2002
What kind of outcome does time-to-event analysis study?
Time-to-event analysis studies the length of time until a specific event occurs, such as death, relapse, or recovery, treating the outcome as a duration rather than a simple yes or no. It needs special methods because some participants have not experienced the event by the end of follow-up, so their times are censored, known only to be longer than the observed period, and these partial observations must be used without being thrown away. Techniques such as Kaplan-Meier and the Cox model do exactly this. Analysing durations with incomplete follow-up is its purpose. Collett (2015) describes these methods.
Source: Collett 2015
Why does time-to-event analysis need special methods?
Time-to-event analysis needs special methods because the data have features that standard approaches cannot handle, chiefly censoring, where the event is not observed for some individuals within the follow-up, so their exact event time is unknown, and the often skewed distribution of times. So time-to-event analysis requires methods that properly incorporate censored observations, using the information that these individuals were event-free up to their censoring time, since ignoring or mishandling censoring would bias the results, which is why specialised survival methods, rather than ordinary analyses of the observed times, are used for time-to-event data.
Source: Kalbfleisch & Prentice 2002
What methods are used in time-to-event analysis?
Methods used in time-to-event analysis include the Kaplan-Meier estimator for the survival curve and the log-rank test for comparing groups, which are non-parametric; the Cox proportional hazards model, a semi-parametric method giving hazard ratios; and parametric survival models assuming a distribution for the times. So time-to-event analysis employs a range of methods, from non-parametric estimation of survival to semi-parametric and parametric regression, all handling censoring, which is why the choice among them depends on the purpose and the assumptions one is willing to make about the distribution of event times.
Source: Kalbfleisch & Prentice 2002
How does time-to-event analysis handle censoring?
Time-to-event analysis handles censoring by using the information that a censored individual was event-free up to their censoring time, rather than excluding them or treating the censoring time as an event. The methods incorporate censored observations appropriately in their estimation, crediting these individuals with survival up to censoring. So time-to-event analysis handles censoring by including censored individuals for the period they were observed, which uses all the available information and avoids the bias that would result from ignoring or mishandling them, a defining feature that distinguishes time-to-event analysis from ordinary analyses of times.
Source: Kalbfleisch & Prentice 2002
When is time-to-event analysis used?
Time-to-event analysis is used whenever the outcome is the time until an event and some individuals may not experience the event during follow-up, producing censoring, as in studies of survival, disease recurrence, or time to device failure. So time-to-event analysis is used for time-to-event outcomes with censoring, common in clinical and epidemiological research, since questions about how long until an event occurs and how factors affect that timing require methods that handle censored data, making it the appropriate framework whenever the interest is in the time to an event rather than only whether the event occurs.
Source: Kalbfleisch & Prentice 2002
Trust Record
Verified by Dr Darrin Baines
British health economist
Professional identity: darrinbaines.org
Verification date: 25 Dec 2025
Content version: 1.0.0
Canonical Identity
- Term code
- HE-ES-SA-217
Stable URI · Machine-readable · Resolvable · CC BY 4.0