Concept Architecture
Concept
Theoretically, Survival Regression is a class of statistical models used to investigate the relationship between explanatory variables and time-to-event outcomes while accounting for censoring. Unlike ordinary regression models, survival regression explicitly models the timing of events such as death, disease progression, treatment discontinuation, or hospital discharge. The methodology is founded on survival analysis and enables estimation of covariate effects on survival time, hazard functions, or event rates.
Mathematically, survival regression may be implemented using semi-parametric models, such as the Cox proportional hazards model, or fully parametric models based on specified survival distributions including Weibull, Exponential, Gompertz, Log-normal, Log-logistic, and Generalised Gamma distributions. Parameters are typically estimated using maximum likelihood estimation for parametric models or partial likelihood estimation for Cox regression. Estimated coefficients are commonly interpreted through hazard ratios or acceleration factors, depending on the modelling framework.
In practice, survival regression is widely used in health economics to analyse overall survival, progression-free survival, time to hospitalisation, treatment persistence, and other time-to-event outcomes. Parametric survival regression is particularly important in health technology assessment because fitted survival functions are extrapolated beyond trial follow-up to estimate long-term costs, quality-adjusted life years, and cost-effectiveness.
Purpose
Used to estimate the effects of explanatory variables on time-to-event outcomes, accommodate censored observations, extrapolate long-term survival, and support decision modelling in health economic evaluations.
Mathematical Formulae
Primary Formula
General proportional hazards model:
h(t|X) = h?(t) ? exp(X?)
where:
- h(t|X) = hazard at time t
- h?(t) = baseline hazard
- X = covariate vector
- ? = regression coefficients
Supporting Formulae
Hazard Ratio:
HR = e?
Survival function:
S(t) = exp[?H(t)]
where:
H(t) = ??? h(u) du
Parametric Weibull hazard:
h(t) = ??t^(??1)
Log-likelihood (parametric models):
ln(L) = ?�?ln[h(t?)] + ?ln[S(t?)]
where:
- �? = event indicator
Related Mathematical Methods
- Cox Proportional Hazards Model
- Accelerated Failure Time Model
- Parametric Survival Model
- Weibull Regression
- Exponential Regression
- Gompertz Regression
- Maximum Likelihood Estimation
- Kaplan?Meier Estimation
- Hazard Ratio
Example
A health economic evaluation compares survival following two cancer treatments.
A Weibull survival regression model estimates:
?Treatment = ?0.42
The hazard ratio is:
HR = e^(?0.42)
= 0.66
This indicates that the treatment reduces the instantaneous risk of death by approximately 34% relative to the comparator. The fitted Weibull model is subsequently extrapolated beyond the trial period to estimate lifetime survival and quality-adjusted life years for cost-effectiveness analysis.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| EXP | =EXP(B2) | Convert regression coefficients into hazard ratios. |
| LN | =LN(A2) | Transform survival times for parametric survival modelling. |
| POWER | =Lambda*Gamma*POWER(Time,Gamma-1) | Calculate Weibull hazard functions. |
| SUMPRODUCT | =SUMPRODUCT(LogLikelihoodRange) | Calculate the total log-likelihood for parametric survival models. |
| Solver | Maximise TotalLogLikelihood by changing ParameterCells | Estimate survival model parameters using maximum likelihood. |
VBA (Optional)
Automate estimation of parametric survival models, hazard ratios, survival projections, and extrapolated outcomes for health economic evaluations.
Sources
- 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.
- Klein JP, Moeschberger ML. Survival Analysis: Techniques for Censored and Truncated Data.
- Collett D. Modelling Survival Data in Medical Research.
- Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation.
- Latimer NR. Survival Analysis for Economic Evaluations Alongside Clinical Trials?Extrapolation with Patient-Level Data. Medical Decision Making. 2013.
- NICE Health Technology Evaluation Manual.
Related Concepts (2)
Library
Publications
1
Bayesian Methods in Health Economics — Gianluca Baio, 1st Edition ed., 2012 (Chapman & Hall / CRC Press)
An overview of Bayesian statistical methods for the analysis of health economic data, covering economic evaluation concepts, statistical cost-effectiveness analysis, Bayesian computation and MCMC, and applied health economic evaluation.
BookView source →
Frequently Asked Questions (6)
What is survival regression?
A modelling approach estimating the relationship between predictor variables and a time-to-event outcome, including the Cox model and parametric survival models.
Source: Cox 1972
What does survival regression relate to a time-to-event outcome?
Survival regression relates predictor variables to a time-to-event outcome, estimating how characteristics such as age or treatment affect the rate at which an event occurs over time. It comes in forms including the Cox model, which leaves the underlying course of risk unspecified while estimating how predictors multiply it, and fully parametric models that assume a particular shape for that course. Throughout, it accommodates censored observations from incomplete follow-up. Linking predictors to the timing of events is its purpose. Collett (2015) describes this approach.
Source: Collett 2015
How does survival regression work?
Survival regression works by relating the predictors to the outcome through the hazard or the survival time: the Cox model expresses the hazard as a baseline hazard multiplied by a function of the covariates, giving hazard ratios without specifying the baseline, while parametric models assume a distribution and relate the covariates to its parameters. Censoring is incorporated in the estimation. So survival regression works by modelling the effect of covariates on the hazard or survival time, using methods that handle censored observations, which yields estimates such as hazard ratios or time ratios quantifying how the predictors influence the risk or timing of the event.
Source: Cox 1972
What is the Cox model in survival regression?
The Cox proportional hazards model is the most common form of survival regression, a semi-parametric method that models the hazard of an event as a baseline hazard multiplied by an exponential function of the covariates, yielding hazard ratios for the covariate effects while leaving the baseline hazard unspecified. So the Cox model is central to survival regression, estimating how covariates affect the hazard through hazard ratios without assuming a form for the baseline hazard, which makes it flexible and widely applicable, and its semi-parametric nature, combining parametric covariate effects with an unspecified baseline, is why it is the standard tool for relating predictors to time-to-event outcomes.
Source: Cox 1972
How are the results of survival regression interpreted?
The results of survival regression are interpreted according to the model: in the Cox model, exponentiated coefficients give hazard ratios, where a value above one indicates the predictor increases the hazard, or rate, of the event and below one decreases it, holding other predictors constant; parametric models may give hazard or time ratios. So survival regression results are interpreted as the effect of each predictor on the risk or timing of the event, most commonly as hazard ratios, which convey the relative rate of the event associated with the predictor, adjusted for the others, providing an interpretable summary of how factors influence survival.
Source: Cox 1972
What assumptions does survival regression make?
Survival regression makes assumptions depending on the model: the Cox model assumes proportional hazards, that the ratio of hazards between groups is constant over time, and that censoring is non-informative; parametric models also assume a specific distribution for the survival times. So survival regression rests on assumptions such as proportional hazards for the Cox model and a correct distribution for parametric models, along with non-informative censoring, and its validity depends on these holding, which is why the proportional hazards assumption is checked and the distributional assumption assessed, since violations can bias the estimates or require methods that relax the assumptions.
Source: Collett 2015
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
- Persistent URI
- https://healtheconomics.wiki/concept/survival-regression
- Term code
- HE-ES-SA-210
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