Concept Architecture
Concept
Theoretically, the Cox Proportional Hazards Model is a semi-parametric regression model used to evaluate the association between explanatory variables and the hazard of an event occurring over time. It is founded on survival analysis and assumes that hazard ratios between individuals remain constant throughout follow-up. In health economics, the Cox proportional hazards model is widely used to estimate treatment effects, identify prognostic factors and generate survival inputs for economic evaluation.
Mathematically, the Cox proportional hazards model expresses the hazard function as the product of an unspecified baseline hazard and an exponential function of covariates. Regression coefficients are estimated using partial likelihood rather than specifying the baseline hazard function, allowing estimation of hazard ratios without assuming a particular survival distribution.
In practice, the Cox proportional hazards model is estimated using patient-level survival data from clinical trials, registries and observational studies. The resulting hazard ratios are incorporated into survival extrapolation, relative effectiveness estimation and health economic decision models, provided the proportional hazards assumption is adequately satisfied.
Purpose
Used to estimate the effect of explanatory variables on survival, quantify hazard ratios, identify prognostic factors and generate survival estimates for health economic evaluations and health technology assessments.
Mathematical Formulae
Primary Formula
h(t|X) = h?(t) ? exp(??X? + ??X? + ? + ??X?)
Supporting Formulae
Hazard ratio:
HR = exp(?)
Partial likelihood:
L(?) = ? exp(?X?) � ???R(t?) exp(?X?)
where:
- h(t|X) = hazard for an individual with covariates X
- h?(t) = baseline hazard
- ? = regression coefficient
- HR = hazard ratio
- R(t?) = risk set at event time t?
Related Mathematical Methods
- Partial likelihood estimation
- Maximum likelihood estimation
- Kaplan?Meier estimation
- Log-rank test
- Schoenfeld residuals
- Parametric survival modelling
Example
A Cox proportional hazards model estimates the treatment coefficient as ? = ?0.357.
Hazard ratio:
HR = exp(?0.357) = 0.70
Patients receiving the intervention have a 30% lower instantaneous risk of death than those receiving the comparator, assuming proportional hazards.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| EXP | =EXP(B2) | Convert regression coefficients into hazard ratios. |
| LN | =LN(B2) | Calculate regression coefficients from hazard ratios. |
| IF | =IF(C2<0.05,"Significant","Not Significant") | Summarise statistical significance of covariates. |
VBA (Optional)
Automate import of Cox regression outputs and calculate hazard ratios, confidence intervals and summary tables for health economic models.
Sources
- Cox DR. Regression Models and Life-Tables. Journal of the Royal Statistical Society: Series B. 1972;34(2):187?220.
- Collett D. Modelling Survival Data in Medical Research. CRC Press.
- Klein JP, Moeschberger ML. Survival Analysis: Techniques for Censored and Truncated Data. Springer.
- Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation. Oxford University Press.
- NICE. Health Technology Evaluation Manual.
Related Concepts (2)
Library
Publications
5
NICE DSU Technical Support Document 14: Survival analysis for economic evaluations alongside clinical trials – extrapolation with patient-level data — Nicholas R. Latimer, TSD 14 ed., 2013 (NICE Decision Support Unit (University of Sheffield))
The reference guidance on survival analysis for economic evaluation: fitting standard parametric models (exponential, Weibull, Gompertz, log-logistic, log-normal) to censored trial data and extrapolating to estimate lifetime survival benefit, with a process guide for model selection and justification.
NICE DSU Technical Support Document 6: Embedding Evidence Synthesis in Probabilistic Cost-Effectiveness Analysis — Software Choices — Dias, Welton, Sutton & Ades, TSD 6 ed., 2011 (NICE Decision Support Unit (University of Sheffield))
Guidance on the software options and practical steps for embedding a Bayesian evidence synthesis directly within a probabilistic cost-effectiveness model so that parameter uncertainty is propagated consistently.
Cost-Effectiveness Analysis in R Using a Multi-State Modeling Survival Analysis Framework: A Tutorial — Williams, Lewsey, Briggs & Mackay, Vol. 37, No. 4 ed., 2017 (Medical Decision Making)
A tutorial on building cost-effectiveness models in R using a multi-state survival-analysis framework, bridging patient-level survival data and decision modelling — a key reference for survival-based economic models in R.
Journal ArticleView source →Survival Analysis: A Self-Learning Text — David G. Kleinbaum & Mitchel Klein, 3rd Edition ed., 2012 (Springer)
A practical introduction to survival data, censoring, Kaplan-Meier methods, Cox regression, proportional hazards and model interpretation.
BookView source →Modelling Survival Data in Medical Research — David Collett, 3rd Edition ed., 2015 (Chapman & Hall / CRC Press)
A medical-research guide to survival modeling, including censored outcomes, parametric models, Cox regression, diagnostics and practical interpretation.
BookView source →
Tools & Resources
1
survHE — Survival Analysis for Health Economic Evaluation (R package) — Gianluca Baio, R package ed., 2023 (CRAN)
An R package for fitting and comparing parametric survival models for health economic evaluation, including Bayesian estimation, and for extrapolating time-to-event data to inform cost-effectiveness models.
Software (R package)View source →
Frequently Asked Questions (6)
What is the Cox proportional hazards model?
A widely used survival model estimating covariate effects, such as treatment, on the hazard rate while leaving the baseline hazard function unspecified.
Source: Cox 1972
Who developed the Cox proportional hazards model?
The model was introduced by the statistician David Cox in the early 1970s and became one of the most widely used methods in medical research. Its innovation was to estimate how covariates such as treatment affect the hazard without having to specify the shape of the underlying hazard over time, which earlier methods required. This semi-parametric approach made survival analysis far more flexible and applicable. The method remains a standard tool for analysing time-to-event data. Cox (1972) set it out.
Source: Cox 1972
How does the Cox model work?
The Cox model expresses the hazard for an individual as a baseline hazard, common to all, multiplied by an exponential function of their covariates, so each covariate multiplies the hazard by a factor. The baseline hazard is left unspecified, and the covariate effects are estimated by partial likelihood, which uses the order of events without requiring the baseline hazard's form. This yields hazard ratios quantifying each covariate's effect, allowing the influence of factors such as treatment on the event rate to be estimated.
Source: Cox 1972
What is the proportional hazards assumption?
The proportional hazards assumption, central to the Cox model, holds that the effect of a covariate multiplies the hazard by a constant factor that does not change over time, so the ratio of hazards between individuals with different covariate values is constant. This means a covariate's hazard ratio is the same at all times. If the assumption fails, for instance if a treatment's effect diminishes over time, the constant hazard ratio misrepresents the effect, so the assumption is checked, and violations are addressed by extending the model.
Source: Cox 1972
What is a hazard ratio in the Cox model?
A hazard ratio in the Cox model is the factor by which a covariate multiplies the hazard, comparing the hazard for individuals differing in that covariate. A hazard ratio above one indicates the covariate increases the hazard, below one that it decreases it, and one that it has no effect. For example, a treatment with a hazard ratio of 0.7 reduces the hazard to 70 per cent of the comparator's. The hazard ratio is assumed constant over time under proportional hazards, summarising the covariate's effect.
Source: Kalbfleisch & Prentice 2002
Why is the Cox model widely used?
The Cox model is widely used because it estimates covariate effects on survival without requiring the baseline hazard to be specified, making it flexible and applicable to many settings where the form of the hazard over time is unknown. It handles censoring, produces interpretable hazard ratios, and accommodates multiple covariates. Its semi-parametric nature avoids strong distributional assumptions about the baseline hazard while still quantifying how factors affect risk. These features make it the standard regression method in survival analysis across medical research.
Source: Cox 1972
Trust Record
Verified by Dr Darrin Baines
British health economist
Professional identity: darrinbaines.org
Verification date: 20 Oct 2025
Content version: 1.0.0
Canonical Identity
- Term code
- HE-EM-SM-008
Stable URI · Machine-readable · Resolvable · CC BY 4.0