Concept Architecture
Concept
Theoretically, Proportional Hazards Assumption is the fundamental assumption underlying the Cox proportional hazards model that the hazard ratio between any two individuals or groups remains constant over time. Rather than requiring the underlying hazard functions to be identical, the assumption specifies that they are proportional by a constant multiplicative factor. This property allows treatment effects and prognostic factors to be estimated without specifying the baseline hazard function.
Mathematically, the proportional hazards assumption is expressed within the semi-parametric Cox regression framework, where covariates act multiplicatively on the baseline hazard. The resulting hazard ratio is independent of time, enabling regression coefficients to be interpreted as constant relative effects throughout the follow-up period. Violation of the assumption indicates that treatment effects or covariate effects change over time.
In practice, the proportional hazards assumption is assessed using graphical methods and formal statistical tests before interpreting Cox regression results. Common approaches include Schoenfeld residual plots, scaled Schoenfeld residual tests, log-minus-log survival plots and models incorporating time-dependent covariates. The assumption is routinely evaluated in health economic survival analyses because violations may require alternative survival models or time-varying treatment effects.
Purpose
Used to determine whether hazard ratios estimated from a Cox proportional hazards model remain constant over time, thereby supporting valid estimation and interpretation of treatment effects in survival analyses used for health technology assessment and economic evaluation.
Mathematical Formulae
Primary Formula
h(t | X) = h?(t) ? exp(??X)
where the hazard ratio between two individuals is:
HR = exp(?)
which remains constant over time.
Supporting Formulae
Hazard ratio between individuals X? and X?:
HR = h(t | X?) / h(t | X?) = exp(??(X? ? X?))
Schoenfeld residual for event i:
r? = x? ? E(X | t?)
Related Mathematical Methods
- Cox Proportional Hazards Model
- Schoenfeld Residuals
- Scaled Schoenfeld Residuals
- Time-Dependent Covariate Models
- Log-Minus-Log Survival Plots
- Survival Analysis
Example
A survival analysis compares a new oncology treatment with standard care. Cox regression estimates ?? = ?0.357.
Hazard ratio:
HR = exp(?0.357) = 0.70
This indicates that the treatment reduces the hazard of death by approximately 30%, provided the proportional hazards assumption holds throughout follow-up. Schoenfeld residual testing produces p = 0.48, providing no evidence against the assumption.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| EXP | =EXP(B2) | Convert regression coefficient to hazard ratio |
| LN | =LN(C2) | Calculate log hazard ratio |
| IF | =IF(P2>0.05,"Assumption Supported","Possible Violation") | Interpret proportional hazards test |
| SCATTER Chart | Residuals versus time | Examine Schoenfeld residual patterns visually |
| TRENDLINE | Applied to residual plot | Assess systematic changes in residuals over time |
VBA (Optional)
A VBA procedure can automatically generate proportional hazards diagnostic reports by summarising hazard ratios, statistical test results and residual plots across multiple survival models.
Sources
- Cox DR. Regression Models and Life-Tables. Journal of the Royal Statistical Society: Series B. 1972.
- Collett D. Modelling Survival Data in Medical Research.
- Kleinbaum DG, Klein M. Survival Analysis: A Self-Learning Text.
- NICE. Health Technology Evaluation Manual.
- Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation.
Related Concepts (2)
Library
Publications
3
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.
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 →
Frequently Asked Questions (6)
What is the proportional hazards assumption?
The Cox model assumption that the hazard ratio between two groups stays constant over follow-up, even as the absolute hazard rate changes.
Source: Cox 1972
What does it mean for the proportional hazards assumption to fail?
The proportional hazards assumption holds that the ratio of risk between two groups stays the same throughout follow-up, so a single hazard ratio summarises their whole comparison. It fails when that ratio changes over time, for example when survival curves cross or a treatment's advantage grows or fades. When it fails, one hazard ratio no longer describes the difference, and quoting it can mislead, since the groups' relative risk depends on when it is measured. The failure signals that a single-number summary is inadequate. Collett (2015) explains this.
Source: Collett 2015
What does proportional hazards mean?
Proportional hazards means that the hazards for different groups or covariate values are proportional to one another over time, so their ratio is constant even as the underlying hazard rises or falls. For example, if a treatment halves the hazard, it does so at every time, giving a constant hazard ratio of one half. The absolute hazards can vary over time, but their ratio does not. This constant proportionality is what the assumption requires and what makes the hazard ratio a meaningful summary of effect.
Source: Cox 1972
Why does the proportional hazards assumption matter?
The proportional hazards assumption matters because the Cox model's hazard ratios are meaningful only if the ratio is constant over time; if it is not, a single hazard ratio misrepresents an effect that changes, potentially misleading. For instance, a treatment whose benefit grows or wanes over time violates the assumption, so its summarised hazard ratio conceals this. Checking the assumption ensures that the model's summary of effect is valid, and violations are addressed by extending the model to allow time-varying effects.
Source: Kalbfleisch & Prentice 2002
How is the proportional hazards assumption checked?
The proportional hazards assumption is checked by methods including plotting the log cumulative hazards for groups, which should be roughly parallel if hazards are proportional; examining scaled Schoenfeld residuals for a trend against time, which indicates non-proportionality; and testing for interactions between covariates and time. Martingale-based and Schoenfeld residuals are used in these checks. A trend or non-parallel curves signals that the assumption fails, prompting model extensions to accommodate the changing effect.
Source: Therneau, Grambsch & Fleming 1990
What is done if the proportional hazards assumption fails?
If the proportional hazards assumption fails, the model is extended to allow the effect to vary over time, for instance by including an interaction between the covariate and time, stratifying on the offending variable so it is not assumed proportional, or fitting separate models for time periods. Alternative models, such as accelerated failure time or flexible parametric models, may be used. Reporting time-specific effects or restricted mean survival can also convey a changing effect. These approaches represent the non-proportional effect rather than forcing a single hazard ratio.
Source: Cox 1972
Trust Record
Verified by Dr Darrin Baines
British health economist
Professional identity: darrinbaines.org
Verification date: 23 Oct 2025
Content version: 1.0.0
Canonical Identity
- Term code
- HE-EM-SM-068
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