Log-minus-log survival plots
Plot ln(−ln S(t)), the log cumulative hazard, for each group against log time.
Look for: roughly parallel curves. Converging, diverging or crossing curves signal non-proportionality.
Proportional, not identical
The assumption does not require the two groups’ hazard functions to be the same. It requires them to be proportional by a constant multiplicative factor. Covariates act multiplicatively on a baseline hazard that is never specified:
h(t | X) = h0(t) · exp(βTX)
For a treatment indicator the hazard ratio is HR = exp(β), and it does not depend on time. That is why one coefficient can summarise the whole comparison. If the ratio changes over follow-up, the treatment or covariate effect is changing over time and a single HR misleads.
Shape the hazardsChoose a baseline hazard and a pattern of treatment effect.
Read the diagnosticsCompare the hazard ratio, log-minus-log curves and residual pattern.
Decide what followsSee whether one HR is adequate or the model must allow the effect to vary.
Start with a pattern of treatment effect
Each case loads a fictional two-arm comparison. Only the first satisfies the assumption. The others show common ways it fails.
What the four charts show
Control (solid) and treatment (dashed) are distinguished by line style as well as colour. The illustration is noise-free and uncensored over a 10-year horizon, so these are the true curves, not estimates from data.
- Control (baseline)
- Treatment
- Hazard ratio
Hazard rates
Absolute hazards may rise or fall freely.
Hazard ratio over time
Flat line = proportional hazards. Log scale.
Survival curves
S(t) = exp(−H(t)), where H is the cumulative hazard.
Log-minus-log plot
ln(−ln S(t)) against ln(t): parallel curves under proportional hazards.
What the diagnostics show
| Measure | 1 year | 3 years | 5 years | 8 years |
|---|
Log hazard ratio per unit of X
e.g. scaled Schoenfeld test
1 = treatment
0 = standard care
HR = h(t | X1) / h(t | X2) = exp(β·(X1 − X2)). The baseline hazard h0(t) cancels, which is why it never needs to be specified.
How the assumption is assessed and handled
The page lists graphical methods and formal tests used before interpreting Cox regression results, and the options available when the assumption fails.
Schoenfeld residuals
For the event at time ti: ri = xi − E(X | ti), the observed covariate minus its expected value among those at risk.
Look for: a trend in scaled residuals against time. A systematic trend indicates non-proportionality.
Formal tests
Scaled Schoenfeld residual tests and tests for interactions between covariates and time.
Look for: evidence against the assumption. A large p-value gives no evidence against it; read it alongside the plots.
Time-dependent covariate models
Include an interaction between the covariate and time so the effect may vary across follow-up.
Look for: whether the time-interaction term changes the estimated effect materially.
If the assumption fails
Extend the model rather than forcing one HR: covariate-by-time interactions, stratification on the offending variable, or separate models for time periods.
Alternatives: accelerated failure time or flexible parametric models.
Reporting a changing effect
Report time-specific effects or restricted mean survival so the summary reflects how the effect changes.
Why it matters in HTA: violations may require alternative survival models or time-varying treatment effects.
The core rule
h(t | X) = h0(t) · exp(βTX), so HR = exp(β) is the same at every time. The absolute hazards can vary over time, but their ratio does not.
Method, safeguards and limitations
This demonstrator illustrates the structure of the assumption. It does not fit a Cox model, run a statistical test or analyse data. The curves are generated, not estimated.
- Illustrative assumptions: the baseline is a Weibull-shaped hazard with cumulative hazard H0(t) = (t/5)k, k = 0.7, 1 or 1.5, over a 10-year horizon. Non-proportional patterns use an exponential approach in log HR with the chosen time constant. Treatment cumulative hazard is integrated numerically (1,000 steps). No censoring, sampling error or competing risks are represented.
- Residual scatter: the points are generated from a fixed random seed around the true log hazard ratio to show the visual signal a trend produces. They are not real residuals and carry no p-value.
- Diagnostics in practice: real assessment combines plots and tests on trial data. A non-significant test provides no evidence against the assumption; it does not by itself demonstrate that hazards are proportional.
- Scope: educational only. No scenario represents a real treatment or makes any claim about clinical effectiveness.
Sources listed on the concept page: 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: Cox Proportional Hazards, Hazard Ratio, Survival Analysis.