Proportional Hazards Assumption

Proportional Hazards Assumption in Survival Analysis and Economic Models

The proportional hazards assumption holds when a treatment or patient characteristic multiplies the event rate by the same factor at every point of follow-up, so a single hazard ratio describes the relative effect for the whole period. The hazard in each group may rise or fall over time, and only the ratio between groups is held fixed. Many economic models in health technology assessment take one hazard ratio from a trial or a network meta-analysis and apply it to a baseline survival curve over a lifetime horizon. When the ratio in fact changes, as with a delayed or fading treatment effect, that single number misplaces benefit in time and can bias the life years and QALYs behind a cost-effectiveness estimate. The NICE manual (PMG36, section 4.6.22) states that the assumption should always be assessed.

What the assumption says about hazards and survival curves

The hazard rate is the event rate among people still event-free. Cox's 1972 paper took the hazard to be a function of the explanatory variables and regression coefficients multiplied by an arbitrary and unknown function of time, the structure of the Cox proportional hazards model. For a treated group compared with a reference group the assumption can be written in four equivalent ways.

$$h_1(t) = \psi\, h_0(t), \qquad H_1(t) = \psi\, H_0(t), \qquad S_1(t) = S_0(t)^{\psi}, \qquad \ln H_1(t) = \ln \psi + \ln H_0(t)$$

where $t$ is time since the start of follow-up, $h_0(t)$ and $h_1(t)$ are the hazards in the reference and treated groups, $H_0(t)$ and $H_1(t)$ are the corresponding cumulative hazards, $S_0(t)$ and $S_1(t)$ are the survival functions, and $\psi$ is the hazard ratio, a positive constant that does not depend on $t$.

Since $\ln H(t) = \ln\{-\ln S(t)\}$, the last form makes the two log-cumulative hazard curves parallel, a vertical distance $\ln \psi$ apart. Bradburn and colleagues state the core property, that the hazard in any group is a constant multiple of the hazard in any other, so the hazard curves cannot cross. Because $S_1(t) = S_0(t)^{\psi}$ with $\psi$ fixed, one survival curve also stays on the same side of the other throughout, so crossing survival curves are evidence against the assumption.

A constant ratio is different from a constant hazard. NICE DSU Technical Support Document 14 (TSD14) notes that the exponential, Weibull and Gompertz models can be fitted as proportional hazards models, while the log-logistic and log-normal are accelerated failure time models that do not produce a single hazard ratio. Technical Support Document 21 (TSD21) shows that a spline-based flexible parametric model with a treatment covariate in its linear predictor is also a proportional hazards model.

Why appraisal models depend on it

TSD14 explains that applying one hazard ratio to a base survival curve for the entire modelled period requires the assumption, meaning the treatment effect is proportional over time and the curves for each group have a similar shape. In its review of NICE appraisals that extrapolated survival, 19 of 32 made some use of this approach, most often where several comparators were included and patient-level data were not available for all of them. The same structure is common in partitioned survival models.

PMG36 ties evidence synthesis to the same check. Section 4.6.23 states that if proportional hazards hold within the trial and are clinically plausible during extrapolation, hazard ratios may be pooled using standard code for treatment differences, and section 4.6.24 points to the alternatives in TSD21 if the assumption does not hold in some of the studies. The assumption therefore has two parts in an appraisal, one that trial data can test and one, about the extrapolated period, that rests on clinical and biological judgement.

How the assumption is checked

PMG36 (section 4.6.22) prefers log-cumulative hazard plots as advised in TSD14, visual inspection of hazard plots or hazard ratios over time, and interpretation of tests for proportional hazards reported in the original trial publications. Each looks for a relative effect that drifts.

The log-cumulative hazard plot shows $\ln\{-\ln S(t)\}$ against log time for each arm. TSD14 recommends these plots as the first step, since they also show whether hazards look constant, monotonic or more complex. Bradburn and colleagues warn that the reading is subjective and that converging or diverging lines may reflect either a lack of proportionality or an important omitted covariate.

Formal tests usually rest on Schoenfeld residuals. The R survival function cox.zph, citing Grambsch and Therneau, plots an estimate of a time-dependent coefficient $\beta(t)$ that would be a horizontal line under proportional hazards and reports a score test for zero slope. Bradburn and colleagues cite a comparison in which the scaled Schoenfeld residuals test, a linear correlation test and Cox's time-dependent covariate test were the most powerful diagnostics, the last also allowing the non-proportionality to be modelled. With few events a test has little power, so a large p-value is weak reassurance, and no test of trial data shows that the ratio stays constant after follow-up ends.

How the assumption fails in practice

Royston and Parmar report that gross breaches do occur, even to the point of crossing survival curves, citing the IPASS and ICON7 cancer trials, and suggest they may arise from different biological modes of action of the treatments or from differentially responsive subgroups. TSD21 notes that delayed responses to immunotherapy and the existence of long-term survivors have been hypothesised to produce complex hazard functions.

Effects can also fade. In ovarian cancer data analysed by Bradburn and colleagues, the hazard ratio for absence of ascites was about 0.74 at 2 years and 0.91 at 5 years. PMG36 (section 4.6.20) asks for scenarios on the duration of treatment effect, including an effect that stops or diminishes gradually, and any such scenario departs from proportional hazards in the extrapolated period.

Worked example with a delayed treatment effect

All numbers in this example are illustrative. A control arm has a constant hazard of death of 0.4 per year. A new treatment has no effect in its first year and halves the hazard thereafter, so the true hazard ratio is 1 and then 0.5, and the effect is assumed to persist to 10 years.

1. Cumulative hazards. For the control arm $H_0(t) = 0.4t$. For the treated arm $H_1(t) = 0.4t$ up to year 1 and $H_1(t) = 0.4 + 0.2(t-1)$ afterwards.

2. Diagnosis. Under proportional hazards the gap would be the same at every time. Here it is zero up to year 1 and then widens, so the plot shows curves that start together and separate, the pattern of a delayed effect.

3. A single ratio fitted to the end of follow-up. A modeller who summarises the three-year trial with the constant ratio that reproduces three-year survival exactly, $\psi = 0.8 / 1.2 = 0.667$, applies a treated hazard of $\psi \times 0.4 = 0.2667$ per year. Then $S_1(3) = \exp(-0.2667 \times 3) = \exp(-0.8) = 0.449$, matching the true value, but at 1 year it predicts $\exp(-0.2667) = 0.766$ against a true 0.670.

4. Life years within the trial. Restricted mean survival time to 3 years, the area under each curve, is $(1-e^{-1.2}) / 0.4 = 1.747$ years for control. For the true treated curve it is $(1-e^{-0.4}) / 0.4 + e^{-0.4} \times (1-e^{-0.4}) / 0.2 = 0.8242 + 0.6703 \times 1.6484 = 1.929$ years, and under the constant ratio $(1-e^{-0.8}) / 0.2667 = 2.065$ years. The true gain is 0.182 years, while the constant ratio implies 0.318 years, about 75% more.

5. Life years over 10 years. For control, $(1-e^{-4}) / 0.4 = 2.454$ years. For the true treated curve, $(1-e^{-0.4}) / 0.4 + e^{-0.4} \times (1-e^{-1.8}) / 0.2 = 0.8242 + 0.6703 \times 4.1735 = 3.622$ years, and under the constant ratio $(1-e^{-2.667}) / 0.2667 = 3.489$ years. The true gain is 1.168 years, while the constant ratio implies 1.035 years, about 11% less.

6. Interpretation. The summary ratio overstates benefit inside the trial and understates it over the longer horizon, because it moves benefit into the first year, when the treatment does nothing, and caps the later effect at 0.667 instead of 0.5. The direction of the error depends on the horizon and the shape of the true effect, so no general correction exists.

What a single hazard ratio means when hazards are not proportional

Uno and colleagues point out that when the assumption is violated, the quantity estimated by the Cox procedure is not simply an average of the true hazard ratio over time and generally depends on the study-specific follow-up times, so two studies of the same population could give different estimates at any sample size. Royston and Parmar add that under proportional hazards the hazard ratio can be applied to the control survival function to approximate the research arm, and that when the assumption is breached this no longer holds. For a cost-effectiveness model, such a ratio summarises one trial's follow-up and is not a parameter to carry across time horizons or to pool with ratios from trials of different length.

Options when the assumption does not hold

The response depends on the data available and on what the model must extrapolate. Patient-level data allow far more choice than published summaries.

• Separate models for each arm. TSD14 notes that with patient-level data it is unnecessary to rely on the assumption, and that if it seems inappropriate the most sensible course is usually separate parametric survival models of the same type for each arm, letting both shape and scale differ.
• Time-varying hazard ratios. TSD21 relaxes the assumption in flexible parametric models through interactions between treatment and a spline of log time, and advises plotting both arms' extrapolated hazards with the implied hazard ratio. In a Cox model the equivalent is a treatment by time interaction, distinct from a time-varying covariate whose value changes during follow-up. TSD14 also lists piecewise modelling.
• Accelerated failure time models. TSD14 notes that log-logistic and log-normal models, which are accelerated failure time models, express the treatment effect as an acceleration factor on the time scale instead of a hazard ratio.
• Stratification. Bradburn and colleagues describe stratifying on a categorical covariate that shows non-proportionality, which gives no estimated effect for it and so suits only covariates not of primary interest.
• Restricted mean survival time. Royston and Parmar propose the difference in restricted means as an alternative measure of treatment effect, illustrated in designs with proportional and non-proportional hazards.

Common misreadings

Most errors come from treating a property of the trial data as a property of the long-term treatment effect. Four recur in economic models.

1. Reading a constant hazard ratio as constant hazards. The baseline hazard can take any shape.
2. Reading a non-significant test as proof. A test with few events can miss a real departure.
3. Assuming proportionality continues after follow-up. PMG36 makes clinical plausibility during extrapolation a separate condition, and waning scenarios break the assumption by design.
4. Confusing a time-varying effect with time dependency of other inputs. A baseline hazard can change with age or time since diagnosis while the hazard ratio stays constant, and the reverse.

Sources

• Bradburn MJ, Clark TG, Love SB, Altman DG. Survival analysis part II: multivariate data analysis, an introduction to concepts and methods. British Journal of Cancer. 2003;89(3):431-436. https://doi.org/10.1038/sj.bjc.6601119
• Bradburn MJ, Clark TG, Love SB, Altman DG. Survival analysis part III: multivariate data analysis, choosing a model and assessing its adequacy and fit. British Journal of Cancer. 2003;89(4):605-611. https://doi.org/10.1038/sj.bjc.6601120
• Cox DR. Regression models and life-tables. Journal of the Royal Statistical Society Series B (Methodological). 1972;34(2):187-202. https://doi.org/10.1111/j.2517-6161.1972.tb00899.x
• Grambsch PM, Therneau TM. Proportional hazards tests and diagnostics based on weighted residuals. Biometrika. 1994;81(3):515-526. https://doi.org/10.1093/biomet/81.3.515
• Latimer NR. NICE DSU Technical Support Document 14: Survival analysis for economic evaluations alongside clinical trials, extrapolation with patient-level data. Sheffield: Decision Support Unit, ScHARR, University of Sheffield; June 2011 (last updated March 2013). https://www.sheffield.ac.uk/nice-dsu/tsds/survival-analysis
• National Institute for Health and Care Excellence. NICE technology appraisal and highly specialised technologies guidance: the manual (PMG36), sections 4.6.20 and 4.6.22 to 4.6.24. London: NICE; 2022 (last updated 13 October 2026). https://www.nice.org.uk/process/pmg36/chapter/economic-evaluation-2
• Royston P, Parmar MKB. Restricted mean survival time: an alternative to the hazard ratio for the design and analysis of randomized trials with a time-to-event outcome. BMC Medical Research Methodology. 2013;13:152. https://doi.org/10.1186/1471-2288-13-152
• Rutherford MJ, Lambert PC, Sweeting MJ, Pennington B, Crowther MJ, Abrams KR, Latimer NR. NICE DSU Technical Support Document 21: Flexible methods for survival analysis. Sheffield: Decision Support Unit, ScHARR, University of Sheffield; 23 January 2020 (updated version uploaded March 2022). https://www.sheffield.ac.uk/nice-dsu/tsds/flexible-methods-survival-analysis
• Therneau TM. cox.zph: Test the proportional hazards assumption of a Cox regression. survival: Survival Analysis, R package documentation (accessed 28 October 2026). https://rdrr.io/cran/survival/man/cox.zph.html
• Uno H, Claggett B, Tian L, Inoue E, Gallo P, Miyata T, et al. Moving beyond the hazard ratio in quantifying the between-group difference in survival analysis. Journal of Clinical Oncology. 2014;32(22):2380-2385. https://doi.org/10.1200/JCO.2014.55.2208