What is the proportional hazards assumption?

The proportional hazards assumption in survival analysis means that the hazards (the event rates) of two groups or individuals are proportional to each other: one is a constant multiple of the other throughout follow-up.

Specifically, it assumes that the hazard ratio, which represents the relative rate of an event occurring between two groups or individuals, is constant over time.

In practice, the proportional hazards assumption means that by multiplying the rate in one group by a constant we can get the rate in another group, and this constant remains the same across follow-up. The constant is the hazard ratio.

For example, if we compare the mortality rate between non-smokers and smokers over 10 years of follow-up, and assume proportional hazards with a hazard ratio of 1.7, the mortality rate in smokers will be 1.7 times that in non-smokers at the start of follow-up, at 6 months, 5 years, 10 years, and at every time point in between.

How to check it

No single test settles it. Look at it in more than one way, and judge how much any departure matters for the question being asked.

• Log–log plot. Plot log(−log S(t)) against log time for each group, using Kaplan–Meier estimates of S(t). Under proportional hazards the curves are parallel, a constant vertical distance apart. Converging, diverging or crossing curves suggest the hazard ratio changes over time. The plot needs a categorical covariate and is noisy early on where there are few events. Drawn from Kaplan–Meier estimates, it does not adjust for other covariates; Stata’s stphplot can, with its adjustfor() option, by basing the curves on Cox models instead. Because it shows cumulative hazards, it reacts to a change in the hazard ratio later and more gently than the hazard ratio itself.
• Schoenfeld residuals. After fitting a Cox model, plot each covariate’s scaled Schoenfeld residuals against time, or a transformation of it such as log time. Under proportional hazards they scatter around a flat line, and a smoothed trend shows how the effect changes with time. The standard software also tests for a non-zero slope (Grambsch and Therneau, 1994), and the result depends on the time scale used: Stata’s estat phtest uses analysis time by default, and R’s cox.zph a Kaplan–Meier scale.
• A time-varying effect. Add an interaction between the covariate and a function of time, such as log time or a spline, and look at how the estimated hazard ratio moves over follow-up. In a Cox model the interaction must be fitted as a time-varying covariate, for example with tvc() in Stata or tt() in R, and not by multiplying the covariate by each person’s own follow-up time, which uses the outcome as a covariate and biases the result. This shows the size of any departure, not only whether there is one.
• A model that doesn’t assume it. Fit a flexible parametric model with a time-varying effect, and compare its hazard ratio over time, and its predicted survival, with those from the proportional hazards model.

With large samples a test can flag departures too small to matter, and with small samples it can miss ones that do. The plots and the estimated hazard ratio over time usually say more than the p-value. A departure can also come from a missing covariate or the wrong functional form for another covariate, not only from an effect that truly changes over time, so check the rest of the model too.

Stensrud and Hernán argue that hazards are rarely proportional in medical studies, conclude that testing the assumption is unnecessary, and suggest reporting differences in survival instead (JAMA 2020;323:1401–1402). Sjölander and Dickman respond that the same is true of any model assumption, and that what matters is whether the assumptions hold approximately, so that the model gives reasonably accurate inference, which is what these checks assess (Am J Epidemiol 2024;193:926–927).

What to do when it doesn’t hold

• Model the change. Estimate the hazard ratio as a function of time, with a time-varying effect in a Cox model (fitted as a time-varying covariate, as above) or in a flexible parametric model, and report how it changes.
• Split follow-up. If a few periods describe the change well, estimate a separate hazard ratio in each, with the cut-points chosen in advance. Hazard ratios in later periods compare the people who survived to them, so in a trial they are no longer randomised comparisons.
• Change the summary. Report differences in survival at chosen times, or in restricted mean survival time, estimated from Kaplan–Meier curves or from a model that does not assume proportional hazards. Both mean the same thing whether or not hazards are proportional. In a trial, choose the summary and the time points in advance, not after seeing whether hazards look proportional.
• Stratify on a categorical covariate whose effect you don’t need to estimate: each stratum gets its own baseline hazard, so no proportional-hazards assumption is made for that covariate. The other covariates are still assumed to have proportional effects, and the same effects in every stratum.

A single hazard ratio reported when the effect changes over time is roughly a weighted average of the changing log hazard ratio over follow-up, and the weights depend on who is still at risk and on how long, and how completely, the study followed people (Struthers and Kalbfleisch, 1986; Xu and O’Quigley, 2000). It can hide a benefit that appears late, or one that wanes. What a hazard ratio means when hazards are not proportional explains that average, and the hazard ratio tool shows it.

Non-proportional hazards are often described as time-dependent (or time-varying) effects: the hazard ratio itself changes over time. This is not to be confused with time-dependent (or time-varying) covariates, where it is the value of the covariate that changes.

What to report

• How the assumption was checked: which plots or residuals, and, for any test, which time scale.
• For the main comparison, the estimated hazard ratio over time with a confidence band, if there is any sign that it changes, so that readers can judge the size of the departure.
• What was done if hazards were not proportional, and whether that approach was chosen before the data were seen.
• Survival curves by group, with the numbers at risk, and an absolute summary alongside any hazard ratio, such as the difference in survival at a stated time or in restricted mean survival time.

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MethodSurvival analysis