Introduction

Time-to-event data are commonly analysed using survival models. The most widely used is the Cox proportional hazards model.

In its simplest form, the model is:

h(t | X) = h₀(t) exp(βX)

Here, h(t) is the hazard at time t, h₀(t) is the baseline hazard, and X represents covariates such as treatment or patient characteristics. The key assumption is that hazards are proportional over time.

What proportional hazards really means

The proportional hazards assumption states that the effect of a covariate on the hazard is constant over time. Formally:

Hazard Ratio(t) = constant

This means that the relative risk between two individuals or groups does not change as time progresses. The baseline hazard may change over time, but the ratio of hazards remains the same.

For example, if a treatment halves the risk of an event, it does so consistently at all times during follow-up. If the hazard ratio changes over time, the assumption is violated.

Why we assume proportional hazards

The Cox model estimates a single hazard ratio for each covariate. This single number is meaningful only if the relative effect truly stays constant over time. If the assumption holds, interpretation is straightforward and clinically intuitive. If it does not hold, the reported hazard ratio represents an average effect that may mask important time-dependent behaviour, potentially misleading clinical conclusions.

How the assumption breaks down

In real studies, treatment or risk factor effects often change over time. Common scenarios include:

• Strong early treatment effects that diminish later
• Delayed treatment benefits that appear only after prolonged follow-up
• Risk factors whose impact increases with disease progression

When the hazard ratio changes over time, hazards are no longer proportional. A visual clue is when Kaplan–Meier curves cross or diverge rather than remaining roughly parallel. This indicates a violation of the PH assumption.

How to see it in your data

Several diagnostic tools exist to assess proportional hazards. The most common is based on Schoenfeld residuals:

• If hazards are proportional, residuals show no systematic trend with time.
• Graphical checks, such as log cumulative hazard plots, help visualize proportionality. Parallel curves suggest the assumption holds; crossing or diverging curves suggest violation.

What to do if hazards are not proportional

A violation does not invalidate the analysis but indicates the model should be adapted:

• Include time-dependent covariates
• Stratify the model by variables that violate proportionality
• Report time-specific hazard ratios
• Use alternative models, e.g., accelerated failure time models

Summary

The proportional hazards assumption states that relative risks remain constant over time. It is fundamental to interpreting Cox model results. When the assumption fails, the analysis should be adapted to ensure valid and clinically meaningful conclusions.