Proportional Hazards Assumption

The proportional hazards assumption is the central premise of the Cox model and related methods: it holds that the hazard ratio between groups or per unit of a covariate is constant over time, so that the effect of a predictor multiplies the underlying hazard by the same factor at every follow-up time. Whether this assumption holds determines whether a single hazard ratio meaningfully summarises an effect.

Definition

The proportional hazards assumption states that the ratio of the hazard functions for any two covariate patterns is constant over time; equivalently, covariates act multiplicatively on a common baseline hazard and do not change that multiplier as follow-up proceeds.

Scope

This topic explains what proportionality means, why it matters for interpreting hazard ratios, and how it is checked — graphically and with formal tests such as those based on Schoenfeld residuals — and what to do when it fails. It is a methodological reference and does not provide clinical recommendations.

Core questions

• What does it mean for hazards to be proportional, and why does a single hazard ratio depend on it?
• How can the assumption be assessed graphically and with formal tests?
• What patterns (such as crossing hazards or time-varying effects) signal a violation?
• What modelling options exist when proportionality does not hold?

Key concepts

• Constant hazard ratio over time
• Baseline hazard and multiplicative covariate effect
• Schoenfeld residuals
• Log-minus-log survival plots
• Time-varying coefficients
• Stratification
• Crossing hazards
• Time-by-covariate interaction

Mechanisms

In a proportional hazards model the hazard for a subject equals an unspecified baseline hazard multiplied by a factor that depends on their covariates but not on time; consequently the log of the hazard ratio is constant and the cumulative hazards of two groups stay in fixed proportion. The assumption is checked by inspecting whether scaled Schoenfeld residuals show a trend against time (a slope indicates a time-varying effect), by examining log-minus-log survival plots for parallelism, or by adding a time-by-covariate interaction and testing it. When proportionality fails — for instance when an early treatment benefit wanes or hazards cross — remedies include stratifying on the offending variable, modelling time-varying coefficients, or restricting the time window (Schoenfeld, 1982; Therneau & Grambsch, 2000; Bradburn et al., 2003).

Clinical relevance

Because a reported hazard ratio assumes a constant effect over time, a violated proportional hazards assumption can make a single hazard ratio misleading — for example averaging over an early benefit and later harm. Recognising this supports careful appraisal of survival analyses; the entry is descriptive of methodology and not clinical guidance.

Epidemiology

Proportional hazards modelling is the dominant approach to covariate-adjusted survival analysis in medical research, so assessing the assumption is a routine, if sometimes neglected, part of analysis and reporting (Bradburn et al., 2003).

Evidence & guidelines

There are no clinical guidelines for the assumption itself; the methodological references are Cox's original model (Cox, 1972), the introduction of partial (Schoenfeld) residuals for diagnostics (Schoenfeld, 1982), and texts that detail checking and extending the model when proportionality fails (Therneau & Grambsch, 2000; Collett, 2015).

History

The assumption is inseparable from Cox's 1972 proportional hazards model, which made covariate-adjusted survival regression practical by leaving the baseline hazard unspecified while assuming a constant multiplicative covariate effect. Diagnostics followed: Schoenfeld's 1982 partial residuals became the basis for the most widely used formal test, later developed into the scaled-residual approach popularised by Therneau and Grambsch (2000).

Debates

How should non-proportional hazards be handled?

When effects vary over time, analysts disagree on whether to report a time-averaged hazard ratio, model time-varying coefficients, stratify, or switch to alternative summaries such as restricted mean survival time, each with trade-offs in interpretability.

Key figures

• David R. Cox
• David Schoenfeld
• Terry Therneau
• Patricia Grambsch

Seminal works

• cox-1972
• schoenfeld-1982

Frequently asked questions

Why does a hazard ratio depend on the proportional hazards assumption?

A single hazard ratio summarises the effect as one constant multiplier of the hazard; if that multiplier actually changes over time, the reported ratio is a time-average that may not describe the effect at any particular point in follow-up.

How is the assumption commonly checked?

Frequently by testing whether scaled Schoenfeld residuals trend with time, by inspecting log-minus-log survival plots for parallel curves, or by adding and testing a time-by-covariate interaction term.