Proportional Hazards Assumption: How to Check It and What to Do When It Fails

1. What Does the Proportional Hazards Assumption Mean?

A standard Cox model can be written conceptually as:

h(t | X) = h₀(t) exp(βX)

where h₀(t) is the baseline hazard and β represents the predictor effect on the log-hazard scale.

The baseline hazard is allowed to change with time. The proportional hazards assumption concerns the relative effect of the predictor. With a fixed coefficient β, exponentiating that coefficient gives a hazard ratio, exp(β), that does not itself depend on time (Steyerberg, 2019).

For two covariate patterns that differ by one unit in X, proportional hazards therefore implies that their hazard ratio remains constant over follow-up, even though the underlying hazards may both rise, fall, or otherwise change with time.

Klein and Moeschberger describe nonproportionality explicitly through extensions in which a predictor is interacted with a function of time. Once such a term is present, the hazard ratio becomes a function of time rather than a single constant (Klein & Moeschberger, 2003).

What does a Cox hazard ratio mean when PH holds?

Under proportional hazards, an HR of 0.70 for treatment versus control means that the modeled treatment hazard is 0.70 times the control hazard throughout follow-up, conditional on the other variables in the model.

For prediction, this distinction becomes especially important because absolute survival predictions require both the predictor effects and the baseline survival function. Steyerberg notes that tables containing only hazard ratios are generally insufficient for calculating absolute risks because survival prediction also requires the baseline hazard or baseline survival (Steyerberg, 2019).

2. Why Nonproportional Hazards Matter

Suppose a treatment has a strong association with the hazard early in follow-up but little association later. Or suppose the direction of the association changes over time.

A standard Cox coefficient compresses that changing relationship into a constant-effect specification.

Harrell demonstrates that interval-specific Cox coefficients can reveal predictors whose importance decreases or otherwise changes with follow-up. Plotting interval-specific log hazard ratios against time can therefore show both whether proportionality is questionable and how the effect changes (Harrell, 2015).

This is why forcing one constant HR onto a clearly time-varying relationship can be scientifically misleading. The problem is not merely that an assumption test has “failed.” The more important problem is that the reported effect summary may conceal temporal structure that matters to the research question.

“Treatment was associated with HR = 0.75.”

Such a statement suggests a stable relative effect unless the time dependence is made explicit.

If the effect is substantially different early and late in follow-up, the analysis and reporting should represent that pattern rather than implying that 0.75 applies uniformly.

3. Do Not Confuse Time-Dependent Covariates With Time-Varying Effects

These concepts are related but different.

One way to represent such nonproportionality is to introduce an interaction between the predictor and a function of time:

X × g(t)

The resulting hazard ratio depends on time when the coefficient for that interaction is nonzero (Klein & Moeschberger, 2003).

4. How to Check the Proportional Hazards Assumption

No single diagnostic should automatically determine the scientific conclusion. The useful strategy is to combine diagnostics that detect departures with methods that help reveal their magnitude and shape.

Graphical checks based on cumulative hazards

Klein and Moeschberger describe several graphical checks.

For categorical groups, plots of the log cumulative baseline hazards should be approximately parallel under proportional hazards. Alternatively, differences between log cumulative hazards can be plotted against time; under PH, those differences should remain approximately constant (Klein & Moeschberger, 2003).

They also describe Andersen plots, in which one estimated cumulative hazard is plotted against another. Under proportional hazards, the resulting relationship should be approximately linear through the origin. Certain departures from linearity can also provide clues about whether the hazard ratio is increasing or decreasing with time (Klein & Moeschberger, 2003).

These methods are useful because they make proportionality a pattern-recognition problem, rather than reducing it immediately to one p-value.

Schoenfeld residual diagnostics

Schoenfeld residuals provide a particularly useful Cox regression diagnostic.

Harrell describes plotting smoothed Schoenfeld residuals against failure time to examine how a predictor's log hazard ratio changes over follow-up. Scaled or weighted Schoenfeld residuals can be put on a scale related to the log relative hazard, making the temporal pattern directly informative about nonproportionality (Harrell, 2015).

The diagnostic can be used both graphically and for formal testing. Harrell describes tests based on association between transformed failure time and Schoenfeld residuals, as well as smoothed residual plots that estimate the predictor effect over time (Harrell, 2015).

The plot is often more informative than the test alone because it can show the direction, magnitude, and timing of the departure.

Interval-specific hazard ratios

Harrell also describes dividing follow-up into intervals and estimating the Cox coefficient separately within each interval.

Plotting those interval-specific log HR estimates and their confidence limits against time can reveal when a predictor is important and how its effect changes. It can also guide the construction of a predictor-by-time interaction when a nonproportional relationship needs to be modeled explicitly (Harrell, 2015).

5. Testing PH Is Not the Same as Understanding the Violation

A formal proportional-hazards test answers a limited question: whether the data provide evidence against a specified constant-effect structure.

It does not, by itself, tell you whether the departure is scientifically important.

Klein and Moeschberger show that PH can be tested by creating an artificial time-dependent term such as Xg(t) and testing whether its coefficient is zero. Importantly, they note that the ability of this test to detect nonproportionality depends on the chosen function g(t) (Klein & Moeschberger, 2003).

Harrell similarly presents formal tests based on Schoenfeld residuals or specified predictor-by-time interactions, while also emphasizing graphical assessment of the effect over time (Harrell, 2015).

Conversely, a nonsignificant PH test should not automatically be interpreted as proof that the effect is exactly constant. The estimated pattern and its uncertainty still matter.

6. What to Do When Proportional Hazards Fails

The appropriate response depends on the research goal and on which variable violates proportionality.

Option 1: Model the effect as a function of time

Klein and Moeschberger describe the general form Xg(t). They also show piecewise specifications in which the HR differs before and after a selected time point. In that setting, separate relative effects can be estimated for different follow-up periods (Klein & Moeschberger, 2003).

Harrell likewise describes predictor-by-time interactions and interval-specific Cox estimates as tools for modeling and understanding nonproportional effects (Harrell, 2015).

The interpretation then changes from:

“The hazard ratio is HR.”

to:

“The hazard ratio varies with follow-up time according to the fitted time-effect relationship.”

That is a time varying hazard ratio, not one universal effect.

Option 2: Use stratification

Klein and Moeschberger explicitly describe stratification as one approach when the proportional-hazards assumption is violated and demonstrate adjustment for a nonproportional factor through a stratified proportional-hazards model (Klein & Moeschberger, 2003).

Option 3: Consider another survival-model framework

Steyerberg discusses parametric survival models and accelerated failure-time formulations as alternatives with different assumptions and interpretations. He also notes that models such as the log-logistic may be useful in situations where predictor effects are concentrated in an early phase, which would appear as nonproportional hazards in a Cox model (Steyerberg, 2019).

Klein and Moeschberger additionally discuss additive hazards models as alternatives to the multiplicative proportional-hazards formulation; these can allow regression effects themselves to vary over time (Klein & Moeschberger, 2003).

The alternative should be selected because its structure matches the research objective and observed time pattern—not simply because the Cox PH test produced a small p-value.

7. A Practical Decision Guide

8. The StatsAlly PH Checklist

Fit

1. Define the event, time origin, follow-up period, censoring structure, and scientific target before fitting the Cox model.
2. Identify which predictors are exposures or treatments of primary interest and which are adjustment variables.
3. Fit a scientifically justified Cox model rather than treating the software's default specification as automatically adequate.
4. Remember that the standard model assumes proportional predictor effects over follow-up.

Diagnose

1. Examine graphical proportional-hazards diagnostics where appropriate.
2. For categorical predictors, consider cumulative-hazard-based graphical checks.
3. Examine Schoenfeld residuals or scaled Schoenfeld residual patterns for important predictors.
4. Use formal PH tests as diagnostic evidence, not as the entire model-checking exercise.
5. Consider interval-specific HR estimates when they help reveal the temporal pattern.

Characterize the violation

1. Identify which predictor shows nonproportionality.
2. Determine whether its estimated effect strengthens, attenuates, reverses, or otherwise changes with follow-up.
3. Examine when the departure occurs.
4. Consider the uncertainty around the apparent time pattern.
5. Distinguish a changing predictor value from a changing predictor effect.
6. Ask whether the departure is large enough to change the substantive interpretation.

Reconsider the effect summary or model

1. Decide whether one constant HR remains an adequate scientific summary.
2. If the changing effect is important, consider a predictor-by-time interaction.
3. Consider piecewise time effects when broad follow-up intervals have distinct relative effects and that specification is scientifically defensible.
4. Consider stratification when the non-PH variable needs adjustment but its HR is not the target of interest.
5. Consider another survival-model framework when the Cox proportional-hazards structure does not adequately represent the research question.
6. For prediction, consider the consequences for absolute survival predictions rather than focusing only on coefficients.

Report transparently

1. State that proportional hazards was assessed.
2. Identify the diagnostic methods used.
3. Report which variables showed important departures.
4. Describe the pattern and magnitude of nonproportionality, not merely the p-value.
5. Explain any time interaction, stratification, or alternative model introduced in response.
6. Do not report one HR as universally applicable if the fitted effect materially changes over time.
7. Report uncertainty for time-specific or time-varying effects where available.
8. Align the interpretation with the final model actually fitted.

9. Common Mistakes

10. What Should a Published Analysis Say?

A defensible methods section should make clear that the proportional hazards assumption was examined and identify the diagnostic approach.

If nonproportionality is detected, the results should go beyond stating that “the PH assumption was violated.” Researchers should explain which predictor was affected, what the time pattern looked like, and how the model or effect summary was changed as a result.

When the final model contains a time interaction, the results should report the resulting time-dependent effect rather than reverting to a single constant HR.

When stratification is used, the report should identify the stratifying variable and explain why its own HR is not being presented.

Bottom Line

The proportional hazards assumption asks whether the relative effect represented by a standard Cox coefficient can reasonably be treated as constant over follow-up.

Checking that assumption should therefore be more than a formal hypothesis test.

Use graphical and residual-based Cox regression diagnostics to investigate the effect over time. Schoenfeld residuals and interval-specific hazard-ratio estimates can help characterize the pattern of departure. If important non proportional hazards are present, reconsider whether the scientific result should be a constant HR, a time varying hazard ratio, a stratified analysis, or another survival-model formulation (Klein & Moeschberger, 2003; Harrell, 2015; Steyerberg, 2019).

The objective is not to make the PH test nonsignificant. It is to represent the relationship between the predictor and the event process in a way that is statistically defensible and scientifically interpretable.

FAQs

What is the proportional hazards assumption?

In the standard Cox model, proportional hazards means that the modeled relative effect of a fixed predictor on the hazard is constant over follow-up. The baseline hazard itself is allowed to change with time (Steyerberg, 2019).

How do I check the proportional hazards assumption?

Supported approaches include cumulative-hazard-based graphical checks, Schoenfeld-residual diagnostics, interval-specific hazard-ratio estimates, and formal tests based on residuals or predictor-by-time interactions (Klein & Moeschberger, 2003; Harrell, 2015).

What are Schoenfeld residuals used for?

In Cox regression, Schoenfeld residuals can be examined against failure time to assess whether a predictor's effect changes systematically over follow-up. Smoothed scaled residuals can help visualize the form of the log hazard ratio over time (Harrell, 2015).

What should I do if the proportional hazards assumption is violated?

The appropriate response depends on the predictor and research objective. Supported options include modeling a predictor-by-time interaction, using piecewise time effects, stratifying on a nonproportional adjustment variable, or considering another survival-model framework (Klein & Moeschberger, 2003; Harrell, 2015; Steyerberg, 2019).

Does a significant PH test mean I must abandon the Cox model?

Not necessarily. Cox-model extensions can explicitly represent effects that vary with time, and stratification can address some nonproportional variables. The diagnostic should be used to understand the departure and select a specification appropriate to the scientific question (Klein & Moeschberger, 2003; Harrell, 2015).

Is a time-dependent covariate the same as a time-varying hazard ratio?

No. A time-dependent covariate changes value during follow-up. A time-varying hazard ratio arises when the effect of a predictor changes with time. A fixed baseline predictor can therefore have a time-varying effect (Klein & Moeschberger, 2003).

Why isn't a PH p-value enough?

A formal test addresses evidence against a specified constant-effect structure but does not fully describe the magnitude, direction, or timing of the departure. Graphical diagnostics and estimates of the effect over time are needed to understand whether the departure is scientifically important (Klein & Moeschberger, 2003; Harrell, 2015).

References

Harrell, F. E., Jr. (2015). Regression modeling strategies: With applications to linear models, logistic and ordinal regression, and survival analysis (2nd ed.). Springer. https://doi.org/10.1007/978-3-319-19425-7

Klein, J. P., & Moeschberger, M. L. (2003). Survival analysis: Techniques for censored and truncated data (2nd ed.). Springer.

Steyerberg, E. W. (2019). Clinical prediction models: A practical approach to development, validation, and updating (2nd ed.). Springer. https://doi.org/10.1007/978-3-030-16399-0