Why Nonproportional Hazards Matter
A hazard ratio is often reported as though it summarizes a treatment or exposure effect throughout follow-up. That interpretation requires care: the ratio may change over time. For example, an intervention might be associated with a lower event hazard soon after treatment begins but a higher hazard later. A single number cannot describe both patterns faithfully.
Survival curves provide a useful first look at this issue. If two estimated curves cross, their observed ordering changes over time. That is descriptive evidence worth investigating, but it does not by itself establish that the population hazards are nonproportional. The crossing may reflect sampling variability, censoring, or a small number of people remaining at risk late in follow-up. Conversely, curves that do not visibly cross do not prove proportional hazards.
The hazard is a rate, not the probability of experiencing the event by time \(t\). A hazard ratio of 0.6 means that, at the specified time and among those still event-free, group 1 has 0.6 times the instantaneous event rate of group 0 under the stated model. It does not mean that 40% fewer people will experience the event over the entire study.
What Crossing Curves Tell Us
Under a proportional-hazards model with a binary group indicator \(z\), the hazards satisfy \(h(t\mid z)=h_0(t)\exp(\beta z)\), where \(h_0(t)\) is the baseline hazard and \(\beta\) is a constant. The corresponding survival functions are related by:
Because \(0\leq S_0(t)\leq1\), this relationship preserves the ordering of the two population survival functions over time. If \(\beta>0\), group 1 has a higher hazard and its survival is no greater than group 0 survival; if \(\beta<0\), the ordering is reversed. Thus, a genuine change in the ordering of the population curves is incompatible with this simple fixed-effect proportional-hazards model.
This is a model implication, not a rule for diagnosing the truth from a graph. Kaplan–Meier curves are estimates subject to sampling variability. They may cross even when the underlying hazards are proportional, especially with few events or sparse risk sets. The plot should prompt questions: where do the curves cross, how many participants remain at those times, and do other diagnostics support a time-varying effect?
Worked Example: A Descriptive Crossing in Kaplan–Meier Estimates
Consider a hypothetical randomized study with 100 participants in each group and no censoring before the two event times shown. In group A, 10 events occur at month 4 and 45 of the 90 remaining participants have events at month 12. In group B, 20 events occur at month 4 and 25 of the 80 remaining participants have events at month 12. Calculate the Kaplan–Meier survival estimates at these times.
At month 4, the estimates are \(S_A(4)=1-10/100=0.90\) and \(S_B(4)=1-20/100=0.80\). At month 12, the conditional survival factors are \(1-45/90=0.50\) for A and \(1-25/80=0.6875\) for B. Therefore:
The estimated curves switch ordering: A has higher estimated survival at month 4, while B has higher estimated survival at month 12. This is descriptive evidence of a changing pattern. The calculation does not establish that the population hazards cross, nor does it quantify uncertainty. In particular, it would be important to inspect the numbers at risk and confidence bands at later times before interpreting the apparent separation.
Reading Time-Varying Hazard Ratios
A direct way to describe nonproportional hazards is to estimate hazard ratios over clinically meaningful periods or to specify a model in which the coefficient varies with time. A simple interval-based summary compares event rates within each period. Such rates are descriptive and do not automatically adjust for covariates or differences in who remains at risk.
Worked Example: Interval-Specific Hazard-Rate Ratios
Suppose a hypothetical study reports the following event counts and total person-time. During months 0–6, group 1 has 4 events over 100 person-years and group 0 has 10 over 100 person-years. During months 6–12, group 1 has 6 events over 30 person-years and group 0 has 3 over 30 person-years. Compute the crude interval event rates and their ratios.
In months 0–6, the rates are \(4/100=0.040\) and \(10/100=0.100\) events per person-year. Their ratio is \(0.040/0.100=0.400\). In months 6–12, the rates are \(6/30=0.200\) and \(3/30=0.100\) events per person-year, giving a ratio of \(0.200/0.100=2.00\).
The crude rate comparison is lower for group 1 in the first interval and higher in the second. This pattern is compatible with a time-varying contrast, but it is not itself a fitted Cox-model result. The calculation assumes that person-time and events are correctly assigned to the stated intervals; it does not adjust for covariates, and the late-period rates are based on less person-time. Without uncertainty estimates and an assessment of the risk sets, the ratios should not be treated as precise evidence of a reversal.
For a model-based illustration, let \(z=1\) identify group 1 and consider a Cox model with a log-time interaction:
Here \(t_*>0\) is a chosen reference time, and \(\beta\) is the log hazard ratio at \(t=t_*\). This model requires positive times for the logarithm and assumes the specified log-time form adequately describes how the effect changes. The baseline hazard remains unspecified, as in the Cox model.
Worked Example: Interpreting a Log-Time Interaction
Suppose, hypothetically, that a fitted model uses \(t_*=3\) months, with \(\hat{\beta}=\log(0.5)\) and \(\hat{\gamma}=1.00\). Calculate the estimated hazard ratios at 3 and 12 months.
At 3 months, \(\log(3/3)=0\), so \(\widehat{HR}(3)=\exp\{\log(0.5)\}=0.500\). At 12 months, \(\log(12/3)=\log(4)\), so:
Under this fitted model, the estimated hazard in group 1 is half that in group 0 at 3 months and twice that in group 0 at 12 months, conditional on the model covariates and being event-free at each time. These are model-based estimates, not direct descriptions of the Kaplan–Meier curves. To assess whether the change is supported by the data, uncertainty for \(\gamma\) and the model’s adequacy must also be considered.
Why a Single Cox Hazard Ratio Can Mislead
The usual Cox model assumes that each covariate has a constant coefficient, so a one-unit increase in a covariate multiplies the hazard by the same amount at every event time, conditional on the other covariates. When the true effect changes over time, fitting that model does not make the changing effect disappear. Rather, the reported coefficient is a single compromise determined by the observed event times and risk sets. Under model misspecification, it is not generally a simple arithmetic average of the true time-specific hazard ratios, nor a universal treatment effect that applies at every time.
The direction of the compromise can depend on when events occur and which participants remain at risk. A study with most events early may yield a summary more reflective of early contrasts; a different follow-up pattern may produce a different estimate. Therefore, a statistically precise single estimate can still be a poor summary of an effect that reverses or changes substantially.
A formal diagnostic can complement the plot. Schoenfeld-residual methods assess whether covariate effects show a systematic time trend; time-varying-effect models estimate a specified changing relationship. These tools answer related but distinct questions. A diagnostic can indicate departure from proportionality without identifying the best alternative model, and a selected time function should be justified rather than chosen solely because it produces a preferred result.
Practical Assessment and Alternatives
Before interpreting nonproportionality, state the time origin, event definition, group comparison, and censoring assumptions. In ordinary survival analysis, inference relies on appropriate risk sets and on censoring being independent of the event time, at least conditional on the modeled covariates. If censoring is informative after conditioning, the observed risk sets may not represent the people who would otherwise remain under observation. Also check whether a few late events drive the apparent crossing.
A useful assessment combines descriptive and model-based evidence. Plot estimated survival curves with numbers at risk and uncertainty bands; examine relevant residual diagnostics; and, when scientifically reasonable, compare the constant-effect model with a prespecified time-varying effect. A formal test is not a substitute for inspecting the magnitude and shape of the estimated effect. Failure to reject proportionality may reflect limited power, not proof that the effect is constant.
| Approach | What it summarizes | Main caution |
|---|---|---|
| Kaplan–Meier curves | Descriptive survival experience by group | Curve crossing is uncertain and does not adjust for covariates. |
| Single-coefficient Cox model | A constant conditional hazard ratio, if proportional hazards is adequate | With nonproportional effects, the estimate is a risk-set-dependent compromise. |
| Time-varying coefficient model | A modeled hazard ratio that changes with time | Results depend on the chosen functional form and time scale. |
| Stratified Cox model | Covariate effects while allowing separate baseline hazards across strata | It does not estimate a coefficient for the stratifying variable. |
These approaches are not interchangeable. Stratification can address a nuisance covariate whose baseline hazard differs across groups, but it does not by itself report a time-varying treatment hazard ratio. A time-varying coefficient directly models a changing contrast, while a descriptive curve comparison avoids asserting a particular hazard model. The choice depends on the scientific question, data support, and whether the assumptions are plausible.
Nonproportional hazards also limit what a single hazard ratio can communicate even when it remains useful as a conventional summary. Report time-specific estimates or curves when the pattern changes meaningfully, and explain their uncertainty. A crossing pattern with very sparse late risk sets may be too unstable for a detailed time-specific interpretation. If the intended question concerns survival probability at a particular time or average time event-free over a fixed interval, a hazard ratio may not be the most direct estimand.
Common Mistakes
- Calling every crossing curve proof of nonproportional hazards. Estimated curves fluctuate. Check uncertainty, risk-set sizes, and model diagnostics before drawing a population-level conclusion.
- Reading a hazard ratio as a cumulative risk ratio. A hazard ratio compares instantaneous rates among those still event-free; it is not the ratio of event probabilities by a stated time.
- Reporting one Cox estimate as if it held at every time. That interpretation requires proportional hazards. If effects vary, describe the time dependence or explicitly qualify the single-number summary.
- Treating a nonsignificant diagnostic as confirmation of proportionality. A test may have low power, particularly with few events. Consider the size and pattern of departures as well as the test result.
Check Your Understanding
Answer each question, then compare your reasoning with the concise solutions.
- If a fitted proportional-hazards model has \(HR=0.70\), what does that say about instantaneous hazards, and what does it not say about cumulative event probabilities?
- Two estimated survival curves cross when only a few participants remain at risk. Give two reasons to avoid treating the crossing as conclusive evidence of nonproportional hazards.
- A time-varying model estimates \(HR(t)=\exp\{\beta+\gamma\log(t/t_*)\}\). What does \(\beta\) represent, and what does \(\gamma\) control?
- Why might a single Cox-model coefficient fitted when the true hazard ratio changes over time differ between studies with different event-time distributions?
Solutions. (1) Under the model, at each time the event-free members of the comparison group have 0.70 times the instantaneous hazard, conditional on modeled covariates. It does not imply 30% fewer cumulative events. (2) Sampling variability can cause estimated curves to cross, and sparse late risk sets make late estimates unstable. (3) \(\beta\) is the log hazard ratio at \(t=t_*\); \(\gamma\) controls its change per unit increase in log time. (4) The misspecified constant coefficient is a risk-set- and event-time-dependent compromise, so studies emphasizing different portions of follow-up can yield different summaries.