Why a Hazard Ratio Needs Care Before an E-Value Calculation
An E-value calculation uses an effect measure on the risk-ratio scale. A hazard ratio (HR), however, compares instantaneous event rates among individuals who have not yet experienced the event; it is not generally the ratio of cumulative risks over a follow-up period. Treating an HR as a risk ratio without checking the outcome frequency and model assumptions can therefore give a misleading sensitivity analysis.
For a rare outcome over a specified follow-up period, the HR may be a useful approximation to the risk ratio (RR). When events are not rare, the relationship depends on the reference-group risk at a particular time. Proportional hazards (PH) is central to the usual conversion: without it, one constant HR does not describe how the hazard ratio changes over follow-up.
The conversion below describes risks at a specified follow-up time. It assumes that the PH model holds for the two groups being compared and that the reference-group survival at that time is known or estimated appropriately. If the HR is adjusted, the reference risk must correspond to the same covariate distribution and estimand as the HR; an unrelated crude risk is not automatically suitable.
Deriving the Fixed-Time Risk Ratio
Let \(H_0(t)\) be the cumulative hazard in the reference group through time \(t\). Under the PH model, the exposed group's cumulative hazard is \(H_1(t)=\theta H_0(t)\). Since survival is the exponential of minus the cumulative hazard, the risks are \(R_0(t)=1-\exp[-H_0(t)]\) and \(R_1(t)=1-\exp[-\theta H_0(t)]\). Consequently:
This equation gives the risk ratio implied by a constant HR and a specified reference-group risk. It also shows why an HR alone is insufficient to determine the risk ratio for a non-rare outcome: the conversion depends on \(R_0(t)\) and the selected time \(t\).
For a harmful association with \(RR\geq1\), the E-value for a point estimate is \(RR+\sqrt{RR(RR-1)}\). If the effect is protective, use the reciprocal of the RR before applying the formula. The E-value is calculated from an RR-scale estimate; an HR should not be inserted merely because it is above 1. If a confidence limit is being assessed, use the limit closest to the null after expressing it on the appropriate RR scale.
Worked Example: A Rare Outcome and the HR Approximation
Suppose a hypothetical study reports an HR of 1.50 through one year. The one-year reference-group risk is 0.020, and assume the PH model is appropriate for the comparison. Is the HR a reasonable RR approximation, and what E-value follows from the converted risk ratio?
The reference-group survival is \(S_0(1)=1-0.020=0.980\). Under PH, the one-year exposed-group risk is:
Thus the risk ratio, retaining unrounded values for the division, is:
The two risks, 2.0% and about 2.985%, are small, so the RR is close to the HR of 1.50. It is still not identical. The E-value is:
On the risk-ratio scale, the point-estimate E-value is approximately 2.350. This calculation does not show that the HR is causal or establish PH; those assumptions must be assessed separately.
When the Outcome Is Not Rare
As the reference risk increases, the risk ratio implied by a given harmful HR generally moves closer to 1. The HR compares hazards in the changing risk sets, whereas the RR compares probabilities accumulated across the entire period. For a common outcome, use the fixed-time conversion if its assumptions and inputs are defensible, or use an appropriate risk ratio estimated directly for the target time.
Worked Example: Converting an HR With a Common Outcome
Suppose a hypothetical PH model gives an HR of 1.50, and the reference-group risk at the selected follow-up time is 0.20. Assume the reference risk and HR refer to the same population and comparison. Calculate the implied risk ratio and its point-estimate E-value.
The reference survival is \(S_0(t)=1-0.20=0.80\). The exposed-group risk implied by PH is:
Using the unrounded risk in the division:
This is materially below the HR of 1.50. The E-value, calculated from the converted RR, is:
Using 1.50 directly in the E-value formula would instead give an E-value for an HR treated as though it were an RR. That is not justified by these data: the reference risk is 20%, so the rare-outcome approximation is not appropriate.
If a confidence interval for the HR is converted, apply the same risk transformation to its relevant limits using a compatible reference risk. Uncertainty in the reference risk also matters; treating an estimated risk as fixed can understate uncertainty in the converted quantity. When risk estimates are available directly, they may be preferable to a conversion that relies on a model and a separate baseline-risk estimate.
Proportional Hazards and Other Limits
The conversion is only as credible as the assumptions behind it. Under PH, the hazard ratio is constant over the chosen follow-up: \(\lambda_1(t)/\lambda_0(t)=\theta\). If the effect changes over time, there is no single constant \(\theta\) to substitute into the formula for all \(t\). A fitted Cox HR in that setting may summarize a time-varying contrast, but it does not by itself determine the two groups' risks at a chosen time.
Assess PH using substantive knowledge and suitable diagnostics, such as plots of scaled Schoenfeld residuals or tests for time-varying effects. A diagnostic that fails to reject non-proportionality does not prove PH, especially with limited data. Also check that the time horizon is clinically meaningful, that censoring is handled appropriately, and that the reference risk corresponds to the HR's target population. With competing events, a cause-specific HR alone does not determine the cumulative incidence of the event of interest: the competing-event process also affects that probability.
Worked Example: Why a Cumulative-Hazard Ratio Is Not Always an HR
Consider hypothetical piecewise cumulative hazards through time \(t\). In the reference group, the cumulative hazard is 0.05 in the first interval and 0.15 in the second, for a total of \(H_0(t)=0.20\). In the exposed group, the corresponding increments are 0.15 and 0.15, for a total of \(H_1(t)=0.30\). The interval-specific hazard ratios are 3.00 and 1.00, so PH does not hold.
The ratio of total cumulative hazards is \(0.30/0.20=1.50\), but this is not a constant HR. It must not be substituted for \(\theta\) in the PH conversion. The risks computed from the total cumulative hazards are:
The actual risk ratio in this hypothetical setup is:
Because both risks are specified, this RR can be used for an E-value calculation if it is the appropriate effect estimate and estimand. The example does not justify calling 1.50 an HR or using it as the RR. The point is that a ratio of cumulative hazards does not repair a violation of PH.
Choosing and Reporting the Scale
The main alternatives are to use an HR as an approximation when risks are genuinely small, to convert it at a specified time under a defensible PH model, or to calculate a risk ratio directly from suitable time-specific risks. The direct-risk approach avoids equating hazard and risk, but it still requires the risk estimates to address the intended population and follow-up. A risk difference is another useful effect measure for reporting absolute impact, but it cannot be inserted into the standard RR-based E-value formula without a justified conversion.
A clear sensitivity analysis should identify the reported HR, the time horizon, the reference-group risk, whether PH is assumed and assessed, and whether the E-value uses an approximate or converted RR. It should distinguish the point-estimate E-value from one based on a confidence limit. E-values address the strength of potential unmeasured confounding required to explain an association under their framework; they do not test PH, correct model misspecification, remove selection bias, or establish causality.
State the follow-up horizon and identify the reference-group risk at that time.
Consider whether PH is credible and whether the HR and baseline risk describe a compatible population and estimand.
Use the HR as an approximation only for sufficiently small risks; otherwise convert under PH or use a suitable direct risk ratio.
Apply the RR formula in the correct direction and state the scale, assumptions, follow-up time, and units or risk context.
Common Mistakes and Practical Checks
- Wrong: “The HR is 1.5, so the risk is 50% higher.” Correction: An HR of 1.5 means the instantaneous hazard is 50% higher under the stated model; the risk ratio may differ.
- Wrong: Use a rare-outcome approximation because the event sounds uncommon. Correction: Check the actual risks over the specified follow-up in both groups. Rarity depends on the time horizon and population.
- Wrong: Convert an adjusted HR with any available reference risk. Correction: The reference risk must be appropriate to the adjusted comparison and target population.
- Wrong: Treat a non-significant PH diagnostic as proof that hazards are proportional. Correction: Diagnostics provide evidence, not proof; assess plots, uncertainty, and plausible time variation.
- Wrong: Assume the HR-to-RR conversion accounts for competing risks or all sources of bias. Correction: The formula maps risks under its survival and PH assumptions; it does not resolve other design or identification problems.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- Under PH, what information in addition to the HR is required to calculate the risk ratio at time \(t\)?
- A harmful HR is 1.4, but the event is common over the study period. Why is inserting 1.4 directly into the RR-based E-value formula questionable?
- If the hazard ratio changes substantially over follow-up, what does that imply for a single HR-to-RR conversion?
- At a specified time, the reference risk is 0.10 and the HR is 2.00. Under PH, calculate the implied exposed risk and RR.
- Why might a cause-specific HR be insufficient to calculate a cumulative-incidence ratio when competing events occur?
Solutions. (1) The reference risk \(R_0(t)\), or equivalently the reference survival \(S_0(t)\), and the assumption that the HR applies proportionally through \(t\) are needed. (2) For a common outcome, the cumulative risk ratio can be substantially closer to 1 than the HR; the E-value formula requires an RR-scale input. (3) A single constant HR does not describe the effect at every time, so the PH conversion is not justified without a suitable time-specific model. (4) \(R_1(t)=1-(1-0.10)^2=1-0.81=0.19\), so \(RR(t)=0.19/0.10=1.90\). (5) Cumulative incidence depends on both the event-of-interest hazard and the competing-event process, so the cause-specific HR alone does not determine the event probability by time \(t\).