Interpreting Hazard Ratios as Absolute Event Risks

Why a Hazard Ratio Is Not an Absolute Risk

A hazard ratio summarizes the relative instantaneous event rate between exposure conditions among individuals who remain event-free at a given time. It does not, by itself, state how many individuals will experience the event by a clinically relevant time. That absolute probability also depends on the baseline event experience and on the time horizon.

For example, the same hazard ratio can correspond to very different absolute event probabilities in populations with different baseline survival. A hazard ratio of 2 may have a modest absolute impact when baseline event risk is low and a much larger impact when it is high. To communicate predicted event risk, specify the time point, the baseline survival for the relevant covariate profile, and the assumptions connecting the exposure conditions.

Definition: Let \(S_0(t)\) denote the probability of remaining free of the event through time \(t\) under a reference exposure condition, and let \(S_1(t)\) denote the corresponding survival probability under a comparison exposure condition. The event risks by time \(t\) are \(F_0(t)=1-S_0(t)\) and \(F_1(t)=1-S_1(t)\). A hazard ratio compares instantaneous hazards; it is not generally equal to \(F_1(t)/F_0(t)\).

Converting a Constant Hazard Ratio

For a single event type, the hazard \(\lambda(t)\) is the instantaneous event rate at time \(t\), conditional on having remained event-free up to that time. The cumulative hazard from time 0 to \(t\) is \(H(t)=\int_0^t \lambda(u)\,du\), and survival is \(S(t)=\exp\{-H(t)\}\). These relations connect the hazard scale to the probability scale.

Suppose the comparison condition has a constant hazard ratio \(\theta\) relative to the reference condition over the interval from 0 to \(t\), so that \(\lambda_1(u)=\theta\lambda_0(u)\) throughout that interval. This proportional-hazards assumption implies that the comparison cumulative hazard is \(\theta H_0(t)\). Therefore its survival and event risk can be calculated from the reference survival.

H_1(t)=\int_0^t \lambda_1(u)\,du =\int_0^t \theta\lambda_0(u)\,du =\theta H_0(t)

S_1(t)=\exp\{-H_1(t)\} =\exp\{-\theta H_0(t)\} =\left[\exp\{-H_0(t)\}\right]^\theta =S_0(t)^\theta, \qquad F_1(t)=1-S_0(t)^\theta

The conversion requires a baseline survival estimate for the same time origin and a compatible population or covariate profile. It also requires the hazard ratio to apply over the entire interval being summarized. If the reported hazard ratio is conditional on covariates, the baseline survival used in the conversion must correspond to the same conditioning. A hazard ratio estimated for one covariate profile cannot automatically be paired with an unrelated population-average survival.

Worked Example: Converting an Elevated Hazard to Absolute Risk

Hypothetical data. At 12 months, the estimated reference-condition survival is \(S_0(12)=0.800\), and the hazard ratio for the comparison condition is \(\theta=1.50\). Assume proportional hazards holds through 12 months and that both estimates refer to the same covariate profile.

The comparison survival is

S_1(12)=0.800^{1.50} =\exp\{1.50\log(0.800)\} =\exp(-0.334715) \approx 0.7155.

Thus the comparison event risk by 12 months is \(F_1(12)=1-0.7155=0.2845\), or about 28.5%. The reference event risk is \(F_0(12)=1-0.800=0.2000\), or 20.0%. The absolute risk difference, comparison minus reference, is \(0.2845-0.2000=0.0845\), about 8.45 percentage points.

The event-risk ratio is \(0.2845/0.2000\approx1.422\), not 1.50. The hazard ratio of 1.50 describes the relative instantaneous hazards under the model, whereas the risk ratio compares cumulative probabilities by 12 months.

Baseline Risk Determines the Absolute Meaning

The power conversion shows directly why a hazard ratio is insufficient on its own: both \(\theta\) and \(S_0(t)\) are needed. At a fixed hazard ratio, a change in the baseline survival changes the absolute comparison risk. The same hazard ratio can therefore have different clinical implications across populations or time horizons.

Worked Example: A Hazard Ratio Below One

Hypothetical data. At two years, reference-condition survival is \(S_0(2)=0.700\), and the comparison-to-reference hazard ratio is \(\theta=0.65\). Assume a constant hazard ratio over the two-year interval and a matching covariate profile.

The comparison survival is calculated as

S_1(2)=0.70^{0.65} =\exp\{0.65\log(0.70)\} =\exp(-0.2318387) \approx 0.793066.

The comparison event risk is \(F_1(2)=1-0.793066=0.206934\), or about 20.7%. The reference risk is \(F_0(2)=1-0.700=0.3000\), or 30.0%. The absolute risk difference, comparison minus reference, is \(0.206934-0.3000=-0.093066\). Thus the model predicts about 9.31 percentage points lower event risk by two years under the comparison condition.

This is an association implied by the specified model and estimates. A causal interpretation requires additional design and modeling assumptions; the arithmetic conversion alone does not establish that changing exposure would cause the predicted difference.

It is useful to report the time horizon and both risks alongside the hazard ratio. A statement such as “the hazard is 35% lower” does not tell a reader whether the predicted absolute difference is small or large. Conversely, an absolute difference without its time horizon or reference risk is incomplete.

When the Hazard Ratio Changes Over Time

If proportional hazards does not hold, a single constant \(\theta\) cannot represent the exposure effect throughout follow-up. Let the time-varying hazard ratio be \(\theta(u)=\lambda_1(u)/\lambda_0(u)\). The comparison cumulative hazard is then the integral of the time-varying ratio weighted by the baseline hazard, and comparison survival is the exponential of its negative. In general, it is not valid to take a single hazard ratio and raise baseline survival to that power unless that ratio is constant over the interval.

H_1(t)=\int_0^t \theta(u)\lambda_0(u)\,du, \qquad S_1(t)=\exp\left\{-\int_0^t \theta(u)\lambda_0(u)\,du\right\}

The weighting matters: periods with more baseline cumulative hazard contribute more to the final exposure-specific cumulative hazard. A simple piecewise calculation illustrates this. It is exact when the hazard ratio is constant within each interval and the supplied baseline cumulative-hazard increments apply to the relevant profile.

Worked Example: A Piecewise-Changing Hazard Ratio

Hypothetical data. Consider follow-up through two years. The reference cumulative-hazard increment is 0.10 from year 0 to year 1 and 0.40 from year 1 to year 2. The comparison hazard ratio is 2.00 in the first year and 0.50 in the second year.

The reference cumulative hazard is \(0.10+0.40=0.50\). Its survival and event risk at two years are

S_0(2)=\exp(-0.50)\approx0.6065, \qquad F_0(2)=1-0.6065=0.3935.

For the comparison condition, multiply each interval's baseline increment by that interval's hazard ratio and add the results:

$$ H_1(2)=(2.00)(0.10)+(0.50)(0.40)=0.20+0.20=0.40. $$

Therefore \(S_1(2)=\exp(-0.40)\approx0.6703\), and \(F_1(2)=1-0.6703=0.3297\). The comparison-minus-reference risk difference is \(0.3297-0.3935=-0.0638\), or about 6.38 percentage points lower by two years. The early hazard increase and later hazard decrease cannot be summarized by either hazard ratio alone; their contributions depend on the baseline hazard accumulated in each interval.

Assumptions, Diagnostics, and Alternatives

Before presenting converted risks, check that the inputs describe the same event definition, time origin, population, and covariate profile. The baseline survival should be estimated or calibrated at the time horizon of interest, and the hazard-ratio model should be appropriate over the interval. Earlier material on diagnosing proportional hazards and calibration for event-time models addresses the model checks in more detail; for this conversion, their practical implication is that an apparently precise risk can still be misleading if either the hazard pattern or baseline survival is poorly represented.

Check proportional hazards. If the estimated exposure effect changes over time, use a time-varying hazard ratio and integrate it against the baseline hazard, rather than applying the constant power formula.
Check baseline calibration. Compare predicted survival with observed survival at clinically relevant times for an appropriate target population. Good relative-effect estimation does not guarantee accurate absolute risks.
Check the endpoint structure. With competing events, \(1-S(t)\) for an event-specific cause-specific hazard is not generally the cumulative incidence of that cause. The competing-event process must be incorporated to obtain the cause-specific absolute probability.
Check the time range. Predictions far beyond the observed follow-up rely on extrapolation of the baseline hazard and exposure effect, and may be unstable even if the fitted model appears adequate within the data range.

The constant-hazard-ratio conversion is compact and transparent when its assumptions hold. A direct model for survival or event probability, or a model with time-varying effects, may be more suitable when proportional hazards fails. Those alternatives still require a sound baseline-risk model and calibration; changing the model form does not remove the need to define the population and time horizon. For population-average predictions from a covariate-adjusted model, predictions can be calculated for individuals with their covariates and then averaged, rather than treating one conditional prediction as the population average.

Uncertainty also comes from both the hazard-ratio estimate and the baseline survival estimate. Transforming a point estimate gives a point prediction, not a complete uncertainty interval. Because these inputs may be statistically related, separately transforming the endpoints of two marginal intervals does not generally give a valid interval for absolute risk. Joint model-based simulation or resampling can propagate their uncertainty together and produce an interval for the predicted risk or risk difference.

Common Pitfall

Common mistake: “The hazard ratio is 0.65, so the event risk is 35% lower.” This treats a relative instantaneous rate as if it were a relative cumulative probability. The correction is to combine the hazard ratio with baseline survival at a specified time: under constant proportional hazards, \(F_1(t)=1-S_0(t)^\theta\). Then report the resulting absolute risks and their difference in context.

Another mistake is reporting an absolute risk without saying when it applies. A 20% risk by one year and a 20% risk by five years are different statements, even if the numerical value is the same. Always name the horizon, event, exposure contrast, and population or covariate profile.

Check Your Understanding

Answer each question using the stated assumptions. Solutions are included for checking your reasoning.

If \(S_0(t)=0.90\) and a constant hazard ratio is 2.00 through \(t\), what is the comparison event risk by \(t\)?
If \(S_0(t)=0.60\) and the constant hazard ratio is 0.50, calculate the comparison risk and the comparison-minus-reference risk difference.
Why is a single hazard ratio insufficient to calculate absolute event risk?
If the hazard ratio changes across follow-up, what quantity must be accumulated to calculate comparison survival?

Solutions. (1) \(S_1(t)=0.90^{2}=0.8100\), so \(F_1(t)=1-0.8100=0.1900\), or 19.0%. (2) \(S_1(t)=0.60^{0.50}=\sqrt{0.60}\approx0.7746\), so the comparison risk is \(1-0.7746=0.2254\), or 22.5%. The reference risk is \(1-0.60=0.4000\); the risk difference is \(0.2254-0.4000=-0.1746\), or 17.5 percentage points lower. (3) The hazard ratio supplies a relative instantaneous effect, but absolute risk also depends on baseline survival and the time horizon. (4) Accumulate the exposure-specific hazard \(\theta(u)\lambda_0(u)\) over time to obtain \(H_1(t)\), then calculate \(S_1(t)=\exp\{-H_1(t)\}\) and \(F_1(t)=1-S_1(t)\).