Rate Ratios Versus Event-Hazard Ratios

Two Ratios, Two Questions

A rate ratio and an event-hazard ratio may both compare how often recurrent events occur in two groups, but they do not generally answer the same question. A marginal recurrent-event rate ratio compares event counts relative to accumulated person-time at the population level. A conditional event-intensity hazard ratio compares instantaneous event intensities for individuals at risk with specified observed histories. The denominator and the conditioning set determine which interpretation is justified.

This distinction matters when participants have different numbers or timings of earlier events. Their histories can affect later event intensity, and the groups being compared may spend different amounts of follow-up time in different histories. Consequently, a population-average rate ratio can differ from a history-specific intensity ratio, even when the latter is constant across histories.

Here, “at risk” means that another event can be counted at that time; it does not mean that a participant must be event-free. A recurrent-event intensity is not simply the hazard of a first event. A participant can remain at risk after an event, with their updated history included in the conditioning information.

What Is in Each Denominator?

For a group followed over an interval, the descriptive rate is total events divided by total observed person-time. Someone observed for two years contributes twice as much person-time as someone observed for one year. An individual can contribute multiple events, and each event contributes to the numerator. The resulting group rate describes the average event frequency per unit of observed time; it does not, by itself, specify how event intensity varies with an individual’s past history.

A conditional intensity has a different denominator in its interpretation: it describes the instantaneous event rate among participants under observation and at risk, conditional on their specified histories and covariates. A model might compare two covariate values for people with the same modeled history. For example, a proportional-intensity model can be written as follows, where \(X_i\) is a group indicator and \(\lambda_0\) may depend on the history:

Under this model, \(\exp(\beta)\) is the conditional event-intensity ratio for \(X_i=1\) versus \(X_i=0\), at the same time and modeled history. Its interpretation requires the model’s intensity structure to be appropriate; it is not automatically a ratio of total events per person-time.

A marginal model for recurrent-event counts instead relates a population-average count, or a mean count per unit of exposure, to covariates. For instance, a log-mean model with a person-time offset may use \(\log E[N_i]=\log(T_i)+\alpha+\beta X_i\), where \(T_i\) is observed person-time. In that model, \(\exp(\beta)\) compares modeled mean event rates for the covariate groups, conditional on any other covariates included in the mean model. It does not condition on each participant’s full event history in the way an event-intensity model does. GEE methods address the correlation among repeated observations when estimating a marginal mean model; that correlation adjustment does not turn its rate-ratio coefficient into a conditional intensity ratio.

Why Averaging Can Change the Ratio

The distinction can be made precise using the defining property of an intensity. Let \(Y_i(t)\) indicate that participant \(i\) is under observation and at risk, and write \(\lambda_i(t\mid\mathcal{H}_i(t-),X_i)\) for the intensity when \(Y_i(t)=1\). For a sufficiently small time interval \(dt\), the expected event increment, conditional on the observed history, is approximately the intensity multiplied by \(dt\). Averaging over histories in a group gives:

Dividing by the expected person-time contribution, \(E[Y_i(t)\mid X_i=x]dt\), gives the group’s marginal rate at that time:

Thus, the marginal rate is an average of conditional intensities over the histories represented among the group’s observed person-time. If those history distributions differ between groups, the two averages need not retain a common history-specific ratio. In particular, a constant conditional ratio \(\exp(\beta)\) does not guarantee that \(r_1(t)/r_0(t)=\exp(\beta)\) when the groups average over different histories. If the histories are standardized to the same distribution and the conditional ratio is constant over those histories, then the standardized marginal rates have that same ratio. This is a special condition, not an automatic property of crude rates.

Assumptions, Diagnostics, and Alternatives

A rate comparison is interpretable only if its numerator and denominator refer to compatible populations and periods. Check that recurrent events are defined and counted consistently, person-time is calculated consistently, and the rules for starting and ending follow-up are clear. If censoring or loss to follow-up depends on event risk, a simple events-per-observed-person-time calculation may not represent the intended population rate; appropriate modeling or adjustment may be needed. A model-based rate ratio also relies on an adequate mean model for the covariates and exposure time. Robust standard errors can address some misspecification of within-person correlation, but they do not repair a misspecified mean or biased person-time.

An event-intensity ratio requires a clearly defined risk process and a history set that is available just before each event time. Covariates used to predict the intensity must be appropriately timed, and the model must describe the conditional intensity adequately. If a constant ratio is claimed, the model must support that constancy over the relevant times and histories. A large difference between a crude rate ratio and a conditional ratio can be a prompt to inspect history distributions, follow-up patterns, and model definitions; it is not, by itself, proof that either estimate is wrong.

The main alternatives target different summaries. A marginal rate model, including a GEE count model with a person-time offset, targets a population-average mean rate and accommodates repeated observations through its estimating approach. A conditional event-intensity model targets instantaneous rates given specified histories. Neither is universally preferable: choose according to whether the scientific question concerns average event burden per unit follow-up or event intensity among people with comparable histories. They may also differ in covariates, time scales, risk sets, and assumptions, so comparing their coefficients as though they were interchangeable can be misleading.

Common Mistakes and Limitations

• Calling every event-rate ratio a hazard ratio. Events divided by person-time give a rate comparison; a conditional event-intensity ratio additionally specifies an instantaneous risk set and history.
• Assuming equal conditional ratios force equal marginal ratios. Different history distributions can give the groups different weights when conditional intensities are averaged.
• Ignoring the person-time definition. Different follow-up or censoring rules can change both the denominator and the population represented by the rate.
• Interpreting an intensity ratio as a causal effect without justification. A conditional model coefficient is an association under its model unless the study design and assumptions support a causal interpretation.
• Using a robust variance estimator as a cure-all. Robust variance estimation may protect uncertainty estimates against some correlation misspecification; it does not make an inappropriate estimand, biased follow-up, or incorrect mean model valid.

These ratios are summaries, not full descriptions of recurrent-event experience. A single rate ratio can hide when events occur, how event histories evolve, and how much participants differ. Conversely, a conditional intensity ratio is tied to its conditioning set and may not summarize the population’s average burden. Good reporting makes those boundaries explicit.