Fixed-Horizon Recurrent-Event Burden Contrasts

Why Fix a Horizon?

A recurrent-event study may follow people for different lengths of time, and the group difference in event burden can change as follow-up continues. A contrast at a prespecified horizon gives a direct answer at a time point that matters to the study: for example, how many hospitalizations per person have occurred by 12 months on average? The contrast is about total accumulated events through that horizon, not the rate at an instant.

The mean cumulative function (MCF) summarizes the expected number of recurrent events accumulated by time \(t\). This tutorial uses group-specific MCF estimates to form differences and ratios at a fixed time \(\tau\). The emphasis is on defining the population quantity, estimating its contrast, and interpreting the result in context.

The mean includes all events counted under the study’s recurrent-event definition; it is not the proportion of people with at least one event. A difference is expressed in events per person by \(\tau\). A ratio is dimensionless and describes relative accumulated burden. Both depend on the chosen horizon, population, event definition, and follow-up start.

Estimation and Uncertainty

For each group, estimate the MCF at \(\tau\), denoted \(\widehat{\mu}_g(\tau)\). The fixed-horizon estimates are plug-in contrasts: subtract the estimated group means for a difference, or divide them for a ratio. In complete follow-up with independent subjects, the group MCF estimate at \(\tau\) is the sample mean of the event counts through \(\tau\). With censoring, use the MCF estimator appropriate to the observation process; its variance estimate must account for censoring and recurrent events as required by that procedure.

For independent groups, the estimated variance of a difference is the sum of the two estimated group variances. For a ratio, inference is usually performed on the logarithmic scale because the ratio is positive and its sampling distribution is often more nearly symmetric there. The delta method gives the approximate variance of the log ratio. These standard errors rely on a large-sample normal approximation and valid variance estimates.

Here \(\widehat{V}_g(\tau)\) is the estimated variance of the group-specific MCF estimate. If the groups are not independent—for example, if observations are paired or clustered—the covariance between group estimates must also be handled; simply adding variances is then generally insufficient. In a difference, a large-sample 95% confidence interval is \(\widehat{\Delta}(\tau)\pm1.96\,SE\). For a ratio, form the interval on the log scale and exponentiate both endpoints. Such an interval describes values compatible with the data and model assumptions under repeated sampling; it is not a probability statement about the fixed parameter.

Conditions and a Short Justification

The contrast is meaningful only after defining the target population, the time origin, the event-counting rule, and the horizon. The horizon should be selected before inspecting group differences when it is intended to support a confirmatory comparison. The same time scale and event definition must apply to both groups.

With complete follow-up through \(\tau\), independent sampling within each group makes the sample mean a direct estimator of \(E\{N(\tau)\}\). With right censoring before \(\tau\), the simple sample mean among those still observed at \(\tau\) generally does not estimate that population mean: those people may differ from those censored earlier. An MCF estimator can use the observed event and follow-up histories, but its interpretation requires censoring to be independent of future event experience, at least conditional on variables incorporated by the estimation procedure. There must also be adequate follow-up support near \(\tau\); an estimate at a horizon with almost no one under observation is unstable and may not be reliably estimable.

The variance addition for independent groups follows from the variance of a difference. If \(A\) and \(B\) are independent estimates, then \(\mathrm{Var}(A-B)=\mathrm{Var}(A)+\mathrm{Var}(B)\). For a ratio, apply a first-order Taylor expansion to \(h(a,b)=\log(a/b)\). Its partial derivatives are \(1/a\) and \(-1/b\). Therefore, when the estimates are independent, the approximate variance is \(V_1/\mu_1^2+V_0/\mu_0^2\), with unknown means and variances replaced by their estimates. This approximation is unreliable if either estimated mean is close to zero or the sample is too small for the normal approximation.

Choosing and Interpreting a Contrast

The difference and ratio answer distinct questions. A difference focuses on the additional expected number of events per person by \(\tau\), which can be useful for service planning. A ratio expresses relative burden and can be easier to compare across settings with different baseline event frequencies. Neither is universally preferable; choose based on the scientific question and report the horizon with the estimate.

A fixed-horizon contrast differs from the rate ratios and event-hazard ratios treated in the earlier material on those subjects: it compares cumulative mean counts by a specified time. It also differs from a comparison of MCF curves over the entire follow-up period, because it reduces the curves to a single prespecified time point. A pointwise contrast can be clinically interpretable, but it does not describe when during follow-up the group means diverged. If results at many horizons are selected after viewing the data, the resulting uncertainty statements do not automatically account for that selection.

The population definition also matters. In an observational comparison, groups may differ in baseline characteristics, so a contrast describes their observed population means and does not by itself show that group membership caused the difference. If the estimand is intended to describe a standardized population, the estimation procedure must target that population explicitly. Likewise, if death ends the opportunity to experience further events, the mean count through \(\tau\) reflects both recurrent-event experience and the time people remain alive and observed; interpretation should state this clearly rather than treating it as a pure event intensity.

Diagnostics and Common Pitfalls

• Inspect support at the horizon. Check how many subjects remain under observation near \(\tau\), the number censored before it, and whether the MCF estimate becomes erratic near the end of follow-up. A nominally computable estimate can still be too weakly supported for reliable inference.
• Check censoring plausibility. Compare follow-up patterns across groups and relevant covariates. Strong evidence that censoring is related to event experience can invalidate an estimator that assumes independent censoring; use an approach that addresses the dependence or qualify the result.
• Check the scale and denominator. A ratio near a zero reference burden can be extreme and imprecise even when the absolute difference is small. Report group-specific means and usually the difference as well as the ratio when both are relevant.
• Do not confuse a fixed-horizon mean with a rate. Dividing an accumulated count by time does not automatically produce a valid rate contrast, especially with varying follow-up or changing event patterns.
• Do not interpret a nonsignificant result as equality. A wide interval may include both meaningful increases and decreases. Report the point estimate and interval, not only a test conclusion.

A common mistake is to say, “The ratio is 2, so group 1 has twice the instantaneous risk of an event.” That is not what the calculation establishes. The ratio compares expected accumulated counts by \(\tau\); it does not assert a constant relationship over time or compare event hazards at each instant. The correction is to state the horizon and say, for example, “the estimated mean number of events per person by 18 months was twice as large in group 1.”