Common Risk-Set And Interpretation Errors

Why Recurrent-Event Interpretations Go Wrong

A recurrent-event data set can contain many event records but relatively few independent subjects. The same subject may contribute several events, changing risk status over follow-up and carrying a history that helps define the next event's intensity. Errors arise when the rows of the data set are mistaken for independent people, when the analysis clock is unclear, or when a model's conditional event intensity is reported as though it were a population-average rate.

These are not merely wording problems. They can change the risk set, the estimand, the uncertainty calculation, or all three. Before interpreting a coefficient or rate, identify who is at risk at each time, which clock measures time, and whether the reported quantity is conditional on a subject's history or averaged across subjects.

The exact meaning of “at risk” depends on the event process. A subject may return to the risk set after an event, or may be temporarily not at risk during recovery. Censoring or death may end observation or prevent further events. The risk indicator must reflect the scientific process and the data collection rules; counting every enrolled subject as at risk for all elapsed time is not automatically correct.

Risk Sets Are About Subjects, Not Event Rows

A recurrent-event model generally uses each subject's event history and at-risk time. Event records from the same person are related observations, not new independent people. A model may specify a mean or intensity structure for the event process, while the variance calculation must account for dependence among events contributed by one subject. Treating each event as an independent subject can make the apparent sample size much larger than the number of independent units and can make standard errors too small.

This does not mean every analysis that uses event records is invalid. The records are needed to describe when events occurred. The error is to assign each record independent-subject status without a model and variance calculation that justify it. Subject-clustered robust variance is one established approach for inference under a specified marginal model, provided subjects are independent clusters and there are enough clusters for its large-sample approximation. It does not repair an incorrect risk set, clock, or mean model.

A useful diagnostic is to compare the number of subjects with the number of event records and to inspect how many events each subject contributes. If the analysis file has one row per event, check that the likelihood or estimating equations and variance method still recognize the subject identifier. Also verify that subjects re-enter the risk set only when the process says they are at risk. An event can increase \(N_i(t)\) without necessarily implying immediate return to risk.

Total-Time and Gap-Time Clocks

A recurrent event can be indexed by time since a common origin, such as enrollment, or by time since the preceding event. These clocks answer different questions. With a total-time clock, a subject who experiences events at times \(T_{i1},T_{i2},\ldots\) is indexed by elapsed time from the common origin. With a gap-time clock, the time for event \(j\) is measured from the previous event: \(U_{ij}=T_{ij}-T_{i,j-1}\), where \(T_{i0}=0\) if the first gap begins at the origin.

A total-time intensity can be written as \(\lambda_i(t\mid\mathcal{H}_i(t-))\). A gap-time intensity for event \(j\) can instead be written as \(\lambda_{ij}(u\mid\mathcal{H}_i)\), where \(u\) is elapsed time since the previous event. Both intensities have units of events per unit of their respective time clock. Changing from one clock to the other changes the baseline time pattern and often the risk-set organization; it is not just relabeling an axis.

For example, a model with \(\lambda_0(s)=0.02s\), where \(s\) is measured in years, has intensity in events per year. The coefficient \(0.02\) has units of events per year squared, so multiplying it by \(s\) in years yields events per year. This unit check matters: the intensity itself is not measured in events per year squared.

Conditional Intensities Are Not Automatically Marginal Rates

A conditional intensity describes the instantaneous event rate for a subject who is at risk at time \(t\), conditional on the specified covariates and observed history. A marginal rate summarizes event occurrence averaged over a target population or group, usually relative to that group's person-time. These quantities need not agree because the at-risk subjects can differ in covariates and histories, and because a conditional model can describe a different comparison from a crude population rate.

For a population with individual conditional intensities \(\lambda_i(t\mid\mathcal{H}_i(t-))\), the instantaneous average rate among those at risk is risk-set weighted:

The numerator averages intensity while weighting subjects who are currently at risk; the denominator is the proportion at risk. Thus this is not generally the intensity for a fixed “typical” subject. A marginal rate over an interval is instead expected event count divided by expected person-time under the specified population and follow-up definition. A conditional hazard or intensity ratio should not be renamed a marginal rate ratio without showing that the relevant averaging and risk-time definitions make them coincide.

A Short Derivation: Why the At-Risk Average Uses Risk-Set Weights

For a small interval of length \(dt\), the definition of conditional intensity gives, for an at-risk subject, \(E\{dN_i(t)\mid\mathcal{H}_i(t-)\}=Y_i(t)\lambda_i(t\mid\mathcal{H}_i(t-))dt+o(dt)\). Average this relation over subjects. Linearity of expectation gives \(E\{dN_i(t)\}=E\{Y_i(t)\lambda_i(t\mid\mathcal{H}_i(t-))\}dt+o(dt)\). The expected at-risk person-time in that interval is \(E\{Y_i(t)\}dt\). Dividing expected events by expected at-risk person-time and taking the limit as \(dt\) tends to zero yields the risk-set weighted expression above. Subjects not at risk contribute neither events nor at-risk time at that instant.

This argument also identifies an assumption behind the calculation: the intensity must correctly describe the conditional mean event increment given the included history, and the risk indicator must correctly represent exposure to the event process. It does not claim that a conditional intensity ratio equals a marginal rate ratio, or that an average conditional intensity remains constant over time.

Checks, Alternatives, and Limitations

Before interpreting a recurrent-event analysis, use a short audit:

• Unit of independence: identify the independent sampling units. If one person contributes multiple events, inference must account for within-person dependence.
• Risk set: state who is at risk at each time, when subjects enter or leave, and whether they can re-enter after events.
• Clock: identify the origin and time scale. Verify that event times, risk intervals, and baseline functions all use that same clock.
• Target quantity: label a result as conditional intensity, rate ratio, event-hazard ratio, or marginal rate, as appropriate. Include the population and time interval for a marginal summary.
• Variance and model adequacy: inspect event counts by subject and compare model-based and subject-clustered uncertainty where appropriate. A variance correction does not validate a misspecified mean or the wrong estimand.

Main alternatives depend on the question. Marginal rate models target population-average event rates or rate contrasts and require an appropriate approach to within-subject variance. Mean cumulative functions describe average accumulated event burden over time without requiring an event-level hazard interpretation. Conditional intensity models describe event occurrence given the modeled covariates and history. These are not interchangeable labels for one answer: the chosen analysis should match the intended contrast and follow-up process.

Diagnostics include plotting event times both from the common origin and as gaps, tabulating risk time by event order, checking whether subjects have substantial variation in event counts, and comparing observed and model-predicted event rates over the stated clock. A sharp change after an event, systematic discrepancies by event order, or a changing risk-set composition may signal that a single interpretation is inadequate. Such checks flag concerns but do not by themselves identify the correct alternative.

No single risk-set convention, clock, or summary is universally best. A marginal model can be unsuitable if the target is explicitly a history-conditional intensity; a conditional model can be unsuitable if the intended result is a population-average burden. Both can be difficult to interpret when observation ends informatively or when the at-risk process is misspecified. These choices should be stated before results are summarized, not inferred after seeing a preferred coefficient.