Choosing A Recurrent-Event Model By Estimand

Start With the Target, Not the Model Name

Three recurrent-event models can be fitted to the same cohort and produce coefficients that look like hazard ratios, yet those coefficients need not answer the same question. The choice should begin with the scientific contrast: do you want an event intensity that updates with a person’s event history, a hazard among people who have reached a particular event order, or an event-order-specific contrast that averages over earlier histories? These are different estimands, not interchangeable ways of describing one universal treatment effect.

The Andersen–Gill (AG), Prentice–Williams–Peterson (PWP), and Wei–Lin–Weissfeld (WLW) approaches differ chiefly in their risk sets and conditioning. Their assumptions and coefficient interpretations must be considered along with the research question. A good fit statistic cannot, by itself, make a model’s estimand match the question.

Three Targets and Their Risk Sets

Let \(N_i(t)\) be the number of recurrent events experienced by subject \(i\) by time \(t\), and let \(Y_i(t)\) indicate that the subject is under observation and eligible for the event process at \(t\). Let \(Z_i\) be a treatment indicator. The event history just before \(t\) is denoted by \(\mathcal{H}_i(t-)\). The conditional event intensity describes the instantaneous event rate given that history:

In an AG model, a common specification is \(\lambda_i\{t\mid\mathcal{H}_i(t-)\}=\lambda_0(t)\exp(\beta Z_i)\). The coefficient \(\beta\) describes a conditional intensity ratio, \(\exp(\beta)\), for subjects with different \(Z_i\) values at the same modeled time and with the histories represented by the model held fixed. The model uses a recurrent-event counting process and does not require a person to have reached a specified event order before contributing to its risk set.

A PWP model instead targets an event-order-specific hazard conditional on reaching that order. For the \(k\)th event, a subject enters that event’s risk set after experiencing \(k-1\) events and remains eligible until the \(k\)th event or censoring. A PWP model can use total time since study entry or gap time since the preceding event. These clocks define different time comparisons; neither changes the defining conditioning on having reached event order \(k\).

A WLW model specifies marginal event-order-specific hazards. For the \(k\)th event outcome, subjects can contribute from the common time origin until that event or censoring, including subjects who have not yet experienced \(k-1\) events. Thus, the event-order contrast is not restricted to people who have reached the preceding order. “Marginal” here refers to not conditioning the event-order-specific contrast on having reached the preceding order; it does not mean that all dependence among a subject’s event outcomes disappears.

Assumptions Before Interpretation

For any of these models, the recorded event times, event order, censoring times, and covariates must support the specified risk sets. Censoring must be independent of the relevant event process conditional on variables included in the analysis, or the analysis must use a justified method that accounts for informative censoring. The proportional-hazards or proportional-intensity form must be reasonable for a single constant coefficient over the modeled time range. If that constancy is not plausible, the coefficient may be an inadequate summary.

The AG intensity interpretation requires that the conditioning history be adequately represented for the intended comparison. PWP requires that event order and eligibility be defined correctly; its conditional target is specifically about subjects who reach that order. WLW requires meaningful event-order outcomes and correct handling of censoring for each one. Because a subject can contribute multiple event-order outcomes, within-subject dependence matters for uncertainty estimates. A robust variance estimator can address dependence in standard errors under suitable conditions; it does not change the estimand or repair a mismatched risk set.

The treatment coefficient in an observational study is generally an adjusted association unless the design and analysis support a causal interpretation. Even in a randomized study, a hazard ratio is a contrast of hazards or intensities, not automatically a contrast in mean event counts or in the probability of ever having an event.

Why the Distinction Matters

The choice changes who contributes to a comparison and what the coefficient summarizes. Conditioning on having reached an event order can be scientifically appropriate, but it changes the target population. A WLW event-order model avoids that eligibility restriction, but its marginal event-order contrast should not be described as the effect among people who have reached the preceding order. The AG model targets a history-conditioned intensity, which is not identical to either of those event-order-specific questions.

A short derivation clarifies why the PWP condition is part of the estimand. Let \(T_{ik}\) be the time of subject \(i\)’s \(k\)th event. A PWP \(k\)th-event risk indicator is zero before the subject has experienced \(k-1\) events, and after that point is one only while the subject remains observed and has not yet had event \(k\). Thus the conditional hazard has the form

The event \(T_{ik}\) can occur in the interval only after \(T_{i,k-1}<t\). Restricting the conditional probability to people who have reached order \(k\) therefore defines the PWP risk set and target. In a WLW model for the same \(k\)th event, the risk set is instead defined by not yet having the \(k\)th event and remaining observed; it is not narrowed to subjects with \(k-1\) prior events. Changing that condition changes the quantity being modeled.

These three approaches also do not automatically answer a question about the average number of events accumulated by a fixed time. An event intensity, an event-order-specific hazard, and an average event count are distinct summaries. If the primary question is about average burden or a marginal event rate, specify that target directly and use a method designed for it rather than relabeling one of these hazard ratios.

Selection, Diagnostics, and Limitations

Useful diagnostics include tabulating the number of participants and events contributing to each event order, checking that risk-set entry and exit times match the intended definition, and examining whether the fitted treatment contrast is plausibly stable over time. Sparse higher-order events can make event-order estimates imprecise or unstable. A marked change in the composition of participants across event orders also warns that a conditional PWP contrast and a marginal WLW contrast may differ substantially for substantive reasons.

A frequent mistake is to select whichever model gives the most favorable or smallest \(p\)-value, then describe its coefficient as the treatment effect on recurrent events generally. The correction is to specify the estimand before looking at the result and to report the actual risk-set and conditioning interpretation. Robust standard errors do not make different estimands equivalent, and agreement in point estimates would not prove that the models answer the same question.

No one approach is universally best. AG is natural when the scientific target is a history-conditioned event intensity over the observed recurrent-event process. PWP is natural when event order and the experience of prior events define the population of scientific interest. WLW is useful when event-order-specific marginal outcomes are the target and the comparison should not be restricted to prior-event attainment. All can be limited by small numbers of later events, informative censoring, misspecified time effects, or a mismatch between the hazard target and the research question.