PWP Gap-Time Model

Why Reset the Clock After Each Recurrence?

For recurrent events, elapsed time since study entry and elapsed time since the most recent event answer different questions. A subject who has just experienced a recurrence may have a different short-term recurrence hazard from someone whose previous event was years ago, even if both have been followed for the same amount of calendar time. The Prentice-Williams-Peterson gap-time model uses time since the previous event as its analysis clock and models each event order conditionally: a subject is eligible for the \(k\)th-event analysis only after experiencing the first \(k-1\) events.

This specification combines two choices. The clock resets to zero after each recurrence, and the risk set for event order \(k\) is restricted to subjects who reached that order. Neither choice says that recurrent events are independent. Rather, the model describes the \(k\)th recurrence among subjects who have already experienced \(k-1\) recurrences, under a specified hazard model and censoring assumptions.

Model Specification and Conditions

For each event order \(k\), define a gap-time risk indicator \(Y_{ik}(u)\). It equals one while subject \(i\) is observed, has experienced at least \(k-1\) events, and has not yet had event \(k\); otherwise it equals zero. For \(k=1\), the clock starts at study origin. For \(k>1\), it starts at \(T_{i,k-1}\). A subject censored before event \(k\) contributes at-risk time for that order only up to censoring.

A common PWP gap-time specification is a stratified proportional-hazards model. Write \(X_i(t)\) for covariates known just before time \(t\), and let \(\boldsymbol{\beta}_k\) denote the regression coefficients for order \(k\). Conditional on the subject's observed event history up to the start of that gap and on the covariate history, the event-order-specific intensity is

Here \(\lambda_{0k}(u)\) is the unspecified baseline hazard for the \(k\)th event order as a function of gap time. The indicator makes the intensity zero when the subject is not eligible or no longer observed for that order. Covariates must be measured or defined consistently at the corresponding calendar time \(T_{i,k-1}+u\), and time-dependent covariates must be predictable (known just before the event they help model). A simpler specification uses a common coefficient vector \(\boldsymbol{\beta}\) for all event orders while retaining separate baselines \(\lambda_{0k}(u)\).

The model requires a proportional-hazards form within each event-order stratum: covariate effects enter multiplicatively and do not vary with gap time unless the model explicitly includes such variation. The baseline hazard may differ freely across event orders. The observed event history determines eligibility, but this alone does not eliminate the need to account for informative censoring. Censoring should be independent of the future event process conditional on the covariates and observed history included in the analysis, or the analysis should use an appropriate correction. Subjects are ordinarily treated as independent of one another; within-subject recurrence dependence is represented through the conditional event-order structure and can also matter for variance estimation.

Eligibility Across Event Orders

The event-order restriction is essential to the PWP specification. At order \(k\), a subject cannot enter the risk set before experiencing event \(k-1\). Once the subject enters, gap time runs from zero until event \(k\) or censoring. A subject who never experiences the required prior events contributes no follow-up to that order. The risk sets therefore differ across orders, even though the same subjects may appear in several order-specific analyses.

At an observed \(k\)th event, the subject is in the order-\(k\) risk set immediately before that event and leaves it at the event. If multiple subjects have events at the same recorded gap time, the software's tie-handling method applies to that event time. The following example illustrates eligibility; it is not a derivation of a partial likelihood.

From the Model to an Analysis

A standard implementation represents the data in long form, with a separate record for each subject and event order for which the subject became eligible. The analysis time for that record begins at gap time zero; its endpoint is the observed gap to the event or censoring, and an event indicator records whether the event of that order occurred. Event order is used to define separate baseline hazards, and the risk set for each order is formed only from eligible records. When covariates change during follow-up, their values must be aligned with the gap interval and the corresponding calendar time.

For a stratified Cox implementation, the order-specific unspecified baselines are handled by stratifying on event order. If coefficients are allowed to differ by order, the covariate effects must also be specified by order; stratifying the baseline alone does not make coefficients order-specific. The model can accommodate different baseline shapes across event orders without requiring a parametric form for those shapes.

Justification for the Reset Clock

The gap-time clock is a re-expression of time within each event-order interval. During the \(k\)th interval, calendar time \(t\) and gap time \(u\) satisfy \(u=t-T_{i,k-1}\). The prior event time is fixed for that interval, so \(du/dt=1\). Consequently, an instantaneous rate per unit calendar time has the same numerical scale per unit gap time; the model changes the time at which the baseline hazard is indexed, not the unit of time. It assigns the \(k\)th event's baseline hazard according to elapsed time since the preceding event, rather than according to time since study entry.

This clock is useful when recurrence risk is plausibly related to how long it has been since the last event. For example, a short-term elevated risk following an event can appear as a feature of \(\lambda_{0k}(u)\). The model does not require the same baseline hazard at every order: \(\lambda_{01}(u)\), \(\lambda_{02}(u)\), and later-order baselines may differ.

Diagnostics, Alternatives, and Limitations

A careful analysis checks the event times, interval starts, event-order labels, and censoring indicators before fitting the model. Each \(k\)th-event record must start at \(T_{i,k-1}\), and no subject may contribute to order \(k\) before having \(k-1\) observed events. An event accidentally assigned to the wrong order or a gap calculated from study origin rather than the previous event changes both the clock and the risk sets.

After fitting, assess whether the proportional-hazards structure is plausible within event orders, for example by examining suitable residual-based diagnostics and time-by-covariate patterns. Inspect the number of subjects and events at each order: later-order strata may have few observations, sparse covariate groups, or no events, leading to unstable or inestimable effects. Check that censoring patterns are compatible with the stated conditional censoring assumption. If subjects with a high unobserved recurrence propensity are also more likely to leave observation, the conditional model may not remove the resulting selection bias.

The principal alternatives use different time origins or target different summaries. In the PWP total-time model, event-order risk sets are likewise conditional on prior event order, but the baseline hazard is indexed by time since study entry. The Andersen-Gill model uses a common total-time framework and does not impose the same event-order conditioning. A marginal rate model instead targets population-level event rates. These approaches are not interchangeable: clock choice and eligibility rules determine which comparisons the model represents.

The gap-time PWP model is not automatically preferable whenever recurrence is important. It may be difficult to estimate effects for late event orders because few subjects reach them, and a separate baseline by order can consume information. It also conditions on reaching each order, so the order-specific comparison is not the same as a population-average treatment effect across all follow-up. The model's assumptions and estimand should be considered together rather than inferred from the clock alone.