Worked Comparison Of PWP Clock Choices

Why the PWP clock choice matters

A Prentice–Williams–Peterson (PWP) model conditions each event-order analysis on having experienced the preceding events. For example, the model for a second recurrence concerns individuals who have had a first recurrence and remain under observation without yet having a second. Once this conditional structure is set, there is still a choice of time scale: measure time from a common origin, or reset the clock at the preceding recurrence. That choice changes the time at which people are compared, the resulting risk sets, and what a fitted hazard ratio describes.

The total-time and gap-time clocks are not interchangeable labels for an otherwise identical analysis. They can produce different partial-likelihood comparisons and different fitted baseline hazards. A coefficient from one clock should therefore be interpreted on that clock’s scale, not treated as a direct estimate of the other clock’s effect.

For a fixed covariate \(Z_i\), a typical event-order-stratified proportional-hazards specification is

for total time, and

for gap time. Here \(\mathcal{H}_{i}\) denotes the observed event history up to the relevant time, and each event order \(k\) has its own baseline hazard. The formula for gap time uses \(u\), time since the previous event, rather than \(t\), time since the common origin. These specifications assume a correctly recorded ordered history, suitable risk-set construction, and proportional covariate effects within the event-order stratum. Censoring must be non-informative conditional on the modeled information for the intended likelihood-based inference. Tied censoring times are permitted; tied event times require an appropriate tie-handling method rather than an assumption that all event times are distinct.

One cohort, two clocks

The following small, hypothetical cohort makes the clock change visible. Times are in years from study entry. A dash in the second-event column means no second recurrence was observed before censoring. The table records first and second recurrence times where observed, followed by each person’s censoring time if they remain under observation that long.

All five people enter the second-event process when their first recurrence occurs. At any given time, a person is included only if they have entered that process, have not yet had a second recurrence, and have not been censored. For gap time, the same eligibility conditions apply, but a person’s second-event clock starts at their first recurrence. In the tables below, risk sets are evaluated immediately before the listed event time. The tied censoring times at year 11 do not create a problem for the second-event comparisons shown here.

What each clock compares

Under total time, a second-event comparison at time \(t\) is among individuals whose first recurrence has already occurred by \(t\), who remain observed, and who have not had a second recurrence by \(t\). Someone who experiences the first recurrence late enters this risk set late. Thus a second-event hazard ratio describes the relative instantaneous rate of a second recurrence at the same time from study origin, conditional on the required event history and continued eligibility.

Under gap time, a comparison at time \(u\) is among people who have been in the second-event process for at least \(u\) years since their first recurrence and remain eligible. Individuals can therefore be at the same modeled time despite having experienced their first recurrences at different calendar times. A gap-time hazard ratio describes the relative instantaneous rate at the same elapsed time since the preceding recurrence, conditional on event history and continued eligibility.

Predicted timing depends on the clock

The baseline hazard is indexed by the chosen time scale, so predictions must use the fitted baseline from the corresponding model. For a specified covariate value, conditional survival over an interval is obtained from the cumulative hazard over that interval. With a constant hazard \(\lambda\), the probability of no event during an additional \(s\) years is \(\exp(-\lambda s)\); the median additional waiting time solves \(\exp(-\lambda s)=0.5\).

Derivation: why the partial-likelihood comparisons change

For an untied event at time \(t_{jk}\) in event-order stratum \(k\), the Cox partial-likelihood contribution compares the covariate value for the person \(i_j\) who has the event with the covariate values for all individuals in that stratum’s risk set \(R_k(t_{jk})\). For a single covariate, that contribution is

This follows by dividing the event person’s modeled hazard by the sum of modeled hazards over the eligible risk set. The common baseline hazard cancels because it multiplies every hazard in that risk set. Changing the clock changes which individuals are in the risk set at the event’s modeled time, and may change which event times coincide. Therefore it changes the denominator, and generally changes the partial likelihood being maximized. The same records can consequently yield different coefficients and baseline-hazard estimates under the two clocks.

The formula above is for an untied event. When event times tie, the likelihood contribution needs an appropriate tie method; the tied gap times for A and B in the example cannot be treated as though one event occurred first without justification. A software default is not a substitute for reporting or checking how ties were handled.

Diagnostics, alternatives, and limitations

Before fitting either model, verify the event history and the time transformations from the original records. In particular, check that every gap time is measured from the immediately preceding event, that a person enters the \(k\)th-event risk set only after event \(k-1\), and that censoring ends observation rather than being recorded as an event. Compare the number at risk at selected event times with a hand-constructed risk set. Unexpectedly large gaps, negative gaps, or a person at risk before their qualifying recurrence often indicate a data-structure error.

After fitting, assess proportional hazards on the chosen time scale and examine whether the modeled covariate effect appears to vary with that scale. A reasonable fit on total time need not imply a reasonable fit on gap time, or vice versa. Also inspect the number of events and risk-set sizes by event order; later-order strata can be sparse, making estimates unstable. The assumed conditional censoring mechanism deserves attention: if follow-up ends for reasons related to the unobserved recurrence process after accounting for modeled information, ordinary likelihood-based interpretation may fail.

The main alternative is not automatically a better clock. The Andersen–Gill approach uses a different recurrent-event formulation and does not impose the same event-order conditioning. The marginal WLW model uses another risk-set construction, which is a distinct modeling target. These alternatives should be chosen to match the scientific estimand and data structure, rather than used to avoid interpreting a clock-specific result.

PWP clock choice has limitations. Neither clock removes the need to define the event process, account for relevant covariates, and assess model assumptions. A hazard ratio is conditional on eligibility and the modeled time scale; it is not a rate ratio for the total number of recurrences, nor does it directly state the probability that an individual will recur by a given date. Predicted event timing further depends on the estimated baseline hazard and the conditioning history. When the scientific question concerns calendar time since study entry, total time may be more directly aligned; when elapsed time since a prior recurrence is central, gap time may align better. This is a design and estimand decision, not a choice to make solely by comparing which coefficient is larger.