What a WLW Coefficient Describes
The defining feature of the Wei-Lin-Weissfeld (WLW) approach is that it models a separate failure-time outcome for each event order. The coefficient for the second event, for example, concerns the marginal distribution of the time to a person’s second event in the original cohort. Its interpretation is not restricted to people who had already experienced a first event at the time under consideration. This distinction separates WLW coefficients from conditional effects in PWP models.
Let \(T_{ij}\) be the time from the common study origin to the \(j\)th ordered event for subject \(i\), and let \(X_i\) be that subject’s covariate vector. A \(j\)th-event outcome is observed if the subject experiences that event before censoring; otherwise, it is right-censored at the end of observed follow-up. In WLW risk sets, subjects enter the risk set for each event order from the common origin, including those who have not yet experienced earlier events. Thus a subject who has not had a first event can still be in the risk set for the second-event time.
The model allows the baseline hazard and covariate coefficients to differ by event order. A common coefficient across orders is a restriction, not an automatic property of WLW. The model is a proportional-hazards model for each event-order outcome: within an order, a covariate contrast has a hazard ratio that does not vary with time. It also requires appropriate handling of censoring. In particular, censoring of each event-order outcome must be non-informative conditional on modeled covariates for the usual interpretation of the Cox model to apply. The analysis also relies on correctly recorded event order and a clearly defined time origin.
From Coefficient to Marginal Hazard Ratio
For two covariate vectors \(x_a\) and \(x_b\), the model implies the event-order-specific hazard ratio
The baseline hazard cancels in this ratio. For a one-unit increase in a single continuous covariate \(x\), with other covariates held fixed, the hazard ratio is \(\exp(\beta_{jx})\). A positive coefficient means a higher instantaneous hazard of the \(j\)th event among subjects still free of that event at time \(t\); it does not mean that the event is more likely by every specified follow-up time, nor does it directly give a difference in event probabilities.
Worked Example: Interpreting a Second-Event Coefficient
Suppose, in a hypothetical cohort, treatment is coded 1 for the treatment group and 0 for the comparison group. The fitted WLW coefficient for the second event is \(\hat\beta_2=0.405\), with a 95% confidence interval for the coefficient of \((0.095,0.715)\). Assume the event-order-specific proportional-hazards model and censoring assumptions are reasonable.
The estimated treatment-to-comparison hazard ratio for the second event is
Thus, at a given time \(t\), the estimated instantaneous hazard of the second event is 1.50 times as high in the treatment group as in the comparison group, among subjects in the respective marginal second-event risk sets and conditional on any other modeled covariates. Those risk sets include subjects who have not yet had a first event.
Exponentiating the confidence limits gives \((\exp(0.095),\exp(0.715))=(1.10,2.04)\), to three significant figures. Under the model and confidence-interval procedure, this is a 95% confidence interval for the treatment hazard ratio for the second-event time. It does not mean there is a 95% probability that the fixed hazard ratio lies inside this particular interval.
Marginal Effects Versus Conditional Recurrence Effects
A conditional PWP model for event order \(j\) restricts its risk set to subjects who have experienced the preceding \(j-1\) events and remain under observation. Its coefficient therefore describes an association with the next event among recurrence survivors eligible for that event order. In contrast, the WLW risk set for \(T_{ij}\) is formed from the original cohort and includes subjects who have not yet experienced earlier recurrences. A WLW coefficient is consequently not an effect “among people who reached event \(j\).”
| Feature | WLW event-order coefficient | Conditional PWP coefficient |
|---|---|---|
| Target outcome | Marginal time to the \(j\)th event from the common origin | Time to the next event among those who experienced the preceding events |
| Risk set at time \(t\) | Subjects still free of the \(j\)th event, including some with no earlier event | Subjects eligible for event order \(j\), having reached the required preceding order |
| Interpretive emphasis | Event-order-specific association in the original cohort | Association conditional on recurrence history and eligibility |
The distinction matters because reaching a later recurrence order selects a subset of the cohort. If covariates are associated with earlier recurrence or with remaining under observation, the people who reach event order \(j\) can differ systematically from the original cohort. The two models then address different questions and need not yield similar coefficients. Neither coefficient is automatically causal: causal interpretation requires additional design and assumptions beyond the regression model.
Worked Example: Effects That Differ by Event Order
Consider a hypothetical study with a binary exposure and separate WLW coefficients \(\hat\beta_1=0.182\) for the first event and \(\hat\beta_2=0.588\) for the second. Assume that each event-order model is adequately specified and its censoring assumption is plausible.
For the first event, the estimated exposure hazard ratio is \(\exp(0.182)=1.20\). For the second event, it is \(\exp(0.588)=1.80\). The second estimate describes a larger relative hazard contrast for the marginal second-event time than the first estimate describes for the marginal first-event time. It is not an estimate of an 80% increase among only those who have already had a first event.
The ratio of the two estimated hazard ratios is \(1.80/1.20=1.50\). This is a descriptive comparison of the point estimates, not a test that the coefficients differ. Since both outcomes come from the same subjects, their estimates can be dependent. A formal comparison needs an appropriate estimate of the variance of \(\hat\beta_2-\hat\beta_1\), including their covariance. Separate confidence intervals alone do not provide that comparison.
Scaling, Coding, and Interpretation
A coefficient is tied to how its covariate is coded. For a continuous covariate, \(\exp(\beta_{jx})\) refers to a one-unit increase; rescaling the covariate changes the numerical coefficient but not the fitted contrast when reported on the same scale. For a categorical covariate, interpretation depends on the stated reference category. With multiple covariates, the ratio compares two specified covariate profiles, holding the remaining terms fixed.
Worked Example: Reporting a Contrast on a Useful Scale
Suppose a hypothetical WLW model for the third event uses medication dose in milligrams, with \(\hat\beta_3=0.030\) per milligram. Assume dose enters linearly on the log-hazard scale and the proportional-hazards assumption is reasonable over the study period. A 1 mg contrast has estimated hazard ratio \(\exp(0.030)=1.03\), which may be difficult to interpret clinically.
For a 10 mg contrast, the log-hazard difference is \(10(0.030)=0.300\). Therefore,
The estimate is a 1.35-fold marginal hazard of the third-event time for two otherwise comparable profiles differing by 10 mg, at a given time among subjects still free of their third event. It is not a 35 percentage-point difference in the probability of having a third event. The model’s linear dose effect on the log-hazard scale is an assumption; it should not be presumed merely because dose is numeric.
Justification and Practical Checks
The hazard-ratio interpretation follows directly from the model. For two profiles \(x_a\) and \(x_b\), their event-order-\(j\) hazards are \(\lambda_{0j}(t)\exp(\beta_j^{\mathsf T}x_a)\) and \(\lambda_{0j}(t)\exp(\beta_j^{\mathsf T}x_b)\). Dividing cancels the common baseline hazard:
This calculation also shows why the interpretation is order-specific: both the baseline hazard and coefficient vector carry the index \(j\). It gives a relative instantaneous hazard, not an absolute risk or a cumulative event count.
Interpretation can fail when the event-order outcome is constructed incorrectly, when censoring is informative, or when a single proportional effect is implausible. For example, if exposure has a strong early association with the second event but little association later, a single \(\beta_2\) averages over a time-constant log-hazard contrast that may not describe either period well. Event-order-specific estimates and appropriate plots or time-varying-effect checks can reveal patterns that a single pooled coefficient would conceal. Conclusions should also reflect sparse later-event data: a point estimate may be unstable when few subjects experience high-order events.
A common mistake is to read \(\exp(\hat\beta_2)\) as the hazard ratio for recurrence among people who have already had a first event. That interpretation silently changes the risk set and estimand. The correction is to state that it compares the marginal hazards of the second-event time in the original cohort, among subjects still free of that event, unless a conditional model was actually fitted.
WLW is useful when the scientific question concerns event-order-specific failure-time associations for the cohort and when retaining subjects in each order’s risk set from the common origin matches that question. PWP models are more appropriate when the target is explicitly conditional on having reached a recurrence order. Neither approach makes the other redundant; choosing between them is a choice of estimand, not merely a computational preference.
Check Your Understanding
Answer each question, then compare with the solutions below.
- A WLW coefficient for a binary exposure is \(-0.223\) for the first event. What is the estimated hazard ratio, and how should it be interpreted?
- Why can someone who has not yet experienced a first event be included in a WLW risk set for the second-event time?
- A model reports a hazard ratio of 1.40 for a third event. Is this necessarily the hazard ratio among subjects who have experienced two events? Explain.
- Why is comparing separate event-order confidence intervals not, by itself, a formal test that the coefficients differ?
Solutions. (1) \(\exp(-0.223)=0.800\). At a given time, the estimated first-event hazard is 0.800 times as high for the exposed profile as for the unexposed profile, among subjects still free of the first event and conditional on other modeled covariates. (2) WLW models the second-event time from the common origin as an event-order-specific marginal outcome; its risk set includes subjects still free of that second event, whether or not they have had the first. (3) No. WLW’s third-event risk set is not restricted to people who have experienced two events, so 1.40 is not inherently a conditional hazard ratio among those people. (4) The estimates use outcomes from the same subjects and can be dependent. A formal test requires the variance of their difference, which depends on the covariance between estimates.