Event-Order Stratification In PWP Models

Why Stratify by Event Order?

A person’s hazard of a first recurrence need not have the same shape over time as the hazard of a second or third recurrence. For example, treatment or recovery after an initial event may change the underlying recurrence pattern. A PWP model represents this possibility by assigning a separate baseline hazard to each event order. That choice is distinct from deciding whether a covariate has the same effect at every order: the model can use separate baselines with a shared covariate effect, or separate baselines and event-order-specific covariate effects.

The risk sets in a PWP model are conditional on event order. A subject can contribute to the risk set for event $k$ only after experiencing event $k-1$. The earlier material on PWP event-order risk sets establishes this eligibility rule. Here the focus is how event-order strata enter the hazard model and its partial likelihood.

Definition: Let $k=1,2,\ldots$ index event order. An event-order-stratified PWP model specifies a separate baseline hazard $\lambda_{0k}(u)$ for each event order $k$, where $u$ is the model’s time scale. In a shared-effect model, $$ \lambda_{ik}(u\mid \mathcal{H}_i(u^-),X_i(u)) = \lambda_{0k}(u)\exp\{X_i(u)^\top\beta\}, $$ for subject $i$ while that subject is at risk for event $k$. Here $\mathcal{H}_i(u^-)$ denotes the observed event history just before $u$, and $X_i(u)$ is the covariate vector. An event-specific-effect version replaces $\beta$ with $\beta_k$.

The hazard is conditional on being eligible for the $k$th event and still under observation. The baseline $\lambda_{0k}$ describes the hazard for the reference covariate value within that event-order stratum; it is not required to match the baseline for any other $k$. The clock $u$ may be total time or gap time, as specified for the PWP model. Stratifying by order does not choose between those clocks.

What the Model Assumes

The model requires correctly constructed event-order risk sets and accurate event times, event order, covariates, and censoring information. For each $k$, the hazard is proportional in the chosen clock: under the shared-effect version, a one-unit covariate contrast has the same hazard ratio at all times within every event-order stratum. Under the event-specific version, that proportional-hazards condition still applies within each stratum, but the covariate coefficient may differ across $k$.

The baseline hazard is left unspecified and may have a different shape and level at every order. The partial-likelihood argument also requires that, conditional on the model covariates and history represented in the risk sets, the event-time ordering is modeled appropriately; ties require a suitable tie-handling method. Censoring must not selectively remove subjects in a way that invalidates the risk-set comparisons, conditional on the variables used in the analysis. Since one subject can contribute records at several event orders, the model’s coefficient specification does not itself make those records independent. Variance estimation should account for within-subject dependence when required by the analysis.

These are conditional associations, not automatically causal effects. A causal interpretation would require additional design and identification assumptions. Sparse risk sets at high event orders can also make estimates unstable even when the model is otherwise correctly specified.

Why Separate Baselines Lead to Separate Likelihood Strata

For a single event at time $u$ in order-$k$ stratum, suppose subject $i$ experiences the event and the eligible risk set is $R_k(u)$. Under the shared-effect model, the probability that $i$ is the event subject at that instant, conditional on one event occurring in this risk set, is proportional to that subject’s relative hazard. The common baseline factor $\lambda_{0k}(u)$ cancels from the ratio:

\frac{\lambda_{0k}(u)\exp\{X_i(u)^\top\beta\}} {\sum_{j\in R_k(u)}\lambda_{0k}(u)\exp\{X_j(u)^\top\beta\}} = \frac{\exp\{X_i(u)^\top\beta\}} {\sum_{j\in R_k(u)}\exp\{X_j(u)^\top\beta\}}.

Multiplying these conditional contributions over event times gives the Cox partial likelihood. Event times in different order strata have different baselines and different eligible risk sets, so their contributions are formed within their own strata. With $D_k$ denoting the event times for order $k$, the shared-effect partial likelihood is

L(\beta) = \prod_k\prod_{u\in D_k} \frac{\exp\{X_{i(u)}(u)^\top\beta\}} {\sum_{j\in R_k(u)}\exp\{X_j(u)^\top\beta\}},

where $i(u)$ is the subject with the event at $u$. In an event-specific-effect model, the corresponding contribution uses $\beta_k$ for every event in stratum $k$. The baseline hazards still cancel within each contribution, but the coefficients are no longer constrained to be equal across strata.

Key distinction: Stratifying the baseline allows $\lambda_{0k}(u)$ to vary freely with event order. It does not by itself allow $\beta$ to vary. A shared-effect stratified model uses one coefficient vector in all strata; an event-specific-effect model uses a different coefficient vector for each order.

Worked Examples

Worked Example: A Partial-Likelihood Contribution at the Second Event

Suppose three subjects are eligible for their second event at a particular total time. Their binary treatment indicators are $X_A=0$, $X_B=1$, and $X_C=0$; subject B has the event. Consider a shared treatment coefficient $\beta$, with a separate order-2 baseline hazard.

The order-2 partial-likelihood contribution is $$ \frac{\exp(\beta X_B)} {\exp(\beta X_A)+\exp(\beta X_B)+\exp(\beta X_C)} = \frac{\exp(\beta)}{1+\exp(\beta)+1}. $$

For illustration, if $\beta=\log(2)$, then $\exp(\beta)=2$, so the contribution is $$ \frac{2}{1+2+1}=\frac{2}{4}=0.500. $$

This is one contribution, evaluated at a specified coefficient; it is not by itself an estimate of $\beta$. The order-2 baseline does not appear because it cancels from the risk-set ratio. An order-1 event would contribute through its own risk set, not by adding an order-1 subject to this order-2 denominator.

Worked Example: Same Covariate Effect, Different Baseline Hazards

Suppose a hypothetical PWP model uses a treatment indicator and estimates a shared coefficient $\beta=\log(1.5)=0.4055$. At a specified point on the chosen clock, let the estimated order-1 baseline hazard be $0.040$ events per person-month and the order-2 baseline hazard be $0.120$ events per person-month.

For an untreated subject, the fitted hazards at that point are the baselines: $0.040$ per person-month for event 1 and $0.120$ per person-month for event 2. For a treated subject, multiply each by $\exp(\beta)=1.5$: $$ \widehat{\lambda}_{1}(u\mid X=1)=0.040(1.5)=0.060 \quad\text{and}\quad \widehat{\lambda}_{2}(u\mid X=1)=0.120(1.5)=0.180 $$

per person-month. Thus the treatment hazard ratio is 1.5 in both strata, while the baseline hazard for event 2 is three times the event-1 baseline at this clock value. These are conditional hazards among subjects eligible for the respective event orders; they do not imply that an individual’s unconditional probability of a second event is three times the probability of a first event.

Worked Example: Event-Specific Treatment Effects

In a hypothetical event-specific-effect fit, suppose the treatment estimates are $\hat{\beta}_1=0.400$ with standard error $0.200$ for the first event and $\hat{\beta}_2=-0.100$ with standard error $0.250$ for the second. The estimated conditional hazard ratios are $$ \exp(0.400)=1.492 \quad\text{and}\quad \exp(-0.100)=0.9048. $$

At event order 1, the fitted treatment hazard is about 1.49 times the untreated hazard, conditional on eligibility and the model covariates. At order 2, the fitted ratio is about 0.905. The different estimates motivate checking whether a shared coefficient is reasonable; separate estimates alone do not establish that the effects differ.

For illustration, suppose the fitted covariance between the two coefficient estimates is $0.0100$. The estimated contrast is $\hat{\beta}_1-\hat{\beta}_2=0.500$, and its variance is $$ 0.200^2+0.250^2-2(0.0100)=0.0825. $$

Its standard error is $\sqrt{0.0825}=0.2872$, giving a Wald statistic $z=0.500/0.2872=1.741$. The two-sided $p$-value is approximately $0.0818$. At the 0.05 level, this is not strong evidence of a treatment-effect difference between these orders. It does not prove the effects are equal. In practice, the covariance must come from the fitted model’s variance estimate, not an assumption that the coefficient estimates are independent.

Choosing and Checking the Effect Structure

A shared-effect, stratified-baseline model is useful when event-order baseline patterns may differ but a common covariate hazard ratio is scientifically plausible. An event-specific-effect model is more flexible about covariate effects, but estimates more parameters and can be imprecise when later-order events are few. The decision about baseline structure and the decision about coefficient structure should therefore be stated separately.

Specification	Baseline by event order	Covariate effect by event order	Main implication
Shared-effect PWP with order strata	Separate	Shared	Allows different baseline shapes while constraining hazard ratios to be common
Event-specific-effect PWP with order strata	Separate	Separate	Allows both baseline hazards and covariate effects to differ by order
Pooled-baseline PWP specification	Common or otherwise restricted	Depends on the chosen model	Imposes a restriction on baseline recurrence patterns across orders

A pooled-baseline model is not merely a simpler way to write the stratified model: it makes an additional assumption that the baseline patterns across orders are linked. Conversely, separate baselines do not require separate covariate effects. The Andersen–Gill model uses a different risk-set construction and answers a different conditional-hazard question; it should not be treated as interchangeable with PWP order strata. These structural choices are also distinct from the choice of total-time or gap-time clock.

Useful checks include inspecting event counts and risk-set sizes by order, verifying that subjects enter each risk set only after the required prior events, and examining proportional-hazards diagnostics within event-order strata. Time-varying covariate effects or systematic residual patterns can indicate that a constant coefficient within a stratum is inadequate. If effects plausibly differ by event order, compare the shared-effect model with a model containing event-order-by-covariate interactions or separate coefficients, using a valid variance estimate and a prespecified inferential approach.

A common warning sign is a very small number of subjects or events in later strata. Then coefficients may have large standard errors, estimates may be sensitive to individual records, and standard asymptotic tests may be unreliable. Report the number of events and the uncertainty, and avoid interpreting a noisy order-specific estimate as a stable biological pattern.

Common Mistake and Limitations

Wrong reasoning: “I stratified the PWP model by event order, so treatment is allowed to have a different effect on every recurrence.” Correction: Baseline stratification changes $\lambda_{0k}(u)$, not the coefficient vector. A model with one $\beta$ still imposes one conditional hazard ratio across event orders. To allow differences, specify $\beta_k$ or event-order-by-covariate interactions and estimate the contrast using the fitted covariance matrix.

Another pitfall is comparing the order-specific baseline hazards as if they were unconditional recurrence probabilities. Each hazard applies to a different conditional risk set, and its numerical value depends on the chosen time scale. The model’s conditional hazard ratios likewise do not by themselves describe marginal event rates, cumulative event counts, or causal effects. Those quantities require their own estimands and assumptions.

Finally, separate baselines protect against forcing the same baseline shape across event orders, but they do not repair incorrect event ordering, informative censoring, omitted time-varying effects, or poor covariate measurement. They also cannot create information in sparse higher-order risk sets.

Key takeaway: In a PWP model, event-order stratification gives each recurrence order its own baseline hazard and risk-set contributions. Whether covariate effects are shared or event-specific is a separate modeling decision that must be made explicit.

Check Your Understanding

Answer each question, then compare with the solutions below.

A model has separate baseline hazards for event orders 1 and 2 but one treatment coefficient. What treatment hazard-ratio restriction does it impose?
At an order-2 event time, three eligible subjects have covariate values 0, 1, and 1; the subject with value 0 experiences the event. Write the partial-likelihood contribution for coefficient $\beta$.
Why does an event-order-specific baseline hazard not appear in the partial-likelihood contribution?
A treatment hazard ratio estimate is 0.80 for event 1 and 1.10 for event 2. Does this alone demonstrate different effects by order? Explain.
Name one risk-set check and one model diagnostic relevant to an event-order-stratified PWP analysis.

Solutions. (1) The model imposes the same conditional treatment hazard ratio, $\exp(\beta)$, at every event order, despite allowing different baseline hazards. (2) The contribution is $1/\{1+\exp(\beta)+\exp(\beta)\}=1/\{1+2\exp(\beta)\}$. (3) At an event time within a stratum, the same $\lambda_{0k}(u)$ multiplies every eligible subject’s hazard and cancels between the numerator and denominator. (4) No. The estimates may differ by sampling variation; assess an event-order contrast using its estimated covariance and uncertainty. (5) Confirm that a subject enters the order-2 risk set only after the first event; inspect proportional-hazards diagnostics within each order stratum. Other valid checks include reviewing higher-order event counts and risk-set sizes.