Checking Proportional Effects For Recurrent Events

What Proportional Effects Mean for Recurrent Events

A proportional effect means that a covariate’s relative effect is constant over the analysis time scale. In a recurrent-event model, the precise interpretation depends on the model: an intensity model describes an instantaneous event rate conditional on the subject’s observed history, while a marginal rate model describes a population-average rate. The diagnostic question is similar in either case: does the covariate coefficient remain constant as time passes?

For example, an Andersen–Gill-type multiplicative intensity model may be written as follows. Here \(N_i(t)\) counts events for subject \(i\) through time \(t\), \(Y_i(t)\) indicates that the subject is under observation and at risk, and \(X_i\) is a covariate that does not change over time.

The history \(\mathcal{H}_i(t)\) contains the observed event and covariate history up to time \(t\). In this model, comparing subjects with \(X_i=1\) and \(X_i=0\), at the same time and under the stated conditioning, gives an intensity ratio of \(\exp(\beta)\). Proportionality means that this ratio does not vary with \(t\). For a marginal rate model, the analogous statement concerns the specified marginal rate ratio, not necessarily a subject-specific intensity ratio.

A constant coefficient is a modeling restriction, not a claim that individual subjects experience constant event rates. The baseline rate \(\lambda_0(t)\) may vary freely with time. The restriction is that the covariate multiplies that baseline by the same relative amount throughout follow-up. Also distinguish a changing covariate value from a changing covariate effect: the earlier material on time-varying covariates concerns the former, whereas this tutorial checks whether the coefficient itself changes.

Checking the Restriction With a Time Interaction

A direct check augments the model with a covariate-by-time interaction. For a positive time scale, one useful specification is

where \(\tau\) is a chosen reference time, and \(\mathbf{Z}_i(t)\) denotes other covariates. The coefficient for \(X_i\) at time \(t\) is \(\beta+\gamma\log(t/\tau)\). Thus, for a one-unit difference in \(X_i\), the model’s relative effect at time \(t\) is \(\exp\{\beta+\gamma\log(t/\tau)\}\). At \(t=\tau\), it is \(\exp(\beta)\). The proportional-effects null hypothesis is \(H_0:\gamma=0\).

The log-time form is one choice, not a universal law. It is sensible when a smooth trend across follow-up is plausible and time is positive. Other prespecified functions, such as a centered linear time interaction or a small number of time intervals, answer different questions about how the effect changes. Choose the time function based on scientific context and the time scale before inspecting results where possible. Trying many forms and reporting only the most significant one makes the nominal p-value misleading.

For inference, estimate the interaction coefficient and use a variance estimator that accounts for clustering of recurrent events within subjects. In a marginal estimating-equation analysis, for example, the sandwich variance is built from subject-level score or estimating-function contributions. Its validity relies on independent subjects, an adequate number of independent clusters, and a correctly specified mean structure for the estimand; it does not require events from the same subject to be independent. A Wald test uses the estimated coefficient divided by its cluster-robust standard error. For several time-interaction terms, test their joint null with a cluster-robust Wald test rather than interpreting each coefficient in isolation.

The basic assumptions still matter. Subjects should be independent of one another, and the model’s risk sets, event process, and covariate history must be constructed consistently with the chosen estimand. Censoring and loss to follow-up must be noninformative conditional on the modeled information, or handled with an appropriate method. There must be enough events across the time range to estimate a trend. Robust standard errors address within-subject dependence in uncertainty estimation; they do not repair an incorrect risk set, informative censoring, a misspecified mean model, or a poorly chosen time scale.

Residual-Based Diagnostics

A time-interaction test evaluates a particular functional form. A residual diagnostic can reveal patterns that the chosen form misses. For a Cox-type multiplicative intensity model, an event-time score residual for a covariate compares the observed covariate value at an event with its risk-set weighted mean at that time. In the simple unweighted form, its contribution is \(x_{\text{event}}-\bar{x}_{R}(t)\), where \(\bar{x}_{R}(t)\) is the risk-set mean using the model’s event-time weights. Recurrent events contribute repeatedly, and each event’s risk set must follow the fitted model’s definition.

Under a correctly specified constant-coefficient model, such residuals should not show a systematic pattern with time. Plot them against event time, often with a smoother and a zero reference line. A sustained increase or decrease can indicate a changing coefficient, though it may also reflect an omitted time-varying covariate, an incorrect baseline or risk-set specification, or sparse data at the edges of follow-up. Because a subject can contribute many residuals, ordinary event-level independence bands are inappropriate. Use a cluster-aware uncertainty calculation, such as a subject-level sandwich procedure or a subject-level bootstrap, when quantifying uncertainty in a residual trend.

Residuals must match the fitted model. A Cox-type score residual should not be transferred uncritically to a marginal rate model with a different estimating equation. For such a model, use its corresponding score, estimating-function, or deviance residuals and account for subject clustering. A residual plot is a diagnostic, not a substitute for a clearly specified estimand or a formal test.

Choosing a Diagnostic and Recognizing Failure Modes

The interaction test and residual plot are complementary. The interaction test gives an interpretable estimate and test for a specified time function. Residuals provide a broader check for systematic time structure but can be noisy and are sensitive to how they are smoothed or grouped. A formal test of one interaction does not rule out other forms of time variation. Conversely, a visually irregular residual plot is not sufficient evidence to reject proportionality, especially with few independent subjects or few late events.

A stratified model can allow different baseline rates across a prespecified grouping variable while retaining a common covariate effect within strata. This addresses baseline differences, not a smoothly changing covariate coefficient. It is therefore not a general substitute for a time-interaction check. Likewise, choosing separate effects for event order addresses a different source of heterogeneity; it should not be used to explain away a time trend without a substantive reason.

Common failure modes include a sparse late risk set, an incorrectly defined at-risk indicator, an unsuitable time transformation, unmodeled time-varying covariates, and dependence among subjects that invalidates the cluster structure. Check risk-set counts and event counts over time, inspect whether the effect changes under a reasonable prespecified time scale, and verify that the number of independent subjects—not merely the number of events—is adequate for robust inference. If conclusions depend strongly on one subject or one time band, report that instability rather than presenting a precise-looking test as decisive.

A nonsignificant test is also not confirmation of proportionality. It may reflect limited precision, few late events, or low power to detect the true pattern. Report the estimated interaction and its interval, describe the diagnostic plot and the available follow-up information, and state the model scale. If substantial nonproportionality is plausible, that finding motivates a model that explicitly represents the changing effect; the diagnostic alone does not select the best alternative.