Time-Dependent Covariates in Cause-Specific Cox Models

Why covariate timing matters

A baseline covariate records a subject’s characteristics once, before follow-up. In many studies, important characteristics change: a laboratory value is measured repeatedly, a treatment begins or stops, or a patient’s clinical status is updated. A cause-specific Cox model can use such information, but each event-time comparison must use the covariate information that was available immediately before that event. Using a later measurement to explain an earlier event introduces future information into the risk set and can distort the estimated association.

This tutorial concerns changing covariate values, not a coefficient whose effect changes with time. In the model below, the covariate vector \(X_i(t)\) may change during follow-up while its coefficient vector \(\beta_k\) remains fixed. The earlier material on time-varying effects in cause-specific Cox regression addresses the distinct question of whether the coefficient itself varies with time.

The cause-specific Cox model and its risk sets

For cause \(k\), the cause-specific Cox model with time-dependent covariates is

where \(\mathcal{H}_i(t^-)\) is subject \(i\)’s observed history just before \(t\), \(\lambda_{0k}(t)\) is an unspecified baseline cause-specific hazard, and \(\beta_k\) is a constant coefficient vector. The model assumes the covariate’s multiplicative effect on the cause-specific hazard is constant over time, conditional on the covariate value and the model’s other terms. It does not require the covariate itself to remain constant.

For a subject who enters at time \(L_i\), define the risk indicator as \(Y_i(t)=1\) when \(L_i<t\leq T_i\), and \(0\) otherwise. At a cause-\(k\) event time \(t\), the risk set consists of subjects who have entered and remain event-free immediately before \(t\). A subject whose competing event occurred earlier is no longer in this risk set. For each subject still at risk, the model uses \(X_i(t)\) as known just before the event time.

A standard data layout expresses each subject’s follow-up as consecutive start-stop intervals, commonly written \((\text{start},\text{stop}]\). Split a subject’s record whenever a covariate value changes, and end the final interval at the event or censoring time. Each interval carries the covariate value applicable throughout it. The event indicator is marked only on the final interval, with the event’s cause recorded separately or coded so that only events of cause \(k\) count as failures in the cause-\(k\) fit.

Worked Example: Building records and identifying the risk set

Partial likelihood: why the pre-event values enter

Assume distinct cause-\(k\) event times for this derivation. At a cause-\(k\) event time \(t\), the model’s instantaneous event rate for subject \(i\) in the risk set is proportional to \(\lambda_{0k}(t)\exp\{\beta_k^\mathsf{T}X_i(t)\}\). Conditional on one cause-\(k\) event occurring among that risk set at \(t\), the probability that subject \(i\) is the one who experiences it is therefore proportional to that rate. Normalizing across the risk set cancels the common baseline hazard:

Here \(R_k(t)\) is the set of subjects at risk immediately before \(t\). Multiplying these conditional contributions over the observed cause-\(k\) event times gives the partial likelihood. In particular, every denominator uses the time-specific covariates of the people in the risk set at that event time, not a baseline value carried forward regardless of updates or a value recorded later.

The score contribution from an event is the event subject’s covariate vector minus the risk-set weighted average covariate vector. For a single scalar covariate, this is \(X_i(t)-\bar X_k(t;\beta_k)\), where the weighted average gives more weight to subjects with larger \(\exp\{\beta_k X_j(t)\}\). This shows directly why incorrect timing matters: assigning a subject a different \(X_j(t)\) changes the event’s comparison and hence the coefficient estimate.

This simple expression assumes no tied cause-\(k\) event times. When ties occur, use an appropriate tie method, such as the Breslow or Efron method, as in the earlier material on handling tied failures in cause-specific Cox models. Start-stop records do not remove the need to handle ties correctly.

Worked Example: An update after the event is not eligible

Worked Example: Interpreting a fitted coefficient

Assumptions, diagnostics, and common failures

Start-stop coding is a data representation, not a remedy for violations of the model. A meaningful fit requires accurate event times and causes, valid covariate timing, and a correctly specified relationship between the covariates and the cause-specific hazard. The usual cause-specific Cox assumptions include a proportional effect for each included covariate as parameterized, and censoring that is conditionally independent of the event process given the modeled history. The data must also provide adequate information and variation in the risk sets to estimate the coefficients.

The covariate process must be observed in a way that supports the intended analysis. In particular, a value should not be assigned retrospectively using information learned after an event. If updates occur only when a patient’s condition worsens, the observation process may itself be informative; carrying the last observed value forward does not automatically resolve that problem. A time-dependent covariate can also be affected by a subject’s evolving health. The fitted coefficient remains a conditional association unless additional causal assumptions and methods justify a causal interpretation.

Useful checks include verifying that each person’s intervals are ordered, have no unintended gaps or overlaps, and end at the correct event or censoring time. Confirm that covariate changes begin only when recorded, that the event indicator appears only on the final interval, and that competing events remove a subject from subsequent cause-specific risk sets. At each event time, manually inspect a few risk sets and confirm the covariate value used for every included person. Boundary rules matter: with \((\text{start},\text{stop}]\) intervals, the interval ending at an event time contains the pre-event value. If an update and event share a recorded time, the analysis needs a defensible ordering rule based on measurement timing; absent evidence that the update preceded the event, do not silently use the updated value.

Also examine whether the assumed functional form and proportional effect are plausible, and whether there are enough cause-\(k\) events across the relevant covariate histories. Sparse updates, long periods of carrying forward a value, extreme values, or abrupt changes in estimates under reasonable coding alternatives can signal instability or measurement problems. A large number of interval rows does not create more independent subjects or more event information.

How this approach differs from alternatives

A baseline value may be simpler and more robust when updates are sparse or poorly timed, but it answers a different question and can discard relevant history. Conversely, updating covariates is not automatically preferable: it requires trustworthy timing and a clear interpretation of the conditional hazard being modeled. The introductory material on competing risks provides the broader distinction between cause-specific hazards and cumulative incidence; here the key operational point is that a prior competing event removes a subject from the cause-specific risk set.

Common mistake: treating rows as independent or using future values

• Wrong reasoning: “Each start-stop row is one observation, so all rows can be treated as separate subjects.” Correction: Rows represent intervals for the same person. The risk set at each event time is formed from people, using the one interval applicable to each person at that time.
• Wrong reasoning: “The most recent value in the completed data file is the best value to use for every event.” Correction: Use only the value known immediately before that event. A later measurement cannot be carried backward.
• Wrong reasoning: “A competing event is just censoring at the time it occurs, so the subject can remain in all later risk sets.” Correction: In the cause-specific analysis, the subject leaves the event-free risk set at the competing event and contributes no later risk time.

These errors can produce a numerically converged fit that is nevertheless based on the wrong comparisons. Inspecting the records and selected event-time risk sets is therefore part of the statistical analysis, not merely a software check.