External Time-Varying Covariates in Transition Regression

Why Covariate Timing Matters

A covariate need not be fixed at baseline. A participant's work schedule, a region's air-quality measure, or a treatment policy assigned independently of a participant's disease history may change over follow-up. In transition regression, the relevant question is not merely what the covariate value is, but which value was available just before the transition event. Using a value recorded after the event can reverse the temporal order and produce an invalid risk-set comparison.

This tutorial focuses on externally measured time-varying covariates in transition-specific Cox models. The earlier material on transition-specific Cox regression and risk sets supplies the basic regression framework: each transition has its own risk set, baseline hazard, and potentially its own covariate effect. Here the additional task is aligning each member of that risk set with the correct covariate value at each event time.

For a transition from state \(r\) to state \(s\), let \(Y_i^{rs}(t)=1\) when individual \(i\) is in state \(r\), under observation, and eligible to make that transition just before \(t\); otherwise \(Y_i^{rs}(t)=0\). A transition-specific proportional-hazards model is

Here \(\lambda_{0,rs}(t)\) is the transition-specific baseline hazard and \(\beta_{rs}\) is the vector of coefficients for that transition. The model says that, among individuals eligible for the same transition at time \(t\), covariate contrasts multiply the instantaneous transition hazard by an exponential factor. The coefficient vector is constant over time in this model, even though \(Z_i(t)\) can change.

Assumptions and the Event-Time Value

Correct use of the model requires more than attaching a covariate column to a data set. The transition definition and the state-occupancy history must determine who is at risk; the covariate value must be known just before the event time; and the conditional transition hazards must have the proportional-hazards form above. For a simple partial-likelihood argument, assume one event at each distinct event time. If tied events occur, a suitable tied-event likelihood approximation or exact method is needed.

• Temporal ordering: covariate information used at \(t\) must be available before the event at \(t\). This is the predictable, no-look-ahead convention.
• Correct risk set: a subject contributes to transition \(r\to s\)'s denominator only while eligible for that transition, not merely because they remain under observation.
• Conditional proportional hazards: the covariate multiplies the transition hazard through \(\exp\{\beta_{rs}^{\mathsf T}Z_i(t-)\}\), with a coefficient that does not change over time.
• Noninformative observation, conditional on the modeled information: censoring or loss to follow-up must not selectively remove people with different unaccounted-for transition risks. Appropriate entry and censoring rules must be reflected in the risk sets.
• Information in the risk sets: covariate values must vary among individuals at risk in a way that allows the coefficient to be distinguished from the baseline hazard.

Suppose a measurement is taken at time 2 and the next measurement is taken at time 5. If the scientific protocol supports treating the measured value as applying from time 2 up to (but not including) time 5, then an event at time 4 uses the time-2 value. This is a data convention grounded in the measurement process, not a universal license to carry every last observation forward. If a measurement is taken at exactly the event time, the value is usable only if it was recorded before the event and its timing is documented. A post-event measurement cannot be substituted.

Partial Likelihood With Time-Updated Covariates

Let \(t_k\) be a distinct event time for transition \(r\to s\), and let \(i_k\) be the individual who makes that transition. Define \(\mathcal R_{rs}(t_k)\) as the set of individuals eligible for transition \(r\to s\) immediately before \(t_k\). Conditional on one transition event in that risk set, the probability that individual \(i_k\) is the one who transitions is

The common baseline hazard cancels. Multiplying these conditional probabilities over event times gives the transition-specific partial likelihood:

This derivation explains why the event-time value is used for every person in the risk set, not just the person who transitions. The comparison is between the transitioning individual's covariate vector and the contemporaneous vectors of all eligible non-transitioning individuals. A subject can contribute a different covariate value at each event time while remaining at risk.

For several modeled transitions, construct the appropriate risk set and partial likelihood contribution for each transition. A person who has moved out of state \(r\) no longer belongs to a risk set for transitions out of \(r\), though they may later enter a risk set for a different transition from their new state. The risk-set definition follows the state process; time-updating a covariate does not alter it.

When the Covariate Cannot Be Estimated

Time-varying does not automatically mean estimable. If an external covariate has exactly the same value for every person in a transition-specific risk set at every event time, its exponential term cancels from each partial-likelihood contribution. The data then contain no partial-likelihood information about its coefficient. This can happen with a calendar-time exposure shared by everyone in the study: at any one event time, all members of the risk set face the same exposure value.

Diagnostics, Alternatives, and Limitations

Before interpreting a fitted model, inspect how the time-dependent covariate history was constructed. A useful audit lists, for each transition event time, the event individual's value, the values for the other members of that transition risk set, and the measurement times that justify each value. This often catches data errors that ordinary model summaries will not reveal.

• Look-ahead or stale values: compare event times with measurement timestamps. A value recorded after an event is not a valid predictor at that event. A value carried forward across a long gap may cease to represent the intended exposure.
• Incorrect transition risk sets: verify that all denominator members are in the relevant origin state and under observation immediately before the event. A person in another state cannot be included just because their covariate is observed.
• Little or no within-risk-set variation: tabulate covariate values among people at risk at event times. A common calendar-time covariate can be inseparable from the baseline hazard; sparse variation can also yield unstable estimates or very wide uncertainty intervals.
• Inconsistent measurement schedules: if measurements are more likely at times related to a person's worsening condition, observed covariate histories may not represent the intended external process. The measurement and observation mechanisms need scrutiny.
• Wrong functional form or transition effect: assess whether the selected covariate scale and transition-specific coefficient are scientifically plausible. A time-updated covariate does not by itself allow its coefficient to vary over time or guarantee that its relationship is linear on the log-hazard scale.

A baseline-only Cox model uses a fixed covariate value and is appropriate only when the scientific target is baseline adjustment or the covariate is genuinely fixed for the analysis. Updating the covariate can better represent changing exposure, but requires correctly timed measurements. Stratifying by transition permits different baseline hazards but does not solve incorrect covariate timing. In practice, start-stop records are a common data representation: each row covers an interval during which state, observation status, and covariate values are constant, with rows split at covariate updates and transition events. The data format is only a representation; the event-time risk sets and the \(t-\) convention still determine the model.

An external covariate is not necessarily randomized, accurately measured, or causally interpretable. Nor does the model account automatically for uncertainty in a noisy covariate, informative missingness, or dependence among repeated transition contributions from the same person. When individuals contribute multiple events or transitions, variance estimation should reflect the sampling structure when needed. These issues affect inference, but they do not change the basic rule: compare the event individual's pre-event covariates with the pre-event covariates of the eligible risk set.