Time-Dependent Covariates After Competing Events in Fine-Gray Models

Why a Competing Event Creates a Covariate Problem

A time-dependent covariate can change while a person is followed. In a cause-specific Cox model, a competing event ends that person’s time at risk for the event of interest. Fine–Gray regression uses a different risk-set construction: people who have had a competing event remain in the subdistribution risk set. If their covariate is only defined or observed while they are event-free, what value should the model use for them after that event?

This is not just a software detail. A Fine–Gray model needs a covariate value for each person represented in its risk set at each time. Assigning a value after a competing event therefore defines the predictor being modelled. Different defensible definitions can answer different scientific questions; an arbitrary value can make the fitted model difficult to interpret.

Definition: Let $T$ be the time to the first event, $J$ its event type, and $k$ the event type of interest. The cumulative incidence function for event $k$ is $F_k(t)=P(T\leq t,J=k)$. Its subdistribution hazard is $$ \widetilde{\lambda}_k(t)=\frac{dF_k(t)/dt}{1-F_k(t)}. $$ In the Fine–Gray risk-set construction, people who have experienced a competing event remain represented after that event, whereas people who have experienced event $k$ do not. With censoring, censoring adjustment is needed; the precise implementation may use inverse-probability-of-censoring weights.

Ignoring censoring for the moment, the corresponding risk-set indicator is

\widetilde{Y}_i(t) = I(T_i\geq t)+I(T_i<t,\ J_i\neq k).

The second term is the crucial difference: someone who had a competing event before $t$ is still represented. By contrast, the cause-specific Cox risk set for event $k$ ends at that competing event. The earlier material on time-dependent covariates in cause-specific Cox models concerns covariates while people are actually at risk; it does not remove the need to define values for Fine–Gray pseudo-risk-set members after a competing event.

The Covariate Must Be Defined on the Pseudo-Risk Set

A common Fine–Gray proportional subdistribution hazards specification is

\widetilde{\lambda}_k(t\mid \mathbf{Z}(t)) = \widetilde{\lambda}_{k0}(t) \exp\!\left\{\boldsymbol{\beta}^{\mathsf T}\mathbf{Z}(t)\right\}.

Here $\mathbf{Z}(t)$ is the chosen covariate vector at time $t$, and $\boldsymbol{\beta}$ is the coefficient vector. A time-dependent covariate must be available and defined for every person represented in the risk set at the times used to fit the model. A value measured only up to a competing event does not meet that requirement unless a rule is specified for later times.

That rule should also be predictable: the value used at time $t$ must not depend on information that would only become available after $t$. For event times, covariates are conventionally evaluated just before the event, denoted $Z(t-)$. These requirements concern the definition and timing of the predictor; they do not by themselves guarantee that a proportional subdistribution hazards model is appropriate. The event coding, censoring assumptions and adjustment, and functional form must also be defensible.

Worked Example: Who Needs a Covariate Value at Time Six?

Suppose the event of interest is disease progression and death is a competing event. In a hypothetical cohort without censoring, person A has not yet had an event at time 6 and has a current biomarker value of 0. Person B died at time 3, with a last measured biomarker value of 2 just before death. Person C remains event-free at time 6 and has a current value of 1. Person D had progression at time 5. Which people are represented in the Fine–Gray risk set at time 6, and what covariate values could be used?

A and C are represented because they have not yet had an event. B is also represented because death is a competing event. D is not represented because D has already had the event of interest. Thus, the model needs a covariate value for B at time 6 as well as for A and C. The observed biomarker value for B after death is not available.

If the analysis specifies freezing the biomarker at its last pre-death value, the time-6 values are $Z_A(6)=0$, $Z_B^\dagger(6)=2$, and $Z_C(6)=1$. This defines one usable predictor process. It does not claim that B’s biomarker was actually measured as 2 at time 6; it defines the value assigned to B in the pseudo-risk set.

Defensible Strategies for Defining the Predictor

There is no universal post-competing-event value for a time-dependent covariate. Choose a rule that is scientifically interpretable, state it before fitting the model, and use it consistently. Common strategies include the following.

Use a baseline covariate. Define the predictor at baseline and keep that value fixed for every person. This avoids needing a post-event measurement. It answers a question about association with the baseline characteristic, not about the effect of later changes in that characteristic.
Freeze the covariate at the competing event. Use the last value observed just before the competing event for that person’s later pseudo-risk-set representation. This is a clearly specified extension of the observed history, not an observed post-event value. It is most plausible when the last pre-event value is a scientifically meaningful summary for the model’s risk-set comparison.
Use an externally defined post-event value or process. This may be defensible when the predictor has a meaningful value after the competing event and is available for the relevant people. Its measurement and interpretation must be consistent across the risk set; a value that exists only for selected survivors or is defined differently after the competing event can create a different, difficult-to-interpret predictor.
Use a model-based post-event trajectory. A specified model may predict values after the competing event, but those values are predictions, not observations. The trajectory model and its assumptions must be justified, and uncertainty from prediction should be accounted for where relevant. Simply inserting predicted values into an ordinary Fine–Gray fit does not automatically propagate that uncertainty.
Change the analysis question. A landmark analysis, for example, can restrict attention to people who are event-free at a specified time and define covariates from information available then. This changes the target population and question; it is not a way to retain the original Fine–Gray estimand while avoiding its post-event covariate issue.

Worked Example: Interpreting a Frozen Covariate

Suppose a hypothetical Fine–Gray model for progression includes a biomarker whose values are frozen at death, a competing event. The fitted coefficient is $\hat{\beta}=\log(1.5)=0.4055$ per biomarker unit. What does this coefficient imply for a one-unit difference, and why does the freezing rule matter?

Under the specified proportional subdistribution hazards model, a one-unit difference in the defined biomarker value corresponds to a subdistribution hazard ratio of

\exp(\hat{\beta}\times 1) = \exp(0.4055) \approx 1.500.

Thus, at a given time, the model associates a one-unit higher value of this defined predictor with a 1.50-fold subdistribution hazard, conditional on the other model covariates and the model specification. This is not a cause-specific hazard ratio, nor by itself a causal effect or a direct statement that progression probability is 50% higher.

For a person who died, the predictor after death is the last pre-death value by definition. If a different rule assigned that person a value three units higher, the corresponding model contrast would be $\exp(0.4055\times 3)\approx 3.375$, rather than 1.500. The calculation illustrates why the assigned post-event value is consequential: it changes the predictor contrast represented by the model. It does not establish that either assignment is correct for a particular scientific question.

Justification: Why the Post-Event Value Is Needed

The risk-set issue follows directly from the subdistribution hazard’s denominator. Since $F_k(t)$ is the probability that event $k$ has occurred by $t$, its complement includes both people who are event-free at $t$ and people who had a competing event before $t$:

1-F_k(t) = P(T>t)+P(T\leq t,\ J\neq k).

The second probability is not zero in general. Accordingly, the Fine–Gray risk-set construction retains people with competing events to represent the population in the denominator of the subdistribution hazard. A regression model that includes a time-dependent predictor must evaluate that predictor for those retained people too. The model cannot resolve an undefined value by silently treating the person as absent without changing the risk-set construction.

This reasoning establishes why a value is required; it does not establish which value should be assigned. That depends on the scientific definition of the predictor and the intended interpretation of its coefficient.

Worked Example: Comparing Two Explicit Definitions

In a hypothetical study of progression with death as a competing event, a biomarker is measured repeatedly while participants are alive. Person R has a value of 4 immediately before death at time 2. At time 5, a Fine–Gray risk set still represents R. Compare a baseline-value definition with a freeze-at-death definition if R’s baseline biomarker was 3.

Under the baseline definition, R’s value at time 5 is $Z_R(5)=3$: the predictor is baseline biomarker throughout follow-up. Under the freeze-at-death definition, the assigned value is $Z_R^\dagger(5)=Z_R(2-)=4$. Both values are specified without using information from after time 5, but they represent different predictor definitions.

For a hypothetical coefficient $\hat{\beta}=0.20$ per unit, the model’s relative subdistribution hazard contrast for a one-unit increase is $\exp(0.20)\approx 1.221$. For R, the linear predictor contribution is $0.20(3)=0.60$ under the baseline definition and $0.20(4)=0.80$ under freezing. The difference is $0.20$, so the corresponding multiplicative contrast is $\exp(0.20)\approx 1.221$. These calculations do not say which definition is preferable. The choice depends on whether the scientific predictor is baseline level or the last observed pre-death level carried forward by rule.

Assumptions, Diagnostics, and Alternatives

Before using a chosen strategy, state the assumptions that support the analysis. The event of interest and competing event must be distinguished correctly. Censoring must be handled under an appropriate independent-censoring assumption, possibly conditional on measured variables, with valid censoring weights when used. The covariate definition must be predictable, consistently applied, and available for the people represented in the risk set. The model must also provide a reasonable description of the subdistribution hazard; for the proportional model above, the covariate effects are assumed constant on the log subdistribution hazard scale unless the model explicitly allows time-varying effects.

Useful checks include auditing individual event and measurement timelines, tabulating how often each covariate is unavailable after a competing event, and verifying that the software retains competing-event participants as intended. Confirm that the data construction follows the declared rule—for example, that freezing uses the last value before, not after, the competing event. Check the fitted model’s functional form and proportionality assumptions using suitable diagnostics. Compare estimates under scientifically plausible definitions, such as baseline values versus frozen values, as a sensitivity analysis. Large differences signal that conclusions depend on the post-event rule and should be reported as such.

Approach	What it defines	Main caution
Fine–Gray with baseline covariates	Association with characteristics measured at baseline	Does not represent later covariate change
Fine–Gray with a declared post-event rule	Association with a predictor defined over the full subdistribution risk set	Interpretation depends on the rule after competing events
Cause-specific Cox model	Cause-specific hazard association among people still event-free and at risk	Answers a different question; competing events end the risk time
Landmark analysis	Association in a defined event-free population at a chosen time	Changes the population and time-specific question

Removing a person from the Fine–Gray risk set at the competing event merely because the covariate is no longer observed is not a neutral missing-data fix. It changes the risk-set construction and therefore the analysis. The cause-specific Cox approach is a defensible alternative when the scientific question concerns the cause-specific hazard among people who remain event-free; it should not be presented as an equivalent implementation of a Fine–Gray model.

Common Pitfalls and Limitations

Using the last observed value without declaring it. A carried-forward value is an analysis rule, not an actual post-event measurement. State the rule and the interpretation it supports.
Using information from the future. Assigning values based on measurements after time $t$ can introduce look-ahead. Define the process using information available by the time of prediction.
Treating the coefficient as a cause-specific effect. Fine–Gray coefficients describe associations on the subdistribution hazard scale under the model and covariate definition. They are not cause-specific hazard ratios and do not establish causality.
Assuming a plausible fill-in is automatically valid. A frozen or predicted value makes the model computable, but it does not prove that the chosen post-event definition is scientifically appropriate or that the model is correctly specified.
Ignoring sensitivity to the rule. If reasonable predictor definitions lead to materially different estimates, the result is definition-dependent. Report this limitation rather than selecting a rule based only on which estimate is preferred.

Fine–Gray regression is especially useful when the target is the cumulative incidence of a particular event, but its pseudo-risk-set construction makes post-competing-event predictor definitions central for time-dependent covariates. When the post-event values have no meaningful interpretation, a baseline predictor or a different analysis question may be more defensible than an elaborate but arbitrary extension.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions.

Why does a person who experiences a competing event remain relevant to the Fine–Gray risk set after that event?
A time-dependent biomarker is observed only until a competing event. Name one explicit strategy for defining its later pseudo-risk-set values and state what those values mean.
A fitted Fine–Gray coefficient is $0.30$ per unit of a defined time-dependent predictor. Calculate the model’s subdistribution hazard ratio for a two-unit contrast and interpret it.
Why is removing competing-event participants from a Fine–Gray risk set because their covariates are unavailable not a neutral solution?
When might a cause-specific Cox model be more appropriate than Fine–Gray regression?

Solutions. (1) They remain represented because $1-F_k(t)$ includes people who had a competing event before $t$, as well as people still event-free. (2) One option is to freeze the value at the last measurement just before the competing event; the later value is then an assigned extension of the observed history, not a post-event observation. (3) The ratio is $\exp(0.30\times2)=\exp(0.60)\approx1.822$. Under the specified model, the two-unit contrast is associated with a 1.82-fold subdistribution hazard, conditional on the other covariates; this is not a causal effect or a cause-specific hazard ratio. (4) Removing those people changes the Fine–Gray risk-set construction and thus changes the analysis rather than simply resolving missing covariate values. (5) It may be more appropriate when the scientific question concerns the event-specific hazard among people who remain event-free and at risk; competing events then end risk time for that cause.