Censoring Assumptions in Fine-Gray Regression

Why Censoring Requires Special Attention

Fine–Gray regression models the subdistribution hazard for a specified event cause. Its risk-set construction is unusual: people who experience a competing event remain represented in the subdistribution risk set, while people censored before the event time cannot be followed in the same way. Estimation therefore needs an assumption about why follow-up ends. If people lost to follow-up tend to have different unobserved event outcomes from those who remain, the observed data can misrepresent the event process the model is intended to describe.

Administrative censoring and loss to follow-up can look identical in the data: in each case, an individual’s event status becomes unknown after a recorded time. Their causes may differ substantially. A common study-end date can produce administrative censoring that is independent of an individual’s event process under the study design. By contrast, people may leave follow-up because their health is worsening, they move to another care system, or they face barriers to continued contact. Such loss can depend on prognosis.

The conditioning matters. Censoring may occur more often for one predictor group than another and still satisfy the assumption, provided that within each covariate pattern, censoring does not further select people according to their event process. Conversely, censoring that appears unrelated to outcome overall may be informative within covariate groups. In practice, the covariates used to condition on should include relevant predictors of both follow-up and the event process, measured before censoring occurs.

What the Assumption Supports in Fine–Gray Regression

For cause 1, the cumulative incidence function is \(F_1(t\mid X)=P(T\leq t,J=1\mid X)\). The cause-1 subdistribution hazard is the instantaneous rate of cause-1 events among people who have not yet had cause 1, with people who have already had a competing event retained in the corresponding risk set. Fine–Gray regression commonly specifies a proportional subdistribution-hazards model,

Here \(\tilde\lambda_{10}(t)\) is an unspecified baseline subdistribution hazard and \(\beta\) describes covariate associations with the cause-1 subdistribution hazard under this model. Independent censoring is not the proportionality assumption: it is a separate condition needed to estimate the model from incomplete follow-up. Correct model form, suitable handling of ties, and adequate covariate support are additional concerns, but they do not repair informative censoring.

Fine–Gray estimation must account for the fact that some individuals are censored before a given event time. In broad terms, its modified risk-set calculations use information about the censoring process so that the individuals still observed represent the relevant population. The details of censoring weights depend on the estimation method. A crucial practical qualification is that a method assuming a single censoring distribution may not adequately handle censoring that varies with \(X\). Conditional independence alone does not guarantee that every software implementation has modeled the censoring distribution appropriately.

A Short Justification: Why Conditioning Helps

The role of conditional independence can be seen by considering the event process among people who remain uncensored to a time \(t\). If \(C\perp (T,J)\mid X\), then, for covariate values \(x\) with \(P(C\geq t\mid X=x)>0\),

To justify this equality, conditional independence gives \(P(T\leq t,J=1,C\geq t\mid X=x) =P(T\leq t,J=1\mid X=x)P(C\geq t\mid X=x)\). Dividing both sides by \(P(C\geq t\mid X=x)\), which must be positive, gives the displayed result. Thus, within covariate values, surviving censoring does not preferentially select people with or without the cause-1 event by \(t\). Fine–Gray methods also have to account for people who have already experienced competing events in their modified risk sets; the same censoring assumption underlies that adjustment. If censoring depends on the event process even after conditioning on \(X\), this justification fails.

Administrative Censoring Versus Predictor-Dependent Loss

Administrative censoring is often independent because the end date is determined by the study calendar rather than by an individual’s evolving health. But the label “administrative” is not itself a guarantee. For example, if enrollment dates vary, participants enrolled later may have shorter potential follow-up. If enrollment timing is associated with predictors or prognosis, censoring can be associated with the event process unless that association is addressed. Conditioning on relevant variables, such as enrollment period, may make the assumption more plausible.

Loss to follow-up may depend on recorded predictors without violating conditional independence. The question is whether censoring remains associated with \(T\) or \(J\) after conditioning on those predictors. For example, a predictor may be associated both with a higher chance of moving away and with the event process. That marginal association does not, by itself, invalidate conditional independence if the predictor is measured and appropriately accounted for. However, using a method that assumes censoring is unrelated to predictors can still yield biased results.

Assessment, Alternatives, and Limitations

Independent censoring is generally not verifiable from observed data alone, because the event outcomes after censoring are unknown. Diagnostics can reveal warning signs, but a lack of warning signs does not prove the assumption. Useful checks include comparing censoring patterns across measured predictors, examining when and why participants leave follow-up, and checking whether there is sufficient follow-up across covariate groups. These checks should also consider study calendar and recruitment timing.

A particularly important condition is positivity: at times and covariate patterns relevant to the analysis, there must be a nonzero probability of remaining under observation. If an entire predictor group is always censored before a time of interest, the data cannot directly support estimation of its event process at that time. Very low probabilities of remaining uncensored can also make estimates unstable, even when they are not exactly zero.

Fine–Gray regression and cause-specific Cox regression address different event-process quantities, but neither makes informative censoring harmless. Cause-specific Cox analysis removes a competing-event subject from the risk set at that event; Fine–Gray analysis retains that subject in its subdistribution risk-set construction. Both require appropriate assumptions about censoring, though the precise estimation machinery differs. Switching from one regression approach to the other is not, by itself, a solution to predictor-dependent or prognosis-dependent loss.

When censoring depends on measured predictors, possible responses include conditioning on those predictors or using a method that models the conditional censoring distribution. The choice must match the data and the estimator’s assumptions. When censoring may depend on unmeasured prognosis, investigators should describe the concern and assess how conclusions could change under plausible departures from independent censoring. No diagnostic based only on the observed outcomes can settle that question.