Inverse-Probability Censoring Weights for Fine-Gray Models

Why Fine–Gray Risk Sets Need Censoring Weights

A Fine–Gray model concerns the subdistribution hazard for a selected event cause. Its risk set is deliberately different from an ordinary event-free risk set: a person who has experienced a competing event remains in the subdistribution risk set. A person censored before a later event time, however, has an unknown subsequent event status. Weighting lets the estimating procedure account for that incomplete follow-up rather than simply treating every early-censored person as if their status were known.

The weights use an estimate of the censoring distribution. Their construction depends on the convention used for that distribution, especially at censoring times. This tutorial uses a right-continuous survival function throughout and states explicitly how it enters the weights.

Definition: Let $T_i$ be subject $i$'s event time, $J_i$ the event cause, and $C_i$ the censoring time. The observed time is $X_i=\min(T_i,C_i)$, and $\Delta_i=I(T_i\leq C_i)$ indicates whether an event was observed. For censoring covariates $W_i$, define $$ G(t\mid W_i)=P(C_i>t\mid W_i). $$ This is the probability of remaining uncensored beyond $t$. The strict inequality determines the value at a censoring-time jump. Its right-continuous Kaplan–Meier estimate is $$ \widehat G(t\mid W_i)=\prod_{s\leq t}\left(1-\frac{d_C(s\mid W_i)}{r_C(s\mid W_i)}\right), $$ where $d_C(s\mid W_i)$ is the number censored at $s$, and $r_C(s\mid W_i)$ is the number still at risk for censoring immediately before $s$. For this censoring-distribution calculation, observed events of any cause are treated as censoring of the censoring-time observation; censoring events are the failures.

The Weighted Subdistribution Risk Set

Fix cause $k$. Ignoring censoring for a moment, the Fine–Gray subdistribution risk set at time $t$ contains subjects whose event has not yet occurred and subjects who have already had an event of a cause other than $k$. In observed data, a subject with an observed competing event before $t$ is known to belong to that risk set. A subject censored before $t$ is not: their later event status is unknown.

For a target-event time $t$, define the Fine–Gray weighted risk contribution $Y_i^w(t)$ by

Y_i^w(t)=I(X_i\geq t) +I(X_i<t,\Delta_i=1,J_i\ne k) +I(X_i<t,\Delta_i=0)\frac{\widehat G(t-\mid W_i)}{\widehat G(X_i\mid W_i)}.

Here $\widehat G(t-\mid W_i)$ is the left limit immediately before $t$. The ratio for a subject censored at $X_i<t$ is evaluated only when its denominator is positive. An observed competing event before $t$ contributes one, not this censoring ratio: its occurrence and cause are already known. A subject with an observed cause-$k$ event before $t$ contributes zero. A subject still observed at $t$ contributes one.

At a target-event time, the Fine–Gray risk-set sum for covariate vector $Z_i$ is $\sum_iY_i^w(t)\exp(\beta^\top Z_i)$. With distinct event times and no event–censoring ties, the corresponding partial-likelihood contribution for a cause-$k$ event in subject $i$ at $t_i$ is

\frac{\exp(\beta^\top Z_i)} {\sum_jY_j^w(t_i)\exp(\beta^\top Z_j)}.

The censoring covariates $W_i$ need not be identical to the Fine–Gray model covariates $Z_i$. They should include variables needed to make censoring independent of event time and cause, conditional on the information used in the censoring model.

Why the Survival Ratio Appears

For $c<t$, among subjects who remained uncensored beyond $c$ (so $C_i>c$), the conditional chance of remaining uncensored through just before $t$, given $W_i$, is represented by

P(C_i>t-\mid C_i>c,W_i) =\frac{G(t-\mid W_i)}{G(c\mid W_i)}.

For a subject actually censored at $c$, this conditional probability is not conditioned on that subject's observed $C_i=c$. The Fine–Gray IPCW risk-set construction separately assigns the ratio $\widehat G(t-\mid W_i)/\widehat G(c\mid W_i)$ as the subject's remaining risk-set contribution at $t$, under conditional independent censoring, to account for unknown post-censoring event status. This is a time-dependent risk-set weight; it is not simply $1/\widehat G(t)$. It decreases as the estimated chance of continued observation decreases. Under the stated censoring assumptions, this construction supports estimation of the subdistribution-hazard regression while accounting for the unknown post-censoring status.

The ratio’s time limits matter. With $G(t)=P(C>t)$, a Kaplan–Meier product through time $t$ estimates the right-continuous value after the censoring jump at $t$. Risk sets at event time $t$ are evaluated just before that event, so the numerator uses $\widehat G(t-)$. In the numerical examples, no censoring time coincides with a target-event time. If ties occur, the analysis must state and apply a consistent tie-ordering convention; substituting a right-continuous value at one point and a left-limit value at another can change the weight.

Worked Example: Estimate the Censoring Survival and a Risk-Set Weight

Hypothetical data. Eight subjects have observed times 1 through 8. At times 1, 5, and 8, cause $k$ occurs; at times 3 and 6, a competing cause occurs; censoring occurs at times 2, 4, and 7. Assume no tied times and a common censoring distribution.

For the reverse Kaplan–Meier calculation, the cause-$k$ and competing-event observations leave the censoring risk set without being counted as censoring failures. At time 2, one subject has already had an event at time 1, so $r_C(2)=7$ and $d_C(2)=1$. At time 4, the subjects with observed times 1, 2, and 3 have left, giving $r_C(4)=5$ and $d_C(4)=1$. Thus

\widehat G(2)=1-\frac{1}{7}=\frac{6}{7}, \qquad \widehat G(4)=\frac{6}{7}\left(1-\frac{1}{5}\right) =\frac{24}{35}=0.6857.

At $t=5.5$, the censoring jump at 4 is included, but the jump at 7 is not, so $\widehat G(5.5-)=24/35$. The subject censored at 2 contributes

\frac{\widehat G(5.5-)}{\widehat G(2)} =\frac{24/35}{6/7} =\frac{4}{5}=0.8000.

The subject with a competing event at time 3 contributes one at 5.5, because the competing event was observed. The subject censored at time 4 contributes $(24/35)/(24/35)=1$. Subjects still observed beyond 5.5 also contribute one. The subject with a cause-$k$ event at time 5 contributes zero after that event.

A Weighted Partial-Score Calculation

The next calculation shows how the weights affect a Fine–Gray risk-set denominator. It illustrates one event-time contribution, not a complete fitted model. Assume a single binary covariate $Z$, no ties, and evaluate the partial score at $\beta=\log 2$.

Worked Example: Calculate a Fine–Gray Risk-Set Denominator

Use the same hypothetical subjects as above. At the cause-$k$ event time $t=5$, let $Z=1$ for the subjects observed at times 2, 4, 5, and 8; let $Z=0$ for those observed at times 3, 6, and 7. The subject with an event at time 1 has already had a cause-$k$ event and is excluded. Since no censoring jump occurs at 5, $\widehat G(5-)=24/35$. The subject censored at 2 has weight $0.8$; the one censored at 4 has weight 1. The observed competing event at time 3 remains in the risk set with weight 1.

At $\beta=\log 2$, $\exp(\beta Z)$ equals 2 when $Z=1$ and 1 when $Z=0$. The $Z=0$ contributions are 1 each for the subjects observed at 3, 6, and 7, summing to 3. The $Z=1$ contributions are $0.8(2)$ for the subject censored at 2 and $2$ each for the subjects observed at 4, 5, and 8. Therefore

\sum_jY_j^w(5)\exp(\beta Z_j) =3+0.8(2)+2+2+2 =10.6.

The weighted covariate sum is $0.8(2)+2+2+2=7.6$, so the risk-set weighted mean of $Z$ is $7.6/10.6=0.7170$. The event subject has $Z=1$, giving a score contribution of $1-0.7170=0.2830$ at this value of $\beta$. This is positive: at $\beta=\log 2$, this event’s contribution favors a somewhat larger coefficient, considered on its own.

When the Censoring Model Must Be Conditional

A marginal censoring estimate is suitable only when censoring is independent of event time and cause without conditioning on additional covariates, or when the chosen model otherwise adequately represents the censoring mechanism. If censoring differs across measured groups that also carry information about event outcomes, estimate $G(t\mid W)$ conditionally. A stratified reverse Kaplan–Meier estimate is one option for categorical $W$; a model for the censoring distribution can be used for richer covariate patterns, provided its assumptions are defensible.

Worked Example: Conditional Censoring Weights by Group

Hypothetical data. Consider two groups, A and B, with four subjects each. In each group, one subject has an event at time 1. In group A, one is censored at 2 and the remaining subjects have events at 3 and 4. In group B, subjects are censored at 2 and 3, and the remaining subject has an event at 4. For simplicity, use a separate reverse Kaplan–Meier estimate within each group and evaluate the risk set at $t=3.5$.

In group A, the censoring risk set at time 2 has three people, because the event at time 1 has left it. Thus $\widehat G_A(2)=2/3$, and there is no further censoring jump before 3.5. The subject censored at 2 has weight

\frac{\widehat G_A(3.5-)}{\widehat G_A(2)} =\frac{2/3}{2/3}=1.

In group B, the censoring jump at time 2 also gives $\widehat G_B(2)=2/3$. At time 3, two subjects remain in the censoring risk set and one is censored, so $\widehat G_B(3.5-)=(2/3)(1-1/2)=1/3$. The subject censored at 2 therefore has weight

\frac{\widehat G_B(3.5-)}{\widehat G_B(2)} =\frac{1/3}{2/3}=0.5000.

A pooled estimate would combine the groups’ censoring jumps and give the time-2-censored subjects the same weight, despite the later censoring at time 3 occurring only in group B. If group membership is needed to make censoring independent of event outcomes, the pooled weights do not reflect the relevant conditional censoring distribution. This small example is illustrative; with such sparse groups, the estimates themselves would be unstable.

Assumptions, Diagnostics, and Limitations

The weighting procedure depends on several assumptions. Most importantly, censoring must be independent of the event time and cause conditional on the covariates $W$ used to model $G$. The censoring model must also be estimated adequately, and the data must provide support for remaining uncensored over the times and covariate patterns being analyzed. In practical terms, $\widehat G(t\mid W)$ should not be zero where the analysis needs a risk-set contribution. This support condition is often called positivity.

Useful checks include examining censoring patterns over time and across important covariate groups, checking whether a chosen conditional model describes those patterns, and inspecting $\widehat G$ near the latest target-event times. Near-zero survival estimates make ratios sensitive to small changes in the censoring model and leave little information about later follow-up. Sparse strata, misspecified covariate relationships, and unmeasured predictors of both censoring and event outcomes can undermine the analysis. The ratio in this Fine–Gray construction is bounded by one when estimated survival is nonincreasing and the denominator is positive, but that does not eliminate instability in the estimated ratios or uncertainty in the regression estimates.

The weights themselves are estimated quantities. Inference should account for that estimation rather than automatically treating every fitted weight as fixed. The weights also do not repair dependent censoring caused by unmeasured factors, nor do they create information beyond the observed follow-up.

The main alternative in this regression setting is a cause-specific Cox model. Its risk set removes a person when a competing event occurs; the Fine–Gray risk set retains that person for the subdistribution-hazard analysis. These models answer different questions and use different risk-set constructions. Censoring weights in Fine–Gray regression are not a substitute for choosing the model that matches the scientific target.

Common Mistakes

Mixing survival conventions. If $G(t)=P(C>t)$, the Kaplan–Meier product through $t$ includes the censoring jump at $t$. Use the left limit $\widehat G(t-)$ for the risk set immediately before an event at $t$.
Giving a known competing event a censoring weight. An observed competing event before $t$ contributes one to the subdistribution risk set. It is not an unknown post-censoring outcome.
Using $1/\widehat G(t)$ as the Fine–Gray risk-set weight. A subject censored before $t$ contributes the time-specific ratio $\widehat G(t-)/\widehat G(X_i)$ under the convention used here.
Assuming a pooled censoring model is always adequate. If censoring depends on covariates, estimate or model the censoring distribution conditional on the relevant covariates.
Interpreting weights as a cure for informative censoring. They rely on conditional independent censoring and adequate support; they cannot correct an unmeasured censoring mechanism by themselves.

Key takeaway: With $G(t\mid W)=P(C>t\mid W)$, use a right-continuous censoring-survival estimate and the left limit at the target-event time. In Fine–Gray risk sets, observed competing events remain with contribution one, while subjects censored earlier contribute the censoring-survival ratio $\widehat G(t-\mid W)/\widehat G(X_i\mid W)$.

Check Your Understanding

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

If $\widehat G(c)=0.70$ and $\widehat G(t-)=0.42$ for a subject censored at $c<t$, what is the subject’s Fine–Gray risk-set weight at $t$?
At time $t$, what contribution does a subject with an observed competing event before $t$ make to the Fine–Gray risk set, and why?
Suppose a censoring jump occurs exactly at a target-event time $t$. Which censoring-survival value is used for the risk set immediately before the event under this tutorial’s convention?
What assumption is needed to use a pooled censoring distribution when censoring differs across measured groups?

Solutions. (1) The weight is $0.42/0.70=0.6000$. (2) The contribution is 1 because the competing event and its cause are known; the subject remains in the Fine–Gray subdistribution risk set. (3) Use $\widehat G(t-)$, the left limit before the censoring jump, consistent with $G(t)=P(C>t)$ and a product through times $s\leq t$. (4) The pooled censoring model must adequately represent censoring without conditioning on group, such as when censoring is independent of event outcomes unconditionally; otherwise estimate censoring conditionally on the relevant group or covariates.