The Weighted Partial Likelihood for Fine-Gray Regression

Why Weight the Fine–Gray Pseudo Risk Set?

Fine–Gray regression models the subdistribution hazard for one specified cause, say cause $k$. Its risk set is not the ordinary cause-specific risk set: a person who has already experienced a competing event is retained in the pseudo risk set. When follow-up is censored, however, the competing-event history needed to construct that set is not fully observed for everyone. Censoring weights adjust the contributions of observed competing events to the risk-set sums.

The resulting partial likelihood resembles the Cox partial likelihood, but the denominator is a weighted pseudo risk-set sum. The weights depend on the censoring distribution and on the time of the earlier competing event. They are not event counts, and a weighted risk-set total need not be an integer.

Definition: Let $T_i$ be subject $i$'s event time, $C_i$ their censoring time, $\tilde T_i=\min(T_i,C_i)$, and $\Delta_i=I(T_i\le C_i)$. If $\Delta_i=1$, let $J_i$ denote the event cause. For a chosen cause $k$, the weighted pseudo risk-set contribution at time $t$ is $$ Y_{ik}^{w}(t)=I(\tilde T_i\ge t)+I(\tilde T_i<t,\ \Delta_i=1,\ J_i\ne k)\, \frac{\widehat G(t-)}{\widehat G(\tilde T_i-)}. $$ Here $\widehat G$ estimates the censoring survival function, and $t-$ denotes the left limit just before $t$. A subject with an observed competing event before $t$ remains represented, with the displayed weight. A subject censored before $t$ contributes zero under this form of the weighted risk set.

For the model, write the subdistribution hazard for cause $k$ as $\lambda_k^*(t\mid Z_i)=\lambda_{k0}^*(t)\exp(\beta^\top Z_i)$, where $Z_i$ is the covariate vector and $\lambda_{k0}^*(t)$ is an unspecified baseline subdistribution hazard. At a target-cause event time, the weighted partial-likelihood denominator combines each pseudo risk-set contribution with its relative-risk term.

From Weighted Risk Sets to Partial Likelihood

Suppose $t_1,\ldots,t_D$ are the distinct observed event times for cause $k$, with one cause-$k$ event at each time. Let $i_r$ be the subject with the target event at $t_r$. That subject is observed at $t_r$, so $Y_{i_r k}^{w}(t_r)=1$. Conditional on an event at $t_r$, the model assigns the event to subject $i_r$ in proportion to their relative risk, compared with the weighted relative risks in the pseudo risk set.

L_k(\beta)=\prod_{r=1}^{D} \frac{\exp(\beta^\top Z_{i_r})} {\displaystyle\sum_{j=1}^{n}Y_{jk}^{w}(t_r)\exp(\beta^\top Z_j)}.

The unspecified baseline hazard cancels from each event-time comparison, as in Cox partial likelihood. The distinctive feature is the denominator: it uses $Y_{jk}^{w}(t_r)$, not simply an indicator that subject $j$ has not yet had any event. The log partial likelihood is the sum of the log contributions. Differentiating it gives the score equation used to estimate $\beta$.

\ell_k(\beta)=\sum_{r=1}^{D} \left[ \beta^\top Z_{i_r}- \log\left\{\sum_{j=1}^{n}Y_{jk}^{w}(t_r)\exp(\beta^\top Z_j)\right\} \right].

U_k(\beta)=\sum_{r=1}^{D} \left[ Z_{i_r}- \frac{\sum_{j=1}^{n}Y_{jk}^{w}(t_r)Z_j\exp(\beta^\top Z_j)} {\sum_{j=1}^{n}Y_{jk}^{w}(t_r)\exp(\beta^\top Z_j)} \right].

The fraction in the score is a weighted mean covariate vector at that event time. Thus each event contributes its observed covariates minus the risk-set mean implied by the fitted relative risks and the weighted pseudo risk set.

Assumptions and conditions: This construction requires correctly recorded event causes and times, and a censoring mechanism that is independent of the event process under the conditioning used to estimate $G$. If censoring depends on covariates, the censoring survival model generally needs to condition on those covariates (or otherwise account for that dependence). The censoring survival must be positive over times where the model is fit, and the Fine–Gray proportional subdistribution-hazards model must be an appropriate working model for the intended analysis. Estimated censoring weights add uncertainty; inference should use a variance procedure appropriate to the fitting method and weight estimation.

Why the Censoring-Survival Ratio Appears

For a subject with a competing event observed at time $u<t$, the pseudo risk set retains that subject at time $t$. The ratio $\widehat G(t-)/\widehat G(u-)$ adjusts that retained contribution for the change in censoring survival between the observed competing-event time and the current event time. It is at most one when $\widehat G$ is nonincreasing, and it equals one if there is no estimated censoring loss over that interval. It is not the same as assigning the subject a new event at $t$.

A short algebraic check clarifies the role of the weight. Define $S_r^{(0)}(\beta)=\sum_jY_{jk}^{w}(t_r)\exp(\beta^\top Z_j)$ and $S_r^{(1)}(\beta)=\sum_jY_{jk}^{w}(t_r)Z_j\exp(\beta^\top Z_j)$. Then the event-time log contribution is $\beta^\top Z_{i_r}-\log S_r^{(0)}(\beta)$. Since the derivative of $S_r^{(0)}$ is $S_r^{(1)}$, differentiation gives $Z_{i_r}-S_r^{(1)}/S_r^{(0)}$, exactly the event contribution in the score above. Therefore, at a maximum, the total observed covariate contribution balances the model-weighted covariate contributions across the event times.

Worked Example: Calculating a Retained Competing-Event Weight

In hypothetical data, a subject has an observed competing event at time $u=2$ months, and a target-cause event occurs at $t=5$ months. Suppose the estimated censoring survival values are $\widehat G(2-)=0.70$ and $\widehat G(5-)=0.30$. The competing-event subject is retained in the pseudo risk set at month 5 with weight

\frac{\widehat G(5-)}{\widehat G(2-)} =\frac{0.30}{0.70}=0.4286.

If this subject has $Z=0$, their contribution to the denominator at a fitted value $\beta=\log(2)$ is $0.4286\exp\{(\log 2)0\}=0.4286$. The subject is present in the pseudo risk set but contributes less than one full unit. The calculation uses the censoring survival just before each time, consistent with the stated left-limit convention.

Worked Example: Two Weighted Event-Time Contributions

Consider hypothetical risk-set summaries for a binary covariate $Z$. At the first target event time, let $\beta=\log 2$. The target event has $Z=0$, so its numerator is $1$. Another risk-set subject has $Z=1$ and weight $1$, contributing $2$ to the denominator. A retained competing-event subject has $Z=0$ and weight $0.429$, contributing $0.429$. The event contribution is

\frac{1}{1+2+0.429}=\frac{1}{3.429}\approx 0.2916.

At a second target event time, let $\beta=\log(1.2)$. The event subject has $Z=1$, giving numerator $1.2$. The weighted denominator contributions are $1.2$ from that subject, $1.2$ from another $Z=1$ subject of weight 1, $1$ from a $Z=0$ subject of weight 1, and $0.364$ from a retained $Z=0$ competing-event subject. Thus

\frac{1.2}{1.2+1.2+1+0.364} =\frac{1.2}{3.764}\approx 0.3188.

The two-event partial-likelihood contribution at these specified coefficient values is approximately $0.2916\times0.3188=0.09297$, using the unrounded event contributions. This is a likelihood value at the stated $\beta$, not a p-value or a probability that the model is true.

Worked Example: A One-Event Score Contribution

Suppose a hypothetical target event occurs for a subject with $Z=1$. At $\beta=0$, the event subject has weight 1 and contributes $1$ to the denominator. A $Z=0$ subject contributes $1$, and a retained competing-event subject with $Z=1$ has weight $0.5$, contributing $0.5$. The denominator is $2.5$, and the weighted mean covariate is

\frac{(1)(1)+(0)(1)+(1)(0.5)}{2.5} =\frac{1.5}{2.5}=0.6.

The event's score contribution is observed $Z$ minus this mean, or $1-0.6=0.4$. It is positive at $\beta=0$: this event contributes information pushing the coefficient upward, although the final estimate depends on all target events and all risk sets.

Implementation Checks and Failure Modes

Before interpreting a fitted model, check how the software constructs the censoring distribution, handles ties, and estimates uncertainty. With tied target events, the simple product displayed above assumes one event per distinct time. A fitting procedure must specify a tie approximation or exact method; weighted denominators should not be combined with a tie rule as though the weights were integer subject counts.

Check the direction and range of the weight. Under a nonincreasing $\widehat G$, $\widehat G(t-)/\widehat G(u-)$ for $u<t$ lies between zero and one. A ratio above one often signals that the numerator and denominator have been reversed or the survival estimates indexed incorrectly.
Check support and stability. If $\widehat G(u-)$ is zero or extremely small, the ratio is undefined or numerically unstable. Sparse follow-up and heavy censoring can make late risk-set contributions unreliable.
Check censoring assumptions. If censoring differs systematically by covariate but a marginal censoring survival is used without justification, the resulting weights may not adequately adjust the risk sets. Compare the assumed censoring model with observed follow-up patterns.
Check event coding and time ordering. A competing event before $t$ belongs in the retained part of the pseudo risk set; a target-cause event before $t$ does not. A subject censored before $t$ is not treated as an observed competing event.

This approach differs from cause-specific Cox regression, where a competing event removes the subject from the cause-specific risk set at its event time. Fine–Gray regression instead retains competing-event subjects through the weighted pseudo risk set to model the subdistribution hazard. An unweighted Fine–Gray-style denominator would omit the censoring adjustment, while a full likelihood would require modeling more of the event-time distribution than the partial likelihood does. The weighted partial likelihood is useful, but it does not remove the need to assess model form, censoring assumptions, and data support.

Common Mistake

A frequent error is to use $1/\widehat G(u-)$ as the weight for a competing event at $u$, or to put a censored subject into the retained-competing-event term. These operations do not match the weighted pseudo risk-set definition used here. For an observed competing event before $t$, use $\widehat G(t-)/\widehat G(u-)$; for a subject censored before $t$, neither indicator in the stated definition is active. Always identify the person's observed status and event time first, then apply the matching term.

Key takeaway: Fine–Gray’s weighted partial likelihood has the Cox-like event numerator, but its denominator sums relative risks over a censoring-adjusted pseudo risk set. The ratio of censoring-survival estimates modifies the retained contribution from an observed competing event before the current target-event time.

Check Your Understanding

Answer each question using the weighted risk-set definition above; brief solutions follow the questions.

A competing event occurs at $u$ before a target event at $t$. If $\widehat G(u-)=0.80$ and $\widehat G(t-)=0.40$, what is the retained risk-set weight?
At a target event time, what does a subject who was censored earlier contribute under the displayed risk-set definition?
For one target event with $Z=0$, the weighted denominator is 4. What is its partial-likelihood factor?
Why should an estimated censoring survival that is near zero prompt caution?

Solutions. (1) The weight is $0.40/0.80=0.5000$. (2) Zero: the subject was censored before the current time and does not satisfy either indicator in the definition. (3) The numerator is $\exp(\beta^\top 0)=1$, so the factor is $1/4=0.25$. (4) A near-zero denominator in a censoring-survival ratio can make weights unstable or undefined, so risk-set contributions and estimates may be highly sensitive to sparse late follow-up.