Sampling Weights in Fine-Gray Regression

Why Fine-Gray Regression Needs Two Kinds of Weights

Fine-Gray regression models the subdistribution hazard for a specified cause. Its risk set is unusual: people who experience a competing event remain represented in the risk set after that event. With right censoring, inverse-probability censoring adjustments are used to account for the information lost when follow-up ends before event status is known. Those adjustments do not, by themselves, make a sample representative of a target population.

If people have unequal probabilities of being sampled, a second kind of weight may be needed: a sampling weight that represents how much of the target population each sampled person stands for. The two weights address different mechanisms. A population-representation weight is attached to a sampled person; a censoring adjustment modifies that person's contribution to the Fine-Gray pseudo risk set at a given time. Combining them is not a matter of choosing one or the other.

The aim here is to construct and interpret the weighted Fine-Gray estimating equation. It is not to re-teach the Fine-Gray model or the general use of inverse-probability censoring weights. The essential point is to keep the two weighting roles explicit throughout the calculation.

Model and Combined Estimating Equation

For covariate vector \(Z_i\), the proportional subdistribution-hazards model for cause \(k\) is

Here \(\lambda_k^*(t\mid Z_i)\) is the subdistribution hazard, \(\lambda_{k0}^*(t)\) is its baseline function, and \(\beta\) is the regression coefficient vector. At time \(t\), the Fine-Gray pseudo risk set represents people who have not yet had the event of interest, including people already known to have had a competing event.

One common inverse-censoring-adjusted representation uses the following risk contribution, where \(\hat G(t\mid L_i)\) estimates the probability of remaining uncensored through time \(t\), conditional on relevant variables \(L_i\). Let \(\Delta_i=1\) indicate an observed event and \(\Delta_i=0\) indicate censoring. The left-limit notation accommodates event times at which censoring also occurs.

An observed person still under observation and event-free at \(t\) has \(q_i(t)=1\). A person with an observed competing event before \(t\) remains represented, with the censoring adjustment shown. A person censored before \(t\) without a known competing event contributes zero at \(t\). This expression presumes a censoring model and timing convention appropriate to the data; implementations can differ in details, so the definition used by the analysis software must be checked.

For \(r=0,1\), define the weighted risk-set sums as follows, with \(Z_i^{\otimes 0}=1\) and \(Z_i^{\otimes 1}=Z_i\).

The factor \(1/n\) is optional: it cancels from the ratio below. For a single covariate, the weighted risk-set mean is \(S_w^{(1)}/S_w^{(0)}\). A corresponding weighted score equation is

where \(N_{ik}(t)=I(\tilde T_i\le t,\ J_i=k)\) counts an observed event of the target cause. Thus the event contribution is multiplied by its sampling weight, while the risk-set mean uses both \(d_i\) and \(q_i(t)\). Tied target-cause events require a tie-handling convention consistent with the Fine-Gray procedure being used.

Assumptions and a Short Justification

Interpretation as a target-population analysis requires more than entering weights into software. The sampling weights must be appropriate for the target population and the sampling design; in particular, units with relevant target-population membership must have positive probability of representation. Censoring must be conditionally independent of the event process given the variables in the censoring model, with adequate positivity so that \(\hat G(t\mid L_i)\) does not approach zero over the times analyzed. The proportional subdistribution-hazards model must be an adequate description for the covariates and time range of interest. If weights are estimated, their estimation and the design features that generated them can affect uncertainty estimates.

The combined form follows by starting with the Fine-Gray weighted partial-likelihood contribution. At a target-cause event time \(t\), the risk denominator sums relative hazards across the pseudo risk set. Each person's contribution to that sum is multiplied by \(d_i\), to represent the target population, and by \(q_i(t)\), to adjust the pseudo risk-set contribution for censoring. Differentiating the log of the weighted event contribution gives an event covariate minus the risk-set weighted mean.

Summing this expression over observed target-cause failures yields the score above. This is a justification of the estimating equation, not a claim that a complex weighted analysis has an ordinary independent-observation likelihood. The sampling design and estimated weights may require design-based or robust variance calculations.

Diagnostics, Alternatives, and Limitations

Before interpreting a fitted model, check both weighting mechanisms separately and then inspect their combined contributions. Useful checks include:

• Confirm that sampling weights match the intended target population and account for the design, rather than treating them as censoring weights.
• Inspect the range and distribution of \(d_i\), \(\hat G(t\mid L_i)\), and combined risk contributions \(d_iq_i(t)\). Very large or unstable contributions can make estimates sensitive to a few observations.
• Check censoring-model fit and positivity over the analysis time range. Small estimated \(\hat G\) values can produce extreme adjustments or make late-time estimates unreliable.
• Verify event coding and the handling of competing events and ties. The event indicator in the score must identify the target cause, not all failures.
• Use variance estimation appropriate to the sampling design, including clustering or stratification when relevant, and account for estimated weights where required. A naive model-based standard error may not reflect the actual uncertainty.

A sampling-weighted cause-specific Cox model also uses population-representation weights when the target is a population parameter, but it concerns a cause-specific hazard and uses a different risk-set construction. Fine-Gray regression retains competing-event cases in its pseudo risk set; it is therefore not enough to transfer the cause-specific risk-set calculation unchanged. Neither model's coefficient should be described as a causal effect solely because sampling weights were used. The choice depends on the scientific estimand and the interpretation sought.

Weight truncation or stabilization may reduce the influence of extreme values, but changes the estimating procedure and can introduce bias for the original target. Such choices should be justified and accompanied by sensitivity analyses. Weighted Fine-Gray regression also cannot repair a misspecified subdistribution-hazards model, an incorrect censoring model, or a sample that gives inadequate representation to important target-population groups.