Sampling Weights in Cause-Specific Cox Regression

Why Weight a Cause-Specific Cox Model?

A study may sample some members of a target population more heavily than others. For example, a survey might deliberately oversample a small demographic group. An unweighted cause-specific Cox fit then describes the sampled risk sets as if their composition represented the target population. Sampling weights can correct that imbalance when the sample design and analysis assumptions support such a correction.

The purpose of these weights is not to change what counts as a cause-specific event. For cause \(k\), a person contributes to the risk set until the event of interest, a competing event, or censoring; a competing event removes that person from the cause-\(k\) risk set. The weights instead adjust how much each sampled person represents in the target population. This is distinct from the earlier material on inverse-probability censoring weights, whose purpose is to adjust for censoring.

Weighted Partial Likelihood and Its Score

Suppose \(T_i\) is the observed time for person \(i\), \(\delta_{ik}=1\) indicates an observed event of cause \(k\), and \(X_i(t)\) is the covariate vector. Let \(R(t)\) be the set still at risk immediately before time \(t\) for cause \(k\). Assume the cause-specific hazard model \(\lambda_k(t\mid X_i(t))=\lambda_{0k}(t)\exp\{\beta^\mathsf{T}X_i(t)\}\). For now, assume distinct event times and write \(t_i\) for the event time of person \(i\).

Each event is weighted by the event person's sampling weight. Every person in the risk set contributes to its weighted denominator using that person's sampling weight. Using weights only for events, or only in the risk-set sums, does not implement this weighted estimating procedure.

To see the estimation equation, define the weighted risk-set mean \(\bar X_w(t,\beta)=\sum_{j\in R(t)}w_j e^{\beta^\mathsf{T}X_j(t)}X_j(t)/\sum_{j\in R(t)}w_j e^{\beta^\mathsf{T}X_j(t)}\). Differentiating the objective gives the weighted score:

The estimate \(\hat\beta_w\) solves \(U_w(\hat\beta_w)=0\), when a finite solution exists. This equation shows the basic comparison at each event time: the event covariate is contrasted with the sampling-weighted covariate average among those at risk. The model concerns the cause-specific hazard; it does not by itself give a risk ratio or a cumulative-incidence contrast.

Assumptions and What the Weights Accomplish

A weighted fit is not automatically representative merely because a weight column is present. Its use requires a defensible sampling and outcome-observation framework. Important conditions include:

• Known or consistently estimated inclusion probabilities. The weights must reflect the sample design, including relevant unequal selection probabilities and, when needed, nonresponse adjustments. If inclusion probabilities are estimated, their estimation may also need to be reflected in variance calculations.
• Positivity of selection. Every part of the target population relevant to the estimand must have a nonzero probability of selection. Very small inclusion probabilities create large weights and can make estimates unstable.
• Appropriate risk-set observation. Conditional on the variables used in the analysis, censoring must not selectively remove people in a way that invalidates the cause-specific risk-set comparison, unless an appropriate censoring adjustment is also used. Competing events are handled as exits from the cause-specific risk set, not as ordinary censoring for the competing-risk process.
• A suitable hazard model and covariate history. The assumed log-linear proportional-hazards relationship must be adequate for the target population and the time-varying covariates must be correctly represented. Weighting does not repair nonproportional hazards, omitted confounding in an associational model, or incorrect event coding.
• Correct use of the sampling design in inference. If the design has strata, clusters, or finite-population features, uncertainty calculations should reflect them. Treating a complex survey as a simple random sample can give misleading standard errors even when the point estimate is weighted.

The target is usually a cause-specific regression relationship in the population represented by the weights. Interpretation depends on the model and covariates: for example, \(\exp(\beta_r)\) compares cause-\(k\) hazards for a one-unit increase in covariate \(r\), holding the other modeled covariates fixed, under the proportional-hazards specification. It is not automatically a causal effect.

Weight Scaling and Variance

A useful check is what happens if every sampling weight is multiplied by a positive constant \(c\). The weighted risk-set sum becomes \(c\) times its original value. Consequently, each logarithm in the objective gains \(\log(c)\), while its event contribution is multiplied by \(c\). If \(W_E=\sum_{i:\delta_{ik}=1}w_i\), then:

The second term is constant in \(\beta\). Therefore the score and observed information are multiplied by \(c\), but the coefficient estimate is unchanged, provided the unpenalized objective has a finite maximizer and the same risk sets are used. The inverse observed-information standard error, if used as a model-based calculation, is multiplied by \(1/\sqrt{c}\). That scaling behavior is a feature of the pseudo-likelihood curvature, not evidence that changing the arbitrary weight scale changed the information in the survey.

For a complex probability sample, a design-based or robust variance is usually more appropriate than relying on the inverse Hessian alone. Such methods use the sampling units and design structure—for example, primary sampling units and strata—or use valid replicate weights. In a sandwich calculation, the score contributions are aggregated according to the sampling design; the variance is not obtained by simply treating each weighted record as \(w_i\) independent people. A global rescaling of all weights generally cancels between the sandwich “meat” and “bread,” unlike the inverse-Hessian-only calculation.

Design Weights Are Not Censoring Weights

The two kinds of weights answer different questions. A sampling weight represents how many target-population members a sampled person stands for. An inverse-probability censoring weight represents the inverse probability of remaining uncensored up to a particular time, conditional on modeled predictors. The former addresses the sample’s representation of the population; the latter addresses selective observation caused by censoring under its own conditional-independence and positivity assumptions.

The weights may be combined when both sampling and censoring need adjustment, but that does not make their assumptions interchangeable. The combined estimating procedure and its variance must account for the sampling design and the estimation of censoring probabilities. In particular, applying a censoring weight does not correct unequal sampling, and applying a design weight does not generally correct informative censoring.

Diagnostics, Alternatives, and Limitations

Begin by checking the weight distribution and the survey design documentation. Verify the event indicator, cause coding, time origin, and risk-set membership, then compare weighted and unweighted covariate summaries to see what population representation changes. Inspect whether a small number of observations dominate weighted event contributions or risk-set sums. Large weights, sparse weighted risk sets, or highly variable fitted results under reasonable weight truncation or calibration choices are warning signs, not automatic justification for trimming. Trimming changes the weighting procedure and can introduce bias; its rationale and sensitivity analysis should be reported.

Assess the proportional-hazards and functional-form assumptions using suitable residual-based or time-interaction diagnostics, interpreted with the weighted design and variance in mind. Check convergence and whether the weighted score is near zero at the reported estimate. A finite estimate may fail to exist when covariate patterns separate events from risk sets or when risk sets are too sparse. Weighted estimation does not fix model misspecification, lack of overlap, or poor data quality.

The main alternatives depend on the target and design. An unweighted Cox fit can be reasonable for a model explicitly targeting the sampled cohort under a suitable sampling mechanism, but it may not represent the population when inclusion is informative. A design-weighted fit targets a population-representative estimating equation under its assumptions; survey-robust or replicate-weight variance addresses uncertainty from the sampling design. A censoring-weighted approach addresses a different observation mechanism. Stratification or calibration can be part of the survey design or analysis, but does not remove the need to define the target population and variance method.

For tied event times, the distinct-time expression above is not enough: a tie-handling method must be specified, and weighted implementations can differ. The Breslow or Efron treatment should be chosen and implemented consistently with the software’s weighted procedure; results can differ when ties are substantial. Finally, a cause-specific hazard model alone does not directly estimate cumulative incidence. For that population risk question, the competing event process also matters.