Weighting for Hypothetical Competing-Event Elimination

Why Weight for a Hypothetical Elimination?

When a competing event occurs, it prevents a later target event from being observed in the usual way. A target-event risk under the observed course of events therefore answers a different question from the risk that would arise if the competing event could be eliminated. Inverse-probability-of-censoring weighting (IPCW) addresses the latter question by treating the competing event as censoring, then upweighting people who remain free of it to represent similar people who did not.

This reweighting is not a mechanical way to turn an observed cumulative incidence into a hypothetical risk. It identifies the hypothetical risk only if measured histories suffice to account for the selection induced by the competing event, and if people with relevant histories have a positive chance of avoiding that event. The weights correct selection from competing-event occurrence; they do not, by themselves, remove confounding of treatment.

Use discrete intervals \(1,\ldots,K\). A person is at risk at the start of interval \(j\) if neither event has occurred earlier. Let \(H_j\) denote the measured history available by the start of interval \(j\), including baseline covariates and any relevant prior covariates. For a person still at risk and target-event-free through interval \(j\), define \(g_j(H_j,A)=P(D_j=0\mid A,H_j,\text{at risk at }j,\text{target-event-free through }j)\). In practice this probability is estimated from the observed data, often with a regression model. If other loss to follow-up occurs, it requires its own censoring model and corresponding weights.

Constructing the Weights

For an outcome assessed in interval \(k\), only competing-event censoring before interval \(k\) can have removed someone whose outcome is needed. The unstabilized competing-event weight is therefore \(W^D(k-1)=\prod_{j=1}^{k-1}1/g_j(H_j,A)\), with \(W^D(0)=1\). It is evaluated along the person’s observed history while that person remains eligible. A target event in interval \(k\) is assessed before the competing-event decision at the end of that interval, so its contribution uses the weight through \(k-1\), not through \(k\). By contrast, a contribution requiring the person to remain uncensored through the end of interval \(k\) includes the interval-\(k\) factor.

The product is a sequentially evaluated probability construction. Its factors are conditional probabilities based on the history available at each interval, evaluated along the observed, eligible history. It is not generally the probability of avoiding the competing event conditional on the person’s entire future covariate path: conditioning on future covariates changes the conditional probabilities and need not yield these factors.

The exchangeability condition is strong. It says that, after conditioning on measured history, people who experience the competing event would have had the same distribution of future target-event outcomes under elimination as comparable people who remain free of it. It is not verified merely because a censoring model fits well. If important common causes of the competing event and target outcome are unmeasured, weighting cannot remove the resulting selection bias.

Why the Weighted Risk Formula Works

For each interval \(k\), let \(I_k\) indicate a target event in that interval, observed before the competing-event decision at its end, with no earlier target event. Among people who have not been removed by a competing event before \(k\), weighting by the inverse probability of that continued eligibility compensates for the histories lost to competing events. Under sequential exchangeability and positivity, the weighted interval-\(k\) event contribution represents the corresponding event contribution in the population under elimination. Summing the interval contributions gives a risk.

This expression is an identification formula; an empirical estimate replaces the conditional expectation with an average and the probabilities \(g_j\) with fitted probabilities. If treatment is randomized, or if the target is the risk within treatment arm \(a\), that arm-specific average illustrates the competing-event weighting alone. For a marginal treatment contrast in an observational study, treatment weighting or standardization is also needed to address treatment assignment and target-population covariate distribution. The resulting risks are then contrasted, for example by subtracting the risk under \(a=0\) from that under \(a=1\).

A short justification follows by considering an eligible history at each interval. Conditional on that history, the probability of remaining free of the competing event through interval \(j\) is represented sequentially by the relevant \(g\)-factors. An observed contribution from someone who remains eligible is multiplied by the reciprocal of those factors, cancelling the probability of that observed eligibility in expectation. Applying this argument successively for \(j=1,\ldots,k-1\) gives the inverse product for an interval-\(k\) outcome. The exchangeability condition permits the resulting weighted distribution of observed histories to stand in for the histories that would have been represented under elimination. No factor \(1/g_k\) is needed for \(I_k\), because \(I_k\) is recorded before the competing-event decision at the end of interval \(k\).

Worked Calculations

Diagnostics, Alternatives, and Limitations

Inspect the fitted \(g_j\) probabilities and the resulting weights by interval and treatment arm. Probabilities near zero create large inverse weights; a few people can then dominate the estimate. Useful checks include weight summaries, the largest weights, the number of eligible people remaining at each interval, and whether measured histories with large weights have adequate representation. Compare weighted and unweighted covariate distributions over the histories used to model competing-event censoring. Severe instability or poor overlap indicates a problem; it is not repaired just by reporting a weighted estimate. Truncating weights can reduce variability but changes the estimator and may introduce bias, so any truncation rule should be justified and its effect reported.

The main alternatives answer different questions or make different modeling choices. The observed cumulative incidence describes target events under the competing events as they actually occurred. Treating competing events as ordinary censoring in a standard survival analysis without a causal justification does not, by itself, identify the elimination risk. The earlier material on the competing-risk g-formula provides a model-based standardization route to a hypothetical risk; IPCW instead models the competing-event process and reweights observed outcomes. Both require assumptions about measured history and positivity. A weighted discrete-time hazard approach can also be used: weight the interval-specific target-event experience by eligibility weights, estimate hazards, and combine the hazards into a cumulative risk. Its risk-set and denominator conventions must match the interval ordering; the outcome contribution for interval \(k\) still does not include the interval-\(k\) competing-event factor.

The hypothetical intervention itself may be clinically or biologically difficult to define: eliminating a competing event could change later covariates or treatment pathways. The estimand therefore needs a clear interpretation as a specified hypothetical regime, not as an automatically realizable intervention. Further, IPCW corrects only selection due to the competing event under the stated assumptions. It does not fix unmeasured confounding, misspecified probability models, lack of positivity, or ambiguous treatment versions. In finite samples, the weighted risk can even fall outside the interval from zero to one; this signals that the empirical weighted sum is unstable or that model choices need scrutiny, not that a probability outside that range is meaningful.