Constructing Inverse Probability of Censoring Weights

Why censoring weights depend on contribution time

Censoring removes information about what would have happened later, not necessarily at random. If people with different observed characteristics have different chances of remaining under follow-up, an unweighted summary of the observed histories may overrepresent those who are easier to follow. Inverse probability of censoring weighting (IPCW) gives greater weight to observed histories that represent people with a lower estimated chance of still being observed.

The central practical question is which probability belongs in the denominator for each observed contribution? The answer is the estimated probability of remaining uncensored through the time of that contribution, conditional on the history used in the censoring model. This is a time-specific rule. In a first-event analysis, an absorbing event ends the trajectory being analyzed. In a multi-state model, a transition need not end follow-up, so later observed contributions need their own weights through their later times.

The censoring probability is usually built from conditional probabilities of remaining observed over successive intervals. Write \(q_{ij}\) for the probability that individual \(i\), uncensored through visit \(j\), remains uncensored up to visit \(j+1\), conditional on the measured history available for that interval. With \(G_i(0)=1\),

Thus an IPCW is the inverse of a cumulative probability, not generally the inverse of just the censoring probability for the current interval. If the censoring model uses only baseline covariates, the same subject-specific model can supply probabilities at successive visits. If it uses time-varying covariates, each interval probability can depend on the measured history available before that interval. Do not use information recorded only after censoring or after the contribution time to construct that contribution's weight.

Assumptions behind the weight

Weighting is not a correction that works merely because a censoring model has been fitted. Its interpretation depends on the assumptions linking censoring, observed history, and the event process.

• Conditional independent censoring: given the measured history \(H_i(k)\) used in the model, remaining observed through visit \(k\) is independent of the full-data contribution of interest at that visit. In practical terms, the included history must capture the predictors needed to make censoring conditionally unrelated to the outcome contribution.
• Positivity: for histories represented in the target population, the probability \(G_i(k)\) must be greater than zero at the times of interest. If some histories have essentially no chance of remaining observed, the corresponding inverse weights are undefined or extremely large.
• Correct estimation of \(G_i(k)\): the censoring model must adequately estimate the relevant conditional probabilities. A model that omits important predictors or misrepresents their time pattern can leave residual selection bias.
• Consistent observation and timing: censoring and event dates must follow the analysis's stated ordering convention. The censoring model must also use covariate history measured in the way assumed when it was fitted.

Conditional independent censoring is an assumption about the information included in the model; fitting a flexible model does not establish that the assumption is true. The earlier material on independent censoring and censoring survival probabilities gives the setting for estimating \(G_i(k)\). Here the focus is how to turn those estimates into weights and attach them to observed histories.

Assigning weights to observed histories

At a given visit \(k\), an observed risk-set or event-history contribution is multiplied by \(1/G_i(k)\). For example, if \(D_i(k)\) indicates a particular observed event at visit \(k\), the weighted event contribution is \(R_i(k)D_i(k)/G_i(k)\). Since \(D_i(k)\) can be observed only when \(R_i(k)=1\), an actually observed event contributes \(1/G_i(k)\); an event not observed because the person was censored contributes no recorded event term at that visit. This weighting does not mean that the censored person is known to have had an event.

For a first-event competing-risks analysis, the first event of any type ends the first-event trajectory. Weight the observed event contribution through its event visit, and do not assign that trajectory later first-event risk-set contributions. A person censored before a visit contributes only at visits for which the required history was observed. By contrast, in a multi-state model a transition can lead to a new non-absorbing state, with follow-up continuing. The transition at one visit and later state or transition contributions are distinct contributions: a later observed contribution is weighted using the censoring probability through its own later visit, conditional on the history available then.

Why inverse weighting can recover a full-data contribution

The basic justification is an expectation identity. Let \(Z_i(k)\) be a contribution that would be available at visit \(k\) under complete follow-up, and suppose censoring through \(k\) is conditionally independent of \(Z_i(k)\) given the measured history \(H_i(k)\). Also suppose \(G_i(k)=P\{R_i(k)=1\mid H_i(k)\}>0\). Then, conditional on that history,

The second equality uses conditional independence; the last uses \(E\{R_i(k)\mid H_i(k)\}=G_i(k)\). Averaging over histories gives \(E\{R_i(k)Z_i(k)/G_i(k)\}=E\{Z_i(k)\}\). This explains the inverse: an observed contribution from a history with probability \(G_i(k)\) represents, on average, \(1/G_i(k)\) such full-data contributions. If conditional independence fails, the factorization in the second line is invalid and inverse weighting alone need not remove bias.

Checks, alternatives, and limitations

Before applying weights, verify that the time indexing agrees across the event-history data and the censoring model. In the convention used here, \(G_i(k)\) is the probability of being uncensored through visit \(k\), so it includes the conditional interval probabilities leading up to that visit. If an analysis uses a different event/censor tie convention, the definition of the at-risk indicator and the probability in the denominator must be aligned with it. A common implementation error is to use the interval probability \(q_{i,k-1}\) alone instead of the cumulative probability \(G_i(k)\).

Also check whether the covariates in the censoring model are genuinely available before the interval they predict. A model that uses future state or covariate information can create look-ahead bias. Weights that become very large are a warning that some observed histories had small estimated probabilities of remaining under observation; they can make estimates unstable. Their distribution and the adequacy of the censoring model deserve scrutiny. Weight stabilization, truncation, and formal positivity diagnostics are separate procedures; they do not change the basic time-specific construction given here.

IPCW is one way to address censoring under conditional independent censoring. When censoring is independent without conditioning on covariates, an unweighted analysis designed for independent censoring may suffice. When censoring depends on measured history, conditional weighting can target the selection induced by that history, provided the model and positivity assumptions are credible. Neither weighting nor a more elaborate censoring model repairs dependence on unmeasured predictors of future outcomes. Model-based approaches make different assumptions about the event process, while IPCW makes its key adjustment through the observation process; neither is assumption-free.

• Wrong: “A participant's weight is the inverse of the probability of avoiding censoring in the current interval.” Correction: for a contribution at visit \(k\), use the inverse of the cumulative probability \(G_i(k)\) of remaining uncensored through that visit.
• Wrong: “Any observed transition ends the trajectory, so no later weights are needed.” Correction: that rule applies to an absorbing endpoint or a first-event analysis; in a multi-state model with continuing follow-up, weight each later observed contribution through its own time.
• Wrong: “A large weight proves that the person was censored informatively.” Correction: it indicates a small estimated probability of remaining observed under the fitted model. It calls for investigation, but by itself does not establish why censoring occurred.