Why Censoring Errors Can Change the Question
Censoring is not just a data-formatting detail. It determines which outcomes are observed, which individuals contribute to a risk set, and what assumptions are needed to infer beyond observed follow-up. A mistake can change the estimand, introduce bias, or leave the target quantity unidentified. These problems are especially consequential in competing-risk analyses, where a competing event is an observed outcome—not ordinary loss to follow-up.
The focus here is diagnosis. The earlier material on censoring weights and their diagnostics provides the construction and routine checks; the key question in each case is whether the analysis still represents its intended target and whether the required assumptions are defensible.
For a target event type \(k\), a common target is the cumulative incidence function \(F_k(t)=P(T\le t,J=k)\): the probability that the event of interest has occurred by time \(t\) as the first event. A different quantity, the cause-specific hazard, concerns the instantaneous rate of type-\(k\) events among people still event-free immediately before a time. Those quantities answer different questions. Treating competing events as censoring in a Kaplan–Meier calculation does not generally estimate \(F_k(t)\).
Failure 1: Coding Competing Events as Censored
Suppose the analysis targets the real-world probability of experiencing event type \(k\) by time \(t\). The competing event is part of the outcome process and must reduce the probability of remaining event-free. A Kaplan–Meier calculation that instead censors competing events estimates a “net” quantity under a hypothetical interpretation in which those competing events do not prevent the event of interest, rather than the observed cumulative incidence.
Here is a small hypothetical illustration. Assume everyone is initially observed, there is no loss to follow-up, event times are ordered without ties, and the event counts below are complete. These conditions let us compare the arithmetic directly; they do not make the two estimands interchangeable.
Worked Example: Competing Events Mistakenly Censored
In a hypothetical cohort of 100 people, 10 experience the event of interest at time 1. At time 2, 20 of the remaining 90 experience a competing event. At time 3, 7 of the 70 people still event-free experience the event of interest. Find the cumulative incidence of the event of interest by time 3, then compare it with the Kaplan–Meier calculation that censors competing events.
Cumulative incidence. The event-of-interest probability increases by 10/100 at time 1. After that, 90 people remain event-free. The 20 competing events at time 2 reduce the event-free group to 70, so the event-of-interest increment at time 3 is 70/100 multiplied by 7/70:
Thus, the observed cumulative incidence is 0.170, or 17.0% by time 3.
Incorrect Kaplan–Meier treatment. If the 20 competing events are censored at time 2, the Kaplan–Meier risk set at time 3 contains 70 people. The estimated event-free probability for the event of interest is
The 19.0% result is not the observed 17.0% cumulative incidence. It arises because the calculation removes competing events from the risk set as if they were censoring, then estimates the event-of-interest risk among the remaining group. The difference is not a rounding error; it reflects a different treatment of the event process.
For a cause-specific hazard analysis, a person who experiences a competing event leaves the risk set because that person is no longer event-free. That risk-set operation is appropriate for the hazard’s definition. It does not mean the event should be relabelled as loss to follow-up when the target is a cumulative incidence. Always state the estimand before deciding how event types enter the analysis.
Failure 2: Using Information After Censoring
Censoring marks the limit of the observed event history for the analysis. A common failure is to use a covariate measured after that limit as though it were known at the censoring time—for example, using a six-month clinical status to model censoring for someone whose follow-up ended at month three. This can leak future information into a model and make the proposed adjustment unavailable for people censored at the relevant time.
There is an important distinction between later ascertainment and information genuinely unavailable. If a valid registry linkage discovers an event after the recorded administrative end date, and the linkage establishes that event follow-up actually continued through that date, the observation window may need to be corrected. It is not valid simply to treat an event beyond a true end of observation as if it had been observed under the original follow-up scheme. The time origin, ascertainment process, and censoring definition must agree.
Worked Example: A Predictor Recorded After Follow-Up Ends
In a hypothetical study, follow-up ends at month 3 for one participant. A censoring model is then fit using a health score recorded at month 6, and that score is used to form a weight for the participant’s contribution through month 3. Explain the problem and a defensible correction.
The month-6 score is not part of the information observed by month 3. It may itself depend on events, treatment, or health changes occurring after the participant’s follow-up ended. Using it as a predictor of censoring at month 3 conditions on future information and can introduce selection bias or leakage. It also cannot support a prospective prediction of who remains observed at month 3 when it is unavailable for people whose observation ended earlier.
A defensible analysis defines the information history at each time using variables measured no later than that time, then models continued observation using that history. If the month-6 score is actually recovered for everyone through a separate, valid data source, the analyst must explain why it is available and whether the estimand and censoring mechanism justify its use; its later measurement alone does not make it a valid month-3 predictor. If event follow-up was in fact available after month 3, correct the follow-up records and censoring times rather than pretending that future information had been observed under the original design.
Failure 3: Ignoring Positivity
Inverse-probability-of-censoring weighting requires more than a well-fitting censoring model. At each relevant time and for histories represented in the target population, there must be a positive probability of remaining uncensored. In notation, if \(H\) denotes the observed covariate and event history, the conditional probability of remaining under observation through time \(t\), \(G(t\mid H)=P(C\ge t\mid H)\), must be positive wherever the target population has support. If \(G(t\mid H)=0\), the outcomes after \(t\) for that history are never observed, so weighting cannot recover them from the data.
Very small probabilities are a practical warning even when they are not exactly zero. A contribution weighted by \(1/G(t\mid H)\) can dominate the estimate, so the nominal sample size may greatly overstate the effective information. This is why positivity checks and assessments of extreme influence belong alongside model fitting, not after the result appears implausible.
Worked Example: Zero Observation Probability in a Subgroup
In a hypothetical study, all participants in subgroup A are censored before month 6, while participants in subgroup B remain under observation. The target is the month-6 event risk in the full population, including subgroup A. Can censoring weights identify the target?
For subgroup A, \(G(6\mid H)=0\). The required weight \(1/G(6\mid H)\) is therefore undefined. No observed month-6 outcomes exist for this subgroup, so the data provide no direct information about its month-6 risk. A censoring model cannot manufacture that information. Unless additional assumptions or external information identify the missing outcomes, the full-population target is not identified by inverse-probability weighting.
The analyst should report the support failure and reconsider the target population, time horizon, or data source. Restricting the target to a population with observation support changes the estimand and must be justified; it is not a numerical fix that preserves the original target.
Failure 4: Applying Weights Without Checking Stability
Even when positivity holds mathematically, a few very large weights can make an estimate unstable. The following hypothetical data show how a single observation can dominate. Suppose four observed individuals have censoring weights \(1,1,2,20\), and their binary event indicators are \(0,0,1,1\). For illustration, calculate the weighted event proportion as the sum of weighted indicators divided by the sum of weights. This simple calculation is not a general competing-risk estimator; it isolates the influence of the weights.
Worked Example: Extreme Influence and Truncation
The weighted event proportion is
The effective sample size diagnostic is \((\sum_i w_i)^2/\sum_i w_i^2\). Here,
Although four people are present, the weights carry roughly 1.42 observations’ worth of equal-weight information by this diagnostic. The person with weight 20 contributes 20/24, or 83.3%, of the total weight.
If a prespecified truncation caps the weight at 5, the same illustrative proportion becomes \((2+5)/(1+1+2+5)=7/9=0.7778\). This substantial change signals sensitivity to the extreme weight. Truncation can reduce variance and influence, but it generally introduces bias relative to the untruncated weighted estimator; it does not restore positivity or prove that the censoring model is correct. Report sensitivity to the chosen cap and explain the trade-off.
Stability checks should examine the weight distribution, the largest weights and their associated histories, effective sample size, and the contribution of high-weight observations to the estimate. A model may produce finite weights yet still be practically unstable. Conversely, a large weight is a warning to investigate, not proof by itself that the estimate is biased. The cause may be sparse support, model misspecification, coding errors, or a genuinely rare observation pattern.
A Diagnostic Sequence and the Main Alternatives
A disciplined review starts with the target and data-generating structure, then asks whether the assumptions and computations match them. Conditional independent censoring means that, given the included observed history, censoring provides no additional information about the relevant unobserved event outcomes. Positivity requires a positive chance of remaining observed for the histories and times needed by the target. Correctly specified censoring probabilities are also needed for the corresponding weighting estimator to work as intended. These conditions are substantive and cannot be established solely by a good model fit.
Specify the event type and time horizon. Check that competing events are represented as observed outcomes when estimating cumulative incidence, not silently recoded as loss to follow-up.
For each predictor used in a censoring model, verify that it is measured by the time at which continued observation is being modelled. Reconcile later event discoveries with the actual ascertainment window.
Look for histories with zero or near-zero estimated observation probability. Summarize weight tails, effective sample size, and influential observations; repeat key estimates under defensible weight-handling choices.
If support is absent, or conditional independent censoring is implausible, acknowledge that weighting alone does not solve the problem. Consider a different target, additional data, or a sensitivity analysis with explicit assumptions.
The alternatives depend on the failure. For cumulative incidence, an estimator that accounts for competing event types is appropriate when the competing events are observed; a cause-specific hazard answers a different, rate-based question. For loss to follow-up, inverse-probability weighting can adjust under its assumptions, while augmented approaches combine weighting with an outcome model and may offer protection if one of the models is correctly specified under the relevant conditions. Imputation or explicit informative-censoring sensitivity models can examine departures from conditional independent censoring, but they do not turn unobserved outcomes into observed facts. No method automatically repairs absent positivity.
| Warning sign | Why it matters | First response |
|---|---|---|
| Competing event coded as censoring | May replace observed cumulative incidence with a different quantity | Verify event coding and state the target estimand |
| Predictor measured after censoring | Can leak future information or be unavailable for the relevant risk set | Rebuild time-specific histories and verify ascertainment dates |
| Zero or near-zero observation probability | Weights are undefined or extreme; target may lack support | Identify unsupported histories and reconsider what is identifiable |
| Estimate changes sharply under reasonable weight caps | A small number of observations may dominate | Report influence and sensitivity; investigate the source of extremes |
Common Mistakes and Practical Limits
- Wrong: “A competing event prevents the target event, so it is censoring.” Correction: It is an observed event that changes the event-free population. Whether it exits a particular risk set depends on the estimand; it is not ordinary loss to follow-up.
- Wrong: “A value recorded later is useful because it predicts censoring well.” Correction: Predictive association does not make future information valid for an earlier time. Respect the measurement and observation timeline.
- Wrong: “All weights are finite, so positivity is satisfied.” Correction: Estimated probabilities can hide sparse support, and very large finite weights can still make inference unstable. Inspect support and influence.
- Wrong: “Truncation fixes extreme weights and therefore fixes the analysis.” Correction: Truncation trades some variance for possible bias; it cannot create information for an unsupported subgroup or verify the censoring assumptions.
Diagnostics can reveal contradictions and fragility, but they do not prove conditional independent censoring. Unmeasured predictors of both censoring and outcome remain a concern even if measured covariates are balanced. When the data cannot support the target without strong assumptions, say so rather than presenting a stabilized number as if the identification problem had disappeared.
Check Your Understanding
Answer each question, then compare with the solutions.
- A study estimates the probability of cause 1 by two years but censors participants when cause 2 occurs. What is the central interpretive problem?
- A censoring model uses a biomarker measured after a participant’s follow-up ended. What must be checked before that biomarker can be used?
- For a subgroup, the estimated probability of remaining observed through the target time is zero. Can finite-sample weight truncation identify that subgroup’s outcomes?
- Weights are \(1,1,2,20\). Calculate their effective sample size diagnostic.
- A capped-weight estimate differs substantially from the uncapped estimate. Does this establish that the capped estimate is correct?
Solutions. (1) Censoring cause 2 treats an observed competing outcome as loss to follow-up and generally does not estimate the observed cumulative incidence of cause 1. (2) Check when and how the biomarker was measured, whether it is available for the histories being modelled, and whether later ascertainment changes the true follow-up window; do not use it as though it were known earlier without justification. (3) No. Truncation cannot supply outcomes where observation probability is zero; the target is unsupported without additional identifying assumptions or data. (4) \(n_{\mathrm{eff}}=(1+1+2+20)^2/(1^2+1^2+2^2+20^2)=576/406=1.419\). (5) No. The difference indicates sensitivity to extreme weights; truncation may reduce variance but can add bias, so the assumptions, support, and sensitivity results must be assessed.