Why Weighting Changes a Competing-Risk Risk Estimate
Loss to follow-up can make an observed competing-risk estimate unrepresentative of the population of interest. If people who leave follow-up differ in their future event risks from otherwise similar people who remain, the observed risk sets may give a biased picture of who is still at risk. Inverse probability of censoring weighting (IPCW) corrects this selection by giving more weight to observed people who represent similar people who were lost.
The target here is an absolute risk: the probability of experiencing a specified event type by time \(t\), before any competing event. A competing event is an observed outcome, not a loss to follow-up. It removes a person from the event-free risk set, but it remains part of the event process that determines the absolute risk.
With no loss to follow-up, the Aalen–Johansen estimator combines estimates of the cause-specific hazards to estimate the CIF. With loss to follow-up, IPCW can correct its risk sets and event counts. The essential idea is to recover, in expectation, the full-data population that would have been observed had censoring not removed people from follow-up.
Assumptions and the Weighted Estimator
Let \(H(u)\) denote the measured history available just before time \(u\), including baseline covariates and, when appropriate, time-varying covariates. Define \(G(u-\mid H(u))\) as the conditional probability of remaining uncensored up to just before \(u\), given that history. In a baseline-covariate model, this is written \(G(u-\mid X)\), where \(X\) is the measured baseline information. The weight for an observed person still under observation just before \(u\) is the reciprocal of this probability.
For example, conditional independent censoring says that, among people with the same relevant measured history, those lost to follow-up do not have systematically different future event experience from those who remain. This is an assumption, not something established by fitting a censoring model. A model cannot recover information for histories with no chance of remaining observed, which is why positivity matters.
For observed data, let \(Y_i(u)\) indicate that person \(i\) is event-free and uncensored just before \(u\), and let \(dN_{ik}(u)\) indicate that this person experiences cause \(k\) at \(u\). Write \(\widehat G_i(u-)\) for the estimated conditional probability of remaining uncensored. The weighted cause-specific hazard increment and the corresponding weighted CIF estimate are:
Here \(\widehat G_i(u-)\) is evaluated using person \(i\)'s measured history before \(u\). In discrete time, the sums run over event times. The weighted risk set uses the same weights in its event totals and its denominator. The event-free survival estimate \(\widehat S^{\,w}\) decreases for events of all causes; the CIF for cause \(k\) accumulates only cause \(k\)'s hazard increments.
The justification for weighting is direct. Conditional on the measured history and the event process, conditional independent censoring gives
Thus an observed person who remains uncensored represents, on average, the people with the same history who would otherwise have been censored. Applying the argument to event indicators and risk-set indicators recovers their full-data totals in expectation. Their ratio estimates the full-data cause-specific hazard increment; combining those increments by the product and sum above gives the competing-risk CIF. This is an identification and large-sample justification, not a claim that a finite-sample ratio is exactly unbiased.
Worked Example: Calculating a Weighted CIF Across Two Event Times
Suppose a hypothetical cohort has weighted event-free risk-set totals and weighted cause-specific event totals as shown below. Each total is the sum of the relevant IPCW contributions. Assume both causes are fully observed when they occur, the conditional censoring model is adequate, and positivity holds through time 2.
| Time | Weighted risk set | Cause 1 events, weighted | Cause 2 events, weighted |
|---|---|---|---|
| 1 | 10.0 | 2.0 | 1.0 |
| 2 | 7.0 | 1.4 | 0.7 |
At time 1, the cause-specific hazard increments are \(2/10=0.20\) for cause 1 and \(1/10=0.10\) for cause 2. Therefore event-free survival just after time 1 is \(1-0.20-0.10=0.70\), and the cause 1 CIF is \(0.20\).
At time 2, the increments are \(1.4/7=0.20\) for cause 1 and \(0.7/7=0.10\) for cause 2. The additional cause 1 risk is the event-free survival just before time 2 multiplied by the cause 1 increment: \(0.70(0.20)=0.14\). Consequently,
The estimated probability of cause 1 by time 2 is \(0.34\), or 34% in this hypothetical population. The competing events contribute to the reduction in event-free survival: excluding cause 2 from that reduction would incorrectly leave more people at risk of cause 1.
Worked Examples: Covariate Adjustment and Weight Instability
Worked Example: Correcting Unequal Loss Across Risk Groups
Consider a hypothetical cohort of 100 people at risk at one event time. A measured baseline risk group has 40 people in the high-risk group and 60 in the low-risk group. By that time, the estimated probabilities of remaining uncensored are 0.50 and 1.00, respectively. Among observed people, the high-risk group has 20 in the risk set, including 4 cause 1 events and 2 cause 2 events; the low-risk group has 60 in the risk set, including 6 cause 1 events and 6 cause 2 events. Assume censoring is conditionally independent given risk group and the group-specific probabilities are correctly estimated.
The high-risk group receives weight \(1/0.50=2\); the low-risk group receives weight \(1/1.00=1\). The weighted risk-set total is \(20(2)+60(1)=100\). The weighted cause 1 event total is \(4(2)+6(1)=14\), and the cause 2 total is \(2(2)+6(1)=10\). The hazard increments are therefore \(14/100=0.14\) and \(10/100=0.10\). At this single event time the cause 1 CIF is \(0.14\), or 14%.
Without weighting, the observed risk set is \(20+60=80\), and the cause 1 increment is \(10/80=0.125\). The weighted and unweighted results differ because half of the high-risk group is lost, and the weighting restores that group's representation under the stated assumption. The difference does not itself prove that weighting is correct: the conclusion depends on the conditional independence, positivity, and model assumptions.
Worked Example: How Truncation Changes a Risk Estimate
Suppose a hypothetical one-time weighted risk set has total weight 100. Its cause 1 events contribute total weight 20 and cause 2 events contribute weight 10, so the cause 1 increment is \(20/100=0.20\). Imagine one cause 1 event has weight 10, while all other contributions together are 90 in the risk set, 10 for cause 1, and 10 for cause 2.
If weights are capped at 5, that event's contribution falls from 10 to 5. The revised risk-set total is \(100-10+5=95\); the cause 1 total is \(20-10+5=15\); and the cause 2 total remains 10. The truncated cause 1 increment is \(15/95=0.1579\). Truncation has reduced this estimate from 0.20 to about 0.158 because it changes the relative contributions to the risk set and event totals.
This calculation shows why truncation is a bias–variance trade-off, not a cost-free repair. It may reduce the influence of an extreme weight, but it does not make a history with near-zero observation probability well supported. Report the rule used and compare results across reasonable choices.
Diagnostics, Alternatives, and Limits
Before interpreting a weighted CIF, inspect the censoring probabilities and weights over time and across histories. Look for very small estimated \(\widehat G\), large or concentrated weights, sparse histories, and abrupt changes in the weighted risk-set size. Check whether the censoring model captures the measured predictors of loss to follow-up, and assess whether weighting improves balance in those predictors among the weighted observed risk sets. These checks can identify instability or model problems; they cannot verify conditional independent censoring with respect to unmeasured information.
A useful comparison is with the unweighted estimate and, where appropriate, estimates under alternative censoring models or weight truncation rules. Large differences signal sensitivity that deserves investigation, not an automatic reason to prefer one result. Inference must also reflect that the weights are estimated; uncertainty procedures that treat estimated weights as fixed can understate uncertainty.
| Approach | What it does | Key limitation |
|---|---|---|
| Unweighted Aalen–Johansen | Estimates CIFs from observed event and risk-set data. | Does not correct selection from dependent loss to follow-up. |
| IPCW Aalen–Johansen | Weights observed risk sets and events to account for measured predictors of censoring. | Requires conditional independent censoring, positivity, and a suitable censoring model. |
| Cause-specific analysis that censors competing events | Estimates a cause-specific hazard under its own risk-set definition. | That hazard alone is not the absolute risk of the cause; a CIF must account for competing events. |
A common mistake is to treat a competing event as if it were loss to follow-up and then interpret a cause-specific hazard or a single-cause survival calculation as the probability of that event. The correction is to retain competing events as observed outcomes and include all cause-specific hazard increments in the event-free survival factor used to construct the CIF. IPCW addresses censoring selection; it does not turn competing events into censoring or remove them from the risk calculation.
The method is limited by the information available. If important predictors of both loss and event risk are unmeasured, weighting on recorded history may not remove selection bias. If some histories have little or no chance of remaining observed, weights can become unstable or the target risk may not be identified from these data. These limitations should be reflected in the interpretation, rather than hidden by a numerically precise estimate.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- In a competing-risk CIF, should a competing event be treated as censoring when constructing the event-free survival factor? Why?
- At one event time, the weighted risk-set total is 50, with weighted cause 1 and cause 2 event totals of 5 and 10. What are the two hazard increments, and what is the cause 1 CIF at that time?
- What does positivity require of the conditional probability of remaining uncensored?
- Why can truncating an extreme censoring weight change the point estimate?
Solutions. (1) No. A competing event is an observed event that removes a person from the event-free risk set and contributes to the total hazard decrement; omitting it would distort the survival factor used in the CIF. (2) The increments are \(5/50=0.10\) for cause 1 and \(10/50=0.20\) for cause 2. At this single time the cause 1 CIF is \(0.10\); the remaining event-free fraction is \(1-0.10-0.20=0.70\). (3) For histories represented in the target population and times being analyzed, the conditional probability of remaining uncensored must be greater than zero. (4) Truncation changes the relative weighted contributions to the risk set and event totals, so their ratio—and therefore the estimated hazard increment—can change.