Competing Events as Censoring and Estimand Consequences

Why the Event Treated as Censoring Matters

In a competing-risks study, a person may experience the event of interest or a different event that makes the event of interest impossible to observe afterward. For example, death from another cause prevents a later diagnosis from being observed. If the competing event is recorded as censoring, an inverse probability of censoring weighting (IPCW) analysis can correct for selective removal from the risk set. But this choice does not preserve the original, observed-world question: it changes the quantity being estimated.

The key distinction is between the probability of the event of interest in the world where competing events occur and the probability of that event in a hypothetical setting where competing events are eliminated. The first is a cumulative incidence. The second is often called a net risk, or a hypothetical risk under elimination of the competing event. IPCW that censors competing events is directed toward the latter only under suitable assumptions; it does not, by itself, estimate the former.

Definition: Let $T$ be the time to the first event and $J$ its cause, with $J=1$ denoting the event of interest and $J=2$ a competing event. The observed-world cumulative incidence of cause 1 by time $K$ is $P(T\leq K,J=1)$. A net risk by time $K$ is the risk of cause 1 in a hypothetical event process in which cause 2 cannot occur. Interpreting net risk causally requires assumptions linking that hypothetical process to the observed data.

Two Different Risk Calculations

For clarity, consider discrete intervals $k=1,\ldots,K$, with no tied events and a fixed event ordering. Among people still free of either event at the start of interval $k$, define $a_k$ as the conditional probability of the event of interest during that interval and $b_k$ as the conditional probability of the competing event. Thus $a_k+b_k\leq 1$. The probability of remaining free of both events through interval $k$, conditional on being event-free at the start of follow-up, is

S_{\mathrm{all}}(k)=\prod_{j=1}^{k}(1-a_j-b_j).

To have the event of interest in interval $k$, a person must first remain free of both events through interval $k-1$, then experience cause 1 in interval $k$. Therefore the observed-world cumulative incidence is

F_1(K)=\sum_{k=1}^{K} \left\{\prod_{j=1}^{k-1}(1-a_j-b_j)\right\}a_k,

where the empty product for $k=1$ equals 1. By contrast, if competing events are removed and the target-event hazards remain $a_k$, the hypothetical survival probability is $\prod_{k=1}^K(1-a_k)$, and the corresponding net risk is $1-\prod_{k=1}^K(1-a_k)$. The two formulas differ because actual cumulative incidence accounts for people who experience a competing event and can no longer experience cause 1.

Result: Under the discrete-time definitions above, the observed cumulative incidence is $F_1(K)=\sum_{k=1}^K S_{\mathrm{all}}(k-1)a_k$, whereas the net risk is $1-\prod_{k=1}^K(1-a_k)$. They are generally unequal. They coincide if $b_k=0$ for every interval, but need not otherwise.

Derivation. The events “cause 1 occurs in interval $k$” are mutually exclusive across intervals. The probability of that event is the probability of reaching interval $k$ without either event, multiplied by the conditional probability $a_k$. Summing gives the cumulative-incidence formula. In the hypothetical process with cause 2 removed, survival through interval $k$ is the product of the conditional probabilities $1-a_j$ for $j\leq k$; subtracting survival through $K$ from 1 gives net risk. If all $b_k=0$, then each factor $1-a_j-b_j$ equals $1-a_j$, so the formulas agree. Otherwise equality is not guaranteed. $\square$

Worked Example: Same Target Hazard, Different Competing Risk

Consider two groups, each followed for two intervals. Both have constant target-event hazards $a_1=a_2=0.20$. Group A has $b_1=b_2=0$; Group B has $b_1=b_2=0.50$. In each interval the two causes are mutually exclusive, and the remaining probability is no event.

For Group A, the observed cumulative incidence is $0.20+(1-0.20-0)(0.20)=0.20+0.16=0.36$. The net risk is $1-(1-0.20)^2=1-0.64=0.36$.

For Group B, only $1-0.20-0.50=0.30$ remain event-free after interval 1. Its observed cumulative incidence is therefore $0.20+0.30(0.20)=0.26$. Its net risk is still $1-(1-0.20)^2=0.36$, because the hypothetical calculation removes the competing event while retaining the target hazards.

The observed-world target-event risk is lower in Group B because many people experience the competing event before they could experience cause 1. The net risks are equal. Thus a difference in cumulative incidence can reflect differences in the competing-event process, even when the target-event hazards are identical.

What IPCW Estimates When Competing Events Are Censored

Treating cause 2 as censoring means that a person is removed from later cause-1 risk sets at the competing event. If this removal is informative, IPCW gives greater representation to people who remain under observation but resemble those removed. For a simple formal statement, suppose $L$ is a baseline covariate, $T^0$ is the hypothetical time to cause 1 if cause 2 were eliminated, and $C$ is the interval in which cause 2 would occur under the observed process, with the convention that within an interval cause 2 is assessed before the target-event opportunity. Write $G_k(L)=P(C>k\mid L)$, the probability that cause 2 has not occurred by the target-event opportunity in interval $k$, given $L$.

Result: Suppose (i) consistency holds, so the observed cause-1 event time agrees with $T^0$ whenever cause 2 has not occurred first; (ii) $T^0$ and $C$ are conditionally independent given $L$; and (iii) $G_k(L)>0$ for every relevant $k,L$. Then $$ E\left[\frac{I(C>k)I(T^0=k)}{G_k(L)}\right]=P(T^0=k). $$ Consequently, summing the weighted event contributions over $k\leq K$ identifies $P(T^0\leq K)$, the net risk, not the observed-world cumulative incidence.

Proof. Conditional on $L$, conditional independence gives $E[I(C>k)I(T^0=k)\mid L]=P(C>k\mid L)P(T^0=k\mid L)$. The first factor is $G_k(L)$, which is positive by assumption. Dividing by $G_k(L)$ and taking expectations conditional on $L$ therefore gives $P(T^0=k\mid L)$. Averaging over $L$ yields $E[I(C>k)I(T^0=k)/G_k(L)]=P(T^0=k)$. Summing these equalities over $k=1,\ldots,K$ gives $P(T^0\leq K)$, since the mutually exclusive events $T^0=k$ partition the event $T^0\leq K$. Consistency connects the weighted observed contributions to the potential event indicators in the expression. $\square$

In practice, $G_k(L)$ is replaced by a fitted probability of remaining free of the competing event. With time-varying measured history, the analogous argument requires sequential conditional independence at each interval and positive conditional probabilities of avoiding cause 2. These conditions are not mere technicalities: weighting can correct selection only for the histories represented in the censoring model. Even if those conditions hold, the hypothetical interpretation also requires that the target-event process under elimination of cause 2 is meaningfully represented by the assumed $T^0$. IPCW cannot verify that counterfactual claim from the observed data alone.

Worked Example: A One-Interval IPCW Calculation

Suppose half the population has $L=H$ and half has $L=L$. In a hypothetical setting without competing events, the cause-1 event probabilities are 0.40 for $H$ and 0.10 for $L$. Before the target-event opportunity, the competing-event probabilities are 0.50 for $H$ and 0.10 for $L$. Assume conditional independent censoring given $L$.

The net risk is $0.5(0.40)+0.5(0.10)=0.20+0.05=0.25$. The observed cause-1 event probability, which counts only people not removed by the competing event, is $0.5(1-0.50)(0.40)+0.5(1-0.10)(0.10)=0.10+0.045=0.145$.

The probability of avoiding the competing event is 0.50 in stratum $H$ and 0.90 in stratum $L$. The weighted event contribution from $H$ is $0.5(1-0.50)(0.40)/0.50=0.20$. From $L$, it is $0.5(1-0.10)(0.10)/0.90=0.05$. Adding gives $0.20+0.05=0.25$, the net risk. It does not give 0.145, the observed-world probability of cause 1. The calculation illustrates exactly what the inverse probability weights undo: the differential removal associated with $L$, under the stated conditional-independence assumption.

Choosing the Estimand Before the Estimator

If the scientific question concerns the actual probability of cause 1 by a given time, competing events are part of the outcome process, not ordinary loss to follow-up. The Aalen–Johansen estimator estimates the observed-world cumulative incidence by combining the event-free survival and cause-specific event increments. A cause-specific hazard analysis can also be used to obtain cumulative incidence when the hazards for all relevant causes are incorporated. These approaches preserve the competing event in the observed-world quantity.

If the question instead concerns what the cause-1 risk would be if cause 2 were eliminated, treating cause 2 as censoring can be appropriate, with IPCW used to address informative removal. That net-risk interpretation needs conditional exchangeability, positivity, consistency, and a defensible specification of the hypothetical intervention. A subdistribution-hazard analysis is another approach often used to model cumulative incidence, but its hazard and regression coefficients are not net risks.

Analysis choice	Target quantity	Role of competing event
Aalen–Johansen cumulative incidence	Observed-world probability of cause 1 by time $K$	Retained as a competing event
Cause-specific hazards combined across causes	Observed-world cumulative incidence	Contributes to event-free survival and cause-specific increments
Cause 2 censored with IPCW	Net or hypothetical cause-1 risk, under required assumptions	Treated as censoring and weighted for selection
Subdistribution-hazard analysis	Model for the observed-world cumulative incidence	Handled through the subdistribution risk-set construction

Worked Example: A Changing Competing Hazard

Consider two groups with constant target-event hazards over two intervals. In Group A, $a_1=a_2=0.20$ and $b_1=b_2=0$. In Group B, $a_1=a_2=0.25$, $b_1=0.60$, and $b_2=0$. All interval probabilities are valid: in Group B, $0.25+0.60=0.85$ in interval 1 and $0.25+0=0.25$ in interval 2.

Group A has observed cumulative incidence $0.20+(1-0.20)(0.20)=0.36$, and net risk $1-(0.80)^2=0.36$. Group B has observed cumulative incidence $0.25+(1-0.25-0.60)(0.25)=0.25+0.15(0.25)=0.2875$. Its net risk is $1-(1-0.25)^2=1-0.5625=0.4375$.

The target hazards are constant within each group, but Group B’s competing-event hazard is not: it is 0.60 in the first interval and 0 in the second. Its observed cumulative incidence is lower than Group A’s, while its net risk is higher. These are not contradictory results; they answer different questions.

Assumptions, Diagnostics, and Common Pitfalls

Before interpreting a competing-event IPCW analysis, specify whether the target is an observed-world cumulative incidence or a hypothetical net risk. For a net-risk analysis, conditional independent censoring requires that, given the modeled history, those who experience the competing event do not differ in their potential target-event process from those who remain uncensored. Positivity requires a nonzero probability of avoiding the competing event in each relevant history stratum. Consistency requires a clear hypothetical intervention and agreement between observed and hypothetical target outcomes when the competing event has not occurred.

A fitted censoring model cannot establish these assumptions. Still, large or highly variable weights, near-zero predicted probabilities of avoiding the competing event, and sparse histories indicate that the weighted estimate may be unstable or rely heavily on extrapolation. Model checks should include the predicted censoring probabilities and the distribution of the resulting weights. If some covariate histories always experience the competing event, positivity fails there; reweighting cannot reconstruct the missing hypothetical outcomes without additional assumptions.

Common mistake: “A competing event prevents later observation of the target event, so censor it and use IPCW to estimate the target event’s actual cumulative incidence.” The correction is that this procedure targets a hypothetical net quantity under assumptions. The observed-world cumulative incidence requires accounting for the competing event as an event.
Common mistake: Calling net risk the probability of cause 1 “in the observed data.” It is instead a risk in a specified hypothetical process, unless competing events are absent.
Common mistake: Assuming that successful weight fitting proves the hypothetical intervention is meaningful. Weight diagnostics address aspects of model behavior and support; they do not validate the causal interpretation of eliminating a competing event.

The practical limitation is fundamental: the observed data contain no cause-1 outcomes after a competing event. A net risk therefore depends on assumptions about outcomes that are not observed for those people. If the scientific question is about real-world prognosis or resource planning, the cumulative incidence is usually the directly relevant quantity. If it is about an intervention that could prevent the competing event, its causal interpretation requires more than relabeling that event as censoring.

Check Your Understanding

Answer each question, then compare with the concise solutions below.

With $a_1=0.10$, $a_2=0.20$, and $b_1=b_2=0$, calculate the observed cumulative incidence and net risk by interval 2.
With $a_1=a_2=0.20$, $b_1=0.30$, and $b_2=0$, calculate the observed cumulative incidence and net risk by interval 2.
State the conditional-independence and positivity assumptions that justify the baseline-covariate IPCW identity.
Which estimand is addressed by treating a competing event as censoring with IPCW: observed-world cumulative incidence or net risk? What extra interpretive caution is needed?

Solutions. (1) Both are $0.10+(1-0.10)(0.20)=0.28$, since no competing event occurs. (2) The observed cumulative incidence is $0.20+(1-0.20-0.30)(0.20)=0.20+0.10=0.30$. The net risk is $1-(0.80)^2=0.36$. (3) The hypothetical target-event time and competing-event censoring time must be conditionally independent given baseline $L$, and $P(C>k\mid L)$ must be positive in every relevant stratum and interval. (4) It addresses a net or hypothetical risk under those assumptions, not the observed-world cumulative incidence. Interpreting it causally also requires a well-defined competing-event elimination intervention and a defensible link between that intervention and the hypothetical target-event process.