Competing-Event Risk as a Co-Outcome

Why Treat the Competing Event as an Outcome?

When the event of primary interest can be precluded by another event, the competing event is not simply a nuisance in the analysis. It is an outcome that may itself matter clinically and may be affected by treatment. For example, a treatment could lower the risk of disease recurrence while raising the risk of death before recurrence. Reporting only recurrence risk would leave out an important part of the treatment’s effect.

The treatment-specific cumulative incidence of the competing event describes the probability that it occurs by a chosen time under each treatment strategy. Its causal contrast is interpreted alongside the corresponding contrast for the primary event. These two effects provide a joint risk profile, not a decomposition into biological mechanisms.

Definition: Let $A$ denote treatment strategy, with $a=0$ and $a=1$ representing two strategies. Let $T^a$ be the time to the first event under strategy $a$, and let $J^a$ be its cause. For competing-event cause $k$, the treatment-specific cumulative incidence at time $t$ is $$ F_k^a(t)=P(T^a\leq t,\ J^a=k). $$ The causal risk difference for cause $k$ by time $t$ is $F_k^1(t)-F_k^0(t)$; when $F_k^0(t)>0$, the causal risk ratio is $F_k^1(t)/F_k^0(t)$. A positive risk difference means a higher probability of experiencing cause $k$ by time $t$ under strategy 1.

The definition concerns the first event. If a person experiences the primary event first, that person cannot later experience the competing cause as the first event. Likewise, a competing event occurring first rules out subsequently experiencing the primary event as the first event. This mutual exclusivity is central to interpreting the risks.

Assumptions for a Causal Interpretation

Potential outcomes define the target, but they are not all observed for any one person: each person follows only one treatment strategy. A causal interpretation of observed treatment-specific risks therefore requires conditions connecting the observed data to those potential outcomes.

Consistency: a person who follows strategy $a$ has observed event time and cause equal to $T^a$ and $J^a$, with the treatment strategies defined clearly enough to avoid materially different versions being treated as identical.
Exchangeability: in a randomized trial, random assignment makes treatment independent of the potential event outcomes, subject to the trial’s design. In an observational study, identification generally requires that measured baseline covariates $L$ suffice to control confounding: conditional on $L$, treatment is independent of the potential event outcomes.
Positivity: every covariate pattern included in the target population has a positive probability of receiving each strategy being compared. Without this overlap, a treatment-specific risk for some part of the target population cannot be learned from the observed data without extrapolation.
Outcome ascertainment: if follow-up is complete through $t$, the first event and its cause by $t$ must be accurately observed. With censoring before $t$, identification additionally requires an appropriate independent-censoring assumption, possibly conditional on treatment, baseline covariates, and relevant observed history, together with a valid method for handling censoring.

Under randomization, consistency, and complete follow-up, the arm-specific observed risk estimates the corresponding potential-outcome risk. In an observational study, a common identification expression under conditional exchangeability is

F_k^a(t)=\int P(T\leq t,\ J=k\mid A=a,L=l)\,dP_L(l).

This standardization averages the conditional risk under treatment $a$ over the covariate distribution of the target population. It is not enough to compare unadjusted risks when treatment groups differ in prognostic covariates. Censoring must also be addressed: treating people censored before $t$ as if they had no event by $t$ will generally underestimate risk.

How the Co-Outcome Profile Fits Together

Suppose the possible first-event causes are mutually exclusive categories $1,\ldots,K$, and define $F_{\mathrm{any}}^a(t)=P(T^a\leq t)$. By partitioning the event by its cause, the risks satisfy

F_{\mathrm{any}}^a(t)=\sum_{k=1}^{K}F_k^a(t).

The probability of remaining free of any first event through $t$ is $1-F_{\mathrm{any}}^a(t)$. Subtracting the two treatment-specific identities gives

F_{\mathrm{any}}^1(t)-F_{\mathrm{any}}^0(t) =\sum_{k=1}^{K}\{F_k^1(t)-F_k^0(t)\}.

Thus, when the primary event is cause 1 and the competing event is cause 2, the two cause-specific risk differences sum to the risk difference for any first event, provided these are the only event causes being counted. This identity is a bookkeeping relation among risks. It does not say that one treatment effect causes the other, or that the competing-event contrast explains the primary-event contrast.

Worked Example: Opposing Effects in a Randomized Trial

Consider a hypothetical randomized trial with complete two-year follow-up of 200 participants assigned to each strategy. By two years, 40 participants in strategy 0 had the primary event and 20 had the competing event. In strategy 1, 25 had the primary event and 35 had the competing event. The remaining participants had neither first event by two years.

Because treatment is randomized, follow-up is complete, and first-event causes are consistently classified, the observed arm proportions estimate the treatment-specific risks. For the primary event, the risks are $40/200=0.200$ under strategy 0 and $25/200=0.125$ under strategy 1. The risk difference is

$$ 0.125-0.200=-0.075. $$

Strategy 1 has a 7.5-percentage-point lower two-year primary-event risk in this hypothetical trial. Its primary-event risk ratio is $0.125/0.200=0.625$.

For the competing event, the risks are $20/200=0.100$ and $35/200=0.175$. The competing-event risk difference is

$$ 0.175-0.100=0.075, $$

and the risk ratio is $0.175/0.100=1.75$. Strategy 1 therefore has a 7.5-percentage-point higher two-year competing-event risk. The joint profile shows lower primary-event risk alongside higher competing-event risk. It does not establish why either risk changed.

The any-event risks are $(40+20)/200=0.300$ for strategy 0 and $(25+35)/200=0.300$ for strategy 1. Their difference is $0$, which agrees with the sum of the two cause-specific differences: $-0.075+0.075=0$. This equality checks the arithmetic; it does not make the cause-specific outcomes interchangeable.

Standardizing the Competing-Event Risk

For an observational comparison, the same joint reporting can be used after standardizing each treatment-specific risk to a common baseline population. The following example illustrates the arithmetic. Its causal interpretation depends on consistency, conditional exchangeability, positivity, and appropriate handling of censoring.

Worked Example: Standardized Risks Across Two Baseline Strata

Suppose a hypothetical target population is 60% low risk and 40% high risk. In the low-risk stratum, the observed primary-event and competing-event risks are 0.20 and 0.10 under strategy 0, and 0.10 and 0.20 under strategy 1. In the high-risk stratum, the corresponding risks are 0.10 and 0.30 under strategy 0, and 0.10 and 0.40 under strategy 1. Each stratum-specific pair sums to at most one, as required for mutually exclusive first-event causes.

Standardize the competing-event risk for strategy 0 to the target population:

$$ F_2^0(t)=0.60(0.10)+0.40(0.30)=0.06+0.12=0.18. $$

For strategy 1, the standardized risk is

$$ F_2^1(t)=0.60(0.20)+0.40(0.40)=0.12+0.16=0.28. $$

The standardized competing-event risk difference is $0.28-0.18=0.10$, a 10-percentage-point increase, and the risk ratio is $0.28/0.18=1.56$.

For comparison, the standardized primary-event risks are $0.60(0.20)+0.40(0.10)=0.16$ under strategy 0 and $0.60(0.10)+0.40(0.10)=0.10$ under strategy 1. The primary-event risk difference is $0.10-0.16=-0.06$, or a 6-percentage-point decrease. Reporting both outcomes makes the opposing directions visible. The comparison is causal only if the stated identification assumptions hold; standardization by itself does not remove unmeasured confounding.

Time Horizon and Interpretation

Every cumulative-incidence contrast is tied to a specific time horizon. An effect at one year need not have the same magnitude or direction as an effect at three years. Risks are cumulative, so each cause-specific risk can stay the same or increase with time, but its treatment contrast can change as the two strategies’ risks accumulate at different rates.

Worked Example: A Competing-Event Effect That Grows Over Time

In a hypothetical trial, the primary-event and competing-event risks at one year are $0.08$ and $0.04$ under strategy 0, and $0.05$ and $0.06$ under strategy 1. At three years, those risks are $0.18$ and $0.12$ under strategy 0, and $0.10$ and $0.20$ under strategy 1. Assume randomization, consistent event classification, and adequate follow-up or valid censoring adjustment.

At one year, the competing-event risk difference is $0.06-0.04=0.02$, while the primary-event risk difference is $0.05-0.08=-0.03$. The any-event risks are $0.08+0.04=0.12$ and $0.05+0.06=0.11$, for a risk difference of $-0.01$.

At three years, the competing-event risk difference is $0.20-0.12=0.08$, and the primary-event risk difference is $0.10-0.18=-0.08$. Any-event risk is $0.30$ under each strategy, so its difference is zero. The co-outcome profile therefore changes over time. It would be incomplete to report the three-year result as if it described the one-year treatment effect.

Comparisons, Diagnostics, and Limitations

Quantity	Question answered	Interpretive caution
Primary-event cumulative incidence	What is the probability of the primary event by time $t$ under each strategy?	A lower risk does not by itself identify the mechanism of the reduction.
Competing-event cumulative incidence	What is the probability of the competing event by time $t$ under each strategy?	This is an outcome in its own right, not ordinary loss to follow-up.
Any-event risk	What is the probability of any first event by time $t$?	It can conceal opposing changes in the separate causes.
Cause-specific hazard	How does the instantaneous event rate among those still event-free compare?	It is not the same estimand as a cumulative risk contrast.

A useful diagnostic is to check that, within each treatment strategy and time horizon, the cause-specific cumulative incidences are nonnegative and sum to no more than one. If all first-event causes have been included, their sum should equal the any-event risk. Also verify that the treatment groups use the same time horizon and event definitions, and that censoring has not been treated as event-free follow-up without justification.

The competing-event risk is a total effect on the probability of experiencing that cause first by time $t$. A difference can reflect changes in event occurrence, changes in the chance of experiencing another cause first, or both. The contrast alone does not separate these possibilities. Likewise, observing a lower primary-event risk together with a higher competing-event risk does not prove that one change caused the other. Such explanations require additional assumptions and analyses.

A common mistake is to call the competing event “censored” and then interpret the primary-event analysis as though the competing event were merely missing follow-up. That reasoning changes the question: a competing event is an observed outcome that prevents a later first primary event, whereas censoring means the event process is no longer observed. The correction is to report the primary-event and competing-event cumulative incidences under each strategy, with their contrasts at a stated time horizon. A second pitfall is reporting only the primary-event contrast; an unchanged any-event risk can still conceal substantial and opposing cause-specific differences.

These risk contrasts are marginal, time-specific summaries. They do not by themselves describe when events occur within the interval, provide an explanation of treatment pathways, or guarantee precise estimates. In small samples, uncertainty in the separate risks can be substantial, and the two estimates are statistically dependent because event causes are mutually exclusive. Present uncertainty intervals for each contrast when available, and interpret the whole profile in light of the study design and clinical importance of each event.

Key takeaway: Treat the competing-event cumulative incidence as a co-outcome: define its risk under each strategy at a fixed time, calculate a clear risk contrast, and interpret it alongside the primary-event risk. This gives a fuller description of the treatment-policy effect without assigning a causal mechanism to the pattern.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions.

Under strategy 0, the two-year competing-event risk is 0.12; under strategy 1, it is 0.18. Calculate the risk difference and risk ratio, and interpret the difference.
Why is a competing event not equivalent to censoring when estimating the primary-event cumulative incidence?
In a randomized study with complete follow-up, what assumptions support interpreting arm-specific first-event proportions as treatment-specific risks?
If the primary-event risk difference is $-0.04$ and the competing-event risk difference is $0.04$, what is the any-event risk difference when these are the only two first-event causes?
Does a higher competing-event risk alongside a lower primary-event risk establish that the competing event caused the primary-event reduction? Explain.

Solutions. (1) The difference is $0.18-0.12=0.06$, a 6-percentage-point increase; the ratio is $0.18/0.12=1.50$. Strategy 1 has 1.5 times the two-year competing-event risk. (2) The competing event is an observed first event that prevents a later first primary event; censoring is incomplete observation of the event process. (3) Randomization, consistency, correct outcome and cause classification, and complete follow-up through the horizon support the interpretation. (4) The differences sum to $-0.04+0.04=0$. (5) No. The joint risk pattern describes total effects on the two outcomes, but does not identify a causal mechanism connecting them.