Event-Free Probability Across Competing Causes

Why Event-Free Probability Is a Complement

When several event types can occur, a prediction may ask whether a person will experience a particular event by a chosen time. A related question is whether the person will experience any of the event types by then. The probability of avoiding every event type is the remaining probability after accounting for all the mutually exclusive ways the first event could occur.

This complement is useful in absolute risk prediction because it gives a direct, interpretable probability: the chance of having no event of interest, of any listed type, through time \(t\). It also provides an important consistency check. If cause-specific cumulative incidences and event-free probability refer to the same population, event definition, and horizon, their probabilities must account for the whole population without overlap or omission.

Definition: Let \(T\) be the time to the first event and let \(J\in\{1,\ldots,K\}\) identify its cause. For cause \(k\), the cumulative incidence function is \(F_k(t)=P(T\le t,\ J=k)\). The event-free probability through time \(t\) is \(S_{\mathrm{all}}(t)=P(T>t)\): the probability that no event of any of the \(K\) types has occurred by \(t\).

The subscript “all” emphasizes that this is freedom from every event type in the specified set, not just one cause. The event types must describe mutually exclusive possible causes of the first event. For example, if the endpoint is first hospitalization and the causes are cardiac and respiratory hospitalization, one person cannot have both as the cause of the same first hospitalization. The event-free probability depends on which event types have been included: omitting a relevant event type changes the question being answered.

The Identity and Its Conditions

At a fixed time \(t\), the possibilities are that no event has occurred by \(t\), or that the first event occurred by \(t\) and had one of the \(K\) causes. These possibilities are mutually exclusive and exhaustive. In particular, a person who has already had a first event cannot also be counted as event-free at \(t\), or as having a different cause for that same first event.

\{T>t\}\ \cup\ \bigcup_{k=1}^{K}\{T\le t,\ J=k\} \quad\text{is the full sample space, with disjoint events.}

Taking probabilities of this partition gives the central result:

Theorem: If the \(K\) causes exhaust the possible types of first event, then, for every \(t\ge 0\),

S_{\mathrm{all}}(t) =P(T>t) =1-\sum_{k=1}^{K}P(T\le t,\ J=k) =1-\sum_{k=1}^{K}F_k(t).

This identity concerns probabilities in the same population and at the same time point. It does not require that the causes have independent event times. It does require a coherent definition of the first event and an exhaustive cause list. If a relevant first event is left unclassified, or if the cause categories overlap, the terms do not form the stated partition and the identity need not hold.

The definition is about the event-time distribution, not about what was observed for every individual. In a study, some event times may be right-censored, so estimating the probabilities from data requires appropriate handling of censoring. For the usual nonparametric estimates, censoring must be independent of the event process in the relevant population or strata, with sufficient follow-up to estimate the chosen horizon. These are estimation assumptions; they are not extra algebraic conditions on the identity itself.

Worked Calculations

Worked Example: Finding the Probability of No Event

Suppose that in a hypothetical population, by 3 years the cumulative incidence of event A is \(0.18\), and that of event B is \(0.07\). Assume A and B are the exhaustive, mutually exclusive types of first event. Find and interpret the event-free probability at 3 years.

Using the complement identity,

S_{\mathrm{all}}(3) =1-\{F_A(3)+F_B(3)\} =1-(0.18+0.07) =0.75.

Thus, the event-free probability through 3 years is \(0.75\), or 75%. In this hypothetical population, 75% of individuals would have neither event A nor event B by 3 years. It is not the probability of avoiding A alone: that probability would also count individuals who experience B.

Worked Example: Checking the Identity From Estimated Incidences

Consider a hypothetical cohort of 100 individuals initially at risk. By time 1, 10 have a first event of type A and 5 have a first event of type B. Five more are censored before time 2. At time 2, the 80 remaining at risk have 8 type-A events and 4 type-B events. Assume there are no other event types. Calculate the estimated cause-specific cumulative incidences and event-free probability through time 2 using these risk-set contributions.

At time 1, the estimated incidence increments are \(10/100=0.10\) for A and \(5/100=0.05\) for B. The estimated event-free probability after these events is \(1-0.10-0.05=0.85\). The five censored individuals are not counted as having an event; they leave the risk set before time 2.

At time 2, each risk-set event proportion is weighted by the estimated event-free probability just before that time. The increments are

\Delta\widehat F_A(2)=0.85\left(\frac{8}{80}\right)=0.085, \qquad \Delta\widehat F_B(2)=0.85\left(\frac{4}{80}\right)=0.0425.

Therefore, through time 2,

\widehat F_A(2)=0.10+0.085=0.185, \qquad \widehat F_B(2)=0.05+0.0425=0.0925.

The complement calculation gives

\widehat S_{\mathrm{all}}(2) =1-(0.185+0.0925)=0.7225.

So the estimated probability of remaining free of either event through time 2 is 72.25%. As a check, the estimated event-free fraction after time 1 is \(0.85\); at time 2, the event proportion among those at risk is \(12/80=0.15\), leaving \(0.85(1-0.15)=0.7225\). The two calculations agree.

Estimates, Uncertainty, and Checks

For estimated cause-specific cumulative incidences from the same analysis, define the estimated event-free probability by

\widehat S_{\mathrm{all}}(t)=1-\sum_{k=1}^{K}\widehat F_k(t).

This formula gives a coherent event-free estimate when the cause-specific estimates use the same cohort, horizon, event definitions, and compatible estimation procedure. Estimates produced separately from different populations or incompatible models may not satisfy the identity. In that case, taking their sum as though it were a single coherent partition can yield a misleading result, including a sum greater than 1.

The complement also affects uncertainty. Since subtracting a random quantity from 1 does not change its variance, the variance of the event-free estimate is the variance of the sum of the cause-specific estimates. For two causes this is

\operatorname{Var}\{\widehat S_{\mathrm{all}}(t)\} = \operatorname{Var}\{\widehat F_1(t)\} +\operatorname{Var}\{\widehat F_2(t)\} +2\operatorname{Cov}\{\widehat F_1(t),\widehat F_2(t)\}.

For more causes, the variance includes the variance of each estimate and twice each distinct pairwise covariance. Cause-specific estimates from the same cohort are generally dependent because they share the risk sets and event-free population. Ignoring covariance is therefore not generally justified. Earlier material on the Aalen variance estimate and covariance between cause-specific estimates provides tools for estimating this uncertainty.

Worked Example: Uncertainty for the Complement

At a hypothetical horizon, suppose two estimated cause-specific incidences are \(0.185\) and \(0.0925\). Suppose their estimated variances are \(0.000400\) and \(0.000225\), and their estimated covariance is \(0.000050\). Calculate the event-free estimate, its standard error, and a pointwise 95% Wald interval.

The event-free estimate is

\widehat S_{\mathrm{all}}=1-(0.185+0.0925)=0.7225.

Using the covariance as well as the two variances,

\widehat{\operatorname{Var}}(\widehat S_{\mathrm{all}}) =0.000400+0.000225+2(0.000050) =0.000725,

so the estimated standard error is \(\sqrt{0.000725}=0.02693\). A pointwise 95% Wald interval is

0.7225\pm1.96(0.02693) =0.7225\pm0.05278 =(0.6697,\ 0.7753).

This interval describes uncertainty in the event-free probability at this horizon under the estimation and approximation assumptions. In repeated samples, intervals constructed this way would cover the true event-free probability about 95% of the time if the Wald approximation is adequate. It does not mean there is a 95% probability that this fixed parameter lies in this particular interval.

Interpretation, Alternatives, and Failure Modes

Event-free probability is an absolute risk quantity: it refers to the probability of avoiding all listed first-event types through a specified time. It is often useful alongside each cause-specific cumulative incidence. The latter answers “what is the probability that this cause occurs first by \(t\)?” The complement answers “what is the probability that none of the listed causes occurs by \(t\)?”

A related alternative is to estimate survival from the composite endpoint “any first event,” treating every listed cause as an event. With consistent event definitions and censoring handling, this targets the same event-free probability. By contrast, the introductory material on Cumulative Incidence Versus Net Risk distinguishes the actual probability of a cause-specific event in the presence of competing events from a net-risk quantity that removes competing events from consideration. A net risk for one cause is not the event-free probability across all causes.

Several checks help identify a mismatch:

Range: each cumulative incidence should be between 0 and 1, and the sum across exhaustive causes should not exceed 1. The complement should also be between 0 and 1.
Shared definitions: verify that every estimate uses the same time origin, horizon, target population, and definition of first event.
Shared cause set: check that all event types relevant to the “any event” endpoint are included exactly once.
Rounding: rounded values can make the displayed sum differ slightly from 1. Calculate the complement from unrounded estimates when available.
Uncertainty: check that the variance calculation includes covariance terms when cause-specific estimates come from the same data.

A common mistake is to say, “The probability of avoiding event A is \(1-F_A(t)\), so that is the probability of being free of all events.” The correction is that \(1-F_A(t)\) includes people who have experienced a competing event B by \(t\). To obtain event-free probability across the specified causes, subtract the cumulative incidences of all those causes from 1. Another mistake is to treat censoring as evidence of being event-free through the horizon. Censoring means the later event status is unknown; estimation must account for that uncertainty rather than classify the person as event-free indefinitely.

The identity is simple, but its interpretation can fail if the event set is incomplete or the estimates are not compatible. It also does not by itself establish that a prediction model is calibrated or that censoring assumptions are plausible. The result gives the correct relationship among probabilities once the target, event definitions, and estimates are coherent.

Key takeaway: For mutually exclusive and exhaustive first-event causes measured in the same population, event-free probability through \(t\) is exactly \(1-\sum_{k=1}^{K}F_k(t)\). Apply the same complement to compatible estimates, and account for covariance when quantifying uncertainty.

Check Your Understanding

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

At a chosen horizon, two exhaustive cause-specific cumulative incidences are 0.12 and 0.31. What is the event-free probability, and what does it mean?
Why does \(1-F_A(t)\) generally fail to represent freedom from both A and B?
An analysis reports cumulative incidences of 0.62 and 0.43 for two causes that are claimed to be exhaustive. What should be checked?
For two cause-specific estimates, why is the variance of their sum not generally just the sum of their variances?

Solutions. (1) \(1-(0.12+0.31)=0.57\): the probability of neither cause occurring by the horizon is 57%. (2) \(1-F_A(t)\) includes people whose first event was B, so it excludes only cause A, not every event. (3) The sum is \(1.05\), which would imply a negative complement; check event definitions, population, horizon, compatibility of the estimates, and computation before interpreting the result. (4) The estimates can be dependent, so the variance includes \(2\operatorname{Cov}(\widehat F_A,\widehat F_B)\) in addition to their individual variances.