Probability Bounds For Competing-Risk Predictions

Why Probability Bounds Matter

A predicted risk is a probability, so it must be between zero and one. Competing-risk predictions have additional structure: a cause-specific risk is the probability that a particular event is the first event by a specified time, while the probability of any event includes all event causes. The risk for one cause cannot exceed the risk of any event, and the risks for mutually exclusive causes cannot collectively exceed one.

These checks are simple, but they catch serious errors in prediction outputs: invalid values, cause-specific risks that conflict with the overall risk, or cause totals that exceed the probability of any event. Checking each value separately is not enough. A set of individually plausible numbers can still be jointly incoherent.

Definition: For a covariate profile \(x\) and prediction horizon \(t\), let \(F_k(t\mid x)\) be the cumulative incidence of cause \(k\): the probability that an individual with profile \(x\) experiences cause \(k\) as the first event by time \(t\). Let \(S(t\mid x)\) be the probability of remaining free of every event through \(t\). The probability of any event by \(t\) is \(A(t\mid x)=1-S(t\mid x)\).

The bounds below concern risks calculated for the same profile, prediction horizon, and target population. The event causes are mutually exclusive classifications of the first event. Assume cause-specific hazards are nonnegative and define a coherent event-free survival distribution. Censoring is not an event cause: it affects what can be estimated from observed data, but it is not included as an event in these probabilities.

The Bounds and Their Justification

For any one cause \(k\), experiencing cause \(k\) by \(t\) is one way to experience any event by \(t\). This event-set inclusion gives \(F_k(t\mid x)\leq A(t\mid x)\). Since all risks are probabilities, the basic bounds are

0\leq F_k(t\mid x)\leq A(t\mid x)\leq 1.

If the model includes an exhaustive set of possible first-event causes, the cause-specific events partition the event “any event by \(t\).” Their probabilities therefore add to that probability. If the predictions include only a subset of causes, their sum is at most the probability of any event, because other causes may account for some of the total.

\sum_{k=1}^{K} F_k(t\mid x)=A(t\mid x)=1-S(t\mid x) \quad\text{(all causes included),}

0\leq \sum_{k\in\mathcal{K}} F_k(t\mid x)\leq A(t\mid x)\leq 1 \quad\text{(only causes in subset }\mathcal{K}\text{ included).}

A short derivation shows why the equality holds for a coherent competing-risk model. In continuous time, let \(\lambda_k(u\mid x)\) be the cause-specific hazard for cause \(k\) at time \(u\), given survival event-free until just before \(u\). With nonnegative, integrable hazards, event-free survival obeys \(dS(u\mid x)=-S(u\mid x)\sum_k\lambda_k(u\mid x)\,du\). The cumulative incidence for cause \(k\) is the accumulated probability of being event-free just before \(u\) and then experiencing cause \(k\) at \(u\).

F_k(t\mid x)=\int_0^t S(u\mid x)\lambda_k(u\mid x)\,du.

Summing over all causes and using the survival relation gives

\sum_{k=1}^{K}F_k(t\mid x) =\int_0^t S(u\mid x)\sum_{k=1}^{K}\lambda_k(u\mid x)\,du =-\int_0^t dS(u\mid x) =S(0\mid x)-S(t\mid x) =1-S(t\mid x).

Here \(S(0\mid x)=1\), meaning the person is event-free at the prediction origin. The integral also makes the individual bound clear: every integrand is nonnegative, and the cause-specific hazard is no greater than the sum of all cause-specific hazards. Thus \(F_k(t\mid x)\leq\sum_jF_j(t\mid x)=A(t\mid x)\).

This argument is written for continuous event times. With discrete event times or a model with jumps, the corresponding calculation uses the model’s event-free survival and cumulative-incidence increments, including the probability of remaining event-free up to each increment. The same probability bounds and exhaustive-cause identity still apply when those quantities are calculated coherently.

Worked Example: Checking Two Cause-Specific Predictions

Suppose a hypothetical prediction table reports, for one patient profile at 3 years, a risk of 0.16 for cause A, 0.21 for cause B, and 0.37 for any event. Assume A and B are the only possible event causes in the model. Check the bounds and the combined prediction.

Each cause-specific risk must be nonnegative and no larger than the any-event risk. Here, \(0\leq 0.16\leq 0.37\) and \(0\leq 0.21\leq 0.37\). The individual bounds hold.

Because the two causes are exhaustive and mutually exclusive, their risks should sum to the any-event risk: \(0.16+0.21=0.37\). Their sum is also below one, \(0.37\leq 1\). The predictions pass both the individual and joint checks.

The implied probability of remaining event-free through 3 years is \(1-0.37=0.63\). This complements the any-event risk and is consistent with the reported cause-specific risks.

Worked Checks for Incoherence and Subsets

A check of each cause separately is necessary but not sufficient. For a complete set of causes, the sum and any-event risk must agree; for an incomplete set, the sum must not exceed the any-event risk. These are stronger checks than merely confirming that every printed value lies between zero and one.

Worked Example: Individually Valid Risks With an Invalid Total

A hypothetical model reports, at a 5-year horizon, \(F_1=0.44\), \(F_2=0.39\), and \(A=0.72\). The two causes are said to be exhaustive. Determine whether these predictions are coherent.

First check the individual bounds: \(0\leq 0.44\leq 0.72\) and \(0\leq 0.39\leq 0.72\). Both cause-specific predictions pass. But the sum is \(0.44+0.39=0.83\), which is not equal to \(0.72\). It is greater than the reported probability of any event by \(0.83-0.72=0.11\).

The cause predictions are therefore inconsistent with the reported any-event risk under the stated exhaustive-cause assumption. Although the sum \(0.83\) is itself below one, that does not repair the contradiction: the three values cannot all describe one coherent distribution of first events. The predictions should be investigated rather than reported as a coherent set.

If the two causes were only a subset, the requirement would still be \(F_1+F_2\leq A\). The reported sum of 0.83 would also violate that weaker condition.

Worked Example: Checking Predictions for a Subset of Causes

Suppose a hypothetical report gives a 2-year any-event risk of \(A=0.58\). It lists two causes of interest with risks \(F_1=0.24\) and \(F_2=0.19\), but other event causes are not listed. Check whether the displayed subset is possible and calculate the unlisted share of the any-event risk.

The listed risks are nonnegative, and each is no greater than 0.58. Their sum is \(0.24+0.19=0.43\), which satisfies \(0.43\leq 0.58\). Thus the subset does not exceed the probability of any event.

The unlisted causes together must account for \(0.58-0.43=0.15\) of probability by 2 years. The remaining event-free probability is \(1-0.58=0.42\). These quantities add to one: \(0.43+0.15+0.42=1.00\). The check supports coherence of the displayed totals, though it does not by itself validate how the individual predictions were estimated.

How to Diagnose a Failed Check

Start by confirming that the values being compared refer to the same covariate profile, time origin, horizon, and target population. A risk for one subgroup should not be compared as if it were the overall risk for another population. Also confirm that “any event” refers to the same event definition and that the listed causes are either exhaustive or explicitly identified as a subset.

Then apply the checks in order:

Check individual values.
For every reported risk, verify it is between zero and one. For each cause, also verify \(F_k\leq A\) when the any-event risk is reported.

Check the cause total.
Sum the mutually exclusive cause-specific risks. The sum must be at most \(A\); it should equal \(A\) when the cause list is exhaustive.

Check the complement.
Calculate \(1-A\) and compare it with the reported event-free probability, if available. They should agree within numerical precision.

Investigate, do not silently repair.
Recheck the prediction inputs, cause definitions, aggregation, and numerical output. Report discrepancies and their size rather than altering risks just to make the sum fit.

Small differences can arise when displayed predictions have been rounded. For example, values rounded to two decimal places need not sum exactly to another independently rounded value. Check the unrounded outputs where possible, and judge discrepancies against the precision of the reported numbers. Rounding does not explain substantial violations, such as a cause risk larger than the any-event risk by several percentage points.

For predictions standardized over a target population, the same bounds apply when the same nonnegative population weights are used for every cause and for any event. A weighted average of values in a valid interval remains in that interval; summing the standardized cause risks also preserves the equality or inequality when the underlying individual predictions satisfy it.

What These Checks Do—and Do Not—Establish

Probability bounds are structural diagnostics, not measures of predictive accuracy. Passing them does not show that risks are calibrated to observed outcomes, that uncertainty has been adequately quantified, or that the model is appropriate for the intended population. Conversely, a failed check signals an inconsistency in the reported probabilities or in how they have been assembled; it does not identify the precise source of the problem.

The check also depends on the event definition. The cause-specific risks concern which event occurs first by the horizon. They are not probabilities of ever experiencing each event regardless of what happens first. Nor should a censoring outcome be added to the event causes to force the sum to one. The probability not assigned to an event by \(t\) is event-free survival through \(t\), not censoring.

A common mistake is to reason, “Each cause risk is below one, so their total is valid.” That only checks the values individually. For example, two cause risks of 0.62 and 0.51 are each valid probabilities in isolation, but their sum is 1.13, which cannot be the total probability of mutually exclusive first-event causes. The correction is to check both the individual bounds and the joint total: the cause risks must sum to the any-event risk when exhaustive, or remain below it when only a subset is reported.

Key takeaway: For a coherent competing-risk prediction at a shared profile and horizon, each cause-specific risk lies between zero and the any-event risk. The sum over an exhaustive set of causes equals the any-event risk \(1-S(t\mid x)\); a subset sums to no more than that risk, and every such total is at most one.

Check Your Understanding

For each question, state the relevant check and show the arithmetic. Solutions follow each question.

A hypothetical prediction reports \(F_1=0.12\), \(F_2=0.31\), and \(A=0.43\), with two exhaustive causes. Are the values coherent? Solution: Both cause risks are between zero and 0.43, and \(0.12+0.31=0.43=A\). They pass the checks.
A hypothetical model reports \(F_1=0.36\) and \(A=0.29\). What fails? Solution: The individual bound fails because \(0.36>0.29\). A cause-specific first-event risk cannot exceed the risk of any event.
Three exhaustive cause predictions are 0.10, 0.22, and 0.17. What any-event risk and event-free probability should they imply? Solution: The sum is \(0.10+0.22+0.17=0.49\), so \(A=0.49\) and \(S=1-0.49=0.51\).
A report gives \(A=0.70\) and two listed causes with risks 0.25 and 0.30; other causes may exist. Is the subset possible, and how much any-event probability remains for unlisted causes? Solution: The listed sum is \(0.25+0.30=0.55\leq0.70\), so it is possible. The unlisted causes account for \(0.70-0.55=0.15\).
Why does a sum of cause-specific risks below one not by itself establish coherence? Solution: The sum must also agree with the any-event risk for exhaustive causes, or be no greater than it for a subset. For example, a sum of 0.60 is below one but is incoherent if the reported any-event risk is 0.50.