Interpreting Competing-Risk Prediction Tables

What a Prediction Table Is Predicting

A competing-risk prediction table compresses several decisions into each entry: which population is being described, which event is counted, and by what time. A number such as 0.18 is not interpretable on its own. It could be the predicted probability of one event by a particular horizon for a person with specified covariates, or an average risk for a defined population. Those are different prediction targets, even when the numerical values happen to be similar.

The central quantity for event type \(k\) is its cumulative incidence by time \(t\): the probability that event \(k\) is the first event to occur by \(t\). A competing event is not simply an observation that disappears from the target. Its occurrence prevents the event of interest from being the first event later, and therefore affects the absolute risk being predicted.

The phrase “by time \(t\)” includes an event occurring at \(t\). In practice, tables may use horizons such as 1 year or 5 years; the time unit and the precise horizon must be read from the heading or footnote. A cause-specific risk of 0.18 at 3 years means a predicted probability of 18% of experiencing that cause as the first event within 3 years, for the stated target population. It does not mean an 18% instantaneous event rate or an 18% risk per year.

Read the Target Before the Number

A table may report predictions for specified covariate profiles or for a population. A profile-specific entry, \(F_k(t\mid x)\), is conditional: it describes individuals represented by that particular covariate profile under the fitted prediction model. A population-standardized entry averages predictions over the covariate distribution of a named target population. These quantities answer different questions and need not agree.

For a target population with covariate profiles \(x_1,\ldots,x_n\) and specified weights \(w_i\) summing to 1, a standardized prediction is the weighted average of profile-specific predictions:

The weights define the population to which the result applies. Equal weights describe the empirical cohort when every person is included once; other weights may represent a different target distribution. Thus, “standardized risk” is incomplete unless the table or its documentation identifies the population being standardized to. It is not automatically the risk for every person, nor does it necessarily describe the sample used to fit the model.

Why the Probabilities Fit Together

At a fixed horizon, an individual has experienced exactly one of the following: cause 1 first, cause 2 first, continuing through cause \(K\) first, or no event yet. These categories are mutually exclusive and collectively exhaustive, assuming the listed causes include every event type being counted. Therefore, for a given profile and horizon:

This identity is a useful interpretation aid. The event-free probability is not the complement of one cause-specific risk unless there is only one possible event type. Instead, it is the complement of the risks for all listed event types combined:

The identity also explains why the competing events matter for absolute risk. A person who experiences Cause B first is not event-free at a later horizon and cannot subsequently experience Cause A as the first event. A table’s cause-A risk is therefore not interpreted as though Cause B could be removed while leaving all else unchanged.

Intervals, Comparisons, and Checks

A prediction table may give an uncertainty interval alongside each estimated risk. For example, a 95% confidence interval of 0.14 to 0.24 around an estimated 3-year risk of 0.19 indicates uncertainty in the estimated risk under the stated statistical procedure and assumptions. In repeated applications of that procedure, 95% of intervals constructed this way would cover the fixed target quantity. It is not correct to say there is a 95% probability that the fixed risk lies in this particular interval.

Check whether the interval is described as pointwise or simultaneous. A pointwise interval addresses one specified estimate, such as one cause at one horizon. A collection of pointwise 95% intervals does not generally provide 95% coverage for the entire set of causes and horizons together. Also check whether the uncertainty includes model fitting, baseline-risk estimation, and any standardization step; the table’s methods or footnote should clarify what was accounted for.

Intervals for separate cause-specific risks may not add with an event-free interval to exactly 1. The probability identity applies to the point predictions, not necessarily to the endpoints of separately calculated marginal intervals. A mismatch among interval endpoints alone is not evidence that the point estimates violate probability coherence.

Common checks help detect misread or potentially problematic tables. First, verify that units and horizons match: 0.12 at 2 years is not comparable to 0.12 at 5 years without specifying the different horizons. Second, for a single profile or consistently standardized target, each cause-specific cumulative incidence should be nondecreasing across increasing horizons, while event-free probability should be nonincreasing. Third, at each horizon, the point predictions should be between 0 and 1 and should satisfy the probability identity up to rounding. Fourth, confirm that the target population and event definitions are stated; otherwise, a precise interpretation may not be possible. Limited follow-up support, model uncertainty, or an unclear target can make a numerically tidy table misleading.

Different table types answer different questions. A profile-specific prediction answers what the model predicts for that covariate profile. A standardized prediction answers what the model predicts on average for a specified covariate distribution. A table of cause-specific hazards describes instantaneous event rates among those still event-free, rather than the probability of an event by a horizon. A single-cause net-risk quantity also has a different interpretation from a competing-risk cumulative incidence because it does not represent the observed-world probability of that cause occurring first in the presence of competing events. Read the column label and estimand, not just the word “risk.”

For instance, with a 2-year Cause A risk of 0.12 and a 5-year risk of 0.25, the difference is \(0.25-0.12=0.13\), or 13 percentage points of the original target population experiencing Cause A first between those horizons. Among people still event-free at 2 years, the conditional probability of Cause A during the next 3 years would instead divide that increment by the 2-year event-free probability, provided the same target and model apply. Using the example’s \(S(2)=0.80\), this is \(0.13/0.80=0.1625\), or 16.25%. The denominator is essential because the conditional target is the subset event-free at 2 years.