Why Numerical Consistency Matters
A competing-risk analysis can report counts, cumulative incidence curves, horizon-specific risks, and regression estimates in several places. Each result may look reasonable in isolation yet conflict with another result—for example, a cause-specific risk above one, a cumulative incidence curve that falls over time, or cause-specific event counts that do not reconcile with cohort flow. An audit checks both the arithmetic and whether the quantities being compared describe the same population, time horizon, and estimand.
The aim is not to force every number in a report to be identical. A crude observed proportion and an adjusted model-based risk can differ legitimately. A cause-specific hazard ratio and a risk ratio are different effect measures. The audit should identify true mathematical contradictions, clarify differences that have an explanation, and prompt follow-up when the source of a discrepancy is not documented.
Core Probability Checks
Each cause-specific cumulative incidence must also be nondecreasing in time: extending the time horizon cannot undo an event that has already occurred. The sum across causes must likewise be nondecreasing and cannot exceed one. These are basic probability constraints, so a violation in estimates described as joint risks for the same population and horizon calls for investigation.
A useful justification comes from the event-time updates used by the Aalen–Johansen estimator. At an event time, let \(Y\) be the number at risk immediately beforehand and \(d_k\) the number of events from cause \(k\). Write \(S\) for the probability of having had no event yet, and \(F_k\) for the cumulative incidence of cause \(k\). The updates are:
If \(S(t-)+\sum_k F_k(t-)=1\), then after the update:
The identity holds initially, when \(S(0)=1\) and all \(F_k(0)=0\), and therefore continues to hold after each event-time update. In particular, the cause-specific risks cannot sum above one in this coherent joint calculation. This justification assumes correctly classified, mutually exclusive causes and valid risk sets. Estimation also depends on the analysis method’s assumptions, including appropriate handling of right censoring; the arithmetic audit cannot establish those assumptions by itself.
Audit Counts, Curves, and Risks Together
Start by fixing the denominator, event definitions, and follow-up horizon. A cohort-flow summary should make clear how many individuals entered the analysis, how many experienced each mutually exclusive event, and how many were censored or remained event-free at the relevant point. Counts at a particular horizon should not exceed the cohort size. When the data are fully observed through that horizon, the event counts and event-free count should add to the denominator. With censoring before the horizon, however, those raw counts do not by themselves determine the estimated cumulative incidence.
Then check each reported curve and horizon table. Curves should begin at zero (or the explicitly defined starting point), remain within probability bounds, and not decrease. Values in a horizon table should agree with the corresponding plotted or tabulated curve to the precision displayed. If curves are reported for several causes in the same cohort, inspect their sum at each horizon. Tiny discrepancies may be due to rounding; material discrepancies need an explanation such as different populations, estimands, or methods.
Worked Example: Reconciling Counts With Horizon Risks
Hypothetical data. A cohort contains 1,000 people, all followed completely to two years unless they experience one of two competing events. By one year, 60 have experienced cause A, 40 cause B, and 900 have had neither event. By two years, the cumulative counts are 120 for cause A, 80 for cause B, and 800 event-free.
Because follow-up is complete through each stated horizon, the observed proportions equal the cumulative incidence estimates in this example. At one year:
At two years:
The flow totals reconcile at each horizon: \(60+40+900=1000\) at one year and \(120+80+800=1000\) at two years. Each cause-specific risk increases over time, and the sums, 0.100 and 0.200, remain at or below one. The reported values are numerically coherent. This direct count calculation would not be justified if some people were censored before the horizon; in that setting, an estimated cumulative incidence—not simply the event count divided by the original cohort size—is needed.
Check Joint Risks and Their Time Ordering
A second pass focuses on probability constraints. Compare cause-specific risks only when they refer to the same time horizon, population, covariate profile or standardization target, and event definitions. If a report combines values from different models or populations, their sum may not represent a joint probability at all; the report should label that distinction rather than present them as a set of competing outcomes.
Worked Example: Detecting an Impossible Pair of Risks
Hypothetical report. For the same cohort and two-year horizon, a table reports \(F_1(2)=0.62\) and \(F_2(2)=0.43\) as risks of two mutually exclusive competing causes.
Both entries individually lie between zero and one, but their sum is:
That would leave an implied probability of \(1-1.05=-0.05\) for having neither event by two years, which is impossible. The entries therefore cannot both be joint cause-specific risks for the same population and horizon. The appropriate response is to check the underlying calculations, labels, and analysis populations—not to silently alter one value. For instance, confirm whether one entry was copied from a different horizon, whether one is a different estimand, or whether separate model predictions were combined in a way that does not preserve joint probability constraints.
If the values instead were 0.58 and 0.37, their sum would be 0.95 and the implied remaining probability would be 0.05. That pair satisfies the probability constraint. A sum below one establishes numerical possibility, not that the values were estimated correctly.
For time ordering, check each cause separately as well as the total. If a curve reports \(F_1(1)=0.18\) and \(F_1(2)=0.15\), it has decreased by 0.03, which is incompatible with a cumulative incidence function. A swapped time label, transcription error, or mismatched estimation target may explain the display, but the values as labelled cannot both be correct for the same cumulative incidence curve.
Compare Model Results Without Equating Different Estimands
Counts and crude risks describe observed outcomes in a defined sample. Adjusted absolute-risk predictions describe model-based risks for specified covariate values or a specified standardization population. They need not equal crude proportions, especially when covariate distributions differ across groups. Similarly, a cause-specific hazard ratio compares cause-specific event rates among people who remain event-free; a subdistribution hazard ratio uses a different risk-set construction. Neither is generally the ratio of two horizon-specific cumulative incidence risks.
An audit should therefore check labels and calculations before interpreting a numerical difference as an error. For a reported risk difference, subtract the two risks at the same horizon and on the same scale. For a reported risk ratio, divide those risks. For a hazard ratio, confirm that it is labelled as a hazard ratio and interpreted in relation to its model, rather than treating it as a risk ratio. The introductory material on choosing the reported competing-risk estimand is useful context for this check.
Worked Example: Reconciling Crude Counts and Adjusted Predictions
Hypothetical data. Two groups each contain 200 people, with complete follow-up to two years. In group A, 24 experience cause 1, 16 experience cause 2, and 160 experience neither event. In group B, the corresponding counts are 12, 28, and 160.
The crude two-year risks for cause 1 are \(24/200=0.120\) in group A and \(12/200=0.060\) in group B. For cause 2, they are \(16/200=0.080\) and \(28/200=0.140\). Each group’s counts sum to 200, and the cause-specific risks in each group sum to 0.200.
Suppose a model adjusted for baseline covariates reports two-year predicted risks of 0.100 for cause 1 and 0.070 for cause 2 in group A, and 0.070 and 0.120 in group B, for the stated standardization target. The within-group sums are 0.170 and 0.190, both permissible. These adjusted risks do not match the crude proportions, but that difference is not, by itself, an arithmetic contradiction: the model estimates an adjusted quantity, while the counts give crude sample proportions.
From the adjusted cause 1 risks, the risk difference for group B versus group A is \(0.070-0.100=-0.030\), or 3.0 percentage points lower in group B; the corresponding risk ratio is \(0.070/0.100=0.700\). These calculations do not establish that a reported cause-specific hazard ratio is 0.700. That hazard ratio is a separate model estimate and must be reported and interpreted as such. A useful audit would verify the prediction horizon, covariate target, group labels, and cause definitions, then confirm that the reported risk difference and ratio agree with the displayed adjusted risks.
Practical Audit and Common Failure Modes
Record the population, denominator, event definitions, time origin, horizon, and estimand for every number being compared.
Check that cohort counts and exclusions are explained. At a fully observed horizon, reconcile cause counts with event-free counts; with censoring, do not assume raw proportions equal estimated risks.
Verify bounds, nondecreasing cause-specific curves, and a sum across joint competing causes no greater than one.
Substitute displayed risks into reported differences or ratios, allowing for the stated rounding precision.
Check whether a discrepancy reflects adjustment, a different estimand, model specification, population, horizon, or an actual error. State any unresolved discrepancy rather than obscuring it.
A frequent mistake is to label every difference between a crude proportion and an adjusted predicted risk as an inconsistency. The correction is to compare like with like: first match the population and estimand, then verify the arithmetic. Another is to regard individually plausible cause-specific probabilities as jointly plausible. A set of mutually exclusive cause risks must satisfy the sum constraint even when every entry is between zero and one.
Rounding also matters. If risks are displayed to two decimal places, recomputing a difference from the displayed values may not reproduce a more precise model-based difference exactly. Small discrepancies can be consistent with rounding; report precision should be sufficient to distinguish that from a material inconsistency. Confidence intervals for separately reported risks need not have endpoints that sum to one: the key joint probability check applies to the point estimates presented as a set of risks, not to separately constructed interval endpoints.
Numerical auditing has limits. It can detect impossible values, inconsistent labels, and unreconciled arithmetic, but it cannot prove that the event classifications, censoring assumptions, model specification, or data processing are correct. When estimates come from different models, valid-looking values may still fail to describe a coherent joint distribution. Document the method and target population, and resolve discrepancies using the analysis record rather than guesswork.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- At a fixed horizon, two mutually exclusive cause-specific risks are 0.32 and 0.41. Are they jointly possible? What is the implied probability of neither event by that horizon?
- A cause-specific cumulative incidence is reported as 0.24 at one year and 0.21 at two years for the same cohort and estimand. What is the audit finding?
- A report gives a crude two-year risk of 0.10 and an adjusted predicted risk of 0.08. Is this necessarily an inconsistency? Name one reason they might differ.
- Two adjusted cause-specific risks for the same group, horizon, and standardization target are 0.55 and 0.49. What should the auditor conclude?
Solutions. (1) Yes. Their sum is \(0.32+0.41=0.73\), leaving an implied no-event probability of \(1-0.73=0.27\), which is nonnegative. (2) The values are inconsistent with a cumulative incidence curve for the same cause and estimand because it decreases by 0.03 as time increases; check labels, horizons, and source calculations. (3) No. Adjustment for baseline covariates or a different standardization target can make the adjusted model-based risk differ from the crude observed proportion. (4) Their sum is \(0.55+0.49=1.04\), exceeding one. They cannot be joint cause-specific risks for the same target as labelled, so the underlying models, estimands, and reporting must be checked.