Why Tabulate Cumulative Incidence at Fixed Times?
A cumulative-incidence curve shows how the estimated probability of experiencing a particular event changes over time. A table at prespecified horizons complements that curve by making a few clinically useful absolute-risk estimates easy to locate and compare. For example, a clinician may want to know the estimated probability of relapse by 1 year or of a treatment-related complication by 5 years.
A useful table must do more than list percentages. It identifies the event cause, time origin, time unit, estimate, and uncertainty. It also gives enough context to judge how much observed follow-up supports each horizon. The introductory material on cumulative-incidence figures concerns the graphical display; here the emphasis is the precise tabular summary.
In the usual single-cohort setting with one initial state, the cumulative incidence function is estimated with the Aalen–Johansen estimator. It accounts for the timing of events and censoring while treating events from other causes as competing events, not as ordinary censoring. Its value at a reported horizon is an estimated probability, not a crude fraction of the cohort with an observed event.
Specify the Horizon and the Estimand
Before constructing the table, define the time origin and the unit of time. “Two years after randomization” is interpretable; “at 2” is not. If participants enter at different dates or delayed entry applies, state the time origin and analysis population clearly. Prespecified horizons should be clinically meaningful and chosen without looking for the most favorable estimate.
State the event cause in words that match the study definitions. If the estimand is relapse before death, for example, name relapse as the event of interest and death as a competing event. This prevents a reader from confusing the reported probability with the probability of any event or with a cause-specific rate.
The number at risk at a horizon is usually the count still under observation and free of any event immediately before that time. It is useful context, but it is not the denominator used in a simple proportion \(x/n\) to obtain the cumulative-incidence estimate. Earlier events and censoring occur at different times, and the estimator uses that event-time and risk-set information. A table should also clarify whether the displayed risk count is just before or just after events at the listed time.
Confidence Intervals and Their Interpretation
A common large-sample approach starts with an estimated standard error \(SE\{\hat F_k(t)\}\) for the Aalen–Johansen estimate and constructs a pointwise interval on the logit scale. This transformation keeps the limits between 0 and 1. The approximate calculation is:
This delta-method interval is an approximation. Its use assumes the estimator has a sufficiently regular, approximately normal sampling distribution on the transformed scale; very sparse events, estimates near 0 or 1, and little follow-up near a horizon can make the approximation unreliable. Other pointwise interval procedures may be used, but the method should be identified, especially when intervals from different analyses are compared. A pointwise 95% interval is not a simultaneous band covering the entire curve with 95% confidence.
Worked Example: Calculating a Pointwise Interval
Suppose a hypothetical analysis estimates the 3-year cumulative incidence of a specified complication as \(\hat F(3)=0.3736\), with estimated standard error \(0.1188\). Time is measured from treatment initiation in years. Assume the logit approximation is judged suitable for illustration.
First calculate the estimated logit: \(\hat\eta=\log(0.3736/0.6264)\approx -0.5169\). The product in the delta-method denominator is \(0.3736(0.6264)=0.2340\), so \(SE(\hat\eta)\approx 0.1188/0.2340=0.5077\).
The logit-scale limits are approximately \(-0.5169\pm1.96(0.5077)\), or \(-1.512\) and \(0.4781\). Transforming each limit with the expit gives approximately \(0.1807\) and \(0.6173\). Thus the table can report a 3-year cumulative incidence of 37.36% (pointwise 95% CI 18.07% to 61.73%). This interval is wide, reflecting substantial uncertainty; it does not mean there is a 95% probability that the fixed population parameter lies inside this particular interval.
Worked Reporting Examples
The following examples use hypothetical data. They illustrate how to combine the estimate with its interval and the context needed to interpret a horizon-specific result.
Worked Example: A Cohort Table With Two Horizons
A hypothetical cohort of 500 patients is followed from surgery for a specified event, with death before that event treated as a competing event. The outcome is measured in years. Suppose the analysis produces these Aalen–Johansen estimates and pointwise intervals:
| Horizon after surgery | Patients at risk just before horizon | Event of interest observed by horizon | Estimated cumulative incidence (95% CI) |
|---|---|---|---|
| 1 year | 352 | 58 | 12.4% (9.5% to 15.9%) |
| 3 years | 174 | 103 | 24.8% (20.2% to 29.8%) |
The 1-year entry means that the estimated probability of the specified event by 1 year is 12.4%, with a pointwise 95% confidence interval from 9.5% to 15.9%. The 3-year estimate is 24.8%, with an interval from 20.2% to 29.8%. The risk counts are not denominators for these percentages: for example, \(58/500=0.116\), but that crude fraction is not the reported 1-year cumulative-incidence estimate. The estimate accounts for the timing of events and censoring, including competing deaths.
A concise report should also state that 500 patients entered the analysis and give the numbers of competing events and censorings by relevant times, or provide equivalent follow-up context elsewhere in the results. The table alone does not explain why the risk set falls from 352 to 174.
Worked Example: A Less Precise Estimate at a Later Horizon
In a second hypothetical cohort, the 5-year cumulative incidence of a rare cause is estimated as \(0.310\), with standard error \(0.120\). Suppose time is measured in years from diagnosis and a logit interval is appropriate for this illustration.
The logit estimate is \(\log(0.310/0.690)\approx-0.8003\). Since \(0.310(0.690)=0.2139\), the transformed standard error is \(0.120/0.2139\approx0.5601\). The logit-scale interval is \(-0.8003\pm1.96(0.5601)\), or approximately \(-1.898\) to \(0.2975\).
Applying the expit gives limits of approximately \(0.130\) and \(0.5738\). The result is therefore 31.0% (pointwise 95% CI 13.0% to 57.4%) at 5 years. The wide interval should prompt a reader to inspect follow-up support at 5 years, including the number at risk and the counts of each cause. If only a small number remain under observation, the estimate may be highly uncertain even when its point estimate appears clinically notable.
Worked Example: Distinguishing CIF Estimates From Crude Fractions
A hypothetical study enrolls 240 participants and reports results through 2 years. By then, 31 participants have experienced event A and 18 have experienced competing event B; the estimated cumulative incidences are 14% for A and 8% for B. Suppose their pointwise 95% confidence intervals are 8.6% to 19.8% for A and 4.8% to 12.4% for B. There are 96 participants at risk just before 2 years.
The crude observed fractions are \(31/240=0.1292\), or 12.92%, for A and \(18/240=0.0750\), or 7.50%, for B. These are not the reported cumulative-incidence estimates: the CIF calculation uses the event times and risk sets and accounts for censoring. The appropriate table entries remain 14% and 8%, with their stated intervals.
For a single initial state and mutually exclusive first-event causes, the estimated probability of either event by 2 years is approximately \(0.14+0.08=0.22\), leaving an estimated probability of neither event by then of approximately \(1-0.22=0.78\). The 96 at risk just before 2 years is contextual information, not a denominator for either CIF. State the 240 enrolled participants, event counts by cause, the risk count, and the 2-year time unit so readers can interpret the precision and follow-up support.
Building a Clear Table and Checking Its Support
A practical table can include one row per cause and horizon, or one row per horizon with separate estimate columns for each cause. Use a consistent precision, commonly one decimal percentage point for estimates and confidence limits when that precision is justified. Avoid implying that the reported precision is greater than the data support. Include the confidence level and indicate whether intervals are pointwise.
At a minimum, the table or its accompanying note should identify:
- the event cause and competing-event framework;
- the time origin and explicit time unit for every horizon;
- the cumulative-incidence estimate and confidence interval, with the interval method stated;
- the analysis cohort size, event counts by cause, and a clearly defined risk count or other follow-up summary.
Check that estimates are nondecreasing across increasing horizons for a given cause, apart from tiny display discrepancies caused by rounding. Across causes, the CIFs at a common horizon should be coherent: in a single-first-event setting their sum cannot exceed 1. Do not add confidence limits to create a confidence interval for the sum. If such a combined probability is needed, estimate and quantify its uncertainty directly.
Choose clinically meaningful horizons, specify the time origin and unit, and name the event cause.
Obtain the cause-specific cumulative-incidence estimate and its pointwise interval at each horizon, using the stated method.
Report the initial cohort size, event counts by cause, and a defined risk count or other suitable summary of support.
Check time units, cause labels, ordering over time, plausible bounds, interval method, and consistency across causes before rounding.
Common Pitfalls, Alternatives, and Limits
- Using the risk count as the denominator. The estimate at a horizon is not generally the number of observed events divided by the number at risk there. Report the risk count as context, not as the CIF denominator.
- Reporting percentages without a time unit. “Risk was 20%” is incomplete. State, for example, “the 3-year cumulative incidence was 20%.”
- Calling a pointwise interval a curve-wide interval. A pointwise 95% interval applies at one horizon. It does not guarantee 95% coverage across every horizon shown.
- Hiding weak support at a late horizon. A late estimate may be based on few people under observation. Show the risk count or describe follow-up sufficiently for readers to recognize this limitation.
- Overinterpreting rounded entries. Small differences between causes or adjacent horizons may disappear or appear due to rounding. Check calculations before rounding and use consistent precision.
A table is not a replacement for a figure when the timing pattern across the full follow-up period matters. Conversely, a figure alone can make exact values and their uncertainty hard to retrieve. The two displays serve complementary purposes. Horizon-specific tables also do not by themselves explain covariate-adjusted associations or provide adjusted absolute-risk predictions; those require distinct estimands and methods. Finally, no amount of tabular detail compensates for poor follow-up near the horizon or an inappropriate censoring assumption. The estimate and interval should be interpreted in light of the study design and available data.
Check Your Understanding
Answer each question, then compare your reasoning with the brief solutions.
- A table reports a 2-year cumulative incidence but labels the time column only “2.” What information is missing?
- Why should the number at risk immediately before a horizon not be used as the denominator for the cumulative-incidence estimate?
- A cause-specific estimate is 0.18 at 1 year and 0.15 at 3 years. What should be checked?
- What does a pointwise 95% confidence interval at 5 years describe, and what does it not describe?
- At a horizon, two mutually exclusive first-event causes have estimated cumulative incidences of 0.22 and 0.13. What is the estimated probability of either event by that horizon, and what additional uncertainty statement is needed if it is reported?
Solutions. (1) State the time origin and unit, such as 2 years after treatment initiation. (2) The CIF uses event timing and risk-set information over follow-up, accounting for competing events and censoring; it is not a simple event count divided by the horizon-specific risk count. (3) Check the cause labels, time ordering, estimates before rounding, and data or software extraction: a CIF for a fixed cause should not decrease over time. (4) It gives uncertainty for the estimate at 5 years under the stated interval method; it is not a 95% probability statement about the fixed parameter and is not a simultaneous interval for the whole curve. (5) The estimated probability is \(0.22+0.13=0.35\), or 35%. If reported, its uncertainty should be calculated for the combined probability rather than inferred by adding the causes’ confidence limits.