What a Cumulative Incidence Figure Needs to Communicate
A cumulative incidence plot is useful only when a reader can tell which event is being counted, by when, and in what population. A curve without those details may look polished but be scientifically ambiguous. In a competing-risks setting, the cause matters: the cumulative incidence of one event is not the same quantity as the probability of experiencing any event.
For event cause \(k\), the cumulative incidence function (CIF) \(F_k(t)\) is the probability of experiencing cause \(k\) by time \(t\), before experiencing a competing event. A figure usually displays an estimate of this function. Its labels should name the cause in plain language, not merely show a code such as “1” or “event A.”
For example, in a right-censored cohort with accurately classified causes, an Aalen–Johansen estimate of the CIF for cause \(k\) can be written as
Here, \(d_k(u)\) is the number of cause-\(k\) events at time \(u\), \(Y(u)\) is the number at risk just before \(u\), and \(\widehat S(u-)\) estimates the probability of remaining free of every event cause up to just before \(u\). The display is meaningful under the analysis assumptions, including appropriate handling of censoring and reliable event-cause classification. Plot design cannot correct a poorly defined estimand or invalid analysis; it can, however, make the intended quantity transparent.
Labels, Axes, and Uncertainty
The vertical axis should say what the plotted values are. Suitable labels include “Cumulative incidence of cardiovascular death” and “Estimated probability of relapse before death.” If the scale runs from 0 to 1, adding “probability” makes its meaning explicit. If values are shown from 0% to 100%, label the axis “Cumulative incidence (%)” and make sure the tick marks match the plotted values. Do not label the axis simply “Survival” when the curve is a cause-specific cumulative incidence.
The horizontal axis needs both a time unit and a meaningful time origin. “Months since randomization” is more informative than “Time” or “Months.” Other reasonable origins include diagnosis, treatment initiation, or transplant. If entry is delayed or participants have different origins, explain the chosen time scale and population in the caption or methods. A reader should not have to infer whether 12 means 12 days, months, or years.
Uncertainty should be visible or explicitly justified if it is omitted. A figure may show pointwise confidence intervals at selected times, or a confidence band across the plotted time range. A pointwise 95% confidence interval refers to uncertainty at a particular time; it does not claim 95% coverage of the entire curve simultaneously. A simultaneous 95% confidence band is constructed to cover the whole curve over a specified range with the stated confidence level. State which type is shown. Shading can be legible for one curve, but several overlapping ribbons can obscure both the curves and each other; line types or a separate panel may work better.
Use a common probability scale when comparing groups in the same figure. Starting the vertical axis at zero usually makes absolute differences easier to judge. A restricted range can be useful when probabilities are small, but it should be clearly marked and should not visually exaggerate modest differences. State the confidence level and, where relevant, the interval or band method in the caption or legend.
Worked Example: A Single-Cohort Figure
Suppose a hypothetical cohort of 120 patients is followed after randomization. The target event is cardiovascular death, and non-cardiovascular death is a competing event. The estimated CIF for cardiovascular death is 0.18 at 12 months, with a pointwise 95% confidence interval of 0.12 to 0.25. At 24 months, the estimate is 0.31, with a pointwise 95% confidence interval of 0.23 to 0.40. The analysis output gives numbers at risk of 120 at baseline, 91 at 6 months, 60 at 12 months, 38 at 18 months, and 21 at 24 months.
A clear vertical label is “Cumulative incidence of cardiovascular death (probability).” A horizontal label is “Months since randomization.” The caption should state that non-cardiovascular death is treated as a competing event and that the shaded region, if used, is a pointwise 95% confidence interval.
The axis can be displayed as a probability from 0 to 1 or as a percentage. Converting the 12-month estimate gives \(0.18\times100=18\%\); its interval becomes \(0.12\times100=12\%\) to \(0.25\times100=25\%\). At 24 months, the corresponding values are \(31\%\) and \(23\%\) to \(40\%\). Use one scale consistently: do not plot fractions while labeling the axis in percent.
A risk table beneath the plot could show the five supplied counts at 0, 6, 12, 18, and 24 months. Its caption should define the convention—for example, patients under observation and free of all event causes immediately before each displayed time. These counts are not the number who have not experienced cardiovascular death alone: patients who have had the competing event are no longer at risk for the first occurrence of cardiovascular death.
Group Comparisons and Risk Tables
For a group comparison, identify each group directly in a legend and use the same time origin, horizontal scale, and vertical scale. Use distinguishable line colors or patterns that remain readable in grayscale when possible. If curves are close or confidence bands overlap heavily, separate panels with common axes can be clearer than one crowded panel. A caption should say how groups were defined and identify the event cause and competing event.
A number-at-risk table can help readers see how much follow-up supports the late portions of the curves. It is particularly useful when follow-up is long or the cohort becomes small over time. But the table should not dominate the figure: choose a few informative time points, align them with the horizontal axis, and keep group labels consistent with the legend. Define whether counts are measured immediately before each time point or by another convention. In standard first-event analyses, a person who has experienced any event cause is no longer in the event-free risk set.
Worked Example: Comparing Two Treatment Groups
In a hypothetical randomized study, relapse is the event of interest and death before relapse is the competing event. There are 80 patients in the standard-care group and 82 in the new-treatment group. At 12 months, the estimated relapse CIFs are 0.14 (pointwise 95% confidence interval 0.09 to 0.21) and 0.22 (0.15 to 0.30), respectively. At 24 months, the estimates are 0.25 and 0.34. Numbers at risk are 80 and 82 at baseline, 52 and 49 at 12 months, and 25 and 21 at 24 months, for standard care and new treatment, respectively.
The plot should label the vertical axis “Cumulative incidence of relapse” and the horizontal axis “Months since randomization.” The legend should name the two treatment groups. The caption should state that death before relapse is a competing event and identify the intervals as pointwise 95% confidence intervals. Both curves should use the same axes; otherwise, their visual separation could be misleading.
At 12 months, the plotted estimates are \(14\%\) and \(22\%\), because \(0.14\times100=14\) and \(0.22\times100=22\). Their descriptive difference is \(22\%-14\%=8\) percentage points. That difference is a visual comparison of estimated CIFs, not by itself a confidence interval or a statistical test of a group effect. The figure should not imply that the treatment caused the difference unless the study design and analysis support a causal interpretation.
A risk table below the plot can show, at 0, 12, and 24 months, the counts 80/82, 52/49, and 25/21. Separate rows for the two groups and columns aligned to the time axis make the counts interpretable. Because the later risk sets are smaller, the 24-month curve portions should be read with the displayed uncertainty in mind.
Sparse Causes, Diagnostics, and Alternatives
A rare cause can produce a curve that appears nearly flat, even while its uncertainty is substantial. If only a few events occur, show uncertainty when it can be estimated reliably, and avoid smoothing that suggests more information than the data contain. Do not extend a curve beyond the period with meaningful follow-up merely to fill the plotting window. If the late risk set is very small, consider ending the displayed range earlier or clearly marking the limited support with a risk table and caption.
Worked Example: A Sparse Event Cause
Consider a hypothetical cohort of 300 transplant recipients. The event of interest is a specified severe infection, and death before infection is a competing event. By 36 months, there are seven infection events and the estimated CIF is 0.025, with a pointwise 95% confidence interval of 0.009 to 0.066. At 48 months, the estimate is 0.031, with a pointwise interval of 0.010 to 0.084. The numbers at risk at 0, 12, 24, 36, and 48 months are 300, 214, 126, 68, and 39.
Label the vertical axis “Cumulative incidence of severe infection” and the horizontal axis “Months since transplant.” State that death before infection is competing, and report the seven observed infection events in the caption or accompanying results text. In percent, the 36-month estimate is \(0.025\times100=2.5\%\), and its interval is \(0.9\%\) to \(6.6\%\). The 48-month estimate is \(3.1\%\), with an interval from \(1.0\%\) to \(8.4\%\).
The wide intervals communicate that the estimate is imprecise; they should not be hidden because the line looks flat. The decline in the number at risk—from 68 at 36 months to 39 at 48 months—also signals that the late curve is based on a smaller risk set. A figure extending well beyond 48 months would suggest unsupported precision, so the plotted range should stop at or near the period with adequate follow-up.
A useful diagnostic is to compare the labels and caption with the analysis output before finalizing the graphic: does the cause match, do the time origin and units match, and does the probability scale match the values? Also check that the uncertainty limits contain their estimates, that the curve is nondecreasing as a cumulative probability, and that the risk counts do not increase over time in an ordinary first-event cohort. Unexpected increases in a risk table may indicate a different counting convention, delayed entry, or a coding error that needs explanation.
The main alternative is not simply another plotting style, but another quantity. A cause-specific hazard plot describes an instantaneous event rate among those currently at risk; it is not a plot of cumulative probability. A Kaplan–Meier complement that treats competing events as censoring also does not represent the CIF for a particular cause. Labeling either of these as a cumulative incidence plot would misstate what is being displayed. A table of estimates at selected times can complement a figure when exact values are important, but the plot shows the trajectory over time. Keep the figure focused on its defined estimand rather than trying to make one graphic serve every purpose.
Common failures include a generic title such as “Survival,” a time axis without an origin, a legend that says only “Group 1” and “Group 2,” and an unlabelled shaded area. Correct these by naming the cause, time scale, groups, and uncertainty type in the axis labels, legend, or caption. Another mistake is to omit all late risk-set information while asking readers to interpret a long, sparse tail; add informative counts or shorten the displayed range.
Design Checklist
State the event cause in the axis label, title, or caption, and identify the competing event or events.
Give the time origin and units; label the vertical axis as probability or percent and use matching tick marks.
Identify the confidence level and whether the display is pointwise or simultaneous; avoid overlapping ribbons that conceal the curves.
Add a concise, defined risk table when counts help readers assess follow-up, and avoid emphasizing unsupported tails.
Check Your Understanding
Answer each question, then compare with the solutions below.
- A figure shows a CIF for stroke but labels its vertical axis only “Outcome.” Give a clearer axis label.
- A shaded region is described as a “95% confidence band,” but it consists of pointwise intervals. What should be corrected?
- A plot compares two groups using different vertical-axis ranges. Why is this problematic, and what is a better choice?
- At 18 months, a CIF estimate is 0.07 with a 95% confidence interval of 0.03 to 0.13. Express the estimate and interval as percentages.
- A risk table’s count at a given time includes people who already experienced a competing event. Is that count appropriate for an ordinary first-event risk table? Explain.
Solutions. (1) Use “Cumulative incidence of stroke (probability)” or “Cumulative incidence of stroke (%),” with a scale matching the values. (2) Call it a set of pointwise 95% confidence intervals, or replace it with a genuinely simultaneous band; pointwise intervals do not promise coverage of the entire curve. (3) Different ranges can exaggerate or hide the difference between curves. Use common axes so heights and separations are directly comparable. (4) \(0.07\times100=7\%\); the interval is \(0.03\times100=3\%\) to \(0.13\times100=13\%\). (5) No. In an ordinary first-event analysis, a person experiencing any event cause has left the event-free risk set; define counts according to the analysis convention and state that convention.