From Model Output to a Results Paragraph
Competing-risk analyses often produce several kinds of results at once: numbers of each event type, estimated cumulative incidence at selected times, and regression estimates. A useful results paragraph connects these pieces without treating them as interchangeable. It names the event, comparison, time horizon, estimand, estimate, and uncertainty, then states what the results do and do not support.
The earlier material on choosing and naming the reported competing-risk estimand, reporting cumulative incidence, and describing cause-specific and subdistribution effects side by side establishes the main distinctions used here. The focus now is translation: how to turn a compact, hypothetical output into accurate prose, and how to check that the prose matches the numbers.
Before writing, confirm which event is the outcome of interest, which event types are competing events, the analysis population and follow-up horizon, and which model estimate answers the stated question. For a cause-specific Cox model, the proportional-hazards assumption concerns the cause-specific hazard ratio over time. For a Fine–Gray model, the proportional-subdistribution-hazards assumption concerns the subdistribution hazard ratio over time. Both models also require suitable handling of censoring; in standard formulations, censoring must be independent of the relevant event process, at least conditional on modeled covariates. A regression paragraph should not imply these assumptions were verified merely because software returned an estimate. Report diagnostics or limitations when they materially affect interpretation.
The reporting task is not to reproduce every line of software output. It is to select enough information for a reader to identify the sample, event, time scale, estimand, comparison, and uncertainty. Counts describe what was observed; cumulative incidence estimates describe absolute event probability over time while accounting for competing events; regression estimates describe a model-based relative contrast. Each has a different role.
Worked Example: Connecting Counts and Absolute Risk
Worked Example: Connecting Counts and Absolute Risk
Suppose a hypothetical randomized study follows 240 participants for up to two years. There are 120 participants in each treatment group. The event of interest is disease recurrence; death before recurrence is a competing event. By two years, the recurrence counts are 24 in the new-treatment group and 18 in the control group. The corresponding death counts are 14 and 20, and 12 and 10 participants, respectively, are censored before two years. The remaining 70 and 72 participants have no recorded event by two years. Estimated two-year recurrence cumulative incidences are 20.5% and 15.8%.
Check the counts. In the new-treatment group, \(24+14+12+70=120\). In control, \(18+20+10+72=120\). Across groups, there are 42 observed recurrences, 34 deaths before recurrence, 22 censored observations, and 142 participants with no event recorded by two years. These are event and follow-up counts, not estimates of two-year recurrence probability.
Calculate the absolute contrast. The estimated risk difference in two-year recurrence cumulative incidence, new treatment minus control, is
Thus the estimated difference is 4.7 percentage points higher in the new-treatment group. No confidence interval for this difference is supplied, so the paragraph must not imply that this contrast is statistically distinguishable from zero. Because some participants were censored before two years and death competes with recurrence, the cumulative incidence estimates need not equal the crude proportions \(24/120=20.0\%\) and \(18/120=15.0\%\).
Possible results text: “Among 120 participants assigned to each group, 24 new-treatment participants and 18 control participants experienced recurrence during observed follow-up; death before recurrence occurred in 14 and 20 participants, respectively. The estimated two-year cumulative incidence of recurrence was 20.5% with new treatment and 15.8% with control, an absolute difference of 4.7 percentage points. The estimated cumulative incidences account for competing deaths and censoring; the difference is descriptive because an interval estimate was not provided.”
Reporting Regression Estimates Without Changing Their Meaning
For a regression estimate, the sentence should identify the model and its estimand before interpreting its direction. If the model is adjusted, name the adjustment set or refer to the prespecified covariates in a way that can be traced to the analysis. A confidence interval communicates precision on the reported scale; a p-value summarizes compatibility with the null under the model and its assumptions. Neither should be interpreted as the probability that the null hypothesis is true.
For a hazard ratio, the null value is 1. In a standard Wald analysis, the estimate and its confidence interval can be checked using the log scale. If \(\widehat{\beta}\) is the log hazard-ratio estimate and \(SE\) its standard error, then the estimated ratio is \(\exp(\widehat{\beta})\), and a 95% Wald interval is \(\exp\{\widehat{\beta}\pm1.96SE\}\). The Wald statistic is \(z=\widehat{\beta}/SE\), with a two-sided p-value calculated from the standard normal distribution. These checks can reveal whether a paragraph has copied a value incorrectly or paired an estimate with the wrong interval.
Worked Example: Reporting a Fine–Gray Estimate
Suppose the adjusted Fine–Gray output for recurrence gives a subdistribution hazard ratio of 1.31 for new treatment versus control, with a 95% confidence interval of 0.78 to 2.19 and a two-sided p-value of 0.30. The model adjusts for age and baseline disease severity. Assume the output was obtained from a prespecified model and its censoring and proportional-subdistribution-hazards assumptions are judged adequate for this analysis.
Check internal consistency. The log estimate is \(\ln(1.31)=0.270\). The interval’s log-scale endpoints are \(\ln(0.78)=-0.248\) and \(\ln(2.19)=0.784\). Using the rounded displayed endpoints, the implied standard error is approximately
The approximate Wald statistic is \(z=0.270/0.263=1.03\), corresponding to a two-sided p-value of about 0.30. Rounding explains small discrepancies. The interval includes 1, consistent with the p-value being above 0.05.
Interpretation. The point estimate is above 1, but the interval is broad and includes both 1 and values below and above it. The result does not provide strong evidence of a difference in the modeled subdistribution hazard. It does not show that treatment increases recurrence risk by 31%, nor does it establish that the groups have equal recurrence probabilities.
Possible results text: “In the adjusted Fine–Gray model, new treatment was associated with a higher estimated subdistribution hazard of recurrence than control (subdistribution hazard ratio, 1.31; 95% CI, 0.78–2.19; two-sided \(p=0.30\)). The interval included 1, so the data did not provide clear evidence of a difference in the modeled subdistribution hazard after adjustment for age and baseline disease severity.”
The phrase “did not provide clear evidence of a difference” is preferable to “there was no difference.” A nonsignificant result can reflect a small effect, limited information, or both; its interval shows which effect sizes remain compatible with the data. Also avoid “31% higher risk”: the Fine–Gray estimate is a ratio of subdistribution hazards, not a ratio of cumulative incidence probabilities.
Worked Example: Presenting Two Estimands Side by Side
Suppose the same hypothetical study reports an adjusted cause-specific Cox hazard ratio for recurrence of 1.42 (95% CI, 0.79–2.55; \(p=0.24\)) and the adjusted Fine–Gray estimate of 1.31 (95% CI, 0.78–2.19; \(p=0.30\)). Both models adjust for age and baseline disease severity. The estimates answer different questions, so the paragraph must label each rather than presenting them as competing versions of one effect.
A concise report is: “The adjusted cause-specific Cox model estimated a higher cause-specific hazard of recurrence with new treatment than with control (hazard ratio, 1.42; 95% CI, 0.79–2.55; \(p=0.24\)). The adjusted Fine–Gray model estimated a subdistribution hazard ratio of 1.31 (95% CI, 0.78–2.19; \(p=0.30\)). Both intervals included 1; neither analysis provided clear evidence of a treatment-group difference in its respective estimand.”
Why this wording works. “Cause-specific hazard” refers to the instantaneous recurrence rate among participants who remain free of all events. “Subdistribution hazard” refers to the Fine–Gray model’s association with the recurrence cumulative incidence. The paragraph does not say the estimates are equal, does not infer equivalence, and does not claim treatment has no effect. Since both models share an adjustment set, that information can be stated once, provided it is clear that it applies to both.
A useful consistency check is the confidence interval’s relation to the point estimate and null. The cause-specific interval 0.79–2.55 is centered approximately around 1.42 on the log scale, and the subdistribution interval 0.78–2.19 is centered approximately around 1.31 on that scale. Both include 1, in keeping with their reported p-values above 0.05. The two estimates need not match: a cause-specific hazard contrast and a subdistribution hazard contrast are not the same estimand.
A Short Derivation for an Uncertainty Check
The standard error calculation in the Fine–Gray example follows from the usual two-sided Wald interval on the log scale. If \(\widehat{\beta}=\ln(\widehat{HR})\), then the 95% interval endpoints are \(\widehat{\beta}-1.96SE\) and \(\widehat{\beta}+1.96SE\). Subtracting the lower endpoint from the upper gives
Therefore, \(SE\) is approximately the log interval width divided by 3.92. The p-value check follows by comparing \(\widehat{\beta}/SE\) with a standard normal distribution under the null \(\beta=0\), equivalent to a hazard ratio of 1. Because published output is rounded, this is an audit rather than a demand for exact equality. Profile-likelihood intervals or tests that are not Wald-based may not produce this exact relationship.
Common Reporting Failures and Practical Checks
A clear paragraph is not necessarily a long paragraph. Before finalizing it, check whether every number has a named event, group, time horizon, and estimand. Then read each interpretation literally: does it describe the model output, or has a hazard ratio silently become a risk ratio? Does the conclusion respect the interval and p-value? Are observed counts being confused with estimated probabilities?
Name the event of interest, competing event, population, comparison, and time horizon before selecting numbers.
Keep observed counts, cumulative incidence estimates, and model-based relative estimates distinct. Do not substitute a crude event proportion for a cumulative incidence.
Check group counts against the sample sizes, intervals against point estimates and the null, and p-values against the stated test when possible.
Use the named estimand, report units for absolute quantities, and describe uncertainty without claiming proof of no effect or causation.
Common failure modes include omitting the time horizon from a cumulative incidence, reporting “risk increased by 31%” for a subdistribution hazard ratio of 1.31, and describing a p-value of 0.30 as evidence that the null is true. Another is reporting several model estimates without saying which covariates were adjusted for. If a result is imprecise, say so directly; do not hide a wide interval behind a sentence focused only on the point estimate.
There are limits to what a results paragraph can establish. A paragraph cannot repair a model with poor fit, serious assumption violations, informative censoring, or insufficient events. Nor does a well-written adjusted association by itself justify a causal conclusion. The purpose of concise reporting is to make the actual analysis transparent, not to strengthen the evidence through confident wording.
Worked Example: Correcting an Overstated Draft
A draft says: “New treatment increased recurrence risk by 31%; 24 patients had recurrence compared with 18 on control, proving the treatment had no effect because \(p=0.30\).”
Problems. The 31% statement turns a Fine–Gray subdistribution hazard ratio into a risk ratio. The counts alone do not account for differing follow-up or competing deaths. Finally, \(p=0.30\) does not prove no effect; the 95% confidence interval of 0.78 to 2.19 includes a range of possible associations.
Correction. “The estimated two-year recurrence cumulative incidence was 20.5% with new treatment and 15.8% with control, an absolute difference of 4.7 percentage points. In the adjusted Fine–Gray model, the subdistribution hazard ratio was 1.31 (95% CI, 0.78–2.19; \(p=0.30\)); the estimate was imprecise and did not provide clear evidence of a difference in the modeled subdistribution hazard.”
The corrected version reports the absolute estimates and model estimate separately, identifies the estimand, and matches its conclusion to the uncertainty. If a confidence interval for the absolute difference were available, it should be reported as well; it should not be invented from the two point estimates alone.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions below.
- A two-year cumulative incidence is 12% in one group and 9% in another. What is the absolute difference, and what additional information is needed to assess its uncertainty?
- A Fine–Gray model gives a subdistribution hazard ratio of 0.72. Why is “28% lower risk” not a precise interpretation?
- A cause-specific Cox model reports a hazard ratio of 1.10 with a 95% confidence interval of 0.60–2.02 and \(p=0.77\). What is an appropriate conclusion?
- Why might crude event proportions differ from cumulative incidence estimates in a competing-risk study?
Solutions. (1) The absolute difference is \(12\%-9\%=3\) percentage points. A confidence interval or other uncertainty estimate for that difference is needed; the two point estimates alone do not establish its precision. (2) The estimate is a subdistribution hazard ratio, not a risk ratio. Report the model and estimand, and do not translate it directly into a percentage change in cumulative incidence. (3) The interval includes 1 and is wide; the analysis does not provide clear evidence of a difference in the modeled cause-specific hazard. It does not prove no effect. (4) Crude proportions use observed counts divided by group size and do not fully account for censoring times or competing events, whereas cumulative incidence estimates account for event timing and competing causes.