Reporting Gray-Test Design and Results

Why Reporting Details Matter

A Gray-test result is not fully interpretable from a statistic and p-value alone. Readers need to know which cause was tested, which groups were compared, what follow-up period was relevant, and how censoring was handled. They also need cumulative incidence estimates to see the event probabilities underlying the comparison. These details make the analysis reproducible and prevent a global test from being mistaken for a specific difference at one time point.

Reporting has two related parts. The design report states what analysis was planned and the assumptions used to plan it. The results report states what analysis was performed and what it found. Keep planned choices distinct from observed facts: for example, planned follow-up is not the same as the follow-up actually achieved.

What the Test Compares

Definition: Let \(T\) be time from the defined origin to the first event, \(J\) the cause of that event, and \(G\) the group. For cause \(k\) and group \(g\), the cumulative incidence function is \(F_{k,g}(t)=P(T\leq t, J=k\mid G=g)\). It is the probability of experiencing cause \(k\) by time \(t\), before a different event precludes it.

For \(G\) groups, the Gray test assesses the null hypothesis that the cause-specific cumulative incidence functions are equal across groups over the test's time domain:

H_0: F_{k,1}(t)=F_{k,2}(t)=\cdots=F_{k,G}(t) \quad\text{for all tested }t.

The alternative is that at least one group function differs somewhere over that domain. The test is an omnibus comparison: it does not by itself identify which groups differ, when the difference occurs, or its size at a particular time. Therefore, report cumulative incidence estimates at clinically meaningful times alongside the test.

The test and estimates require suitable handling of censoring. In the usual unadjusted setting, censoring must be independent of the event process within groups, at least conditional on any variables used to define the analysis. Subjects should be independent, group and cause classifications should be correctly defined, and the data should support the large-sample approximation used for the test. State the censoring mechanisms and any assumptions about them rather than simply writing “censored observations were included.” Competing events are part of the competing-risks outcome structure; they are not ordinary loss-to-follow-up censoring when estimating the cumulative incidence.

A Reproducible Reporting Checklist

A concise report can still be complete if it identifies the estimand, design choices, and statistical output in a fixed order. Include the following information, as applicable:

Name the target.
Specify the event cause being compared, how that cause is defined, and the groups. State the time origin and eligibility rule if these are not otherwise unambiguous.

Define the follow-up scope.
Give the planned or observed follow-up horizon and identify the time points at which cumulative incidence estimates will be reported. Say whether the test is global over observed event times or restricted to a prespecified interval, and identify the software or implementation if a restriction is used.

Explain censoring and competing events.
Describe administrative censoring, loss to follow-up, and the data cutoff. State the assumption about censoring and clarify that events from other causes are treated as competing events, not as ordinary censoring, in the cumulative incidence analysis.

Specify the planned test.
State that a Gray test compares the cause-specific cumulative incidence functions, give the significance level, and describe any stratification, multiplicity adjustment, or prespecified subgroup analysis that changes the test.

Report the output and estimates.
For the analysis, report the Gray statistic, degrees of freedom, and p-value. Give group-specific cumulative incidence estimates at the chosen times, with confidence intervals where available, and identify the estimator and uncertainty method.

Why the Reference Distribution Has \(G-1\) Degrees of Freedom

The Gray test builds a vector of scores that summarizes departures from the null across the \(G\) groups, accounting for their covariance. Under the null and the regularity conditions for the large-sample test, the standardized score vector is approximately multivariate normal. The test statistic is a covariance-standardized quadratic form:

Q=U^\mathsf{T}\widehat{\Sigma}^{-1}U.

Here \(U\) is the score vector and \(\widehat{\Sigma}\) its estimated covariance matrix. Group comparisons contain only \(G-1\) independent contrasts: adding the same amount to every group does not create a difference between groups. Thus, when the covariance has full rank for those contrasts, \(Q\) is compared with a chi-square distribution with \(G-1\) degrees of freedom. The p-value is the probability, under the null reference distribution, of a statistic at least as large as the observed \(Q\). It is not the probability that the null hypothesis is true.

Worked Example: Reporting a Two-Group Trial Result

Suppose a hypothetical randomized trial compares Treatment A with Treatment B for the cumulative incidence of disease-related hospitalization, with death before hospitalization as a competing event. Follow-up is planned through 24 months. At the analysis cutoff, censoring occurs at the last known event-free contact or at 24 months, whichever comes first. Assume loss to follow-up is independent of the event process within treatment groups. The test output is \(Q=4.12\), with \(df=1\).

Calculate the p-value. For one degree of freedom, the upper-tail probability is approximately 0.0424:

p=P(\chi^2_1\geq 4.12)\approx 0.0424.

At a prespecified \(\alpha=0.05\), this result is statistically significant because \(0.0424<0.05\). In a hypothetical output, the estimated 24-month cumulative incidence is 0.18 in Treatment A and 0.10 in Treatment B. The estimated difference is \(0.18-0.10=0.08\), or 8 percentage points.

A suitable report is: “By 24 months, the estimated cumulative incidence of disease-related hospitalization was 18% in Treatment A and 10% in Treatment B, with death treated as a competing event. The Gray test comparing the groups gave \(Q=4.12\), \(df=1\), \(p=0.0424\).” The test provides evidence against equality of the group cumulative incidence functions over its tested time domain. The estimates describe the 24-month probabilities; the test does not establish that the difference is exactly 8 percentage points at every time, nor does this result alone establish a causal explanation for the difference. Confidence intervals should also be reported when available.

Planning and Results Should Be Reported Separately

A design report should document the choices that made the planned test meaningful: target cause and groups, allocation, total sample size, anticipated event and censoring patterns, follow-up, type I error level, and target power. If more than one cause or comparison is to be tested, state how multiplicity is handled. Include the assumptions or sensitivity scenarios that materially affect the planned operating characteristics. These details let readers distinguish an analysis designed in advance from choices made after seeing the data.

A results report adds the number analyzed in each group, the analysis cutoff, observed follow-up, and the actual test output. Report the cumulative incidence estimates on the same time scale as the design question. If an estimate is not available at the planned horizon because no one remains under observation, say so; do not imply that the data support an estimate there. When a planned test or censoring approach changes, explain what changed and why.

Worked Example: A Multiplicity-Adjusted Report

Suppose a hypothetical study prespecifies two primary Gray tests, one for each of two causes, and uses a Bonferroni familywise level of 0.05. Each test is therefore assessed at \(0.05/2=0.025\). For the first cause, suppose the test gives \(Q=5.02\), \(df=1\), and \(p=0.0251\).

Because \(0.0251>0.025\), this result does not meet the prespecified threshold for that cause. It would be misleading to call it statistically significant just because \(0.0251<0.05\): that comparison ignores the declared multiplicity adjustment. The report should name both primary causes, state the Bonferroni procedure and per-test level, give the statistic and p-value, and include group-specific cumulative incidence estimates. The conclusion is limited to this test; it does not show that the other cause has no group difference.

Worked Example: Separating the Estimate Horizon From the Test Domain

Suppose a hypothetical protocol specifies 18-month cumulative incidence estimates, but the analysis software reports a standard Gray test using all available event times through 30 months. At 18 months, the estimated cumulative incidences for the target cause are 0.12 and 0.09 in two groups. The test output is \(Q=1.44\), \(df=1\), \(p=0.230\).

The 18-month difference is \(0.12-0.09=0.03\), or 3 percentage points. For the test, the upper-tail probability is \(P(\chi^2_1\geq1.44)\approx0.230\), so the test does not reject equality at \(\alpha=0.05\). The report must not say that the test specifically tested the 18-month difference: the test used event times through 30 months. It should state both facts, for example: “The estimated 18-month cumulative incidences were 12% and 9%; the Gray test over available event times through 30 months gave \(Q=1.44\), \(df=1\), \(p=0.230\).” If a test restricted to 18 months is intended, that restriction must be prespecified and supported by the selected implementation.

Diagnostics, Alternatives, and Limits

Before finalizing a report, check that the number of groups agrees with the reported degrees of freedom, that the cause label is consistent across the protocol and results, and that estimates and test refer to the stated population and time origin. Confirm that the horizon is supported by follow-up, and that censoring reasons and the analysis cutoff are described. Sparse events, very small groups, or substantial loss to follow-up can make the large-sample approximation or the censoring assumption questionable; report these limitations rather than treating a p-value as a guarantee of validity.

The main alternative is to present group-specific cumulative incidence estimates at selected times, with confidence intervals, without an omnibus test. This can be more directly focused on a prespecified time, but it does not answer the same global comparison question and requires care if many times are examined. A cause-specific hazard comparison addresses a different quantity and should not be presented as a test of equality of cumulative incidence functions. Gray's test is useful for an overall comparison of the target cause's incidence, but it does not provide a simple effect size, adjust for covariates by itself, prove proportionality, or identify the time pattern responsible for a result. Estimates and plots help describe that pattern; additional analyses should be clearly identified as such.

Common Reporting Mistakes

Reporting only “Gray test, \(p=...\).” Name the cause, groups, time scope, statistic, degrees of freedom, and censoring approach.
Calling a competing event censoring. Distinguish competing events from right censoring due to study end or loss to follow-up.
Claiming a time-specific test from a global result. State the test's time domain separately from the horizon of the displayed cumulative incidence estimates.
Reporting a p-value without the estimates. Include group-specific cumulative incidences at clinically meaningful times and uncertainty where available; the test alone conveys neither the probability scale nor the size of a difference.
Ignoring multiplicity or design changes. Report the adjustment and any departure from the prespecified analysis, rather than silently applying a different threshold or test.

Key takeaway: A complete Gray-test report links a precisely named cause and group comparison to its follow-up and censoring rules, then gives the test specification and output alongside time-specific cumulative incidence estimates. A test result and an estimate answer related, but distinct, questions.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions below.

A study compares three groups for the cumulative incidence of renal failure. What degrees of freedom does the standard Gray test use, assuming the usual full-rank conditions?
A report gives a 12-month cumulative incidence estimate and a Gray-test p-value, but the test used follow-up through 36 months. What clarification is needed?
Two prespecified tests use a Bonferroni familywise level of 0.05. A test has \(p=0.018\). Does it meet the per-test threshold?
Why is a competing event not described as ordinary censoring when reporting the cumulative incidence analysis?

Solutions. (1) \(df=3-1=2\), because there are two independent group contrasts. (2) State that the estimate is at 12 months but the test compares the functions over its specified domain through 36 months; do not imply that the test tests only the 12-month difference. (3) Yes. The per-test threshold is \(0.05/2=0.025\), and \(0.018<0.025\). (4) A competing event is a distinct event that prevents the target event from occurring first; it contributes to the competing-risks outcome and is not merely incomplete observation from censoring.