Why Crossing Curves Need Careful Interpretation
Cumulative incidence curves summarize the probability of experiencing a particular event by each time, in the presence of competing events. When two groups’ curves cross, the group with the higher estimated cumulative incidence may change over follow-up. A single statement such as “Group A has higher incidence” can then conceal important timing differences.
Crossing also matters for testing. Gray’s test forms a global comparison over event times. Its score accumulates signed contributions: contributions favoring one group at some times can be offset by contributions favoring the other group later. The result can be a small test statistic even when the curves differ visibly over parts of follow-up. This is not a defect unique to crossing curves; it reflects that a global test compresses a time-varying pattern into a single summary.
A crossing is a feature of the curves, not itself a statistical test or a formal conclusion. Estimated curves can cross because the underlying functions cross, because of sampling variation, or both. Conversely, curves need not visibly cross for a test to have low power to detect a difference. Interpretation therefore uses the plot, the test, and the design together.
How Opposing Contributions Can Offset
Gray’s test compares groups for a specified event type while accounting for competing events. Its score is built from time-specific differences between observed event counts and their expected counts under the null, using the test’s subdistribution risk-set construction. For a two-group comparison, write the weighted score schematically as
Here \(dU(t_j)\) is the score increment at event time \(t_j\), \(w(t_j)\) is the chosen time weight, and \(V_w\) is the estimated variance of the weighted score. Under the null and usual regularity conditions, \(Q_w\) is compared with a chi-square distribution with one degree of freedom. The sign of \(U_w\) indicates the direction of the score contrast under the chosen group coding; \(Q_w\) measures its standardized magnitude.
The score is not simply the difference between the two cumulative incidence estimates at the end of follow-up, nor is it the unweighted area between the curves. It uses event-time information, a risk-set construction, and a variance estimate. A signed area can help describe a pattern, but it is not a substitute for Gray’s statistic.
A useful way to understand potential cancellation is to inspect score increments by time. If earlier increments favor Group A and later increments favor Group B, their weighted sum may be close to zero. A weight changes how much each period contributes to the numerator, and also changes the variance used for standardization. Thus weighting may reduce cancellation for a particular alternative, but it does not guarantee a more powerful or more interpretable test.
Worked Example: Seeing a Crossing in Estimated Curves
Hypothetical data. Suppose two groups are followed for a target event and a competing event, with independent participants and complete event-type recording. The table gives hypothetical estimated cumulative incidence values for the target event.
| Time | Group A | Group B | Difference, A minus B |
|---|---|---|---|
| 3 years | 0.04 | 0.10 | -0.06 |
| 6 years | 0.16 | 0.22 | -0.06 |
| 9 years | 0.30 | 0.27 | 0.03 |
| 12 years | 0.38 | 0.35 | 0.03 |
At 6 years, Group A’s estimated cumulative incidence is \(0.16\), or 16%, compared with 22% in Group B: the difference is \(-0.06\), or \(-6\) percentage points. At 9 years, the difference is \(0.30-0.27=0.03\), or 3 percentage points. The ordering has reversed between years 6 and 9, so the estimated curves cross in that interval.
The difference at 12 years is 3 percentage points, but it does not summarize the whole pattern: Group A was lower at the earlier listed times. Nor do these values alone establish that the underlying curves truly cross. That requires uncertainty information and a suitable inferential analysis. The usual Gray test provides a global test of equality, not a dedicated test that identifies the crossing time.
Weights Change the Time Emphasis
An unweighted score gives the score increments their standard contribution under the selected Gray test. A weighted test multiplies increments by a function of time. The weight should be chosen for a defensible reason and specified before examining which choice gives the smallest p-value. The earlier material on weight functions covers common forms; here the focus is how they behave when contributions change direction.
One common family uses a power of the remaining target-event incidence, such as \(w(t)=[1-\hat F_k(t)]^\rho\), with the precise pooled or null-based estimate determined by the test implementation. With \(\rho=0\), the weight is 1. For \(\rho>0\), the weight generally decreases as the target-event cumulative incidence increases, giving relatively more weight to earlier times. A weight that increases over time could instead emphasize later contributions. The weight’s values and its effect on the variance both matter.
Worked Example: A Weight Changes the Net Score
Hypothetical score contributions. Suppose a Gray analysis produces two grouped score contributions for the target event: an early contribution of \(-2.4\) score units and a later contribution of \(+2.4\) score units. These hypothetical increments have opposite signs, corresponding to a shift in which group’s event experience is relatively higher over time.
With unit weights, the combined score is
The cancellation is exact in this simplified calculation. Now suppose a prespecified weight assigns 1.00 to the early period and 0.35 to the later period. The weighted score is
The weighted numerator no longer cancels, because the later contribution has been downweighted. This calculation demonstrates the mechanism; it is not by itself a test result. A valid test also needs the variance \(V_w\) calculated for that same weighting scheme. For example, if the corresponding hypothetical variance were \(V_w=4.00\), then \(Q_w=(-1.56)^2/4.00=0.6084\). One must use the variance from the actual weighted analysis, not carry over the unweighted variance.
This example does not show that the chosen weight is preferable. It shows that weights alter the balance between periods. A different pattern, or a weight favoring later times, could yield a different score and variance.
Worked Inference and Its Limits
For a prespecified weighted test, the usual large-sample reference is a chi-square distribution under the null of equal target-event cumulative incidence functions. The analysis requires independent participants, correctly classified event types and times, and censoring that is independent of the event process within the comparison groups (or independent conditional on variables properly accounted for in the analysis). The test also needs adequate information over follow-up for its asymptotic approximation to be credible. A weight must be defined over the relevant times and used consistently in both the score and variance.
If censoring differs substantially between groups or depends on prognosis in an unaccounted-for way, the estimated curves and test may be misleading. Sparse risk sets late in follow-up also make late-time patterns unstable. The study design, event counts, censoring pattern, and estimation diagnostics should therefore be checked before assigning substantive meaning to a crossing or a weighted result.
Worked Example: Interpreting a Hypothetical Gray Test
Hypothetical analysis. In a two-group study, suppose event-time plots show a crossing, and a prespecified weighted Gray analysis reports \(U_w=-1.56\) and \(V_w=4.00\). Assume independent participants, correct event classification, independent censoring within groups, and enough observed events for the large-sample approximation.
The statistic is
For one degree of freedom, the upper-tail p-value is approximately \(0.435\). At a 0.05 significance level, this result does not provide sufficient evidence to reject equality of the target-event cumulative incidence functions over follow-up. It does not establish that the functions are equal, and it does not rule out meaningful differences during particular periods.
The negative score reflects the direction of the weighted contrast under the analysis’s group coding, but the squared statistic and its p-value do not identify when the curves differ. The curve plot, time-specific estimates with uncertainty, and the prespecified weight are needed to describe the pattern. The p-value is not the probability that the null hypothesis is true.
Diagnostics, Comparisons, and Common Pitfalls
When curves appear to cross, first check whether the pattern is supported by enough observations. Examine numbers at risk or their subdistribution-risk-set analogue, event counts by type, censoring by group, and uncertainty around the cumulative incidence estimates. A late crossing based on few participants can be visually striking but highly variable. Check that the target event was defined consistently and that competing events were not treated as ordinary censoring when estimating cumulative incidence.
Gray’s test and the log-rank test answer different questions. Gray’s test assesses equality of target-event cumulative incidence functions in the competing-risks setting. The log-rank test is directed at equality of cause-specific hazards among individuals still free of any event. If competing-event incidence differs between groups, cause-specific hazard comparisons and cumulative-incidence comparisons can lead to different conclusions. The choice should follow the scientific estimand, not whichever test yields a smaller p-value.
A common mistake is to see a crossing and conclude that Gray’s test “must” be nonsignificant because positive and negative differences cancel. The correction is that crossing makes cancellation possible, not inevitable. The score depends on event-time contributions, risk sets, weighting, and variance. Another mistake is to try several weights after seeing the curves and report only the most favorable result. That practice makes the nominal p-value misleading unless the multiplicity and selection are properly addressed. Prefer a prespecified weight, and treat additional weighted analyses as exploratory or account for the selection.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions below.
- At 4 years the estimated target-event cumulative incidence is 12% in Group A and 8% in Group B; at 8 years it is 18% in Group A and 23% in Group B. What feature do the estimates suggest, and what can they not establish by themselves?
- Two hypothetical score increments are \(+1.5\) early and \(-1.0\) late. Calculate their combined score with unit weights and with weights 1.00 early and 0.40 late.
- A weighted Gray test has \(U_w=2.0\) and \(V_w=4.0\). Calculate \(Q_w\). If its one-degree-of-freedom p-value is 0.317, what is the correct conclusion at \(\alpha=0.05\)?
- Why is the unweighted signed area between two estimated cumulative incidence curves not Gray’s test statistic?
- Why should an analyst avoid choosing a time weight solely because it produces the smallest observed p-value?
Solutions. (1) The ordering reverses between the listed times: A is 4 percentage points higher at year 4 and 5 percentage points lower at year 8. This suggests a crossing between those times, but the estimates alone do not establish a crossing in the underlying functions or quantify its uncertainty. (2) With unit weights, \(1.5-1.0=0.5\). With the stated weights, \(1.00(1.5)+0.40(-1.0)=1.5-0.4=1.1\). (3) \(Q_w=2.0^2/4.0=1.00\). Since \(0.317>0.05\), there is insufficient evidence to reject the equality null; this does not prove equality. (4) Gray’s score uses event-time contributions from its subdistribution risk-set construction and its variance, not simply the geometric area between plotted estimates. (5) Selecting a weight after seeing results uses the data to choose the test and can inflate the chance of a small p-value; prespecify the weight or account for selection and multiplicity.