Competing-Event Incidence and Test Power

Why Competing Events Matter for Power

A study can keep its sample size, follow-up period, and target-cause hazard contrast unchanged, yet have fewer observed target events if competing events become more common. Since the Gray test compares target-cause cumulative incidence functions (CIFs), this change can affect both the size of the observed CIF contrast and the information available to detect it. Planning power therefore requires attention to competing-event incidence, not just to the target-cause rates.

This tutorial uses a simple constant-hazard model to show the mechanism and quantify it. The calculations are design illustrations, not a substitute for calculating Gray-test power under the full event-time and censoring distributions. In particular, expected target-event counts alone do not determine Gray-test power.

Definition: For group \(g\), let \(\lambda_{1g}\) and \(\lambda_{2g}\) be the constant cause-specific hazards per unit time for the target event (cause 1) and a competing event (cause 2). With no other event types, the target-cause CIF by time \(t\) is \(F_{1g}(t)=P(T\leq t,J=1\mid g)\), where \(T\) is time to the first event and \(J\) identifies its cause. The target-event contrast may be expressed as a difference in cause-specific hazards, a hazard ratio, or a difference in CIFs; these are distinct quantities.

How a Competing Hazard Changes Target Incidence

Under the constant-hazard model, the probability of remaining free of either event through time \(t\) is \(S_g(t)=\exp[-(\lambda_{1g}+\lambda_{2g})t]\). A target event at time \(u\) requires survival without either event until \(u\), followed by a target event in the next small interval. Thus its CIF is the integrated target-event hazard among people still event-free:

F_{1g}(\tau) =\int_0^\tau S_g(u)\lambda_{1g}\,du =\frac{\lambda_{1g}}{\lambda_{1g}+\lambda_{2g}} \left\{1-\exp[-(\lambda_{1g}+\lambda_{2g})\tau]\right\}.

Holding \(\lambda_{1g}\) fixed, increasing \(\lambda_{2g}\) lowers \(F_{1g}(\tau)\). The competing event removes people from the possibility of subsequently experiencing the target event. This does not mean the target cause-specific hazard has fallen: it is unchanged in this model. It means fewer people remain at risk of experiencing the target event before the horizon.

The same increase can also shrink the CIF difference between groups even when the target cause-specific hazard contrast is unchanged. A fixed hazard difference does not imply a fixed CIF difference, because both CIFs depend on the competing hazard and on time. This distinction is central to interpreting a power scenario.

Worked Example: A Shared Increase in the Competing Hazard

Suppose two hypothetical groups have constant target cause-specific hazards of \(0.07\) and \(0.10\) per year. The target hazard difference is \(0.03\) per year, and the hazard ratio is \(0.70\). Assume the competing-event hazard is the same in both groups, follow-up is two years, and there is no censoring before two years. Compare a competing hazard of \(0.10\) per year with one of \(0.80\) per year.

For competing hazard \(0.10\), the target CIFs at two years are

F_{1A}(2)=\frac{0.07}{0.17}\{1-\exp(-0.34)\} =0.11868, \qquad F_{1B}(2)=\frac{0.10}{0.20}\{1-\exp(-0.40)\} =0.16484.

The CIF difference, group B minus group A, is \(0.16484-0.11868=0.04616\), or about 4.616 percentage points by two years.

For competing hazard \(0.80\), the corresponding calculations are

F_{1A}(2)=\frac{0.07}{0.87}\{1-\exp(-1.74)\} =0.06634, \qquad F_{1B}(2)=\frac{0.10}{0.90}\{1-\exp(-1.80)\} =0.09274.

The CIF difference is \(0.09274-0.06634=0.02640\) using the displayed rounded values; using unrounded CIFs it is approximately \(0.02641\). Across the scenarios, the target cause-specific hazards, their difference, and their ratio are unchanged. The higher competing hazard lowers target incidence in both groups and reduces the target CIF contrast.

From Incidence to Expected Target-Event Counts

If \(n_g\) participants enter group \(g\), and each has the same target-event probability \(F_{1g}(\tau)\) by the specified horizon, the expected number of target events by that horizon is \(n_gF_{1g}(\tau)\). This expectation is useful for planning and for checking whether an assumed event yield is plausible. It is not a power calculation: a Gray test uses event-time information and its variance, not only the final number of target events.

Worked Example: Expected Counts in Equal-Sized Groups

Use the two hazard scenarios above and suppose there are 2,000 participants in each group. Under the lower competing hazard, the expected target-event counts by two years are

2{,}000(0.11868281)\approx237.4 \quad\text{and}\quad 2{,}000(0.16483998)\approx329.7.

Under the higher competing hazard, the expected counts are

2{,}000(0.06633744)\approx132.7 \quad\text{and}\quad 2{,}000(0.09274457)\approx185.5.

For example, \(2{,}000(0.1648)=329.6\) when the rounded CIF is used, whereas the unrounded value gives \(2{,}000(0.16483998)\approx329.7\). The displayed count of 329.7 is therefore based on the unrounded incidence. The sum of expected target events across the two groups falls from about 567.0 to about 318.2 when the competing hazard rises. These are expected counts, not guaranteed counts in a realized study.

What These Counts Say—and Do Not Say—About Gray-Test Power

Power is the probability that a specified test rejects its null hypothesis under a specified alternative and design. For a Gray test, that probability depends on the alternative CIF curves over follow-up, the test’s weight function, group allocation, censoring, and the variance of the Gray score. It is not determined by the number of target events alone. The earlier material on Gray’s score, variance estimation, and censoring conditions supplies the test-specific framework; the design implication here is that competing-event assumptions help determine the CIFs and event-time pattern that enter that framework.

The expected-count comparison is still informative. In many designs, fewer target events mean less information for detecting a target-CIF difference, all else being equal. In the example, however, “all else being equal” cannot be taken literally: the CIF difference also shrank, from about 0.04616 to about 0.02641. Both the available target-event information and the alternative being tested have changed on the CIF scale, despite unchanged target cause-specific hazards. A simple rule based only on the ratio of total target-event counts would miss this change in contrast and the Gray test’s event-time weighting.

A defensible power calculation therefore specifies the group-specific CIF curves over time, not merely one target hazard or one final event count. It also specifies the competing-event process and censoring assumptions. Under a fully specified model, power can be obtained using an appropriate Gray-test approximation or by simulating the complete event and censoring process and applying the intended Gray test to each simulated data set. The approximation should match the planned test, including its weight choice; simulation should reproduce the planned allocation, follow-up, and censoring.

Worked Example: A Count-Based Diagnostic, Not a Gray Power Estimate

In the preceding example, the expected total target-event count across both groups is about 567.0 under the lower competing hazard and 318.2 under the higher one. Their ratio is

\frac{318.2}{567.0}\approx0.561.

If, purely as a rough diagnostic, one assumes that test information is proportional to the total target-event count and that the standardized alternative is otherwise unchanged, the corresponding standard-error ratio would be approximately

\sqrt{\frac{567.0}{318.2}}\approx1.335.

Under those restrictive assumptions, the higher-competing-hazard scenario would have about 33.5% larger standard error in this simple count-proportional approximation. But the assumption that the standardized alternative is unchanged is not met in the earlier calculations: the CIF difference fell as well. Furthermore, Gray-test information is not generally exactly proportional to the total target-event count. The calculation is therefore a warning that event yield may deteriorate, not a numerical estimate of Gray-test power. A Gray-test power calculation must use the specified CIF alternatives and the test’s variance and weighting structure.

Assumptions, Diagnostics, and Limits

The constant-hazard formula assumes that each cause-specific hazard is constant over the modeled follow-up, that the two event types are mutually exclusive as first events, and that the event process is adequately represented by these two causes. The count calculation additionally treats \(F_{1g}(\tau)\) as the target-event probability for each enrolled participant by horizon \(\tau\). If censoring occurs before \(\tau\), the expected observed count is not simply \(n_gF_{1g}(\tau)\) unless the censoring pattern is incorporated appropriately.

For a Gray-test power analysis, the planned censoring assumptions must be compatible with the test’s required conditions, and the hypothesized CIF curves should be examined over the full follow-up period. A single CIF difference at the horizon may conceal curves that cross or differ mainly early or late; such shapes can interact with the test’s weight function. Check that assumed competing-event rates are plausible, that the implied target CIFs are between zero and one, and that calculated event counts do not exceed group sizes. Compare predicted event counts with available historical or pilot information, while allowing for uncertainty in those inputs.

The constant-hazard model is a transparent way to show the mechanism, not a universal description of clinical event times. Hazards may vary over time, and competing hazards may differ between groups. A competing event can also be clinically important in its own right. A lower target CIF caused by more competing events should not be described as evidence that the target cause-specific hazard is lower or that the target outcome has improved. The target cause-specific rate and the probability of experiencing the target event by a horizon answer different questions.

Key takeaway: Holding the target cause-specific hazard contrast fixed does not hold the target CIF contrast fixed. More competing events can lower expected target-event counts and alter the alternative Gray test is meant to detect. Counts help diagnose event yield, but Gray-test power requires the full alternative CIF curves and a test-appropriate variance calculation or simulation.

Common Mistakes

Using the target hazard as if it were the target-event probability. A cause-specific hazard is an instantaneous rate among people still event-free. Calculate the CIF by accounting for survival from every event cause.
Assuming the CIF contrast stays fixed when the target hazard contrast stays fixed. The competing hazard enters both groups’ CIFs, so recompute the CIFs under each competing-event scenario.
Equating expected target-event counts with Gray-test power. Counts indicate event yield, but the Gray score’s variance, time pattern, weighting, censoring, and CIF contrast also matter.
Calling an expected count an observed count. \(n_gF_{1g}(\tau)\) is an expectation under the model; the realized number varies from study to study.

Check Your Understanding

Answer each question using the distinction between cause-specific hazards, CIFs, event counts, and test power.

If the target cause-specific hazards are unchanged but the competing hazard increases in both groups, what happens to each group’s target CIF under the constant-hazard model?
With 1,000 participants and a target CIF of 0.12 by the study horizon, what is the expected number of target events?
Why does a fixed target cause-specific hazard difference not imply a fixed target CIF difference?
Is the ratio of expected target-event counts enough to calculate the ratio of Gray-test powers? Give one reason.

Solutions. (1) Each target CIF decreases, because more participants experience the competing event before they can experience the target event. (2) The expected count is \(1{,}000(0.12)=120\) target events. (3) Each CIF depends on both the target and competing hazards, as well as the time horizon; changing the competing hazard changes the CIF even when the target hazard difference is fixed. (4) No. Gray-test power also depends on the CIF curves over time, the score variance and weight function, and censoring, among other design features.