Why Planning Assumptions Need to Be Tested
A sample-size calculation is conditional on its inputs. In a competing-risks study, the assumed target-event cumulative incidence is especially important: it determines how many participants are expected to experience the event of interest by the analysis horizon. But that incidence also depends on the competing-event process and follow-up. If the assumptions are uncertain, a single enrollment estimate can imply more certainty than the design information supports.
Sensitivity analysis makes this dependence visible. It recalculates the planned enrollment under several plausible sets of assumptions, while holding other design choices fixed when appropriate. The aim is not to select whichever scenario gives a convenient number. It is to find out how much the design changes when uncertain inputs change, and which inputs deserve the most attention before the study begins.
The Approximation Being Varied
For a transparent first-pass calculation, treat experiencing the target event by a fixed horizon as a binary outcome, with probabilities \(p_0\) and \(p_1\) in the control and treatment groups. These are cumulative incidences by that horizon, not cause-specific hazard rates. With equal group sizes, a common normal-approximation calculation for a two-sided comparison is
Here \(n\) is the approximate number of participants per group, \(\alpha\) is the two-sided type I error rate, \(1-\beta\) is the target power, and \(z_q\) is the \(q\)-quantile of the standard normal distribution. Round required group sizes up to whole participants. The formula is a planning approximation for a difference in fixed-horizon event probabilities; it is not an exact Gray-test power calculation. The earlier material on approximate sample-size calculations and Gray-test design establishes its role as an initial planning tool.
Interpret \(p_0\) and \(p_1\) as probabilities of the target event in the presence of competing events. They should be justified for the study’s planned follow-up and event process. A competing event is not simply a censoring event: when it occurs first, it prevents the target event from occurring first. Consequently, changing the competing-event assumptions can change the target-event cumulative incidence, as well as the variability relevant to a design calculation.
The approximation is most useful when the endpoint is defined at a common horizon, the groups are independent, the specified alternative is scientifically meaningful, and the normal approximation is adequate for the anticipated sample size and event probabilities. It does not automatically account for stratification, non-proportional patterns over time, complex censoring, or every feature of Gray’s test. These features may require a more tailored calculation.
A Structured Sensitivity Workflow
State the target-event incidences, follow-up horizon, allocation, significance level, and power. Keep the event of interest and its time horizon explicit.
Vary uncertain inputs over a defensible range. Where target and competing incidences are linked, construct coherent scenarios rather than changing one probability in isolation without explanation.
Use the same approximation and decision rule for each scenario. Change one input at a time to isolate its effect, then compare combined scenarios to see how assumptions may act together.
Report group sizes and total enrollment, identify which changes produce the largest differences, and state which assumptions remain uncertain enough to threaten the design.
Worked Example: Varying the Target-Event Incidences
Worked Example: Varying the Target-Event Incidences
Suppose a hypothetical two-group study plans a two-sided test with \(\alpha=0.05\), 80% power, and equal allocation. At a fixed horizon, the planning incidences are \(p_0=0.16484\) and \(p_1=0.23608\). Use \(z_{0.975}=1.96\) and \(z_{0.80}=0.8416\).
The average incidence is \(\bar p=(0.16484+0.23608)/2=0.20046\), and the difference is \(0.23608-0.16484=0.07124\). Substituting these values gives
Thus the approximate requirement is 495 participants per group, or 990 in total. The rounding is upward because 494 participants per group would be below the unrounded requirement.
Now compare two alternative, hypothetical incidence scenarios using the same \(\alpha\), power, and equal allocation. If \(p_0=0.10\) and \(p_1=0.15\), then \(\bar p=0.125\), and
This scenario requires 686 per group after rounding, substantially more than the reference scenario. If the incidences are instead \(p_0=0.12\) and \(p_1=0.18\), the difference is 0.06 and the same calculation gives approximately 554.8 per group, or 555 per group after rounding. These comparisons show that the assumed incidence contrast matters greatly: a smaller contrast can outweigh the fact that the absolute event probabilities may be higher.
Competing Events and Follow-Up Must Form Coherent Scenarios
The target-event incidence is not freely interchangeable with the competing-event incidence. Both arise from the event process over time. For example, under constant cause-specific hazards \(\lambda_1\) for the target event and \(\lambda_2\) for one competing event, with no other causes, the target-event cumulative incidence by time \(t\) is
This expression follows because the probability of remaining free of either event through time \(u\) is \(\exp[-(\lambda_1+\lambda_2)u]\). The target-event probability accumulated over time is the integral of that event-free probability multiplied by the target cause-specific hazard:
The formula makes two influences clear. A longer horizon allows more events to accumulate, while a higher competing-event hazard can reduce the share of first events that are target events. Real designs may have nonconstant hazards, censoring, or several competing causes, so this constant-hazard expression is an illustration rather than a universal model.
Worked Example: How Follow-Up Changes the Planning Incidence
Consider a hypothetical setting with constant annual target-event hazard 0.08 in the control group and 0.06 in the treatment group. Suppose the competing-event hazard is 0.12 per year in both groups. Assume complete ascertainment through three years for this illustration. The constant-hazard formula gives
The target-incidence difference is about \(0.04139\). With equal allocation, \(\alpha=0.05\), and 80% power, the approximate per-group calculation is
The approximation is about 1,230 participants per group, or roughly 2,460 in total, before any additional enrollment allowance for loss to follow-up. This is not a claim that three years is necessarily preferable: longer observation can increase incidence, but it also changes study duration and may affect censoring or other design constraints. The numerical result depends on the stated constant-hazard model and complete ascertainment assumption.
Allocation and Total Enrollment
When more participants are assigned to one group, the total sample size and the group-specific requirements do not change in the same way. Let \(r=n_1/n_0\) be the treatment-to-control allocation ratio, so that \(n_1=rn_0\). A corresponding unequal-allocation approximation is
The weighted average \(\bar p_r\) reflects the planned group sizes under the null approximation. The formula gives the control-group size first; multiply by \(r\) to obtain the treatment-group size, then round each required group size upward. For a fixed total enrollment, unequal allocation generally provides less information about a between-group contrast than equal allocation, although practical or ethical considerations may justify it.
Worked Example: Comparing Equal and Unequal Allocation
Use the hypothetical incidences \(p_0=0.10\) and \(p_1=0.15\), with two-sided \(\alpha=0.05\) and 80% power. The equal-allocation calculation above requires 686 per group, for 1,372 participants total. Now plan 1.5 treatment participants per control participant, so \(r=1.5\).
The weighted average is \(\bar p_r=(0.10+1.5(0.15))/2.5=0.13\). Substitution in the unequal-allocation formula gives
Round the control requirement up to 579. The unrounded treatment requirement is \(1.5(578.9)\approx868.4\), which rounds up to 869. The resulting planned total is 1,448. This exceeds the equal-allocation total of 1,372, illustrating the enrollment cost of this allocation ratio under these assumptions.
Reading the Sensitivity Results
A compact table helps distinguish the reference estimate from the assumptions that drive it. The following values are hypothetical and use equal allocation, a two-sided \(\alpha=0.05\), and 80% power; group sizes are rounded up.
| Scenario | Control incidence | Treatment incidence | Approximate participants per group | Total after rounding |
|---|---|---|---|---|
| Reference | 0.10 | 0.15 | 686 | 1,372 |
| Lower incidence contrast | 0.08 | 0.11 | 1,499 | 2,998 |
| Higher incidences, wider contrast | 0.12 | 0.18 | 555 | 1,110 |
The lower-contrast scenario requires far more participants even though its incidences are not dramatically different from the reference values. A useful summary therefore names the assumptions that cause the largest enrollment changes, not merely the smallest and largest totals. If an anticipated retention fraction \(q\) is applied separately, a rough enrollment inflation is \(n_{\text{randomized}}\approx n_{\text{analyzable}}/q\); state that additional assumption explicitly and avoid treating losses as though they changed the target-event definition.
Diagnostics, Alternatives, and Limitations
Several checks help reveal when a sensitivity analysis is misleading or when the approximation is strained:
- Check coherence. Target and competing incidences should be compatible with the same follow-up horizon and event process. Incidences derived under incompatible assumptions do not make a meaningful scenario.
- Check the contrast. The approximate requirement grows rapidly as \(|p_1-p_0|\) shrinks because that difference is squared in the denominator. A very large estimate may signal a genuinely hard-to-detect contrast, not an arithmetic error.
- Check direction and scale. Recalculate the contrast, group sizes, and total separately. Confirm that the reported group sizes match the allocation ratio after rounding.
- Check the endpoint and censoring assumptions. Differential follow-up, substantial loss to follow-up, or event patterns that change markedly over time can make a fixed-horizon probability approximation inadequate for the intended Gray-test design.
- Check sensitivity to the scenario range. If plausible values produce a wide range of enrollment, a single “best” estimate hides important design uncertainty. Report that range and explain which inputs are uncertain.
A one-at-a-time analysis is useful for identifying individual drivers, but it can miss interactions: for instance, a different follow-up horizon may alter both target and competing incidences. A set of joint scenarios can show those combined changes. These deterministic comparisons are not a substitute for methods that explicitly represent uncertainty in the inputs. Nor do they establish that a design will attain its nominal power; the approximation can differ from the operating characteristics of the planned test.
The main alternative is to use a more detailed design calculation that represents the event and censoring process and the planned test more directly. The appropriate level of detail depends on how well the simple approximation matches the study design. Sensitivity analysis remains useful either way: it identifies which assumptions need better evidence and communicates how conclusions about enrollment depend on them.
Common Pitfall
Correction: That may be a useful deliberately isolated stress test, but it is not automatically a coherent event-process scenario. The competing event can change the probability of experiencing the target event first. State whether target incidence is held fixed by design or recalculated from a specified event model, and explain the assumption that makes the scenario plausible.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- In a fixed-horizon design, what do \(p_0\) and \(p_1\) represent, and why are they not cause-specific hazards?
- Under equal allocation, suppose a scenario changes from incidences 0.10 and 0.15 to 0.08 and 0.11. Does the smaller contrast imply a larger or smaller approximate sample size?
- For an allocation ratio \(r=1.5\), which group size is calculated first by the unequal-allocation formula, and how is the other obtained?
- Why should a change in follow-up often be accompanied by a reassessment of both target- and competing-event assumptions?
Solutions. (1) They are the probabilities of the target event by the specified horizon in the control and treatment groups, with competing events present; hazards are instantaneous rates, not cumulative probabilities. (2) Larger: the contrast falls from 0.05 to 0.03, and the squared contrast is in the denominator of the approximation. (3) Calculate \(n_0\), the control size, then set \(n_1=1.5n_0\); round both required sizes up. (4) More time at risk changes how many target and competing events can accrue, so the target-event cumulative incidence may change along with the competing-event incidence.