From a Study Question to a Design Alternative
Before planning a Gray test, the investigator needs a precise alternative: what target-cause event probabilities are expected in each group, and how might those probabilities develop over the planned follow-up? “The treatment reduces the event” is not yet a design alternative. It does not identify the event of interest, the comparison groups, the time horizon, or the size and timing of the difference. Those choices determine what curves the test is intended to distinguish.
The aim here is to specify the group-specific cumulative incidence functions (CIFs) that represent a scientifically meaningful alternative. This is not a power or sample-size calculation. It is the step that makes such planning inputs interpretable: the curves should answer the study question, respect the competing-risk structure, and cover the prespecified follow-up period.
The group label may denote randomized assignment or a group defined by an observational study. The probability is an event risk within that group; it is not automatically a causal effect. A causal interpretation requires an appropriate design and assumptions. In either setting, the target event and competing events must be defined as mutually exclusive first events for the analysis.
The alternative for a two-group Gray test is typically that the groups’ target-cause CIFs differ somewhere over the tested period. The design question is more specific: which differences are plausible and important enough to plan around? Define the target cause, the comparison groups, the endpoint of follow-up \(\tau\), and the clinically meaningful pattern of risk differences. A prespecified horizon prevents the design from depending on an unplanned choice of when to compare groups.
Coherent Curves for a Competing-Risk Process
A CIF is an absolute probability, so any proposed curve must obey basic probability constraints. It starts at zero, is nondecreasing, and cannot exceed one. With multiple causes, the sum of all cause-specific CIFs at any time cannot exceed one. The remaining probability is the chance of having had no event by that time. These constraints apply to each group separately.
One way to construct coherent curves is to specify cause-specific hazards. For a first-event process, let \(\lambda_j(t\mid g)\) be the cause-specific hazard for cause \(j\) in group \(g\): the instantaneous event rate for cause \(j\) among people in that group who are still event-free just before time \(t\). If the hazards are specified for all causes, the event-free survival probability is \(S(t\mid g)=\exp[-\int_0^t\sum_j\lambda_j(u\mid g)\,du]\). The target-cause CIF follows by accumulating the target event rate among those who remain event-free.
This relationship provides a check on the logic of a proposed alternative: a target event can occur only among people who have not already had any event. In particular, increasing a competing-event hazard may reduce the number of people still at risk of the target event, and therefore change the target-cause CIF even if the target-cause hazard is unchanged.
For constant cause-specific hazards over an interval, write \(\Lambda=\sum_j\lambda_j\). Over an interval of length \(d\), the probability of any event is \(1-\exp(-\Lambda d)\), and the share attributed to cause \(k\) is \(\lambda_k/\Lambda\). Therefore the target-cause CIF increment over that interval, conditional on being event-free at its start, is \(\{\lambda_k/\Lambda\}\{1-\exp(-\Lambda d)\}\). Multiplying this increment by the probability of being event-free at the interval’s start gives the unconditional increment.
To justify the formula, under constant hazards the probability of remaining event-free for \(u\) time units is \(\exp(-\Lambda u)\). The target-event probability accumulated over a short interval of length \(du\) is therefore approximately \(\exp(-\Lambda u)\lambda_k\,du\). Integrating from zero to \(d\) gives \(\int_0^d\exp(-\Lambda u)\lambda_k\,du =(\lambda_k/\Lambda)\{1-\exp(-\Lambda d)\}\), as claimed. The factor for being event-free at the interval’s start then carries that conditional increment forward to the full study population.
Worked Design Specifications
Worked Example: A Constant Difference in Target-Cause Risk
Suppose a hypothetical randomized study compares two groups over 24 months. The target event is disease recurrence, and death before recurrence is the competing event. For illustration, assume constant cause-specific hazards per month: in group A, recurrence \(0.040\) and competing death \(0.020\); in group B, recurrence \(0.025\) and competing death \(0.020\). These are assumed planning values, not estimates from a real study. The model assumes these rates are constant during follow-up and that recurrence and death are the only first events.
In group A, the total hazard is \(0.040+0.020=0.060\), and the recurrence share is \(0.040/0.060=0.6667\). At 12 months, \(F_k(12\mid A)=0.6667\{1-\exp(-0.060\times12)\} =0.6667(1-\exp(-0.72))\approx0.3422\). At 24 months, \(F_k(24\mid A)=0.6667\{1-\exp(-1.44)\}\approx0.5087\).
In group B, the total hazard is \(0.025+0.020=0.045\), and the recurrence share is \(0.025/0.045=0.5556\). Thus \(F_k(12\mid B)=0.5556\{1-\exp(-0.54)\}\approx0.2318\), and \(F_k(24\mid B)=0.5556\{1-\exp(-1.08)\}\approx0.3669\). The hypothetical 24-month target-risk difference, group A minus group B, is \(0.5087-0.3669=0.1418\), or about 14.2 percentage points. This is a concrete alternative over the full period, not merely a statement that one hazard is smaller.
Worked Example: Making the Competing Event Part of the Story
Consider a hypothetical 12-month comparison of two groups, again with a target event and a competing event. Group A has constant monthly hazards \(0.050\) for the target event and \(0.010\) for the competing event. Group B has a lower target-event hazard, \(0.030\), but a higher competing-event hazard, \(0.040\). Assume the rates are constant and these are the only first events. This specification could represent a setting in which the groups differ in both target-event and competing-event processes.
For group A, the total hazard is \(0.060\), so \(F_k(12\mid A)=(0.050/0.060)\{1-\exp(-0.060\times12)\} =0.8333(1-\exp(-0.72))\approx0.4277\). For group B, the total hazard is \(0.070\), so \(F_k(12\mid B)=(0.030/0.070)\{1-\exp(-0.070\times12)\} =0.4286(1-\exp(-0.84))\approx0.2436\). The target-cause risk is about 18.4 percentage points lower in group B by 12 months. That difference reflects both the target-event rate and the larger competing-event rate. It should not be described as the effect of the target-event hazard alone.
The specified rates imply a coherent alternative because both cause-specific hazards are nonnegative and are defined for the entire period. The example also shows why a design question should name both the target cause and its important competing events: a change in competing-event incidence can affect the target-cause probability that Gray’s test compares.
Worked Example: Describing a Time-Varying Alternative
Suppose investigators consider whether a hypothetical treatment could have a strong early effect on a target event that weakens later. In the control group, assume constant monthly hazards of \(0.040\) for the target event and \(0.010\) for the competing event over 12 months. In the treatment group, assume target-event hazards of \(0.020\) during months 0–6 and \(0.060\) during months 6–12, with a competing-event hazard of \(0.010\) throughout. Treat each interval’s rates as constant and suppose the two stated causes exhaust first events.
At six months, treatment-group event-free survival is \(S(6\mid\text{treatment})=\exp[-(0.020+0.010)\times6]=\exp(-0.18)\approx0.8353\). The treatment-group target CIF at six months is \((0.020/0.030)\{1-\exp(-0.18)\}\approx0.1098\). For control, it is \((0.040/0.050)\{1-\exp(-0.050\times6)\} =0.8000(1-\exp(-0.30))\approx0.2073\).
From month 6 to month 12, the treatment-group total hazard is \(0.060+0.010=0.070\). Its conditional target-event probability in that interval is \((0.060/0.070)\{1-\exp(-0.070\times6)\} =0.8571(1-\exp(-0.42))\approx0.2940\). The unconditional increment is \(0.8353\times0.2940\approx0.2456\), so the 12-month treatment CIF is approximately \(0.1098+0.2456=0.3554\). Control’s 12-month CIF is \(0.8000\{1-\exp(-0.050\times12)\}\approx0.3610\).
This specification has a pronounced early separation but similar 12-month risks. An endpoint-only description would conceal that pattern. The curves implied by the interval-specific rates are an appropriate way to express this design alternative when the scientific question concerns timing as well as risk by the end of follow-up.
Choosing and Checking the Alternative
A useful alternative is a translation of the study question, not an arbitrary mathematical contrast. State which cause is primary, which groups are compared, and how long follow-up will be considered. Then describe the expected group-specific target-cause risks across that period. Risk differences at clinically interpretable times can help communicate the curves, but a few selected values do not by themselves specify the trajectory between those times.
If the intended alternative is represented directly by curve values, check that each group’s values start at zero and do not decrease over time. Check that the values stay between zero and one and that all cause-specific CIFs together remain at most one at every time. If instead curves are generated from cause-specific hazards, state the hazards for all relevant causes and the intervals over which they apply. The assumed rates should be plausible for the intended population and follow-up; a smooth-looking curve is not a substitute for defensible assumptions.
The planned follow-up horizon matters. A curve specified only through a short horizon cannot support claims about longer-term risk. Conversely, specifying differences far beyond planned follow-up does not make them relevant to the planned test. The relevant alternative covers the prespecified analysis period and reflects plausible event experience during that period.
The earlier material on Crossing Cumulative Incidence Curves explains why curves may separate, converge, or cross. For design, the practical implication is to describe the anticipated shape rather than assuming that a single endpoint contrast captures it. Different shapes can represent different scientific questions even if their final risks are similar.
Comparison, Diagnostics, and Limitations
| Design description | What it conveys | What it can miss |
|---|---|---|
| Target-cause CIFs by group over time | Absolute target-event probabilities and their trajectories in the presence of competing events | Requires defensible assumptions over the full follow-up period |
| Cause-specific hazard contrast | Relative event rates among people still event-free | Does not alone determine the absolute target-cause risk difference |
| Risk difference at one endpoint | Absolute difference at a clinically chosen time | Does not describe earlier separation or differences at other times |
The target-cause CIF is the natural object when the question concerns the probability of experiencing a particular event by a given time in the presence of competing events. A cause-specific hazard contrast can help construct a model, but should not be substituted for the absolute-risk curves when stating the alternative. The endpoint risk difference is easy to communicate, but it cannot summarize all possible time patterns. Gray’s test assesses group differences in the CIFs over time; its result does not identify which group has a clinically important advantage at every time point.
Diagnostics for a proposed alternative are primarily checks of definition and coherence. Verify that every cause is assigned consistently, the follow-up horizon matches the planned question, and no curve declines or exceeds one. For a direct multi-cause specification, inspect the sum of CIFs at each time. For hazard-based construction, verify nonnegative rates, complete coverage of the horizon, and that the resulting event-free probability and CIFs agree with the stated event process.
A common mistake is to say, “The treatment halves the target-event hazard, so the target-event risk is also halved.” That conclusion does not generally follow. The CIF depends on the target-event hazard, the competing-event hazards, and the time available for events to occur. The correction is to specify or derive the group-specific CIFs under a coherent competing-risk model, then interpret their absolute probabilities at relevant times.
These design curves remain assumptions, not guarantees about future observations. A single set of curves can be misleading if rates vary over time or differ across the study population. They may also be uncertain when planning information is sparse. The purpose of specifying them is to make the alternative explicit and reviewable, not to claim that the true curves are known exactly.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions below.
- In a first-event study, what probability does \(F_k(t\mid g)\) represent?
- A proposed target-cause CIF has values \(0.10\), \(0.08\), and \(0.20\) at months 3, 6, and 12. Identify the problem.
- Why is a target-cause hazard ratio alone insufficient to specify the target-cause CIF alternative?
- Two groups have equal target-cause CIFs at 12 months but different curves at month 6. Why might an endpoint-only description be inadequate?
- For a direct specification of multiple cause-specific CIFs, name two probability-coherence checks to make at every time.
Solutions. (1) It is the probability that a person in group \(g\) experiences cause \(k\) by time \(t\) as the first event, before a competing event. (2) The proposed curve decreases from \(0.10\) to \(0.08\), but a cumulative incidence function cannot decrease. (3) The CIF also depends on competing-event hazards and on how the hazards develop over time, so the hazard ratio does not uniquely determine the absolute risk curves. (4) The endpoint risks conceal the earlier difference in event probabilities; a design alternative should describe the time pattern relevant to follow-up. (5) Each CIF must be nondecreasing and lie between zero and one, and the sum across causes must not exceed one.