Why Claim Structure Comes Before Testing
A trial may measure several outcomes, but the clinical conclusion depends on how those outcomes are meant to work together. A claim that treatment improves both survival and function is different from two separate claims, one about survival and one about function. Likewise, a claim about a secondary outcome may only be intended after a primary outcome succeeds. These distinctions determine which hypothesis rejections support which conclusions.
The earlier material on familywise error across endpoints and interim looks establishes why repeated testing and multiple endpoint claims require error control. Here the first task is more basic: specify the claims and the rejection pattern they require. The claim structure does not itself provide a multiplicity procedure or determine how alpha is allocated.
The boundary \(0\) is only an illustration. In an actual protocol, the null may use a clinically justified margin, and endpoint effects may be differences in means, risk differences, log hazard ratios, or another suitable scale. State the effect scale, favorable direction, and margin before translating a claim into hypotheses. A statistically favorable estimate is not by itself evidence that a prespecified null has been rejected.
Co-Primary Endpoints: Every Required Component Must Succeed
With co-primary endpoints, the planned clinical conclusion requires evidence of benefit on every designated endpoint. For two endpoints, a claim of benefit on both is supported only when both endpoint null hypotheses are rejected. A small p-value on one endpoint cannot compensate for a failure on the other.
This structure has an important error-control implication. Suppose each endpoint test, including its interim looks, is calibrated so that its probability of rejecting its own null when that null is true is at most \(\alpha\). Under the joint null, at least one component null is true. Rejecting the joint null requires rejecting that true component as well. Thus the probability of a false joint claim is no greater than the probability of rejecting that component null, and hence is at most \(\alpha\). This is the intersection-union logic for a co-primary claim. It does not mean that every possible family of endpoint claims can be tested at level \(\alpha\) without adjustment.
Worked Example: A Co-Primary Benefit Claim
In a hypothetical trial, the protocol defines positive treatment effects as beneficial for both a symptom score and a physical-function score. The clinical claim requires superiority on both endpoints. The protocol’s group sequential tests have been calibrated so that each endpoint test has one-sided type I error \(0.025\) across its interim and final looks. At the analysis, the sequentially adjusted p-values are \(0.018\) for symptoms and \(0.041\) for function.
For each endpoint, the hypotheses are \(H_{0j}:\theta_j\leq 0\) versus \(H_{aj}:\theta_j>0\), with effects measured in the endpoint’s prespecified units. Compare each sequentially adjusted p-value with \(0.025\). For symptoms, \(0.018<0.025\), so reject the symptom null. For function, \(0.041>0.025\), so do not reject the function null.
The required pattern is rejection of both nulls, but only one was rejected. Therefore, the trial does not establish the co-primary claim of benefit on both symptoms and function. It may report the symptom result as an endpoint finding under the stated testing plan, but that result alone does not satisfy the joint clinical success criterion.
Hierarchical Endpoints: Later Claims Require Earlier Success
A hierarchy orders endpoint claims by priority. A common structure has a primary endpoint first, followed by secondary endpoints. A later claim is available only if the required earlier claims have succeeded. This is a logical gate: reaching a favorable result on a later endpoint does not bypass a failed earlier endpoint.
At the level of claims, the null for the claim at position \(j\) is that at least one of the hypotheses \(H_{01},\ldots,H_{0j}\) is true. Thus a false claim at position \(j\) requires rejection of at least one true null among positions \(1,\ldots,j\). The true null that makes the claim false need not be the null at position \(j\); it can be an earlier null. This distinction matters when interpreting the error event.
A fixed-sequence testing plan illustrates the gatekeeping logic: test in order, and proceed to the next hypothesis only after rejecting the current one. With each test calibrated to level \(\alpha\) over its interim looks, the first true null in the sequence cannot be reached unless all preceding false nulls have been rejected; the procedure stops if that first true null is not rejected. This describes the logic of a fixed sequence, not every possible hierarchical multiplicity procedure.
Worked Example: A Fixed Hierarchy With a Failed Gate
Consider a hypothetical trial with three ordered claims: (1) improved time to clinical recovery, (2) improved symptom burden, and (3) improved return to usual activity. The protocol specifies a fixed sequence at one-sided \(\alpha=0.025\), with each listed p-value already adjusted for the interim looks. The values are \(0.012\), \(0.031\), and \(0.009\), respectively.
At position 1, \(0.012<0.025\), so reject the time-to-recovery null and proceed. At position 2, \(0.031>0.025\), so do not reject the symptom-burden null. The sequence stops there. Although the reported value for return to usual activity is \(0.009\), the protocol does not permit that endpoint’s claim to be made after the gate at position 2 failed.
The supported conclusion is the first hierarchical claim only. The second claim is unsupported because its own null was not rejected; the third is unavailable under the specified hierarchy. The third endpoint’s small p-value does not repair the failure at position 2.
Parallel Endpoints: Separate Claims, Shared Error Control
In a parallel structure, each endpoint represents a distinct claim, and no endpoint is a logical prerequisite for another. A rejection for one endpoint supports that endpoint’s claim, not the claims for the other endpoints. If the trial intends to make any of several claims, however, the family of claims must be considered when controlling the probability of one or more false rejections.
Worked Example: Interpreting Parallel Endpoint Decisions
A hypothetical study plans three separate claims: benefit on pain, benefit on mobility, and benefit on sleep. It uses a prespecified group sequential multiplicity procedure. For this example, assume the procedure’s endpoint-level adjusted p-values are \(0.014\), \(0.031\), and \(0.008\), and that its decision threshold is \(0.025\). These adjusted values and the threshold are taken as outputs of the stipulated procedure; they are not raw p-values to which a new adjustment should be applied here.
For pain, \(0.014<0.025\), so reject the pain null. For mobility, \(0.031>0.025\), so do not reject the mobility null. For sleep, \(0.008<0.025\), so reject the sleep null. The supported endpoint claims are therefore pain and sleep, subject to the stated procedure and its assumptions. The mobility claim is not supported.
This pattern differs from a co-primary structure, where failure on one required endpoint would block the joint claim, and from a hierarchy, where failure at an earlier position would block later claims. Here, the mobility result does not logically block the pain or sleep claims; the prespecified multiplicity procedure governs whether those separate rejections are valid.
Comparing the Three Structures
| Structure | Meaning of the clinical claim | Required rejection pattern | Interpretation of a failure |
|---|---|---|---|
| Co-primary | Benefit is required on every designated endpoint. | Reject every component null. | One failed component means the joint claim is not established. |
| Hierarchical | Claims are ordered; later claims depend on earlier success. | For claim \(j\), reject all nulls from position 1 through \(j\). | A failed gate blocks that claim and all later claims in the specified sequence. |
| Parallel | Each endpoint corresponds to a separate claim without a logical gate. | Reject the corresponding null for each claim made, using the prespecified family procedure. | Failure on one endpoint does not, by the claim structure alone, block another endpoint. |
The same endpoints can be organized in different ways depending on the intended clinical conclusion. If the treatment is useful only when both outcomes improve, a co-primary structure may reflect that requirement. If one endpoint is essential and others are of interest only after it succeeds, a hierarchy expresses that priority. If each outcome supports an independently meaningful claim, a parallel structure is more appropriate. The structure should follow the clinical question, not be selected after seeing which endpoints happened to produce favorable results.
Diagnostics, Pitfalls, and Limits
Before interpreting results, check that the endpoint list, favorable direction, effect scale, and claim structure match the protocol. For group sequential testing, also verify that the reported decisions account for all planned interim looks and any endpoint-specific timing. A nominal p-value from one look is not automatically the adjusted evidence required by the decision plan.
- Calling a co-primary result successful after one rejection. The correction is to check every required endpoint; the joint claim requires them all.
- Treating a hierarchy as a ranked list of p-values. The order is a logical testing gate, not a post hoc ranking. A later small p-value cannot override an earlier failure.
- Assuming parallel means no multiplicity issue. Separate clinical claims can still produce a familywise error problem. Use the prespecified endpoint-and-look procedure.
- Confusing non-rejection with proof of no effect. Failure to reject means the evidence did not meet the decision rule; it does not establish that the endpoint effect is zero or harmful.
- Changing the claim structure after results are known. This changes which rejections count as success and can undermine the interpretation and error control of the planned analysis.
These structures clarify what a decision means, but they do not settle every design choice. The appropriate clinical margins, endpoint definitions, correlation among outcomes, handling of missing data, and multiplicity procedure require separate specification. Nor does an endpoint rejection alone establish clinical importance: the effect estimate and its uncertainty should also be interpreted in the endpoint’s units and clinical context.
Check Your Understanding
For each question, identify the claim structure and the rejection pattern needed. Brief solutions follow.
- A trial claims that a treatment improves both exercise capacity and quality of life. What structure is this, and what must be rejected to support the claim?
- In a three-endpoint fixed hierarchy, the first and third nulls are rejected but the second is not. Which hierarchical claims are supported?
- For a claim at hierarchical position 2, \(H_{01}\) is true and \(H_{02}\) is false. If the claim at position 2 is made, which true null has necessarily been rejected?
- In a parallel family, one endpoint does not meet its decision criterion. Does that failure by itself block a claim on another endpoint? What still needs to be checked?
Solutions. (1) This is co-primary: both endpoint nulls must be rejected. (2) Only the first claim is supported; the failed second gate prevents the claim at position 3, even though its null was rejected. (3) The claim requires rejecting both nulls, so the true null \(H_{01}\) has been rejected; \(H_{02}\) is false and is not the true null responsible for the false claim. (4) No logical gate connects the claims in a parallel structure, so that failure alone does not block the other endpoint claim. The analyst must still verify that the prespecified multiplicity procedure controls error across the endpoint family and interim looks.