The All-Criteria Rule for Multiple Primary Endpoints

Why Multiple Primary Endpoints Require a Conjunctive Decision

A bioequivalence assessment can prespecify more than one primary endpoint. For example, a product may require acceptable results for AUC and Cmax, or for those measures together with a partial AUC. Each endpoint addresses a distinct feature of exposure, and its applicable regulatory criterion may differ. When the regulatory question requires all of these endpoints to meet their criteria, the decision is conjunctive: every required endpoint must pass.

This rule is not an averaging procedure. A close match on one endpoint does not compensate for a confidence interval that misses an acceptance limit on another. Nor does a favorable overall impression replace a prespecified endpoint-specific assessment. The regulatory conclusion follows the full set of required criteria.

Definition: Under the all-criteria rule, a study satisfies a multiple-primary-endpoint bioequivalence requirement only if every prespecified primary endpoint meets its applicable acceptance criterion. If even one required endpoint fails, the combined bioequivalence claim is not established by that study.

For a ratio-scale endpoint, the criterion is commonly assessed using a confidence interval for the test-to-reference geometric mean ratio. The earlier material on conventional acceptance limits and ratio-scale interpretation defines how to read such intervals. The essential point here is that each endpoint has its own interval and its own applicable limits. A 90% confidence interval entirely inside the relevant limits passes that endpoint's interval criterion; an interval that crosses either limit does not.

Formal Decision Rule

Suppose there are \(m\) required primary endpoints. For endpoint \(j\), let \(L_j\) and \(U_j\) be its prespecified lower and upper acceptance limits, and let \([C_{Lj},C_{Uj}]\) be the confidence interval used for the regulatory decision. Endpoint \(j\) passes if both inequalities hold:

L_j \leq C_{Lj} \quad\text{and}\quad C_{Uj} \leq U_j.

The all-criteria decision is therefore:

\text{Pass overall} \quad\Longleftrightarrow\quad \bigcap_{j=1}^{m} \left\{L_j \leq C_{Lj}\ \text{and}\ C_{Uj} \leq U_j\right\}.

The limits need not be identical across endpoints. The endpoint definitions, scale, confidence level, and acceptance limits must be the ones specified for the particular regulatory assessment. The rule does not authorize choosing a more favorable margin after observing results.

Worked Example: AUC Passes but Cmax Fails

In a hypothetical assessment, AUC and Cmax are both required primary endpoints. Each uses a 90% confidence interval on the test-to-reference geometric mean ratio scale, with limits of 0.80 to 1.25. The analysis reports log-scale estimates and standard errors. For illustration, use a critical value of 1.70 for both intervals.

For AUC, suppose the estimated ratio is 0.98 and its log-scale standard error is 0.040. The interval is calculated as the exponentiated log estimate plus or minus the critical value times its standard error:

\exp\left[\ln(0.98)\pm 1.70(0.040)\right] = \exp[-0.0882,\ 0.0478] = [0.916,\ 1.049].

Both limits are inside 0.80 to 1.25, so AUC passes.

For Cmax, suppose the estimated ratio is 1.03 and its log-scale standard error is 0.140:

\exp\left[\ln(1.03)\pm 1.70(0.140)\right] = \exp[-0.2084,\ 0.2676] = [0.812,\ 1.307].

The upper confidence limit, 1.307, exceeds 1.25, so Cmax fails. AUC passing does not cancel that failure. Because both endpoints are required, the all-criteria decision is not established.

Why the Rule Controls the Combined Claim

The all-criteria rule has a direct hypothesis-testing interpretation. For each endpoint, define the component null hypothesis as non-equivalence: the endpoint does not satisfy its required equivalence criterion. The component alternative is that the endpoint does satisfy the criterion. The overall alternative requires all component alternatives to hold. Thus, the overall null is that at least one required endpoint is not equivalent.

H_0 = \bigcup_{j=1}^{m} H_{0j}, \qquad H_a = \bigcap_{j=1}^{m} H_{aj}.

This is an intersection-union test: the overall alternative is an intersection of component alternatives, and the overall null is a union of component nulls. To reject the overall null, the analysis must reject every component null. If even one component test does not reject its null, there is insufficient evidence for the all-endpoint claim.

A short justification shows why this procedure controls the probability of a false overall equivalence claim at level \(\alpha\). Under the overall null, at least one component null is true. Choose any true component null, say \(H_{0k}\). The event of falsely declaring overall equivalence requires rejecting every component null, and therefore requires rejecting \(H_{0k}\). Consequently:

P(\text{false overall equivalence claim}) \leq P(\text{reject }H_{0k}\mid H_{0k}\text{ is true}) \leq \alpha.

This argument does not require independence among endpoint estimates. It explains why a multiplicity adjustment such as Bonferroni is not automatically needed for a single prespecified claim that all endpoints meet their criteria, when each component test is conducted at level \(\alpha\). It does not mean that every multiplicity issue in a study disappears: additional claims, endpoints, analyses, or opportunities to select results may require separate consideration.

For the familiar two one-sided tests formulation of equivalence, each endpoint is assessed against its lower and upper limits. Under the usual correspondence between those tests and confidence intervals, requiring the endpoint's 90% confidence interval to lie wholly within its acceptance limits corresponds to conducting the two one-sided component tests at the 0.05 level. The overall conclusion still requires every endpoint's criterion to be met.

Applying Different Criteria to Different Endpoints

Endpoints in one study need not share the same limits. For instance, a product-specific assessment may specify a distinct criterion for a partial AUC or another endpoint. The all-criteria rule governs how the decisions are combined; it does not determine which criterion applies. Each endpoint must be matched to its prespecified definition, analysis scale, confidence interval, and limits before the overall decision is made.

Worked Example: All Three Endpoint Criteria Pass

Consider a hypothetical assessment requiring AUC, partial AUC, and Cmax to pass. Suppose the prespecified limits are 0.80 to 1.25 for AUC, 0.85 to 1.18 for partial AUC, and 0.90 to 1.11 for Cmax. For illustration, use a critical value of 1.70. The following estimates and log-scale standard errors are hypothetical.

For AUC, the estimated ratio is 0.99 with standard error 0.040:

\exp\left[\ln(0.99)\pm1.70(0.040)\right] = \exp[-0.0781,\ 0.0580] = [0.925,\ 1.060].

This interval lies inside 0.80 to 1.25. For partial AUC, the estimated ratio is 0.95 with standard error 0.050:

\exp\left[\ln(0.95)\pm1.70(0.050)\right] = \exp[-0.1363,\ 0.0337] = [0.873,\ 1.034].

This interval lies inside 0.85 to 1.18. For Cmax, the estimated ratio is 0.98 with standard error 0.040:

\exp\left[\ln(0.98)\pm1.70(0.040)\right] = \exp[-0.0882,\ 0.0478] = [0.916,\ 1.049].

This interval lies inside 0.90 to 1.11. All three endpoint-specific criteria pass, so the all-criteria rule is satisfied. The conclusion depends on checking each interval against its own limits, not on comparing the intervals with one common set of limits.

When a Single Endpoint Determines the Outcome

A component endpoint can be the sole reason the overall criterion is not met. That does not make the other endpoints irrelevant: their passing results remain part of the assessment. It means only that the combined claim has a bottleneck. A prespecified all-endpoint requirement is not a vote in which two passes outweigh one failure.

Worked Example: Combining Component Test Results

Suppose three endpoint-specific equivalence tests produce the following valid component p-values: 0.012 for AUC, 0.180 for Cmax, and 0.031 for partial AUC. At a component significance level of \(\alpha=0.05\), the AUC and partial AUC tests reject their component null hypotheses, but the Cmax test does not.

For this intersection-union claim, an overall test can be represented by the largest component p-value:

p_{\text{overall}}=\max(0.012,\ 0.180,\ 0.031)=0.180.

Because \(0.180>0.05\), the overall claim does not pass. The two smaller p-values cannot offset the Cmax result. This p-value summary is valid only when the component p-values correspond to the prespecified component hypotheses and analyses; it is not a substitute for checking the applicable endpoint criteria.

Diagnostics, Alternatives, and Limitations

A defensible decision begins with an audit of what was prespecified and what was analyzed. Confirm that the required primary endpoints are correctly identified; that each endpoint is calculated as defined; that the intended scale and acceptance limits are used; and that the confidence interval corresponds to the prescribed analysis. Check that reported limits have not been rounded in a way that changes a borderline decision. If an interval lies very near a boundary, verify the unrounded analysis output and the applicable decision convention rather than relying on a rounded table alone.

Also check whether the statistical analysis is suitable for the study data and design. The component tests or confidence intervals must have their stated operating properties under the model and assumptions used. Missing or unusable observations, deviations from the planned analysis, or an inappropriate interval calculation can undermine the endpoint decisions. A result that fails because an interval is too wide is still a failure to establish the criterion; it is not evidence that the products are necessarily different.

The main alternative decision structures answer different questions. A co-primary, all-must-pass rule supports a claim only if every required endpoint succeeds. By contrast, a rule that declares success if any one of several endpoints passes is a different testing problem: false-positive control must account for the multiple opportunities to succeed. A composite endpoint combines components into one measure, but that is not interchangeable with requiring each component to meet a separate criterion. A primary endpoint accompanied by supportive endpoints also differs from multiple primary endpoints: supportive findings may inform interpretation, but they do not replace a required primary criterion.

The all-criteria rule is deliberately strict about the specified claim, but it has limits. It addresses only the endpoints and criteria included in the decision, and it does not establish equivalence for unmeasured outcomes or guarantee therapeutic similarity in every respect. Nor does a passing set of intervals make the underlying estimates exact; interval and test conclusions remain subject to the analysis assumptions and study information. The earlier material on supportive endpoints distinguishes supporting evidence from criteria that are themselves required for the primary regulatory conclusion.

Common Mistake

Common mistake: “AUC is comfortably within its limits and Cmax is only slightly outside, so the strong AUC result offsets the small Cmax miss.” This is incorrect when both endpoints are prespecified as required primary endpoints. The correction is to apply the stated criterion to each endpoint separately: if Cmax fails, the combined all-endpoint requirement is not met, regardless of how far inside its limits AUC lies.

A related error is to treat “not significant” in a conventional difference test as proof of equivalence. Equivalence requires the endpoint-specific equivalence procedure and its prespecified limits; failure to detect a difference does not establish that the interval lies within those limits.

Key takeaway: For a multiple-primary-endpoint equivalence claim that requires every endpoint, evaluate each endpoint against its own prespecified criterion. The overall claim passes only when all required criteria pass; one failure cannot be offset by another endpoint.

Check Your Understanding

Answer each question using the all-criteria rule. Solutions follow each problem.

A hypothetical study requires AUC and Cmax to pass. AUC's interval lies wholly within its limits, while Cmax's upper confidence limit exceeds its upper limit. What is the overall decision?
Solution: The overall all-criteria requirement is not met. Cmax is a required endpoint and fails; the passing AUC result cannot offset it.
Why can the all-criteria rule control the false-positive rate at level \(\alpha\) without automatically applying a Bonferroni adjustment to the required endpoint tests?
Solution: Under the overall null, at least one component null is true. A false overall equivalence declaration requires rejecting every component null, including that true one. The probability of the overall false declaration is therefore no greater than the probability of falsely rejecting that true component, which is at most \(\alpha\). This reasoning applies to the single prespecified all-must-pass claim, not automatically to other claims or selection procedures.
A required endpoint's interval is \([0.88,1.16]\), and its applicable limits are 0.90 to 1.20. Does it pass?
Solution: No. Although the upper confidence limit, 1.16, is below 1.20, the lower confidence limit, 0.88, is below 0.90. Both interval endpoints must be within the applicable limits.
Three component p-values are 0.020, 0.049, and 0.051. At \(\alpha=0.05\), what is the intersection-union p-value, and what is the conclusion?
Solution: The overall p-value is the maximum component p-value: \(\max(0.020,0.049,0.051)=0.051\). Since \(0.051>0.05\), the overall all-endpoint equivalence claim does not pass.
A team reports that two of three required primary endpoints pass and calls the study “mostly bioequivalent.” What is the appropriate correction?
Solution: If the prespecified regulatory claim requires all three endpoints, the study does not satisfy that combined criterion. “Mostly bioequivalent” is not a substitute for the required decision; report which endpoints passed and identify the failed endpoint.