Why Co-Primary Endpoints Require a Combined Decision
A bioequivalence analysis may require evidence on more than one exposure measure. For example, a protocol may designate both AUC, which summarizes overall exposure, and \(C_{\max}\), which describes peak concentration, as co-primary endpoints. If the prespecified decision rule requires both to meet their bioequivalence criteria, a favorable result for one cannot compensate for a failure on the other.
The central task is therefore twofold: evaluate every endpoint against its own prespecified requirement, then combine those endpoint-level results according to the protocol. This is not the same as averaging results across endpoints or choosing the endpoint with the most favorable interval. The conclusion must represent the complete set of required evidence.
The endpoints and their limits must be selected before examining the results. Common conventional average-bioequivalence limits for exposure ratios are 80% to 125%, but the applicable limits depend on the product, endpoint, and regulatory framework. This tutorial uses 80% to 125% in its examples and treats those limits as prespecified. A study should use its own protocol-specified criteria, not adopt these example limits after seeing its data.
Endpoint-Level Model and Pass Rule
For endpoint \(j\), suppose the analysis compares test (T) with reference (R) on the log scale. In a conventional crossover analysis, the model accounts for the treatment, period, and sequence structure, with the subject contribution represented as specified for the design. The treatment contrast estimates the mean log exposure difference. Exponentiating that contrast gives the test-to-reference geometric mean ratio. The exact model, analysis population, degrees of freedom, and handling of missing observations must be prespecified and appropriate to the study design.
The interval calculation assumes that the fitted model provides a valid treatment-contrast estimate and standard error; that the model’s error assumptions are adequate for the planned inference; and that the design and data support the specified analysis. For the usual model-based \(t\) interval, inference relies on the relevant residual distribution being adequately modeled, with variance and dependence structure handled as specified. These are assumptions about each endpoint’s analysis. The endpoints themselves do not have to be statistically independent for the all-must-pass rule.
For the usual two one-sided tests (TOST) formulation, equivalence for endpoint \(j\) requires rejection at significance level \(\alpha=0.05\) of both hypotheses that the true ratio is at or beyond a limit. This corresponds to the 90% confidence interval lying within the limits. In practice, compare the unrounded interval limits with the prespecified boundaries; rounding for presentation must not change the decision.
Worked Example: Both Co-Primary Endpoints Pass
Hypothetical data. A crossover study prespecifies AUC and \(C_{\max}\) as co-primary endpoints, with 80% to 125% limits for each. For illustration, the fitted model gives \(df=20\) for both contrasts and \(t_{0.95,20}=1.725\). The estimated geometric mean ratios are 1.015 for AUC and 1.040 for \(C_{\max}\), with log-scale standard errors 0.045 and 0.052, respectively. Assume the prespecified model is appropriate and its residual and design assumptions are adequately supported.
AUC. The log contrast is \(\ln(1.015)=0.01489\). The margin of error is \(1.725(0.045)=0.07763\), so the log-scale interval is \(0.01489\pm0.07763=[-0.06274,0.09252]\). Exponentiating gives \([e^{-0.06274},e^{0.09252}]=[0.9392,1.097]\). Both limits are within 0.80 to 1.25, so AUC passes.
\(C_{\max}\). The log contrast is \(\ln(1.040)=0.03922\). The margin is \(1.725(0.052)=0.08970\), giving the log interval \([-0.05048,0.12892]\). Exponentiation gives \([0.9508,1.138]\), also within 0.80 to 1.25. Thus \(C_{\max}\) passes.
Because both prespecified co-primary endpoints pass, the combined conclusion is that the study demonstrates bioequivalence for these endpoints under the stated criteria. This conclusion is limited to the endpoints and criteria evaluated; it does not imply that every possible measure of exposure is equivalent.
Why the All-Must-Pass Rule Controls the Claim
The combined claim is that all co-primary endpoints satisfy their equivalence requirements. Its null hypothesis is therefore that at least one required endpoint does not satisfy its requirement. This is an intersection-union structure: the alternative consists of the intersection of the endpoint-level equivalence claims, while the null is the union of endpoint-level failures.
A short justification shows why the usual endpoint-level test size controls the error rate for this particular claim, without requiring independence between endpoints. If the combined null is true, at least one endpoint—call it \(k\)—is not equivalent. A false combined conclusion requires the analysis to reject the null for every endpoint, including \(k\). Therefore,
The final inequality holds when the endpoint-level test is conducted at level \(\alpha\) under its stated assumptions. The argument applies regardless of how the endpoint results are correlated. It does not mean that endpoints can be tested or reported selectively: the combined claim still requires every designated endpoint to pass.
This reasoning is specific to a claim that requires all endpoints to succeed. If a study instead intends to claim success when at least one among several endpoints passes, the error-control problem is different and requires an appropriate multiplicity strategy. The claim and decision rule must be specified before the results are known.
Worked Example: One Failed Endpoint Determines the Overall Result
Hypothetical data. A study requires both AUC and \(C_{\max}\) to meet 80% to 125% limits. The model is judged appropriate for both endpoints. For illustration, use \(df=24\) and \(t_{0.95,24}=1.711\). AUC has a geometric mean ratio of 1.020 and log-scale standard error 0.050. \(C_{\max}\) has a ratio of 1.150 and standard error 0.060.
AUC. The log estimate is \(\ln(1.020)=0.01980\), and the margin is \(1.711(0.050)=0.08555\). The log interval is \([-0.06575,0.10535]\); exponentiating gives \([0.9364,1.111]\). AUC passes.
\(C_{\max}\). The log estimate is \(\ln(1.150)=0.13976\), and the margin is \(1.711(0.060)=0.10266\). The log interval is \([0.03710,0.24242]\). Exponentiating gives \([1.0378,1.2743]\). Its upper limit exceeds 1.25, so \(C_{\max}\) fails.
The combined result is bioequivalence not demonstrated under the prespecified all-must-pass rule. The AUC result cannot override the \(C_{\max}\) failure. This decision does not prove that the products are different; it says the required evidence for equivalence was not established for every co-primary endpoint.
Reading the Full Set of Results
A useful report displays every co-primary endpoint, its geometric mean ratio, its confidence interval, its limits, and its pass/fail status. The combined conclusion should follow those individual results directly. In particular, distinguish “not demonstrated” from “demonstrated to be non-equivalent”: an interval that crosses a boundary may reflect an estimated ratio outside the limits, insufficient precision, or both.
| Result pattern | Endpoint-level interpretation | Combined conclusion under all-must-pass rule |
|---|---|---|
| Every interval lies within its limits | Every required endpoint passes | Bioequivalence demonstrated for the co-primary endpoints |
| At least one interval extends beyond a limit | At least one required endpoint fails to demonstrate equivalence | Bioequivalence not demonstrated for the set |
| Point estimates lie within limits, but an interval extends beyond a limit | The point estimate is in range, but the endpoint does not pass | Bioequivalence not demonstrated for the set |
A confidence interval that is too wide can fail even when its point estimate is close to 1.00. Precision depends on factors such as variability, sample size, and the design-supported standard error. Conversely, an interval that lies wholly outside a limit signals that the estimated endpoint is not compatible with the specified equivalence region at the chosen confidence level. In either case, use the prespecified interval rule rather than judging the result by the point estimate alone.
Worked Example: An In-Range Estimate Is Not Enough
Hypothetical data. Suppose \(C_{\max}\) is a required co-primary endpoint with 80% to 125% limits. A suitable prespecified model gives a geometric mean ratio of 0.840, log-scale standard error 0.048, \(df=24\), and \(t_{0.95,24}=1.711\). Assume the model assumptions are adequately supported.
The log estimate is \(\ln(0.840)=-0.17435\). The margin is \(1.711(0.048)=0.08213\), giving the log interval \([-0.25648,-0.09222]\). Exponentiating yields approximately \([0.7738,0.9119]\). Although the point estimate 0.840 lies within the limits, the lower confidence limit is below 0.80, so the endpoint fails. If it is co-primary under an all-must-pass rule, the overall bioequivalence claim also fails, whatever the other endpoint’s result.
Diagnostics, Alternatives, and Limitations
Before finalizing a combined decision, check that the analysis and reporting preserve the prespecified endpoint structure. Confirm the endpoint definitions and units, the analysis population, the model and degrees of freedom, the confidence level, and the limits for each endpoint. Review model diagnostics for influential observations, residual patterns, variance problems, or design irregularities that could undermine the interval calculation. If assumptions are questionable, do not silently substitute a different model or confidence interval; investigate and report the issue using the prespecified analysis and appropriate sensitivity analyses.
The main alternative to an all-must-pass rule is a different claim structure, such as success if at least one endpoint passes. That is not a wording change: it changes the hypothesis and can require multiplicity adjustment. Another tempting alternative is to combine endpoint estimates into an average or emphasize a preferred endpoint. Such summaries do not answer whether every separately required endpoint passed. Where endpoints have distinct scientific roles or distinct prespecified limits, assess each against its own criterion.
Common failures include omitting an endpoint from the final table, describing a failed endpoint as “borderline” while presenting the study as successful, and rounding a confidence limit inward until it appears to meet a boundary. Report all endpoints and apply the rule to unrounded results. If the result is inconclusive for one endpoint, say so plainly; do not replace the prespecified criterion with a post hoc judgment.
Check Your Understanding
For each question, apply the stated limits and decision rule. Solutions follow the questions.
- A study requires AUC and \(C_{\max}\) to pass 80% to 125%. Their confidence intervals are \([0.89,1.18]\) and \([0.81,1.26]\). What is the combined conclusion?
- A co-primary endpoint has an estimated ratio of 1.05 and a 90% confidence interval of \([0.84,1.22]\). Does it pass 80% to 125% limits?
- Why does the all-must-pass rule not require independence between endpoints to control the false-positive rate for the combined claim?
- Suppose a protocol requires three endpoints to pass, and two pass while one interval crosses its upper limit. What should the report conclude, and what should it not claim?
Solutions. (1) AUC passes, but the \(C_{\max}\) upper limit 1.26 exceeds 1.25. The combined conclusion is that bioequivalence is not demonstrated for the required endpoint set. (2) Yes. Both 0.84 and 1.22 lie within 0.80 to 1.25, so the endpoint passes. (3) If the combined null is true, at least one endpoint is not equivalent. A false combined claim requires rejecting that endpoint’s null, so the probability of a false combined claim is no greater than the level-\(\alpha\) error probability for that endpoint. This bound does not depend on endpoint independence. (4) The report should state that the combined bioequivalence criterion was not met because one required endpoint failed. It should not claim overall bioequivalence by emphasizing only the two passing endpoints.