Why the Overall Decision Is Joint
For an NTI drug, the prespecified bioequivalence assessment may require acceptable results for both an AUC measure and \(C_{\max}\). This makes the overall conclusion a joint decision: neither endpoint can compensate for the other. A precise AUC result does not erase an unacceptable \(C_{\max}\) result, and a favorable \(C_{\max}\) result cannot rescue an AUC confidence interval that falls outside its acceptance limits.
The endpoint limits need not be identical. The AUC criterion may be tighter than the \(C_{\max}\) criterion, or the two endpoints may use different AUC measures or endpoint-specific rules. The limits and analysis method must be set before examining the results and must follow the applicable study protocol and regulatory requirements. The calculations below use illustrative limits, not universal NTI limits.
Endpoint-Specific Confidence-Interval Decisions
For endpoint \(j\), where \(j\) denotes AUC or \(C_{\max}\), let \(\theta_j\) be the test-minus-reference contrast in the mean of the log-transformed endpoint. The corresponding test-to-reference geometric mean ratio is \(\exp(\theta_j)\). Write the prespecified acceptable ratio limits as \(L_j\) and \(U_j\), with \(0<L_j<U_j\). On the log scale, those limits are \(\log(L_j)\) and \(\log(U_j)\).
A common equivalence procedure uses a two-sided 90% confidence interval for \(\theta_j\), equivalent to two one-sided tests at \(\alpha=0.05\) when the assumptions and standard testing setup apply. Exponentiating both confidence limits gives the interval for the geometric mean ratio. Endpoint \(j\) passes when its entire interval is within that endpoint's limits:
Here \(L_j^{CI}\) and \(U_j^{CI}\) are the exponentiated lower and upper confidence limits. Equivalently, if \([\ell_j,u_j]\) is the confidence interval for \(\theta_j\), the rule is \(\log(L_j)\leq \ell_j\) and \(u_j\leq \log(U_j)\). Merely having the point estimate inside the limits is not enough: the interval must also show sufficient precision to rule out values beyond either limit.
This interval rule relies on a valid endpoint-specific estimate and standard error from the prespecified analysis, an appropriate confidence-interval method for the study design and data, and the planned endpoint definitions and limits. The model and its handling of the replicate design, subject effects, and incomplete observations must be appropriate for the available data. The endpoint’s log-scale contrast must correspond to the intended test-versus-reference comparison. The interval calculation does not fix a misspecified model, an inappropriate analysis population, or an endpoint definition changed after results are seen.
Worked Endpoint Decisions
Worked Example: Both AUC and Cmax Pass
Suppose a hypothetical NTI study prespecifies illustrative limits of 0.90 to 1.11 for both endpoints and uses 90% confidence intervals. For this example, assume the endpoint-specific analyses are appropriate and provide valid standard errors. The 90% \(t\)-multiplier is 1.729 for each endpoint. The AUC log contrast is \(\hat{\theta}_{AUC}=\log(1.015)=0.01489\), with \(SE=0.028\).
The log-scale interval is:
Exponentiating gives the AUC ratio interval \([e^{-0.03352},e^{0.06330}]=[0.9670,1.065]\). This entire interval lies within \([0.90,1.11]\), so AUC passes.
For \(C_{\max}\), suppose \(\hat{\theta}_{C_{\max}}=\log(0.982)=-0.01816\) and \(SE=0.032\). Then:
Exponentiating gives \([0.9291,1.038]\), also entirely within \([0.90,1.11]\). Both required endpoints pass, so the joint AUC and \(C_{\max}\) criterion is met under these illustrative assumptions and limits.
Worked Example: AUC Passes but Cmax Fails
Consider another hypothetical analysis with the same illustrative limits, \([0.90,1.11]\), and a 90% \(t\)-multiplier of 1.711. Assume again that the endpoint-specific analyses are valid. For AUC, let the estimated ratio be 1.020, so \(\hat{\theta}_{AUC}=\log(1.020)=0.01980\), with \(SE=0.025\). The log interval is:
After exponentiation, the interval is \([0.9773,1.065]\), which is inside the AUC limits. AUC passes.
For \(C_{\max}\), let the estimated ratio be 1.075, so \(\hat{\theta}_{C_{\max}}=\log(1.075)=0.07232\), with \(SE=0.030\). Then:
Exponentiation gives approximately \([1.021,1.132]\). Its upper limit exceeds 1.11, so \(C_{\max}\) fails, although its point estimate, 1.075, is itself inside the limits. Because both endpoints are required, the joint equivalence criterion is not met.
Worked Example: Different Limits and a Borderline AUC Interval
Suppose a third hypothetical study prespecifies AUC limits of \([0.95,1.05]\) and \(C_{\max}\) limits of \([0.80,1.25]\). Use a 90% confidence interval with a \(t\)-multiplier of 1.697, and assume appropriate endpoint-specific analyses. For AUC, let the ratio estimate be 1.030, so \(\hat{\theta}_{AUC}=\log(1.030)=0.02956\), with \(SE=0.015\). The interval is:
On the ratio scale this is approximately \([1.004,1.057]\). The upper confidence limit is greater than the AUC upper limit of 1.05. AUC therefore fails, even though the point estimate 1.030 lies inside the AUC limits.
For \(C_{\max}\), let the ratio estimate be 1.100, giving \(\hat{\theta}_{C_{\max}}=\log(1.100)=0.09531\), with \(SE=0.045\). Then:
Exponentiating gives approximately \([1.019,1.187]\), which is within the \(C_{\max}\) limits of \([0.80,1.25]\). \(C_{\max}\) passes, but AUC does not; the joint criterion is not met. This illustrates why the endpoints must be judged against their own prespecified limits rather than against a single shared rule.
Why the Joint Rule Controls the Equivalence Claim
For each endpoint, the equivalence null hypothesis is that the true ratio is at or beyond at least one of that endpoint’s limits; the alternative is that it is between the limits. For the joint claim, the null is that at least one required endpoint is not equivalent. The alternative is that all required endpoints are equivalent. Thus, the joint alternative is an intersection of endpoint alternatives.
This structure explains why the usual endpoint-by-endpoint tests at \(\alpha=0.05\) do not require an additional Bonferroni adjustment merely because both endpoints are required for the overall claim. If the joint null is true, at least one endpoint, say endpoint \(k\), is truly non-equivalent. Declaring joint equivalence requires rejecting endpoint \(k\)'s non-equivalence null, as well as every other endpoint null. Therefore:
The joint false-equivalence error probability is therefore at most \(\alpha\), under valid endpoint tests and the stated joint decision rule. This argument does not require the AUC and \(C_{\max}\) estimates to be independent. It also does not say that the probability both intervals pass is the same as the probability a joint confidence region covers both true effects. Those are different questions.
Diagnostics, Alternatives, and Limitations
Before combining endpoint results, check that the endpoints, limits, confidence level, model, analysis population, and missing-data rules were prespecified and applied as planned. Confirm that each interval is on the intended scale, that exponentiation was done to both interval bounds, and that each bound is compared with the correct endpoint-specific limit. A useful reporting table makes the separate decisions visible:
| Endpoint | Prespecified limits | Ratio estimate and confidence interval | Endpoint decision |
|---|---|---|---|
| AUC measure | AUC-specific lower and upper limits | Reported on the ratio scale | Pass or fail |
| \(C_{\max}\) | \(C_{\max}\)-specific lower and upper limits | Reported on the ratio scale | Pass or fail |
| Joint conclusion | Both endpoint criteria required | — | Pass only if both pass |
A very wide interval, a limit crossed by only a small amount, or a result that changes substantially under reasonable prespecified sensitivity analyses deserves careful review. Such patterns may indicate limited precision, an influential observation, model or data problems, or an endpoint-specific issue. They do not authorize changing the limits after seeing the result. The joint rule is a decision rule, not a diagnostic for why an endpoint failed.
A simultaneous confidence region for multiple parameters is an alternative when the scientific objective is to make a joint statement with simultaneous coverage. It answers a different inferential question and may be more conservative than separate endpoint intervals. Likewise, multiplicity procedures are relevant when the goal is to claim equivalence if either endpoint passes, or when investigators select endpoints after inspecting results. Those procedures do not replace the required-both-endpoints rule for a prespecified intersection-union equivalence claim.
The conclusion is limited to the required endpoints, definitions, population, design, and acceptance criteria actually assessed. Passing both does not prove identical exposure, guarantee equal clinical outcomes for every individual, or establish equivalence for untested PK measures. Conversely, failing an interval rule means the prespecified statistical criterion was not met; it does not by itself prove that the formulations differ clinically.
Common Pitfall
Wrong reasoning: “The AUC ratio is close to one and passes, so the overall NTI study passes even though \(C_{\max}\) misses its upper limit.” This treats one endpoint as able to compensate for another and ignores the prespecified joint claim.
Correction: Evaluate each required endpoint against its own limits and confidence-interval rule. Report both results and conclude joint equivalence only if every required endpoint passes. Do not replace an interval-based decision with a point-estimate judgment or revise endpoint limits after seeing the data.
Check Your Understanding
Answer each question using the endpoint-specific and joint decision rules.
- Both endpoints are required. AUC passes its interval rule, but \(C_{\max}\) fails. What is the joint conclusion?
- An endpoint has limits 0.90 to 1.11, a point estimate of 1.08, and a 90% confidence interval of 1.02 to 1.12. Does it pass?
- Why is an additional Bonferroni adjustment not required solely because both AUC and \(C_{\max}\) must pass for the prespecified joint equivalence claim?
- AUC and \(C_{\max}\) have different prespecified limits. Which limits should be used to assess each interval?
Solutions. (1) The joint criterion is not met because both endpoints are required and \(C_{\max}\) fails. (2) No. The upper confidence limit, 1.12, exceeds 1.11, despite the point estimate being inside the limits. (3) Under the intersection-union rule, a joint declaration requires rejecting every endpoint’s non-equivalence null. When the joint null is true, at least one endpoint null is true, and the chance of rejecting that true null is at most \(\alpha\); the joint false-equivalence error is therefore at most \(\alpha\). (4) Compare the AUC interval with the AUC limits and the \(C_{\max}\) interval with the \(C_{\max}\) limits, then require both to pass.