Why Two Exposure Measures Can Disagree
AUC and Cmax describe different features of drug exposure. AUC summarizes exposure over time, whereas Cmax describes the observed peak concentration. A formulation may therefore produce an AUC estimate close to that of the reference and a less precise or more distant Cmax estimate, or the reverse. Different estimates and confidence intervals are not, by themselves, evidence of an error: they can reflect distinct pharmacokinetic profiles, endpoint variability, and sampling precision.
The interpretive task is to assess each endpoint on its own scale and against its own bioequivalence criterion. The point estimate alone does not decide the outcome. The confidence interval must also satisfy the prespecified limits, and the overall conclusion must account for every endpoint required by the study’s applicable standard.
Model and Decision Rule
For a standard two-treatment, two-period crossover, the endpoint is usually analyzed after natural-log transformation. A treatment contrast is estimated from a model that accounts for treatment, period, and sequence, with subjects nested within sequence represented appropriately in the model. The endpoint-specific estimate is the adjusted test-minus-reference contrast on the log scale. Exponentiating that contrast gives the test-to-reference geometric mean ratio.
This approach assumes the design and model are appropriate for the data: treatment allocation and period conduct follow the protocol; the planned observations are available or handled according to the analysis plan; the model’s residuals are sufficiently well behaved for the interval procedure; and subject-level dependence and period effects are represented correctly. It also relies on the absence of a material carryover effect that would invalidate the treatment comparison. These assumptions should be considered alongside data-integrity and protocol-deviation reviews, not inferred merely because a model produces an interval.
Let \(\hat{\theta}\) be the estimated log test-to-reference contrast, \(SE\) its standard error, and \(t_{0.95,df}\) the 95th percentile of the \(t\) distribution with the model’s degrees of freedom. The usual two-sided 90% confidence interval on the log scale is then exponentiated endpoint by endpoint.
For the usual limits, the log-scale acceptance interval is \([\ln(0.80),\ln(1.25)]\), approximately \([-0.22314,0.22314]\). The equivalence decision is based on whether the entire 90% confidence interval lies within those limits—not whether the point estimate lies there, and not whether the interval merely overlaps them.
This rule is equivalent to two one-sided tests (TOST). The null hypothesis for an endpoint is that its log contrast is at or beyond at least one equivalence boundary; the alternative is that it lies between both boundaries. At significance level \(\alpha=0.05\), rejecting both one-sided null components corresponds to the 90% interval lying within the usual limits. When both AUC and Cmax are required, the overall conclusion requires both endpoint-specific criteria to be met.
Worked Endpoint Decisions
The following examples use hypothetical study summaries. In each, the degrees of freedom are 27, so the illustrative 90% interval uses \(t_{0.95,27}=1.703\). Calculations retain extra digits until the final reported values.
Worked Example: AUC Meets the Criterion but Cmax Does Not
Suppose the estimated log test-to-reference contrast for AUC is \(-0.1000\), with \(SE=0.0350\). For Cmax, suppose the estimate is \(-0.17433\), with \(SE=0.0604\). Both endpoints are analyzed with the same degrees of freedom, but have their own estimates and standard errors.
AUC. The margin is \(1.703(0.0350)=0.059605\). Its log-scale interval is \(-0.1000\pm0.059605=[-0.159605,-0.040395]\). Exponentiating gives a geometric mean ratio of \(\exp(-0.1000)=0.9048\), with a 90% interval of approximately \([0.8525,0.9604]\). Both limits are within 0.80 to 1.25, so AUC meets the usual criterion.
Cmax. The margin is \(1.703(0.0604)=0.1028612\). The log interval is \(-0.17433\pm0.1028612=[-0.2771912,-0.0714688]\), or \([-0.27719,-0.07147]\) to five decimal places. Exponentiating gives a geometric mean ratio of approximately \(0.8400\), with a 90% interval of approximately \([0.7579,0.9310]\). The lower limit is below 0.80, so Cmax does not meet the usual criterion.
Conclusion. The result is discordant: AUC meets the criterion and Cmax does not. If both endpoints are required, the study does not demonstrate bioequivalence under this criterion. It would be inaccurate to report that the product is bioequivalent overall merely because the AUC interval passes.
Worked Example: Cmax Meets the Criterion but AUC Does Not
Consider a different hypothetical study with an AUC log contrast of \(-0.2000\) and \(SE=0.0550\), and a Cmax log contrast of \(-0.0800\) and \(SE=0.0350\). Again, use \(df=27\) and \(t_{0.95,27}=1.703\).
AUC. The margin is \(1.703(0.0550)=0.093665\), giving the log interval \([-0.293665,-0.106335]\). Exponentiating, the geometric mean ratio is \(\exp(-0.2000)\approx0.8187\), and the interval is approximately \([0.7455,0.8991]\). Although the point estimate is within 0.80 to 1.25, the lower confidence limit is below 0.80. AUC does not meet the criterion.
Cmax. The margin is \(1.703(0.0350)=0.059605\), giving the log interval \([-0.139605,-0.020395]\). The geometric mean ratio is \(\exp(-0.0800)\approx0.9231\), and the interval is approximately \([0.8697,0.9798]\). Both confidence limits are within 0.80 to 1.25, so Cmax meets the criterion.
Conclusion. Cmax meets the criterion but AUC does not. If both endpoints are required, the overall conclusion is again that bioequivalence has not been demonstrated. The direction of discordance has changed, but the decision rule has not.
Worked Example: Checking Rounded Limits Before Deciding
Suppose a hypothetical analysis reports an AUC log contrast of \(-0.18633\) with \(SE=0.0400\), and a Cmax log contrast of \(-0.17433\) with \(SE=0.0604\). With \(df=27\), calculate the intervals before interpreting the endpoint decisions.
AUC. The margin is \(1.703(0.0400)=0.06812\). The log interval is \([-0.25445,-0.11821]\). Exponentiating the limits gives approximately \([0.7753,0.8885]\), or 77.53% to 88.85%. Since the lower limit is below 0.80, AUC does not meet the usual criterion.
Cmax. The margin is \(1.703(0.0604)=0.1028612\). The unrounded log limits are \(-0.17433-0.1028612=-0.2771912\) and \(-0.17433+0.1028612=-0.0714688\). Rounded to five decimal places, these are \([-0.27719,-0.07147]\). Exponentiating gives approximately \([0.7579,0.9310]\); its lower limit is also below 0.80.
Conclusion. Neither endpoint meets the criterion. The arithmetic illustrates why the unrounded margin should be carried through both endpoint calculations: rounding an intermediate value can change the last reported digit of a limit. Here, that correction does not change the decision, but accurate reporting still matters.
Why the Overall Decision Is Endpoint-Specific
When a regulatory or protocol-defined assessment requires both AUC and Cmax to meet the same limits, the overall requirement is an intersection: the AUC criterion must hold and the Cmax criterion must hold. This is not a rule to average the two ratios, pool their confidence intervals, or choose whichever result is more favorable. AUC and Cmax have different meanings and can have different uncertainty; a combined number would not answer whether each required characteristic of exposure satisfies its criterion.
The intersection rule also explains why separate endpoint testing does not mean that one successful result can rescue a failed one. To conclude that all required endpoints meet their criteria, every required endpoint must pass its own equivalence test. If one fails, the evidence does not establish the full set of requirements. A failure does not prove that the products are different; it means the analysis has not demonstrated equivalence under the prespecified criterion.
The usual 80% to 125% limits are a common average bioequivalence standard, not a universal rule for every product or jurisdiction. A protocol or applicable standard may specify different endpoints or limits. The analyst must apply the prespecified requirements rather than select limits after seeing the results.
Diagnostics, Alternatives, and Common Pitfalls
A discordant result should prompt careful review, not an automatic search for a preferred endpoint. Confirm that the AUC and Cmax values were derived and transformed as planned, that treatment labels and period assignments are correct, and that the analysis uses the intended population and model. Review missing or unusual concentration observations, sampling times relevant to peak capture, influential subjects, and protocol deviations. For the model, examine residual behavior and subject-level patterns, and verify that the treatment contrast and its standard error use the intended degrees of freedom. These checks can identify data or modeling problems, but a diagnostic does not justify changing the analysis after results are known without a prespecified basis.
The main alternative to separate endpoint-specific reporting is a joint or multivariate description of exposure outcomes. Such analyses may characterize how AUC and Cmax vary together, but they do not replace the endpoint-specific decision when the standard requires each measure to pass its own limits. Similarly, reporting only geometric mean ratios without confidence intervals describes estimated magnitude but not the uncertainty needed for the equivalence decision.
- Pitfall: judging by the point estimate. A ratio of 0.819 is within 0.80 to 1.25, but if its lower confidence limit is 0.746, the endpoint fails the interval criterion. The correction is to assess the entire interval.
- Pitfall: averaging AUC and Cmax results. A favorable AUC result cannot offset a Cmax interval that crosses a limit. The correction is to report and decide on each required endpoint separately.
- Pitfall: treating failure as proof of inequivalence. A failed interval means equivalence was not demonstrated at the specified precision and criterion; it does not establish that the true ratio is outside the limits.
- Pitfall: overinterpreting rounded values. Use full-precision calculations for endpoints and comparisons, then report suitably rounded results. Do not let a rounded limit obscure whether it is inside or outside a boundary.
The method is limited by the design, endpoint measurement, model assumptions, and chosen limits. A confidence interval that fails because it is wide can arise from imprecision even when the point estimate is near one. Conversely, a narrow interval does not correct a biased estimate caused by design or data problems. Interpret the statistical decision together with the study conduct and the prespecified scientific and regulatory requirements.
Check Your Understanding
Answer each question by applying the endpoint-specific decision rule. Brief solutions follow each question.
- An AUC ratio is 0.96 with a 90% interval of [0.82, 1.12], while the Cmax interval is [0.78, 1.08]. If both endpoints are required, what is the conclusion?
Solution: AUC meets the usual limits; Cmax does not because its lower limit is below 0.80. Overall bioequivalence is not demonstrated. - Why is an AUC ratio of 0.82 not sufficient, by itself, to conclude that AUC meets the usual criterion?
Solution: The ratio is a point estimate. The full 90% confidence interval must lie within 0.80 to 1.25. - For an endpoint, the log-scale 90% interval is \([-0.10,0.12]\). Does it meet the usual criterion?
Solution: Both limits lie between \(\ln(0.80)\approx-0.22314\) and \(\ln(1.25)\approx0.22314\), so the endpoint meets the criterion. - If AUC passes and Cmax fails, does the AUC result prove the products are equivalent in peak exposure?
Solution: No. AUC and Cmax describe different exposure features. AUC passing does not establish that Cmax meets its criterion. - Does a failed endpoint interval prove that the true test-to-reference ratio is outside the equivalence limits?
Solution: No. It shows that equivalence was not demonstrated under the specified interval rule; the interval may fail because of its location, its width, or both.