What a Failed Average Bioequivalence Result Can Tell You
A failed average bioequivalence (BE) conclusion is not, by itself, an explanation. It says that the required confidence interval did not fit entirely inside the prespecified BE limits. To understand what happened, separate two features of the result: where the estimated treatment ratio is centered and how wide its confidence interval is. A treatment estimate displaced from the target can cause failure even with a reasonably precise study; a wide interval can cause failure even when the estimate is close to the target. Both can contribute.
This diagnosis is a way to organize review, not a way to revise the decision rule after seeing the data. The endpoint, analysis model, analysis set, and BE limits must come from the prespecified analysis and applicable requirements. A numerical decomposition can show what contributed to the observed interval, but it cannot establish why a study had that result or justify changing the analysis.
Center and Width on the Log Scale
For a conventional analysis using a \(t\)-based interval, the estimated contrast and its 90% confidence interval are
Here \(h\) is the interval half-width, \(t_{0.95,df}\) is the 95th percentile of the \(t\) distribution with the model's degrees of freedom, and \(df\) denotes those degrees of freedom. The interval endpoints on the GMR scale are \(\exp(d-h)\) and \(\exp(d+h)\). Exponentiation preserves the endpoint ordering, but the resulting interval is generally asymmetric around the GMR.
The BE limits are symmetric on the log scale because \(\log(0.80)=-\log(1.25)\). Write \(m=\log(1.25)\), approximately 0.2231. The interval lies within the limits exactly when
or, equivalently, when \(|d|+h\leq m\). The absolute value \(|d|\) measures how far the estimated contrast is displaced from the target of zero; \(h\) measures uncertainty around that estimate. This provides a compact diagnostic: if \(|d|+h\) exceeds the allowed log-scale distance \(m\), the interval fails containment. It does not mean the two contributions are independent causes; it is a decomposition of the observed decision.
This formulation follows directly from the interval endpoints. The lower endpoint constraint requires \(d-h\geq-m\), or \(h\leq d+m\). The upper endpoint constraint requires \(d+h\leq m\), or \(h\leq m-d\). Both must hold, so \(h\leq\min(d+m,m-d)=m-|d|\). Rearranging gives \(|d|+h\leq m\).
Worked Example: Failure Mainly From an Upward Treatment-Ratio Estimate
Suppose a hypothetical crossover study reports \(d=0.245\), \(SE=0.0550\), and \(df=23\). For a 90% interval, use \(t_{0.95,23}=1.714\). Assume these are outputs from the prespecified model and that the study's planned analysis supports using this interval.
Calculate the half-width.
Find the log-scale interval and GMR.
The point estimate is \(\exp(0.245)\approx1.278\), or a treatment-to-reference GMR of about 127.8%. The confidence limits are \(\exp(0.1507)\approx1.163\) and \(\exp(0.3393)\approx1.404\), giving an interval of about 116.3% to 140.4%. Its upper endpoint exceeds 125.0%, so the interval is not contained within the BE limits.
Here \(|d|=0.245\) already exceeds \(m\approx0.2231\), even before adding interval width. The estimated ratio is displaced above the upper limit; the interval's width adds further uncertainty. A plausible review would therefore examine the treatment contrast and the data and model features that might affect it, as well as precision. The calculation alone does not show which feature of the study produced the displacement.
Interpreting the Shape of the Failure
The interval's position relative to the limits can help distinguish patterns, but it does not prove their causes. An interval entirely below 80.00% is consistent with a strongly downward-displaced estimate; one entirely above 125.00% is consistent with a strongly upward-displaced estimate. If an interval crosses a limit, the estimate may be near that boundary, the interval may be wide, or both. Check the actual endpoints rather than labeling every failure as a treatment-ratio problem or a sample-size problem.
A point estimate inside the BE limits is not enough. The decision requires the entire confidence interval to be contained within them. Conversely, a confidence interval that overlaps the BE region is not a passing interval: overlap is not the same as containment. The 90% confidence interval is interpreted as an interval procedure with its stated long-run coverage under the model assumptions; it is not a 90% probability statement about the fixed, unknown treatment effect for this particular interval.
Worked Example: Failure Mainly From Interval Width
Consider a second hypothetical study with \(d=0.0150\), \(SE=0.135\), and \(df=18\). Use \(t_{0.95,18}=1.734\). Suppose the prespecified endpoint, model, and analysis set are appropriate for the study.
Calculate the interval.
The point estimate is \(\exp(0.0150)\approx1.015\). The interval on the GMR scale is approximately \([\exp(-0.2191),\exp(0.2491)]=[0.8033,\ 1.283]\), or about 80.33% to 128.3%. The lower endpoint is above 80.00%, but the upper endpoint exceeds 125.00%; the interval fails containment.
The estimate is close to the target of 1, while the half-width \(0.2341\) is greater than \(m-|d|\approx0.2081\). Thus the interval is too wide to fit, despite the favorable point estimate. This identifies imprecision as the immediate mathematical contributor to failure. It does not establish that sample size was the cause: variability, model degrees of freedom, missing observations, or other design and data features can also affect the standard error.
Using Study Summaries to Direct a Review
A useful review starts with the reported estimate, interval, standard error, degrees of freedom, and analysis model. Then verify that the interval can be reproduced from those summaries, allowing for rounding. If the report gives only GMR-scale endpoints, take logarithms to recover approximate log-scale endpoints. Their midpoint estimates the center and half their difference estimates the half-width. If the endpoints are rounded, these calculations are approximate; do not claim more precision than the report supports.
When the estimate is the main contributor, examine prespecified diagnostics that could help explain the contrast: treatment-by-period patterns, sequence patterns, data integrity, influential subjects, concentration-profile issues, and protocol deviations. These are prompts for checking source data and model fit, not automatic grounds for excluding observations. Earlier material on sequence and period patterns, influential subjects, and data-integrity diagnostics provides the relevant diagnostic concepts.
When interval width is prominent, review the model-based standard error and degrees of freedom, the number of evaluable subjects, within-subject variability, missing or incomplete profiles, and the balance of observations across periods and sequences. Check whether the planned model and covariance structure were fitted as intended and whether the analysis data are consistent with the prespecified population. A wide interval can have more than one contributor, and summary statistics alone usually cannot identify a single cause.
Worked Example: A Narrow Miss and a Focused Diagnostic Plan
A hypothetical study reports \(d=0.0400\), \(SE=0.110\), and \(df=29\). Use \(t_{0.95,29}=1.699\), and assume the estimate and standard error come from the planned analysis.
Reconstruct the interval.
Exponentiating gives an approximate GMR interval of \([0.8634,\ 1.255]\), or 86.34% to 125.5%. The upper endpoint is above 125.0%, so the result fails, although the exceedance is small. The estimated GMR is \(\exp(0.0400)\approx1.041\), or 104.1%.
For this center, containment would require \(h\leq m-|d|=0.22314-0.0400=0.18314\). With the stated degrees of freedom, that corresponds to \(SE\leq0.18314/1.699\approx0.1078\). The observed \(SE=0.110\) is slightly larger. This comparison shows that the observed width is sufficient to place the upper endpoint just outside the limit; it does not prove that increasing enrollment would have produced a passing result.
A focused review would first confirm the calculation, the prespecified model, and the source of the reported standard error. It would then check data completeness, variability, and model diagnostics, along with any documented deviations that affect the planned analysis. The narrow miss does not justify changing the endpoint, removing a subject, or selecting another model merely to move the interval inside the limits.
Assumptions, Alternatives, and Common Pitfalls
These calculations presume a valid estimate and standard error from the prespecified analysis. For a conventional crossover analysis, the model must represent the planned treatment, period, and sequence structure appropriately; the analyzed observations must be correctly linked to their subjects and periods; and the model-based uncertainty calculation must be suitable for the design and data. The study design and conduct must also support the intended comparison. Inspect residuals and influential observations, and investigate data integrity or incomplete profiles where relevant. Diagnostics can reveal concerns, but a diagnostic result should be interpreted in context rather than used as an automatic exclusion rule.
The standard alternatives answer different questions. The point estimate alone describes the estimated ratio, not the precision needed for the BE decision. A test of equality against a null value is not a substitute for the BE procedure: conventional average BE uses two one-sided tests, equivalently assessed through containment of the corresponding 90% confidence interval. Reference-scaled methods apply only under their applicable prespecified conditions; they are not a post hoc remedy for a failed conventional interval. The earlier material on replicate-data reference scaling addresses that distinct approach.
- Pitfall: “The GMR is within 80%–125%, so BE passed.” Correction: the full confidence interval, not only the point estimate, must be within both limits.
- Pitfall: “The interval overlaps the BE region, so the result is close enough.” Correction: overlap does not satisfy containment; identify the endpoint that crosses a limit.
- Pitfall: “A wide interval proves the study was under-enrolled.” Correction: width depends on the standard error and degrees of freedom, which can reflect several design, variability, and data features. The interval alone does not establish the cause.
- Pitfall: “Removing an unusual subject or changing the model could make the interval pass.” Correction: an analysis change needs a valid, prespecified or otherwise scientifically justified basis, and its impact must be transparently assessed. A desired conclusion is not a statistical justification.
The decomposition is therefore most useful as a disciplined starting point: report the estimate, half-width, and endpoint that failed; verify the planned analysis; and use relevant diagnostics to investigate plausible explanations. It describes the geometry of the result, not a license to optimize the result after the fact.
Check Your Understanding
Answer each question using the log-scale decomposition and the conventional 80.00%–125.00% limits.
- A study has \(d=0.050\) and \(h=0.160\). Does its interval satisfy containment? Which component is larger relative to the available margin?
- A GMR estimate is 0.78, but its confidence interval is narrow. Is the result primarily a precision problem or a displacement problem?
- A confidence interval extends from 1.10 to 1.27 on the GMR scale. Does it pass? State precisely how it relates to the upper limit.
- Name two different features, other than sample size, that could contribute to a wide confidence interval.
- Why is it not valid to delete a subject solely because doing so would make the confidence interval pass?
Solutions. (1) The sum is \(|d|+h=0.050+0.160=0.210\), which is less than \(m\approx0.2231\), so the interval passes. The half-width is larger than the displacement, but both fit within the margin. (2) The estimate 0.78 is below the lower BE limit of 0.80, so the main issue is displacement; a narrow interval does not remove that concern. (3) It fails because the upper endpoint, 1.27, exceeds 1.25. The interval crosses the upper limit: its lower endpoint is below 1.25 and its upper endpoint is above 1.25. (4) Examples include high within-subject variability, reduced degrees of freedom, missing observations, or an unsuitable model-based uncertainty estimate. (5) Exclusion must have a scientifically and analytically justified basis consistent with the prespecified rules; choosing subjects based on whether their removal changes the decision biases the analysis.