What an Inconclusive Result Can Tell You
When a bioequivalence (BE) study does not demonstrate equivalence, the decision alone does not explain why. A confidence interval may be too wide to fit inside the BE limits, the estimated treatment ratio may be near a limit, or both. Distinguishing these cases matters: uncertainty about the ratio is not the same as evidence that the ratio is displaced from 1, and neither diagnosis by itself guarantees that a new study will reach a different conclusion.
For a conventional average BE analysis, the treatment contrast is estimated on the log scale and then expressed as a test-to-reference geometric mean ratio. The usual BE limits are 80.00% and 125.00%, equivalent to log limits \(\log(0.80)\) and \(\log(1.25)\). A 90% confidence interval entirely within those limits supports a conclusion of BE under the specified procedure. If an interval overlaps a limit, the study has not demonstrated BE; the interval’s location and width help describe the result, but do not change that decision rule.
Model and Assumptions
For a standard two-treatment, two-period crossover analysis, let \(Y\) denote the log-transformed pharmacokinetic endpoint for a subject and period. The treatment contrast is the estimated difference between the test and reference means on the log scale, denoted here by \(\hat{\theta}\). Exponentiating \(\hat{\theta}\) gives the estimated geometric mean ratio. A typical analysis uses a model that accounts for treatment, period, and sequence, with subject structure appropriate to the crossover design.
The confidence interval described here assumes the model is appropriate for the endpoint and analysis set: subjects are independent of one another; residual variation is adequately represented by the fitted model; the log-scale residuals are sufficiently well behaved for the model-based \(t\) interval; and the treatment contrast and its standard error are calculated using the planned model and valid observations. The design and conduct must also support the model’s interpretation—for example, a crossover analysis relies on the planned treatment-period structure and absence of a consequential carryover effect. Serious protocol or data issues can undermine the usefulness of an apparently precise interval.
For a model-based contrast with standard error \(SE\) and \(df\) degrees of freedom, the two-sided 90% interval on the log scale is:
The corresponding interval for the geometric mean ratio is obtained by exponentiating both endpoints. Because exponentiation is increasing, the order of the endpoints is unchanged. Equivalence is demonstrated only if the whole ratio interval lies between 0.80 and 1.25, including neither boundary or any values beyond it.
Reading Width and Location Separately
On the log scale, the interval’s full width is \(2t_{0.95,df}SE\), and its half-width is \(t_{0.95,df}SE\). Width is therefore influenced by both the achieved \(SE\) and the critical \(t\) value. When comparing studies or considering design changes, report which scale and summary are being used: ratio-scale interval widths are not symmetric around the point estimate, whereas the log-scale interval is symmetric.
The estimated ratio’s displacement from 1 is readily described by \(\hat{\theta}\), since \(\hat{\theta}=0\) corresponds to a ratio of 1. A ratio close to 1 with a confidence interval crossing a BE limit suggests that width is an important part of the inconclusive result. A ratio farther from 1 with a relatively narrow interval suggests that the estimate’s location matters more. These descriptions are diagnostic, not separate regulatory tests. They do not establish the true ratio, and a point estimate inside the limits is not sufficient to demonstrate BE.
Worked Example: A Near-Central Estimate With a Wide Interval
In a hypothetical crossover study of AUC, suppose the estimated log treatment contrast is \(\hat{\theta}=0.015\), with \(SE=0.130\) and \(df=23\). For a two-sided 90% interval, use \(t_{0.95,23}=1.714\). The model assumptions are taken to be reasonable for this hypothetical example.
The half-width is \(1.714(0.130)=0.2228\). Thus, the log-scale interval is:
Exponentiating gives the estimated ratio and interval:
The point estimate is close to 1, but the interval extends above 1.25. The study therefore does not demonstrate BE. Since the estimate is near 1 and the interval is relatively broad, the interval width is an important feature of the result. This does not prove that the true ratio is 1 or that a larger study would succeed; it indicates that this study’s estimate is too uncertain to establish that the ratio lies within the BE limits.
When the Estimate Is Displaced Toward a Limit
A confidence interval can be comparatively narrow yet still overlap a BE limit because the estimated ratio is close to that limit. In that situation, “more precision” is not a complete diagnosis: the estimate’s location is also consequential. Increasing the sample size may narrow a future interval, but it does not systematically move the treatment estimate toward 1. If the underlying ratio is genuinely outside the BE limits, a more precise study may instead provide clearer evidence against BE.
Worked Example: A Displaced Estimate With a Narrower Interval
In a second hypothetical study, suppose the estimated log contrast is \(\hat{\theta}=0.1823\), corresponding to a ratio of about 1.200. Let \(SE=0.040\), \(df=23\), and use \(t_{0.95,23}=1.714\). Assume the model and data are adequate for the interval calculation.
The half-width is \(1.714(0.040)=0.06856\), so the log-scale interval is:
On the ratio scale, the interval is approximately \((1.120,\ 1.285)\). Its upper endpoint exceeds 1.25, so BE is not demonstrated. Relative to the first example, this interval is narrower, but the estimated ratio is more displaced from 1 and near the upper BE limit. The result is inconclusive under the interval criterion, not proof that the true ratio equals 1.200 or lies outside the limits. A future study should be planned with both the plausible ratio and the desired precision in view.
From Achieved Precision to Design Implications
A post-study assessment can inform design planning, but it is not a substitute for prospective planning. For example, if the observed \(SE\) is larger than anticipated, investigators can examine whether the estimate reflects unexpectedly high within-subject variability, fewer usable observations than planned, or other features of the data or analysis. The explanation should be supported by the design records and diagnostics rather than inferred from interval width alone.
In a balanced design with otherwise comparable conditions, standard errors often decrease approximately in proportion to \(1/\sqrt{n}\), where \(n\) is the number of subjects contributing to the analysis. This is a planning approximation, not a guarantee. A future study may have a different variance, dropout rate, or estimate, and its critical \(t\) value can also change. Sample-size recommendations should therefore be based on an explicit target, justified variability assumptions, and the intended decision procedure.
Worked Example: Using the Observed Width as a Planning Check
Return to the first hypothetical study, with \(n=24\), \(SE=0.130\), and a log-scale 90% interval half-width of \(h=0.2228\). Suppose a design discussion proposes a target half-width of \(0.180\) log units, assuming the treatment contrast remains near its observed value and the variability is similar.
Using the approximate relation \(SE\propto1/\sqrt{n}\), and temporarily holding the critical value fixed, the required sample size is estimated by:
This rough calculation suggests about 37 analyzable subjects, before allowing for dropout or refining the calculation with the planned model and critical values. It is only a precision-planning illustration. With the observed contrast \(0.015\), a half-width of \(0.180\) would give an upper log endpoint of \(0.195\), below \(\log(1.25)\approx0.223\). But the future estimate may differ, so this calculation does not guarantee BE or establish that 37 enrolled subjects are sufficient.
Diagnostics, Alternatives, and Limitations
Before interpreting precision, check that the reported estimate and interval come from the prespecified endpoint, analysis set, model, and transformation. Review the number of subjects and usable observations, missingness, influential observations, residual patterns, and any design or protocol concerns that could affect the model or treatment contrast. If different reasonable analyses give materially different estimates or intervals, the result is sensitive to analysis choices; a single narrow interval should not conceal that instability.
The confidence-interval approach is the direct way to connect estimation with the BE limits. A p-value alone is less informative for post-study precision: failing to reject a null hypothesis does not quantify how wide the plausible treatment-ratio range is. Retrospective “observed power” calculated from the estimated effect and observed variance is also not a replacement for the interval; it largely repackages the same data and does not resolve uncertainty about the true ratio. A prospective power or sample-size calculation can help plan a new study, but it answers a planning question under specified assumptions rather than reclassifying the completed result.
The main limitation is that the observed interval describes uncertainty under the fitted model and its assumptions. It does not diagnose the biological or formulation cause of a displaced ratio, prove that a design change will correct it, or predict a future study with certainty. A defensible recommendation might be to improve precision through a justified increase in analyzable sample size when uncertainty is the main concern, or to investigate the product, dose, conduct, and analysis when displacement or instability is prominent. The evidence should determine which interpretation is plausible.
A Practical Interpretation Sequence
Confirm the endpoint, model, analysis set, degrees of freedom, and assumptions relevant to the contrast and its standard error.
Check whether the full 90% ratio interval lies within 0.80–1.25. Do not substitute the point estimate for this comparison.
Report the interval and estimate, and distinguish a broad interval around a near-central estimate from a narrower interval near a BE limit.
Use evidence about variability, usable sample size, and study conduct to discuss a future design; state the assumptions and avoid promising a different outcome.
Check Your Understanding
Answer each question using the distinction between interval width and estimate location. Solutions follow the questions.
- A hypothetical ratio estimate is 1.01, with a 90% confidence interval of 0.79–1.22. Is BE demonstrated, and what feature of the result is especially notable?
- A ratio estimate is 1.24, with a 90% confidence interval of 1.18–1.31. Why is “the interval is fairly narrow” not enough to conclude BE?
- On the log scale, an estimate is \(0.000\) and its 90% interval is \((-0.10,0.10)\). What is the ratio estimate, and what does the interval say relative to the usual limits?
- Why does a larger sample size not guarantee that a future ratio estimate will move closer to 1?
Solutions. (1) BE is not demonstrated because the lower endpoint, 0.79, is below 0.80. The estimate is near 1, so the interval’s width is a notable contributor to the inconclusive outcome. (2) The upper endpoint, 1.31, exceeds 1.25. Interval width does not override the requirement that both endpoints be within the limits; the estimate is also displaced toward the upper limit. (3) The ratio estimate is \(\exp(0)=1.000\). The exponentiated interval is approximately \((0.9048,1.105)\), entirely within 0.80–1.25, so it meets the interval criterion, assuming the model and analysis are appropriate. (4) Increasing sample size can reduce sampling uncertainty under suitable conditions, but it does not systematically change the underlying treatment ratio. The future estimate can remain displaced or vary in either direction.