Inputs to Average Bioequivalence Sample Size Calculations

Why Sample-Size Inputs Must Be Considered Together

An average bioequivalence sample-size calculation is not determined by a single “typical” number of participants. It depends on how far the expected test-to-reference ratio lies from the equivalence limits, how much within-subject variability is expected, how strictly the tests control type I error, and how much power the design is intended to achieve. These inputs interact: a ratio nearer a boundary or greater variability generally makes the target harder to meet, while a larger sample can improve precision.

The calculation is conducted on the natural-log scale because average bioequivalence is assessed using a treatment ratio and its confidence interval. This tutorial focuses on specifying and interpreting the inputs. The design-specific calculation that converts them into a final sample size depends on the study layout and the chosen power method.

The Five Core Inputs

Equivalence limits. The limits define the acceptable range for the population ratio. They are set from the applicable scientific and regulatory requirements; they are not chosen to make a desired sample size attainable. For the commonly used ratio limits 0.80 and 1.25, the log limits are approximately \(-0.2231\) and \(0.2231\). The bounds are reciprocal, so their log-scale distances from zero are equal.

Expected geometric mean ratio. The expected ratio specifies where the true effect is anticipated to lie within the equivalence interval. If \(G_0=1\), the expected log effect is zero and the distances to the two log boundaries are equal when the limits are symmetric. If \(G_0\ne1\), the distance to the nearer boundary is smaller and often drives the required precision. Planning with \(G_0=1\) without justification can therefore be optimistic.

Type I error level. Average bioequivalence is commonly assessed with two one-sided tests (TOST). Each one-sided test uses significance level \(\alpha\). The equivalence conclusion requires rejecting both non-equivalence hypotheses: that the effect is at or below the lower limit, and that it is at or above the upper limit. Using \(\alpha=0.05\) for each one-sided test corresponds to the usual 90% confidence interval decision rule; it does not mean using a 95% interval. The choice of \(\alpha\) changes the critical value and thus the precision needed for a given power.

Target power. Power is the probability, under a specified planning value for the true effect and variability, that the study concludes equivalence by rejecting both one-sided null hypotheses. It is a design goal, not a guarantee for an individual study. Higher target power generally requires more information, and therefore often a larger sample.

Within-subject variability. Variability determines how precisely the treatment effect can be estimated in the chosen design. It must be expressed on the scale and in the form required by the calculation, typically as a log-scale variance or standard deviation, or as a design-specific variance for the treatment contrast. A variance estimate from a different design or scale should not be inserted without the appropriate conversion. The study layout determines how within-subject variability translates into the standard error; the same variability value does not imply the same precision in every design.

How the Inputs Work Together

A useful way to organize the inputs is to separate the location of the expected effect from its uncertainty. The expected ratio and limits determine the available margins \(d_L\) and \(d_U\). The variability and design determine the standard error at a candidate sample size. The type I error level determines the critical value used to assess the confidence interval, while the target power specifies how often that interval should lie wholly within the limits under the planning assumptions.

For orientation, suppose a normal approximation treats the estimated log effect as \(N(\theta_0,SE^2)\), with a fixed standard error \(SE\). Let \(z_{1-\alpha}\) be the \(1-\alpha\) standard-normal quantile. The approximate confidence interval is the estimate plus or minus \(z_{1-\alpha}SE\). It is wholly within \((L,U)\) precisely when the estimate lies between \(L+z_{1-\alpha}SE\) and \(U-z_{1-\alpha}SE\). Under this simplified approximation, the probability of that event is

where \(\Phi\) is the standard normal cumulative distribution function. This expression is an explanatory approximation, not a substitute for the design-specific power calculation. Its useful message is direct: increasing \(SE\) leaves less room for the estimate to fall inside the interval, and, with fixed limits and \(SE\), when the interval can fit inside the limits, the probability is maximized when \(\theta_0\) is at the midpoint of the limits and decreases as \(\theta_0\) moves farther from that midpoint. In an actual calculation, \(SE\) depends on the sample size, design, and variability; finite-sample procedures may also use a \(t\) critical value and degrees of freedom.

Type I Error and Target Power Are Different Inputs

The significance level and power target answer different questions. The significance level controls the probability of falsely concluding equivalence when the true effect is on or beyond a non-equivalence boundary. The power target concerns the probability of concluding equivalence under specified values that lie inside the limits. Neither quantity determines the other. A calculation therefore needs both, along with the assumed true effect and variability.

For TOST, the two tests are both required for success. The conventional rule uses \(\alpha\) for each one-sided test, rather than dividing \(\alpha\) between the two tests as in a two-sided superiority test. Under the usual interval-test correspondence, the two one-sided tests at level \(\alpha\) conclude equivalence when the corresponding \(100(1-2\alpha)\%\) confidence interval is contained within the limits. This distinction is important when specifying the procedure in software or communicating the planned confidence level.

Variability, Design, and Precision

Variability enters through the uncertainty of the treatment-effect estimate. At a fixed sample size, greater within-subject variability generally produces a larger standard error. To maintain the same probability of placing the confidence interval inside the limits, the study then needs more information, often through a larger sample. Conversely, a low variability assumption can yield an undersized study if it is not supported by relevant evidence.

The variability input should match the planned design. In a crossover study, repeated observations on a participant contribute information about the within-participant treatment contrast. The sequence structure and the model used to estimate the treatment effect determine how the variance components contribute to its standard error. It is not enough to state a variability value without specifying whether it is a log-scale standard deviation, a variance, or a variance for a particular contrast.

The following calculation illustrates the effect of precision without claiming a final sample size. Use symmetric log limits \(-0.2231\) and \(0.2231\), an expected log effect of zero, and \(\alpha=0.05\). Under the fixed-\(SE\) normal approximation above, the probability that the interval is contained within the limits is \(2\Phi(M/SE-z_{0.95})-1\), where \(M=0.2231\), provided the interval can fit inside the limits. If \(SE=0.08\), then \(M/SE-z_{0.95}=0.2231/0.08-1.645=1.144\), giving approximately \(2\Phi(1.144)-1=0.747\). This is below a 0.90 power target. The calculation shows why a candidate design’s standard error must be checked against the margins; the actual sample size is found by linking that standard error to the planned design and sample size using the appropriate method.

Assumptions, Diagnostics, and Common Pitfalls

Before accepting a sample-size result, check that the model and inputs correspond to the proposed study. The log-scale treatment effect should represent the estimand used for the bioequivalence assessment; the equivalence limits must be expressed on the same scale; and the assumed within-subject variability must match the design and analysis. The selected power method should also reflect whether its approximation is adequate for the planned sample size and degrees of freedom.

Useful diagnostics include calculating both log margins explicitly, checking that the expected ratio lies inside the limits, confirming the units and scale of the variability input, and comparing the assumed variability with plausible supporting information. If the expected effect is close to a boundary, investigate the consequences of reasonable shifts in that expectation. If a software result seems unexpectedly small, verify that \(\alpha\) was entered as a one-sided level and that the variability parameter was not confused with its square root.

• Using the wrong distance. A ratio of 1.10 does not have the same margin to each boundary as a ratio of 1.00. Compute \(\theta_0-L\) and \(U-\theta_0\) separately.
• Confusing \(\alpha\) with power. Alpha controls false equivalence conclusions at a boundary; power is the success probability under a specified alternative. They are separate design choices.
• Using a superiority-test critical value by habit. TOST at one-sided \(\alpha=0.05\) corresponds to a 90% interval, not a 95% interval.
• Entering a variability value on the wrong scale. A coefficient of variation, a log-scale standard deviation, and a variance are not interchangeable inputs. Confirm the required form before calculation.
• Treating an approximation as an exact result. A normal illustration clarifies the roles of margins and precision, but the final calculation must suit the design and analysis.

No sample-size formula can compensate for implausible planning assumptions. The number produced is conditional on the specified limits, expected ratio, type I error, power target, variability, and design. Reporting those assumptions alongside the result makes the calculation reviewable and helps distinguish a genuine design decision from a software default.