Distinguishing Population and Individual Bioequivalence Power Targets

Why the Power Target Is Not Just a Mean Difference

Two formulations can have nearly the same average exposure and still pose different bioequivalence planning problems. Population bioequivalence (PBE) concerns a population-level comparison that includes a difference in overall variability. Individual bioequivalence (IBE) also addresses within-subject variability and subject-by-formulation interaction. Consequently, the average formulation effect alone does not determine either criterion or the sample size needed to assess it.

This tutorial compares those power targets, not every regulatory implementation of either criterion. The formulas below give a clear component-based representation and a normal-approximation planning calculation. Actual criteria, estimators, and study designs must follow the applicable protocol and regulatory specification; the approximation is not a replacement for design-specific power calculation.

Both expressions include the squared mean difference, but their variability terms are not interchangeable. PBE uses the difference in population variances. IBE includes a within-subject variance contrast and a subject-by-formulation interaction component. A study may therefore have a small \(d\) while its variance components make one criterion less favorable, less precisely estimated, or both.

A crucial distinction is between the true criterion, such as \(\psi_I\), and the uncertainty in its estimate. A criterion below its limit is not automatically demonstrated by the data: sampling uncertainty may leave its upper confidence bound above the limit. Conversely, increasing sample size can reduce uncertainty, but cannot make a true criterion above its limit become compliant.

Criteria, Limits, and Planning Assumptions

For a scalar upper-bound criterion, the planning question can be framed as testing \(H_0:\psi\ge M\) against \(H_a:\psi<M\), where \(\psi\) is the relevant PBE or IBE criterion. A common one-sided decision rule declares success when the estimated criterion plus a one-sided confidence allowance is below \(M\). The exact allowance and estimator depend on the specified method; the simple approximation used here assumes an estimator with standard error approximately \(c/\sqrt{n_{\mathrm{eff}}}\).

This approximation assumes a correctly specified criterion and estimator, a standard error that decreases approximately as \(1/\sqrt{n_{\mathrm{eff}}}\), and a normal approximation that is adequate at the planned size. It also assumes a single scalar decision. If success requires several endpoints or criteria simultaneously, the power of each separately is not the joint power; the joint requirement needs its own calculation.

The expression separates three influences. The target quantiles depend on the chosen \(\alpha\) and power; the gap \(M-\psi\) measures how far the true criterion is inside its limit; and \(c\) represents estimation uncertainty. As the gap shrinks, required information rises rapidly because it is squared in the denominator. As uncertainty increases, required information rises with \(c^2\).

Why the Sample-Size Approximation Has This Form

The derivation follows directly from the upper-bound decision rule. Write \(SE=c/\sqrt{n_{\mathrm{eff}}}\). Under the normal approximation, success occurs when \(\hat{\psi}+z_{1-\alpha}SE<M\). At the planning value \(\psi\), the standardized estimate \((\hat{\psi}-\psi)/SE\) is approximately standard normal. Therefore the success probability is the probability that this standardized estimate is less than \((M-\psi)/SE-z_{1-\alpha}\), giving the power expression above.

To obtain the target information, set this power equal to \(1-\beta\). By the definition of \(z_{1-\beta}\), the required inequality is \((M-\psi)/SE-z_{1-\alpha}\ge z_{1-\beta}\). Rearranging and substituting \(SE=c/\sqrt{n_{\mathrm{eff}}}\) gives \(\sqrt{n_{\mathrm{eff}}}\ge (z_{1-\alpha}+z_{1-\beta})c/(M-\psi)\). Squaring both sides yields the stated approximation, provided \(M-\psi>0\). If \(\psi\ge M\), this inside-the-limit planning calculation does not apply: adding information cannot create the assumed positive gap.

The derivation also explains why PBE and IBE should not be assigned the same sample size merely because their estimated \(d\) values are similar. Their criteria may differ, and the estimator uncertainty \(c\) may differ. For IBE in particular, estimating additional variance components may require a design that supplies the needed within-subject information. Thus \(n_{\mathrm{eff}}\) is an information scale for this approximation, not automatically the number of randomized participants.

Uncertainty Can Change the Information Requirement

Even when two criteria have the same distance to their respective limits, their estimates need not have the same uncertainty. Variance-component estimation, study design, and the covariance among estimated components all affect precision. The next calculation isolates this effect using the normal approximation. Its \(n_{\mathrm{eff}}\) values are effective information units; converting them to a participant count requires a design-specific model.

Reading Power at a Fixed Information Level

A planned sample size is more interpretable when checked against the power it implies. In the approximation, power depends on the standardized gap \((M-\psi)/SE\). A calculation that reports a target without checking the gap, uncertainty coefficient, and assumed criterion can conceal a mismatch between the scientific target and the proposed study.

Design, Diagnostics, and Limits of the Comparison

IBE planning is especially sensitive to whether the proposed design can identify the variance components in its criterion. A design that does not provide the required repeated treatment information cannot estimate a subject-by-formulation component merely by enrolling more participants. Replicate-design choices and their implications for IBE are addressed separately; here the key planning point is that the uncertainty coefficient must reflect the actual design and estimator.

Several checks help identify a faulty or overconfident calculation:

• Check the criterion, not just \(d\). Recompute every mean and variance component using the specified definitions and scale. Confirm the sign and units of each variance contrast.
• Check the gap. Use \(M-\psi\), not \(\psi-M\). A nonpositive gap cannot support the inside-the-limit sample-size formula.
• Check the uncertainty model. The coefficient \(c\) must correspond to the estimator, design, and variance components being planned. Reusing a PBE uncertainty estimate for IBE without justification is not a valid comparison.
• Check the information-to-participant conversion. Effective information units are not necessarily randomized participants. Account for allocation, repeated measurements, and any specified attrition assumptions in the design-specific calculation.
• Check the approximation. Skewed or constrained variance-component estimators, small samples, boundary values, or correlated endpoints can make a normal scalar approximation unreliable. Use an appropriate estimator-based or simulation-based calculation when these features matter.

The main alternative to this scalar approximation is a calculation based directly on the specified criterion estimator and its sampling distribution, or a design-specific simulation that reproduces the planned analysis. Those approaches can represent correlations among components, boundary behavior, and joint success rules more faithfully, but require a more complete model and reliable planning inputs. The approximation is useful for understanding direction and scale; it should not be mistaken for a universal regulatory formula.