Why Population Bioequivalence Needs Its Own Sample-Size Plan
Population bioequivalence (PBE) concerns whether the distributions of a pharmacokinetic endpoint under two formulations are sufficiently similar in a population. A plan based only on the treatment-reference difference in mean exposure can miss an important part of this question: the formulations may have similar means but different population variances. Sample size must therefore reflect both the mean and variance components of the specified PBE criterion.
For a log-transformed endpoint, let \(\mu_T\) and \(\mu_R\) be the population means for the test and reference formulations, and let \(\sigma_T^2\) and \(\sigma_R^2\) be their population variances. One commonly used PBE criterion compares the quantity \((\mu_T-\mu_R)^2+\sigma_T^2-\sigma_R^2\) with a prespecified limit \(\theta\). The protocol and applicable regulatory framework must specify the criterion and its limit; the limit used below is illustrative.
The criterion is not simply a requirement that the two means be close. A nonzero mean difference and a positive variance difference both use up some of the available margin. Conversely, a negative variance difference can offset part of the squared mean difference under this particular criterion. That arithmetic does not mean that an investigator may choose the variance component after seeing the data; the criterion, estimand, and analysis must be specified in advance.
Planning Model and Approximate Standard Error
A useful planning calculation needs a model for how the criterion will be estimated. For a transparent balanced design, suppose there are \(n\) independent evaluable observations under each formulation, with normally distributed log endpoints and population variances \(\sigma_T^2\) and \(\sigma_R^2\). Let \(\bar X_T\) and \(\bar X_R\) be the sample means and \(s_T^2\) and \(s_R^2\) the sample variances. The natural estimator is \(\hat g=(\bar X_T-\bar X_R)^2+s_T^2-s_R^2\).
For this independent-groups model, the sample means and sample variances are independent within each normal group, and the two groups are independent. The difference in sample means has variance \((\sigma_T^2+\sigma_R^2)/n\). For normal data, the variance of a sample variance is \(2\sigma^4/(n-1)\). Applying the first-order delta method to the squared mean difference gives the planning approximation below. It treats \(n-1\) as approximately \(n\) in the variance-component terms.
The mean term follows because the derivative of \(d^2\) at the assumed true mean difference \(d=\Delta\) is \(2\Delta\), so its first-order variance is \(4\Delta^2(\sigma_T^2+\sigma_R^2)/n\). The two variance estimates contribute approximately \(2\sigma_T^4/n\) and \(2\sigma_R^4/n\); their variances add because the groups are independent. This is a large-sample approximation, not an exact finite-sample distribution for \(\hat g\).
The model is consequential. In a crossover or replicate design, observations on the same participant are not independent in the way assumed here. The variance of the estimator must then reflect the design's covariance structure and the variance components actually targeted by the PBE criterion. The formula above is a planning illustration for independent groups, not a universal formula to insert unchanged into every PBE study.
From the Criterion to a Sample-Size Requirement
For planning, suppose the analysis uses a one-sided upper confidence bound for \(g\), and concludes that the criterion is met only if that bound is below \(\theta\). With significance level \(\alpha\), the approximate rule is \(\hat g+z_{1-\alpha}SE(\hat g)<\theta\), where \(z_q\) is the \(q\)-quantile of the standard normal distribution. If the assumed true value is \(g<\theta\), then the approximate probability of meeting this rule is \(\Phi((\theta-g)/SE-z_{1-\alpha})\).
Thus, to target power \(1-\beta\), the planning standard error should be no larger than \((\theta-g)/(z_{1-\alpha}+z_{1-\beta})\). This calculation makes the assumed true mean and variance effects explicit. It also shows why a criterion limit alone is not enough to determine sample size: the distance from the assumed true \(g\) to \(\theta\) matters.
These expressions use a normal approximation to the estimator and a prespecified target power. They provide a planning calculation; they do not replace the final analysis method or resolve whether the model is suitable for the study design. In particular, assumptions used for planning should be checked against relevant prior evidence and justified for the intended population and endpoint.
Worked Example: Separating the Mean and Variance Contributions
Consider a hypothetical log-scale endpoint with \(\Delta=0.10\), \(\sigma_T^2=0.0546\), and \(\sigma_R^2=0.0471\). Use the illustrative limit \(\theta=(\ln 1.25)^2\). Determine the assumed criterion value and its distance from the limit.
The squared mean contribution is \(\Delta^2=(0.10)^2=0.0100\). The variance contribution is \(\sigma_T^2-\sigma_R^2=0.0546-0.0471=0.00750\). Therefore,
Since \(\ln(1.25)\approx0.22314\), the illustrative limit is \(\theta\approx(0.22314)^2=0.04979\). The planning distance is \(\theta-g=0.04979-0.01750=0.03229\). Under these assumptions the true criterion value is below the limit; the distance \(0.03229\), rather than the limit by itself, informs the sample-size calculation. All quantities are on the log-scale criterion scale.
Worked Example: Finding the Balanced Evaluable Sample Size
Continue with the hypothetical values above. Plan for a one-sided \(\alpha=0.05\) rule and 80% target power. Assume independent normal groups and use the first-order standard-error approximation. Find the smallest balanced group size meeting the planning requirement, then inflate enrollment for 10% expected nonevaluability.
For these inputs, the coefficient in the approximate variance is
Using \(z_{0.95}\approx1.6449\) and \(z_{0.80}\approx0.8416\), the target standard error is
The approximate requirement is \(n\geq0.01446714/(0.01299)^2\approx85.7\), so round up to 86 evaluable participants per group. Checking adjacent integers confirms the boundary: at \(n=85\), \(SE\approx\sqrt{0.01446714/85}=0.01305\), which exceeds 0.01299; at \(n=86\), \(SE\approx\sqrt{0.01446714/86}=0.01297\), which meets the requirement. Thus 86, not 85, is the smallest balanced group size under this approximation.
If the expected evaluability proportion is 0.90, the enrollment target per group is
The resulting target is 96 enrolled participants per group, or 192 in total. The enrollment inflation addresses the assumed 10% loss; it does not change the requirement of 86 evaluable participants per group.
How Assumed Effects Change the Plan
Sample size can be highly sensitive to the assumed true effects. A larger absolute mean difference increases both the criterion value through \(\Delta^2\) and, in the approximate variance coefficient, the contribution \(4\Delta^2(\sigma_T^2+\sigma_R^2)\). If the criterion value moves closer to \(\theta\), the available distance shrinks as well. Either change can increase the required sample size substantially.
Worked Example: A Larger Assumed Mean Difference
Suppose the hypothetical variance components remain \(\sigma_T^2=0.0546\) and \(\sigma_R^2=0.0471\), but planning assumes \(\Delta=0.15\). With the same illustrative \(\theta\), the criterion value is \(g=(0.15)^2+0.0546-0.0471=0.0300\), leaving a distance of \(0.04979-0.0300=0.01979\).
The variance coefficient becomes
At \(\alpha=0.05\) and 80% target power, \(SE_{\mathrm{target}}\approx0.01979/2.4865=0.00796\). The approximate balanced size is \(n\geq0.01955214/(0.00796)^2\approx308.6\), so plan for 309 evaluable participants per group. This is far larger than the 86 per group in the earlier calculation: the assumed mean difference both increases the criterion value and increases the uncertainty contribution in \(C\).
A planning analysis should not select optimistic effects merely to obtain a manageable sample size. Instead, justify the assumed mean difference and variance components using suitable evidence, and examine credible alternative assumptions. A calculation based on a single set of inputs is conditional on those inputs; it is not a guarantee that the study will achieve its target.
Diagnostics, Alternatives, and Limitations
Before relying on an analytic sample size, check that the criterion and model align with the planned design. Confirm that the response scale, formulation-specific variance definitions, and limit \(\theta\) match the intended analysis. Check the arithmetic at the chosen integer and the preceding integer, as in the example, and verify that enrollment inflation is applied to evaluable sample size rather than substituted for it.
| Planning approach | Useful when | Main caution |
|---|---|---|
| First-order analytic approximation | The design and distributional assumptions are simple enough to support a transparent standard-error calculation. | Its accuracy depends on the approximation and on the assumed effects and variance components. |
| Design-specific analytic model | The study uses pairing, crossover sequences, or replicate observations with a known covariance structure. | Every covariance and variance component must correspond to the estimand and analysis model. |
| Simulation-based planning | The estimator, design, or decision rule makes a simple analytic standard error inadequate. | The simulated data-generating assumptions must be justified, and simulation error must be considered. |
The first-order formula is limited by its independent-groups and normality assumptions, its large-sample approximation, and the quality of the planning inputs. Skewness, outliers, small samples, variance-component estimation, or a different decision rule may affect operating characteristics. If the planned trial is a crossover or replicate study, use a design-consistent model rather than treating observations from the same participant as independent. A more complex calculation is not automatically more reliable: its assumptions and implementation still require scrutiny.
Common mistake: “The mean difference is small, so the sample size can be based only on the mean.” That reasoning omits \(\sigma_T^2-\sigma_R^2\), which is part of the PBE criterion, and can also omit variance-component uncertainty. The correction is to calculate the full criterion value and use a variance model that estimates its mean and variance contributions under the planned design.
Check Your Understanding
Answer each question using the criterion and planning logic in this tutorial. Solutions follow the questions.
- If \(\Delta=0\) but \(\sigma_T^2>\sigma_R^2\), which component determines whether the criterion is below \(\theta\)?
- Why does a smaller gap \(\theta-g\) generally require a larger sample size at fixed \(\alpha\) and target power?
- With 86 evaluable participants required per group and 15% expected nonevaluability, how many should be enrolled per group?
- Why should the independent-groups approximation not be inserted unchanged for a replicate crossover design?
Solutions. (1) The squared mean contribution is zero, so \(g=\sigma_T^2-\sigma_R^2\); compare that signed variance difference with \(\theta\). (2) The target standard error is \((\theta-g)/(z_{1-\alpha}+z_{1-\beta})\); a smaller gap requires a smaller standard error, which generally requires a larger \(n\). (3) The evaluability proportion is \(1-0.15=0.85\), so \(\lceil86/0.85\rceil=\lceil101.18\rceil=102\) participants per group. (4) Replicate crossover observations from the same participant are dependent, so their covariance and design-specific variance components affect the estimator's uncertainty; the independent-groups formula does not include those terms.