Worked Population and Individual Bioequivalence Planning

Planning for Two Different Bioequivalence Targets

Population bioequivalence (PBE) and individual bioequivalence (IBE) do not ask precisely the same question. A PBE criterion incorporates the difference in average log exposure and the difference in total variability between formulations. An IBE criterion additionally accounts for subject-by-formulation interaction variability and within-subject variability. Consequently, the sample size that gives adequate power for one criterion does not automatically give adequate power for the other.

A useful planning calculation begins with the assumed true values of the metric and its uncertainty, not with a sample size copied from another study. The calculations below use illustrative log-scale criteria and transparent normal approximations. They are teaching examples, not a universal regulatory decision rule: the prespecified criterion, estimator, study design, and variance-component estimation method must be matched in an actual protocol.

Definition: In the illustrative PBE formulation, let \(\delta=\mu_T-\mu_R\) be the difference in mean log exposure, and let \(\sigma_T^2\) and \(\sigma_R^2\) be the corresponding total variances. The PBE metric is \(g_P=\delta^2+\sigma_T^2-\sigma_R^2\). In the illustrative IBE formulation, \(g_I=\delta^2+\sigma_D^2+\sigma_{WT}^2-\sigma_{WR}^2\), where \(\sigma_D^2\) is subject-by-formulation interaction variance and \(\sigma_{WT}^2,\sigma_{WR}^2\) are within-subject variances for test and reference. For a prespecified upper limit \(\theta\), the planning target is that the metric is below \(\theta\).

All quantities in these examples are on the log scale, so the variances are squared log units. For illustration, take \(\theta=0.0400\), corresponding to a variance-scale limit of \((0.20)^2\) in this chosen formulation. Different standards or protocols can define limits and estimators differently; a numerical limit must never be transferred without checking that it matches the criterion being planned.

From an Upper Bound to Approximate Power

Suppose the planned analysis declares the criterion met when a one-sided upper confidence bound for \(g\) is below \(\theta\). If \(\hat g\) is approximately normal with mean \(g\) and planning standard error \(SE\), that upper bound is \(\hat g+z_{1-\alpha}SE\), where \(z_{1-\alpha}\) is the standard normal quantile. The approximate power at the assumed true metric \(g\) is therefore the probability that this bound is below the limit.

\begin{aligned} \text{Power} &\approx P(\hat g+z_{1-\alpha}SE<\theta)\\ &=\Phi\left(\frac{\theta-g}{SE}-z_{1-\alpha}\right). \end{aligned}

Here \(\Phi\) is the standard normal cumulative distribution function. At a one-sided \(\alpha=0.05\), \(z_{1-\alpha}=1.645\). For a target power of 80%, the required standardized distance is approximately \(1.645+0.842=2.487\). Thus an initial planning requirement is \(SE\leq(\theta-g)/2.487\), provided \(g<\theta\). This relation explains why power depends on the margin between the assumed true metric and the limit, as well as on the uncertainty of its estimator.

Planning assumption: This normal approximation treats the planning standard error as fixed. In real replicated crossover studies, variance components are estimated, their estimators may be correlated, and finite-sample distributions may not be normal. Use a design-matched analytical method or simulation for the final plan; regard the calculations here as explicit approximate plans.

Worked Population Bioequivalence Plan

Worked Example: Approximate PBE Sample Size

Hypothetical planning data. Consider a balanced comparison with \(n\) independent observations per formulation. Assume normally distributed log exposures with \(\delta=0.0400\), \(\sigma_T=0.320\), and \(\sigma_R=0.300\). Use \(\theta=0.0400\), one-sided \(\alpha=0.05\), and target power 80%. The independent-groups setup is used to make the variance calculation explicit; it is not a claim that every PBE study uses this design.

First calculate the assumed metric:

$$ g_P=(0.0400)^2+(0.320)^2-(0.300)^2 =0.001600+0.102400-0.090000 =0.01400. $$

The distance to the limit is \(\theta-g_P=0.02600\). For a normal planning approximation, write \(d=\bar X_T-\bar X_R\), \(q=(\sigma_T^2+\sigma_R^2)/n\), and use the bias-corrected squared mean difference \(d^2-q\). Under independent normal groups, \(d\sim N(\delta,q)\), so \(\operatorname{Var}(d^2)=2q^2+4\delta^2q\). Also, the sample variance in a normal group has variance \(2\sigma^4/(n-1)\), and the two group sample variances are independent. Therefore the approximate variance of the PBE metric estimator is:

\operatorname{Var}(\hat g_P)\approx 2q^2+4\delta^2q+ \frac{2\sigma_T^4+2\sigma_R^4}{n-1}.

Substitute the planning values: \(q=0.1924/n\), \(4\delta^2(\sigma_T^2+\sigma_R^2)=0.00123136\), and \(2\sigma_T^4+2\sigma_R^4=0.03717152\). The sample size must be chosen so the resulting \(SE\) is no more than \(0.02600/(1.645+0.842)\approx0.01045436\); conservatively, require \(SE\leq0.01045\). At \(n=354\), the variance is approximately \(2(0.1924/354)^2+0.00123136/354+0.03717152/353=0.00010937\), giving \(SE=0.01046\) and power just under 80%. At \(n=355\), it is approximately \(2(0.1924/355)^2+0.00123136/355+0.03717152/354=0.00010906\), so \(SE=0.01044\). The approximate power is \(\Phi(0.02600/0.01044-1.645)=\Phi(0.845)\), or about 80.1%. The plan is therefore 355 evaluable observations per group under these assumptions.

Worked Individual Bioequivalence Plan

An IBE plan requires assumptions about more than the means and total variances. It needs assumptions about interaction and formulation-specific within-subject variability, and the replicate design must identify those components. The next example uses a stated balanced-design approximation so the role of each component can be seen directly.

Worked Example: Approximate IBE Sample Size

Hypothetical planning data. Assume \(\delta=0.0300\), \(\sigma_D^2=0.0100\), and \(\sigma_{WT}^2=\sigma_{WR}^2=0.0900\), with \(\theta=0.0400\). Use the same one-sided \(\alpha=0.05\) and 80% target power. For this illustrative balanced replicate plan, assume the mean-difference estimator has variance \((2\sigma_D^2+\sigma_{WT}^2+\sigma_{WR}^2)/n\); assume the three variance-component estimators are mutually independent, independent of the mean estimator, and each has variance \(2\sigma^4/n\). These are design-specific working assumptions, not identities that apply to every replicate design.

The assumed IBE metric is \(g_I=(0.0300)^2+0.0100+0.0900-0.0900=0.01090\). Its distance to the limit is \(0.0400-0.01090=0.02910\). By the delta method, the derivative of \(\delta^2\) with respect to \(\delta\) is \(2\delta\). The resulting first-order variance approximation is:

\operatorname{Var}(\hat g_I)\approx \frac{4\delta^2(2\sigma_D^2+\sigma_{WT}^2+\sigma_{WR}^2) +2\sigma_D^4+2\sigma_{WT}^4+2\sigma_{WR}^4}{n}.

The coefficient of \(1/n\) is \(4(0.0300)^2(0.0200+0.0900+0.0900)+2(0.0100)^2+2(0.0900)^2+2(0.0900)^2=0.000720+0.000200+0.01620+0.01620=0.03332\). The required standard error is at most \(0.02910/2.487=0.01170\), corresponding to \(n\geq0.03332/(0.01170)^2\), approximately 243. At \(n=244\), \(SE\approx\sqrt{0.03332/244}=0.01169\), and power is approximately \(\Phi(0.02910/0.01169-1.645)=\Phi(0.8443)\), or about 80.1%. The approximate plan is 244 evaluable subjects in this assumed design.

The 244-subject result is not a general claim that IBE always needs fewer subjects than PBE. The examples use different metrics, component structures, and variance approximations. A fair design comparison must use a common scientific target and plausible, design-specific assumptions for each candidate study.

Enrollment, Feasibility, and Sensitivity

Worked Example: Attrition and an Infeasible IBE Target

Enrollment adjustment. Suppose the IBE plan requires 244 evaluable subjects and 15% of enrolled subjects are expected not to provide evaluable data. The retained fraction is \(1-0.15=0.85\), so the enrollment target is \(244/0.85=287.06\). Rounding up gives 288 enrolled subjects. This rounds up because enrolling 287 would yield an expected \(287(0.85)=243.95\) evaluable subjects, slightly below the target.

Feasibility check. Now suppose the assumed interaction variance is instead \(\sigma_D^2=0.0500\), with the other IBE assumptions unchanged. Then \(g_I=(0.0300)^2+0.0500+0.0900-0.0900=0.05090\), which exceeds \(\theta=0.0400\). The assumed true metric is outside the criterion. Increasing \(n\) makes the estimate more precise around this value; it does not move the true value below the limit. Under the normal approximation, the probability of meeting the upper-bound criterion tends toward zero as \(SE\) tends toward zero. There is no sample size that provides high power for declaring equivalence under this assumed truth.

A basic diagnostic is therefore to calculate the assumed metric before solving for \(n\). If the metric is at or above the limit, investigate whether the planning assumptions are scientifically plausible and whether the desired claim is appropriate; do not try to repair an infeasible target merely by increasing enrollment. If it is below the limit but close to it, the required sample can grow sharply because the available distance \(\theta-g\) is small.

Derivation, Diagnostics, and Alternatives

The normal power expression follows directly from the assumed sampling distribution. If \(\hat g\sim N(g,SE^2)\), the rule \(\hat g+z_{1-\alpha}SE<\theta\) is equivalent to \(\hat g<\theta-z_{1-\alpha}SE\). Standardizing both sides gives:

P(\hat g+z_{1-\alpha}SE<\theta) =P\left(\frac{\hat g-g}{SE}< \frac{\theta-g}{SE}-z_{1-\alpha}\right) =\Phi\left(\frac{\theta-g}{SE}-z_{1-\alpha}\right).

This also shows why a plan should be checked by substituting its final integer sample size back into the power calculation. Rounding a continuous solution down can miss the target. In a real plan, diagnose the approximation by checking whether the assumed design identifies each variance component, whether the component values are plausible, whether the estimator covariance structure matches the design, and whether small-sample or boundary behavior makes a normal approximation doubtful.

A closed-form or delta-method calculation is fast and useful for understanding which assumptions drive the result. A design-matched simulation can instead generate data under the specified replicate schedule, fit the planned model, apply the exact decision rule, and estimate the fraction of simulated studies meeting the criterion. Simulation is generally preferable when variance components are uncertain, estimators are correlated, or the decision statistic has a non-normal finite-sample distribution. It has its own limitation: estimated power has Monte Carlo error, so simulations should use enough repetitions and report that precision. Robust planning or assurance can account for uncertainty in assumed variance values, but that changes the planning question from power at one fixed parameter set to performance averaged over, or protected across, a range of assumptions.

The main practical mistake is to treat PBE and IBE as interchangeable labels for a single average-exposure comparison. PBE includes a total-variability contrast; IBE includes an interaction component and formulation-specific within-subject variability. Omitting a component can make the planned metric and its uncertainty wrong even if the arithmetic is flawless. A second mistake is to report a modeled power value as guaranteed performance: power is conditional on the assumed true means, variance components, design, estimator, and decision rule.

Key takeaway: Specify the criterion and design first, calculate the metric under plausible assumptions, and then calculate power using an uncertainty model appropriate to that design. Compare required enrollment only after accounting for evaluability, and verify the final rounded plan under the same assumptions.

Check Your Understanding

Answer each question, then compare with the short solutions below.

With \(\delta=0.0200\), \(\sigma_T^2=0.0800\), and \(\sigma_R^2=0.0800\), what is the illustrative PBE metric?
Why can increasing sample size not produce high power to declare the criterion met when the assumed true metric exceeds \(\theta\)?
For an upper-bound rule, state the approximate power formula and identify the role of \(\alpha\).
An IBE plan requires 180 evaluable subjects and assumes 10% attrition. What enrollment target results after rounding up?
Name two reasons a design-matched simulation may be preferable to the normal approximation used here.

Solutions. 1. \(g_P=(0.0200)^2+0.0800-0.0800=0.000400\). 2. Greater \(n\) reduces uncertainty around the assumed true metric; if that metric is above the limit, increasingly precise estimates are less likely to satisfy the upper-bound rule. 3. \(\Phi((\theta-g)/SE-z_{1-\alpha})\); \(\alpha\) sets the one-sided critical value and the nominal false-positive rate at the boundary under the stated model. 4. \(180/0.90=200\), so enroll 200. 5. Simulation may better represent correlated variance-component estimators, non-normal finite-sample behavior, or a complex replicate design and decision rule.