Robust Population Bioequivalence Planning

Why Robust Planning Matters

A population bioequivalence sample size is not a property of the target power alone. It also depends on the treatment mean difference, the variance components that determine the population variances, the covariance induced by the study design, and the inferential procedure used to assess the bioequivalence criterion. A calculation can meet its target under one set of assumptions and fall short if, for example, a variance component is larger than anticipated or the random-effects distribution is misspecified.

The goal of robust planning is not to find a sample size that guarantees success under every conceivable model. It is to make the assumptions explicit, examine a defensible range of alternatives, and determine whether the proposed sample size remains adequate across that range. The earlier material on Sample Size for Population Bioequivalence and Calculating Power for Population Bioequivalence supplies the basic planning framework; here the emphasis is sensitivity to its inputs and assumptions.

The criterion combines a squared mean difference and a variance difference. Consequently, an assumption that affects either component can change the chance of meeting the criterion. The definition of \(n\), the margin, the analysis scale, the estimand, and the planned inferential procedure must all be fixed before comparing scenarios. In a crossover design, for instance, the number of evaluable subjects and the allocation across sequences should be defined consistently with the design.

For a simple illustration of the random-effects structure, suppose a response from subject \(i\) on formulation \(j\) is represented as \(Y_{ij}=\mu_j+b_i+u_{ij}+\epsilon_{ij}\). Here \(b_i\) is a subject effect common to formulations, \(u_{ij}\) is a subject-by-formulation effect, and \(\epsilon_{ij}\) is residual variation. If these components are independent with mean zero, the marginal variance for formulation \(j\) is \(\sigma_j^2=\tau_b^2+\tau_{uj}^2+\sigma_{\epsilon j}^2\). Real replicate designs may have a more detailed covariance structure; planning must reproduce the actual design and its fitted analysis rather than substitute this illustration for the protocol model.

A Scenario-Based Robustness Assessment

A practical approach is to define a baseline planning model and then vary its assumptions one at a time and in coherent combinations. Parameter uncertainty means that the model form is retained but quantities such as residual or subject-by-formulation variances are changed. Model departure means changing the data-generating structure itself, such as introducing heavier tails, a different covariance pattern, or dependence between a random effect and another modeled quantity. These are related concerns, but they should not be conflated.

For each scenario, generate data under the planned design, apply the exact planned analysis, and record whether its prespecified population bioequivalence decision is positive. If \(M\) simulated studies are generated and \(K\) conclude bioequivalence, the estimated power is \(\hat{\pi}=K/M\). Its approximate Monte Carlo standard error is \(\sqrt{\hat{\pi}(1-\hat{\pi})/M}\). This describes simulation noise, not uncertainty in the underlying variance estimates or in the model assumptions.

A minimum-power rule is conservative with respect to the scenarios included: it asks whether the target is achieved in the least favorable one. Assurance instead averages power over a distribution of uncertain parameters or scenarios. The two answer different questions. Neither protects against scenarios that were omitted, and assurance is only as defensible as the probability distribution used to weight them.

Uncertainty About Variance Inputs

Variance estimates used for planning are themselves uncertain. A single point estimate can give a misleadingly precise impression, especially when it comes from limited data or a design that estimates the components imprecisely. The earlier material on Evaluating Variance Estimates for Sample Size Planning and Sample Size Planning Under Uncertain Variance addresses the broader issue of uncertain inputs. In robust population bioequivalence planning, the key step is to connect that uncertainty to the particular components that determine \(\sigma_T^2\), \(\sigma_R^2\), and the variability of the planned estimator.

A sensitivity grid might vary each component over a justified range, while a joint parameter distribution can preserve dependence among component estimates. Varying one component at a time is useful for identifying drivers, but it can miss combinations that are plausible together. Conversely, combining the largest value of every component can create an implausible scenario. The choices should reflect the data and the study context, not a mechanical preference for extreme values.

If the scenario probabilities are uncertain or merely subjective, report the assurance as conditional on those weights and show the scenario-specific powers as well. Reporting only an average can hide an important weak point. If the goal is a guarantee over a defined range, a minimum-power criterion is more directly aligned with that goal, although it may lead to a larger sample size.

Departures from the Planning Model

Parameter sensitivity alone does not test whether the assumed random-effects model is adequate. A normal random-effects model, for example, may be used to generate planning data, but the actual study could have heavier tails or a covariance pattern not represented by that model. Such departures may alter the sampling behavior of the variance estimates and the frequency with which the planned decision rule is satisfied. The consequence cannot be inferred from the direction of a single variance component alone; it depends on the estimator, design, and decision procedure.

Useful diagnostics include checking whether the assumed components are identifiable from the proposed design, whether simulated marginal variances and correlations match the intended values, and whether the analysis converges reliably across simulations. Inspect the distribution of estimated variance components and the frequency of boundary estimates, failed fits, or unusual decision outcomes. A simulation that silently discards failed fits can overstate power; the handling of failures must follow a prespecified rule and be reported.

A sensitivity analysis also has limitations. It cannot compensate for a poor estimand, incorrect implementation of the planned analysis, or unrepresentative scenarios. Simulations based on an incorrect covariance structure may be precise but answer the wrong question. Moreover, more simulation replicates reduce Monte Carlo error but do not reduce uncertainty about the true variance components. Those require better evidence, transparent ranges, or a justified distribution for the uncertain inputs.

Choosing and Communicating a Robust Plan

The most useful report gives the baseline assumptions, the range or distribution of uncertain components, the structural departures examined, and the resulting power at each candidate sample size. State whether selection is based on minimum power, assurance, or another explicit rule. If a target is missed only under a particularly adverse scenario, explain why that scenario is or is not considered plausible rather than hiding it in an average.

A common mistake is to calculate power using the most favorable variance estimate, obtain a value just above the target, and describe the sample size as adequately powered. The correction is to distinguish the baseline calculation from robustness: a baseline result supports a conclusion only under its stated assumptions. Evaluate plausible alternatives, account for Monte Carlo error, and qualify the recommendation according to the robustness rule actually used.

Compared with a single plug-in calculation, scenario analysis makes sensitivity visible but does not assign probabilities unless a distribution is specified. Compared with assurance, a minimum-power rule does not average over scenarios, but it directly identifies the weakest case in the chosen set. Neither approach replaces sound design or careful estimation; they are complementary ways to expose dependence on planning assumptions.