Defining Model-Based Bioequivalence Estimands

Why Define the Estimand First?

A bioequivalence conclusion depends on what is being compared and for whom. A mean contrast for a log-transformed exposure metric answers a different question from a comparison of exposure distributions across people, or from a comparison that includes how consistently the two formulations behave within the same person. These are different estimands, not merely different ways to summarize one number.

A model-based estimand must specify the target population, the formulation contrast, the exposure metric and scale, and which sources of variability enter the target. This tutorial defines average, population, and individual bioequivalence estimands in those terms. It does not prescribe a particular fitted pharmacokinetic model or a universal acceptance rule: the numerical decision criterion depends on the applicable scientific and regulatory context.

A Common Setup for the Three Estimands

Let \(i\) index a subject drawn from a stated target population, and let \(f\) denote formulation \(T\) or \(R\). Choose one exposure metric in advance, such as AUC or Cmax, and let \(Y_{ifk}\) be its logarithm for subject \(i\), formulation \(f\), and occasion \(k\). The metric, its measurement window, and the log transformation must be the same for both formulations. On this scale, a difference in means corresponds to a ratio of geometric means on the original scale.

For defining the targets, write the expected log exposure for subject \(i\) under formulation \(f\) as \(m_{if}\). A useful decomposition is \(m_{if}=\alpha_f+b_i+u_{if}\), where \(\alpha_f\) is a formulation-level location, \(b_i\) is a subject propensity shared across formulations, and \(u_{if}\) is a subject-specific deviation for formulation \(f\). On an occasion, the observed log exposure can be represented as \(Y_{ifk}=m_{if}+e_{ifk}\), where \(e_{ifk}\) is within-subject occasion variation with mean zero. This is a variance decomposition for defining targets, not a claim that every application must use this exact model.

The target population must be explicit: for example, eligible adults with a specified disease, or healthy volunteers meeting stated criteria. The formulations are compared for that population, not automatically for every patient who might receive the drug. The expectation and variances below are taken over that target population and, where included, over the relevant occasions.

Average Bioequivalence: A Mean Contrast

The average bioequivalence estimand is the target-population mean difference in log exposure:

It targets the average formulation contrast, not the spread of individual responses. Its exponentiated form, \(\exp(\delta)\), is the ratio of geometric mean exposures for \(T\) relative to \(R\). A positive \(\delta\) indicates a higher geometric mean for \(T\); a negative \(\delta\) indicates a lower one. The units of \(\delta\) are log-exposure units, while \(\exp(\delta)\) is a dimensionless ratio.

Population Bioequivalence: Comparing Between-Subject Distributions

A population comparison asks whether the formulations have similar distributions of subject-specific expected log exposure, not only similar means. One common model-based variance-sensitive contrast is

The first term penalizes a difference in mean log exposure. The second contrasts the between-subject variances under the two formulations. This makes clear why an average contrast alone cannot describe population-level similarity: two formulations can have the same mean but different spreads across subjects. The variance difference can be negative, so \(\Delta_P\) is not simply a sum of nonnegative distances.

Here the population distribution is specifically the distribution of \(m_{if}\). If the intended outcome instead represents a randomly selected subject-occasion exposure \(Y_{ifk}\), occasion variability also affects the marginal variance. Under the decomposition above,

provided the occasion error is uncorrelated with \(b_i\) and \(u_{if}\). The covariance term cannot be omitted unless it is assumed to be zero. Consequently, a population variance estimand must say whether it concerns subject-specific expected exposures or single-occasion exposures, and must account for the covariance structure in the chosen model.

Individual Bioequivalence: Contrasts Within Subjects

An individual comparison concerns paired formulation responses for the same subjects. Define the subject-specific formulation contrast as \(D_i=m_{iT}-m_{iR}\). Under the decomposition above, the common subject propensity cancels: \(D_i=(\alpha_T-\alpha_R)+(u_{iT}-u_{iR})\). Its population mean is \(\delta\), and its between-subject variance is

Thus, the variation in paired formulation contrasts depends on the covariance between a subject’s responses to \(T\) and \(R\). It is not generally obtained by adding the two formulation-specific variances. A variance-sensitive individual estimand commonly has the form

This form includes a squared mean contrast, heterogeneity in subject-specific formulation contrasts, and a difference in within-subject occasion variances. Its components answer distinct questions: whether the mean responses differ, whether subjects differ in their relative response to the formulations, and whether repeat responses are more variable under one formulation. The within-subject terms require data and design information that can identify the relevant occasion variances; they should not be inferred from a single observation per subject-formulation without additional assumptions.

Deriving the Paired-Contrast Variance

The covariance term in \(\tau_D^2\) follows directly from the variance identity for a difference. Let \(U_T=u_{iT}\) and \(U_R=u_{iR}\). Then

The identity holds whenever the second moments exist; it does not require normality. Assuming independence would set the covariance to zero, but that is an additional assumption, not a consequence of using a paired formulation contrast. In crossover data, the same subjects contribute responses under both formulations, so their joint response structure is central to the individual estimand.

Assumptions, Diagnostics, and Alternatives

Before interpreting a model-based estimand, check that its definition and assumptions match the scientific question. At minimum, the analysis should make the following explicit:

• Population: eligibility criteria and the population to which the contrast is intended to apply.
• Outcome: the exposure metric, measurement window, and transformation. AUC and Cmax are different outcomes, and estimates on the log scale are not interchangeable with estimates on the original scale.
• Pairing and variability: whether the target uses subject-specific expected values, single-occasion outcomes, or paired contrasts; how between-subject, within-subject, and formulation-specific variation are represented.
• Dependence: which effects are assumed independent or uncorrelated. In particular, any marginal variance calculation involving \(b_i\) and \(u_{if}\) must include their covariance or explicitly assume it is zero.
• Data support: whether the study design has sufficient repeated observations and formulation exposure within subjects to estimate the variance components used by the chosen target.

A useful diagnostic is to compare model-implied means and variances with summaries of the observed log-exposure data at the same level of aggregation. Large discrepancies, unstable variance estimates, or substantial changes under plausible covariance specifications indicate that the model-based description may not be robust. A fitted variance of zero can also signal a boundary estimate or weak information; it is not proof that the corresponding source of variability is absent in the target population.

The three estimands are alternatives because they emphasize different aspects of the formulation comparison, not because one is universally superior. Average bioequivalence focuses on the mean contrast and is often the most direct target when the question concerns average exposure. Population bioequivalence adds a comparison of between-subject distributions. Individual bioequivalence additionally addresses paired response heterogeneity and within-subject variability. The latter targets need study designs and data capable of informing their additional variance components.

A common mistake is to conclude that equal average exposure means the formulations are equivalent for every subject. That reasoning confuses \(\delta=0\), a statement about the population mean contrast, with \(\tau_D^2=0\), a statement that subject-specific contrasts do not vary. The correction is to state which estimand is being evaluated and report its components. Another error is to add variances while silently assuming independence; covariance is part of the variance identity and must be included unless zero covariance is justified.