Sample Size for Individual Bioequivalence

Why Individual Bioequivalence Changes the Planning Problem

A sample-size calculation for average bioequivalence is driven chiefly by uncertainty in the mean test-to-reference contrast. Individual bioequivalence (IBE) asks a broader question: whether the formulations are sufficiently similar not only in their average responses, but also in the variability of responses within a person and in how the formulation contrast differs from person to person. Consequently, planning requires assumptions about several variance components, as well as the mean contrast.

The calculation also depends on how success is defined. An IBE criterion combines these components into a scalar quantity and compares it with a prespecified limit. Because the estimated criterion has sampling uncertainty, a study is not planned merely to make its expected value fall below the limit. It is planned to give a specified probability of meeting the decision rule, under a stated set of true parameter values.

Definition: Let $\delta=\mu_T-\mu_R$ be the population mean difference on the analysis scale, commonly the natural-log scale for pharmacokinetic exposure. Let $\sigma_D^2$ denote the variance of the subject-by-formulation contrast, and let $\sigma_{WT}^2$ and $\sigma_{WR}^2$ denote the within-subject variances for test and reference. A commonly used form of the IBE criterion is $$ q=\delta^2+\sigma_D^2+\sigma_{WT}^2-\sigma_{WR}^2. $$ IBE is supported when the applicable upper confidence bound for $q$ is below a prespecified limit $\theta_I$. The exact criterion, limit, analysis scale, and decision procedure must be specified by the governing protocol or regulatory framework; $\theta_I$ is not a universal constant.

The mean term penalizes a difference in average exposure regardless of its direction. The interaction term reflects additional subject-specific variation in the test-versus-reference contrast. The within-subject variance difference allows test variability to be penalized relative to reference variability; a smaller test within-subject variance can reduce $q$. The component $\sigma_D^2$ is not the same as the overall between-subject variance shared by both formulations.

Model and Information Needed for Planning

A useful conceptual mixed model for log-transformed responses is

Y_{ijr}=\mu_j+S_i+D_{ij}+\varepsilon_{ijr}, \qquad j\in\{T,R\}.

Here $i$ indexes subjects, $j$ formulation, and $r$ replicate observations within a subject-formulation combination. The fixed formulation means are $\mu_T$ and $\mu_R$; $S_i$ is a subject effect shared across formulations; $D_{ij}$ represents the subject-specific formulation deviation; and $\varepsilon_{ijr}$ is within-subject error. In this representation, $\sigma_D^2$ is the variance of the subject-specific test-minus-reference deviation, and the within-formulation errors have variances $\sigma_{WT}^2$ and $\sigma_{WR}^2$.

Planning presumes a design and model that can identify the quantities in the criterion. In particular, repeated observations under the formulations are needed to separate within-formulation variability from variation in subject-specific contrasts. The observations must be analyzed under a prespecified model, with assumptions about the random effects and residuals that are reasonable for the chosen analysis. A normal mixed-effects model often assumes mean-zero, approximately normal random effects and residual errors, with the covariance structure correctly specified. Independence is assumed across subjects; within-subject dependence is represented by the model rather than ignored.

These assumptions are not guaranteed by the sample-size formula. A poorly chosen design can leave a variance component weakly identified even with many participants. Before using any approximation, planners should confirm that the intended design supplies the repeated measurements needed for each component, and that the planned estimator and confidence-bound method match the IBE criterion.

Deriving a First Sample-Size Approximation

Suppose the estimator $\hat q$ is approximately normal for a sufficiently large evaluable sample, with standard error $\tau/\sqrt{n}$. Here $n$ is the number of evaluable subjects, and $\tau$ summarizes the per-subject information for estimating the criterion under the assumed model and design. It is not one of the raw variance components; it depends on their estimation, their covariance, and the design.

For a one-sided upper-bound decision, the approximate rule is to conclude IBE when $\hat q+z_{1-\alpha}\tau/\sqrt{n}<\theta_I$, where $z_{1-\alpha}$ is the standard-normal quantile and $\alpha$ is the one-sided type I error level. If the true criterion is $q_0<\theta_I$, then the probability of meeting this rule is approximately

P\left(\hat q+z_{1-\alpha}\frac{\tau}{\sqrt{n}}<\theta_I\right) \approx \Phi\left(\frac{(\theta_I-q_0)\sqrt{n}}{\tau}-z_{1-\alpha}\right).

This follows by writing $\hat q$ approximately as $N(q_0,\tau^2/n)$. The decision event is equivalent to $\hat q<\theta_I-z_{1-\alpha}\tau/\sqrt{n}$. Standardizing that inequality gives the displayed probability. Setting the approximate power equal to $1-\beta$ and solving for $n$ yields

n\approx \left\{ \frac{(z_{1-\alpha}+z_{1-\beta})\tau}{\theta_I-q_0} \right\}^2.

The quantity $\theta_I-q_0$ is the criterion headroom: the distance between the assumed true criterion and the limit. As headroom shrinks, the required sample size rises sharply. The expression is a planning approximation, not a universal IBE sample-size formula. In a final design, its normal approximation and standard-error estimate should be checked against the actual estimator and decision procedure, often using a method that reflects the planned model.

Worked Example: A First Evaluable Sample-Size Estimate

Consider a hypothetical planning scenario on the log scale. A protocol specifies $\theta_I=0.0500$. The assumed components are $\delta=0.0400$, $\sigma_D^2=0.0060$, $\sigma_{WT}^2=0.0200$, and $\sigma_{WR}^2=0.0160$. A pilot-based approximation gives $\tau=0.300$. Find an approximate evaluable sample size for 80% power with a one-sided $\alpha=0.05$.

First calculate the assumed criterion:

$$ q_0=(0.0400)^2+0.0060+0.0200-0.0160 =0.0016+0.0100 =0.0116. $$

The criterion headroom is $0.0500-0.0116=0.0384$. Using $z_{0.95}=1.645$ and $z_{0.80}=0.842$,

n\approx \left\{\frac{(1.645+0.842)(0.300)}{0.0384}\right\}^2 = \left(\frac{0.7461}{0.0384}\right)^2 = (19.43)^2 \approx 377.5.

Rounding up gives an approximate target of 378 evaluable subjects. This is not a claim that 378 is sufficient under every IBE analysis: it relies on the hypothetical $\tau$, the assumed true components, and the normal upper-bound approximation. The model, variance assumptions, and analysis procedure must be checked for the intended study.

Worked Planning Scenarios and Sensitivity

Worked Example: Interaction Variability and Criterion Headroom

In a second hypothetical plan, let $\theta_I=0.0500$, $\delta=0.0300$, $\sigma_D^2=0.0120$, $\sigma_{WT}^2=0.0180$, and $\sigma_{WR}^2=0.0160$. Assume the design and pilot data imply $\tau=0.220$. Calculate the approximate evaluable size for 80% power at one-sided $\alpha=0.05$.

The assumed criterion and headroom are

q_0=(0.0300)^2+0.0120+0.0180-0.0160 =0.0009+0.0140 =0.0149, \qquad \theta_I-q_0=0.0351.

Therefore,

n\approx \left\{\frac{(1.645+0.842)(0.220)}{0.0351}\right\}^2 = \left(\frac{0.5471}{0.0351}\right)^2 = (15.59)^2 \approx 243.0.

The approximation gives a target of 243 evaluable subjects. The interaction component contributes directly to $q_0$, leaving less headroom than a smaller interaction variance would. The calculation assumes that the design can estimate this component and that the pilot-based $\tau$ adequately represents uncertainty under the planned analysis.

Worked Example: Sensitivity to Uncertain Criterion Precision

A third hypothetical plan assumes $q_0=0.0180$, $\theta_I=0.0500$, 90% power, and one-sided $\alpha=0.05$. Because the variance-component estimates are uncertain, consider two plausible values for $\tau$: 0.220 and 0.270. Calculate both targets to show the impact of this planning input.

The headroom is $0.0500-0.0180=0.0320$, and $z_{0.95}+z_{0.90}=1.645+1.282=2.927$. For $\tau=0.220$,

n\approx \left(\frac{2.927(0.220)}{0.0320}\right)^2 = \left(\frac{0.6439}{0.0320}\right)^2 = (20.12)^2 \approx 404.9,

which rounds up to 405 evaluable subjects. For $\tau=0.270$,

n\approx \left(\frac{2.927(0.270)}{0.0320}\right)^2 = \left(\frac{0.7903}{0.0320}\right)^2 = (24.70)^2 \approx 609.9,

which rounds up to 610 evaluable subjects. The difference illustrates why an optimistic estimate of criterion precision can produce an inadequate target. The values of $\tau$ should be justified from relevant variability information and the proposed design, not selected merely to achieve a preferred sample size.

Assumptions, Diagnostics, and Alternatives

The approximation is most credible when the planned estimator behaves approximately normally at the proposed sample size, the model identifies all required variance components, and the assumed component values are plausible. The variance of $\hat q$ must include the covariance among estimated components. Ignoring that covariance or treating the estimated components as known can materially misstate power. The normal approximation can also be weak for variance components near zero, for small samples, or when the estimated criterion has a markedly skewed distribution.

Useful checks include comparing the planned design with the model’s ability to estimate each component, reviewing whether pilot estimates are precise enough to support the assumed values, and examining whether conclusions change over a defensible range of $\delta$, $\sigma_D^2$, $\sigma_{WT}^2$, and $\sigma_{WR}^2$. A resulting sample size that changes drastically under small, plausible input changes is fragile. The calculation also targets evaluable subjects; the number enrolled must account separately for anticipated losses and nonevaluable profiles.

The main alternatives answer different questions. Average bioequivalence sample-size planning focuses on a mean contrast and its uncertainty; it does not by itself control the subject-by-formulation interaction component. Population bioequivalence planning includes population-level variability terms, but a population criterion is not interchangeable with the individual criterion above. A more detailed IBE power calculation can generate data under the planned model, fit the intended analysis, and apply the actual decision rule repeatedly. Such an approach can represent finite-sample behavior and component-estimation uncertainty more directly, but it still depends on correct model and parameter assumptions.

No calculation removes the need to justify the target criterion, model, and assumed true effects. In particular, power is conditional on the values used for planning; it is not a guarantee that the study will pass. The sample size should therefore be accompanied by sensitivity results and a clear statement of which assumptions drive the result.

Common Mistake

A frequent error is to use only the expected mean difference, then apply an average-bioequivalence sample-size formula and call the result an IBE target. That reasoning omits both the subject-by-formulation interaction and the within-formulation variance contrast, even though all enter $q$. The correction is to define the IBE criterion first, specify plausible values for every component, and plan power for the criterion-based decision rule. A second error is to regard $\theta_I-q_0$ as a fixed margin that makes uncertainty irrelevant; the sample-size expression shows that uncertainty, represented by $\tau$, directly controls the required $n$.

Key takeaway: IBE sample-size planning requires assumptions about the mean contrast, subject-by-formulation interaction, and within-subject variances, together with the uncertainty of their combined criterion. A transparent approximation is useful for initial planning, but its assumptions and sensitivity must be checked against the planned design and decision procedure.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions.

Which component of $q$ represents variation in the test-versus-reference contrast across subjects, beyond a shared subject effect?
With $\delta=0.0200$, $\sigma_D^2=0.0080$, $\sigma_{WT}^2=0.0190$, and $\sigma_{WR}^2=0.0150$, calculate $q$.
Holding $\tau$, power, and $\alpha$ fixed, what happens to the approximate sample size as $\theta_I-q_0$ becomes smaller?
Why can a mean-only average bioequivalence calculation not automatically serve as an IBE sample-size calculation?
For $q_0=0.0180$, $\theta_I=0.0500$, $\tau=0.200$, 80% power, and one-sided $\alpha=0.05$, calculate the approximate evaluable sample size.

Solutions. (1) The subject-by-formulation contrast variance, $\sigma_D^2$. (2) $q=(0.0200)^2+0.0080+0.0190-0.0150=0.0004+0.0120=0.0124$. (3) It increases, because $n$ is proportional to $1/(\theta_I-q_0)^2$. (4) The IBE criterion also includes interaction and within-formulation variance terms, and their estimation uncertainty affects power. (5) The headroom is $0.0320$, so $n\approx\{(2.487)(0.200)/0.0320\}^2=(0.4974/0.0320)^2=(15.54375)^2\approx241.6$; round up to 242 evaluable subjects.