Sample Size for Bioequivalence Studies

Introduction

Bioequivalence studies are designed to determine whether two pharmaceutical products have sufficiently similar rate and extent of drug exposure. They are particularly important when comparing a generic or follow-on formulation with a reference product.

Unlike a conventional superiority clinical trial, the objective is usually not to demonstrate that one treatment produces a larger mean response than another. Instead, the objective is to demonstrate that the pharmacokinetic exposure produced by the test product is sufficiently similar to that produced by the reference product.

For immediate-release products, the primary pharmacokinetic measures commonly include maximum concentration (C_max) and area under the concentration-time curve (AUC). Depending on the product and study design, additional pharmacokinetic endpoints may be evaluated.

What Does Bioequivalence Mean?

Suppose the test product is denoted by T and the reference product by R. For a pharmacokinetic endpoint such as AUC, the basic quantity of interest is the ratio of exposure:

If the test and reference products have identical exposure on average, the ratio would be approximately:

In practice, bioequivalence does not require the ratio to equal exactly 1. Instead, the study evaluates whether the confidence interval for the relevant test/reference ratio lies within prespecified equivalence limits.

For many conventional bioequivalence studies, the standard acceptance interval is expressed as:

or equivalently:

Why Bioequivalence Studies Use Logarithmic Transformation

Pharmacokinetic measures such as AUC and C_max are typically right-skewed and are naturally interpreted multiplicatively. Consequently, bioequivalence analyses commonly use the natural logarithm:

The treatment comparison is then performed on the difference between mean log exposures:

Exponentiating this difference gives the geometric mean ratio:

Thus:

where \(GM_T\) and \(GM_R\) are the geometric means for the test and reference treatments.

This symmetry is one of the central mathematical features of conventional bioequivalence sample-size calculations.

The Standard 2 × 2 Crossover Design

A common bioequivalence design is a randomized, two-treatment, two-period, two-sequence crossover study.

The two sequences are typically:

Each participant therefore receives both the test and reference formulations. This is extremely useful because each subject acts as their own control.

The within-subject comparison generally has much less variability than a between-subject comparison. This is one of the principal reasons crossover designs are efficient for bioequivalence studies.

Intra-Subject Coefficient of Variation

A central parameter in bioequivalence sample-size calculations is the within-subject coefficient of variation, often denoted CV_W.

It describes how much the pharmacokinetic endpoint varies within the same subject across repeated administrations.

A commonly used relationship between the within-subject variance on the natural log scale and the coefficient of variation is:

Solving for the log-scale variance gives:

This relationship is useful when published variability is expressed as a coefficient of variation but the sample-size calculation is performed on the logarithmic scale.

Why Intra-Subject Variability Drives Sample Size

Suppose two studies have exactly the same expected geometric mean ratio. The first has a within-subject CV of 15%, while the second has a CV of 35%. The second study will generally require substantially more participants.

The reason is straightforward: greater variability produces a wider confidence interval for the treatment ratio. More participants are therefore required to obtain sufficient power to establish bioequivalence.

Power in a Bioequivalence Study

In a superiority study, power usually refers to the probability of detecting a specified treatment difference. In a bioequivalence study, power is the probability that the confidence interval for the treatment ratio will satisfy the equivalence criterion under the assumed true ratio and variability.

The sample size is therefore chosen so that:

where \(L\) and \(U\) are the equivalence limits on the appropriate scale.

A common target is 80% or 90% power. Higher power requires more participants.

Why the Expected GMR Matters

Suppose the anticipated geometric mean ratio is exactly 1.00. The study is then centered in the middle of the equivalence interval.

Now consider an anticipated GMR of 0.90. The ratio is still inside the equivalence interval, but it is closer to the lower boundary.

Consequently, it becomes more difficult to demonstrate bioequivalence. The sample size increases.

The same principle applies if the expected ratio is 1.10 or 1.15. The farther the anticipated ratio is from 1.00, the more participants are generally required.

The Equivalence Limits on the Log Scale

For conventional bioequivalence limits of 80% and 125%, the log-scale limits are:

Numerically:

The anticipated log treatment difference is:

For a GMR of 0.95:

For a GMR of 1.05:

These values are relatively close to zero, meaning the anticipated ratio remains comfortably inside the equivalence interval.

A Simplified Approximate Sample-Size Calculation

For teaching purposes, a simplified normal-theory approximation can illustrate the major drivers of sample size. Let:

• \(CV_W\) = within-subject coefficient of variation
• \(GMR\) = anticipated geometric mean ratio
• \(L\) = lower equivalence limit
• \(U\) = upper equivalence limit
• \(z_{1-\alpha}\) = one-sided critical value
• \(z_{1-\beta}\) = power-related critical value

Using:

and considering the more restrictive distance from the anticipated log ratio to the equivalence boundaries, a useful approximation is:

where \(\delta\) represents the relevant distance between the anticipated treatment difference and the closest equivalence boundary on the log scale.

Worked Example

Consider a standard two-treatment, two-period crossover bioequivalence study. Suppose the planning assumptions are:

The first step is to express the variability on the log scale.

Step 1: Convert CV to Log-Scale Variability

With:

the log-scale variance is:

Therefore:

Step 2: Convert the Expected GMR to the Log Scale

The anticipated GMR is 0.95:

Step 3: Determine the Distance to the Equivalence Boundaries

The log-scale equivalence limits are approximately:

The distance from the anticipated treatment difference to the lower boundary is:

The distance to the upper boundary is:

The lower boundary is therefore the limiting boundary. The relevant distance is approximately:

Step 4: Understand the Power Requirement

For 90% power:

The equivalence calculation also incorporates the relevant significance level through the critical value used for the confidence interval. The exact implementation depends on the statistical framework and regulatory convention used for the study.

A simplified calculation illustrates the relationship between variability, distance from the equivalence boundary, and sample size, but the final integer sample size should be obtained using an appropriate validated bioequivalence procedure.

Why Exact Sample Size Calculations Are Usually Preferred

The simplified normal approximation is useful for understanding the mathematics, but actual bioequivalence sample-size calculations are commonly based on the distribution of the treatment difference and the corresponding 90% confidence interval.

For a conventional two-sided bioequivalence test using 5% overall Type I error, the confidence interval is typically:

After exponentiation, the interval becomes a confidence interval for the GMR. Bioequivalence is concluded when the entire interval lies within the prespecified limits.

Because the critical value depends on the degrees of freedom, which in turn depend on sample size and the crossover analysis, sample-size determination can require an iterative calculation.

The 90% Confidence Interval

A common point of confusion is why bioequivalence studies frequently use a 90% confidence interval rather than the 95% confidence interval familiar from many clinical studies.

The conventional equivalence testing framework uses two one-sided tests, each at the 5% significance level. This is equivalent to evaluating a two-sided 90% confidence interval.

and:

Both null hypotheses must be rejected for bioequivalence to be established.

Sample Size and Within-Subject CV

Variability is often the dominant driver of bioequivalence sample size. Consider a hypothetical study with an expected GMR of 0.95 and otherwise identical assumptions.

The relationship is nonlinear because the variance enters the standard error and therefore interacts with the distance to the equivalence boundary.

What Happens When the Expected GMR Is 1.00?

Suppose the expected GMR is exactly 1.00. The log treatment difference is:

The distances to the two log-scale boundaries are equal:

This is the most favorable location within the equivalence interval. If the true GMR is expected to be close to 1.00, the study generally requires fewer participants than a study in which the anticipated ratio is close to 0.80 or 1.25.

Sample Size When the Expected GMR Is 0.90

Now suppose:

Then:

The distance to the lower boundary becomes:

This is considerably smaller than the 0.2231 distance available when the expected GMR is 1.00.

The result is a substantially larger sample-size requirement.

Parallel Designs vs. Crossover Designs

Although crossover designs are extremely common in bioequivalence studies, not every study uses a conventional 2 × 2 crossover.

The design should therefore be specified before the sample-size calculation is performed. The same CV does not necessarily produce the same sample-size requirement under every design.

Replicate Crossover Designs

Replicate designs can be particularly useful when within-subject variability is high or when the reference product itself is highly variable.

For example, a partial replicate design may expose subjects to the reference product more than once. This can provide information about reference variability that cannot be obtained from a conventional 2 × 2 crossover.

Replicate designs can therefore support specialized approaches to highly variable drugs and may allow regulatory criteria that differ from the simple 80–125% framework.

Highly Variable Drugs

A conventional bioequivalence study can become difficult when intra-subject variability is high. If the within-subject CV is large, the confidence interval around the GMR becomes wide.

For some highly variable drug products, specialized replicate designs can provide a more informative assessment of bioequivalence.

Depending on the applicable regulatory framework, scaled average bioequivalence approaches may be available for certain highly variable reference products. These methods incorporate reference variability into the equivalence criterion.

Dropout and Non-Evaluable Subjects

Bioequivalence studies require subjects to complete the necessary periods and provide evaluable pharmacokinetic data. A subject who completes only one period of a two-period crossover may not contribute the same information as a fully evaluable subject.

If the required number of evaluable subjects is:

and the anticipated proportion of non-evaluable subjects is \(r\), a simple inflation is:

For example, if 30 evaluable subjects are required and 10% are expected to be non-evaluable:

Therefore:

The actual protocol should also account for the specific handling of incomplete periods and the prespecified analysis population.

Dropout Inflation Is Not the Same as Power

A common mistake is to treat dropout inflation as though it increases the statistical power of the completed study. It does not.

The sample-size calculation first determines the number of evaluable subjects needed for the desired power. Enrollment is then increased so that the study is likely to retain approximately that many evaluable subjects.

Screen Failures and Randomization

Bioequivalence studies may also experience screening failures before randomization. These subjects should not generally be confused with post-randomization non-evaluable participants.

If the protocol requires a particular number of randomized and evaluable subjects, screening must be planned separately.

For example, if:

• 40 evaluable subjects are required,
• 10% of randomized subjects are expected to be non-evaluable, and
• 20% of screened subjects are expected to fail screening,

then the enrollment cascade is:

so approximately 45 subjects should be randomized. If 20% of screened subjects fail screening:

Approximately 57 subjects would therefore need to be screened.

Pharmacokinetic Endpoints and Sample Size

Sample size may differ for different pharmacokinetic endpoints. For example:

AUC and C_max are commonly analyzed using logarithmic transformation in conventional bioequivalence analyses. T_max is typically treated differently because it is a time-based endpoint and may not follow the same analysis framework.

Food-Effect and Other Specialized Studies

The same basic statistical concepts can appear in other pharmacokinetic study designs, including food-effect studies.

However, the estimand and design may differ. A food-effect study might compare fed and fasted conditions rather than a test formulation against a reference formulation.

The sample-size calculation must therefore correspond to the actual scientific question and design.

Bioequivalence vs. Therapeutic Equivalence

Bioequivalence is a pharmacokinetic concept. It concerns whether the systemic exposure from two pharmaceutical products is sufficiently similar under the prespecified criteria.

It should not automatically be interpreted as proof that two products are identical in every clinical respect.

The statistical sample-size calculation should therefore be tied to the bioequivalence endpoint and not to a conventional clinical efficacy effect size.

Average Bioequivalence vs. Individual Bioequivalence

Several different concepts of bioequivalence exist. The conventional framework used for many studies is average bioequivalence.

Average bioequivalence focuses on the population-level difference in mean log-transformed exposure.

Individual bioequivalence asks a different question involving within-subject variability and subject-by-formulation interaction. It requires a different statistical framework and is not simply a larger version of an average bioequivalence study.

The Relationship Between CV and Standard Error

On the log scale, the standard error of the treatment difference decreases as the number of subjects increases. Conceptually:

Therefore:

• Increasing CV increases the standard error.
• Increasing sample size decreases the standard error.
• Moving the expected GMR toward an equivalence boundary reduces the available margin.
• Higher power requires a smaller standard error.

These four relationships explain most of the behavior observed in bioequivalence sample-size calculations.

Why Sample Size Can Increase Dramatically Near the Boundary

Suppose the lower equivalence limit is 0.80. An anticipated GMR of 0.95 has a reasonable distance from the boundary. An anticipated GMR of 0.85 has much less room.

whereas:

On the log scale the corresponding relationship is even more appropriate because bioequivalence is evaluated multiplicatively.

As the anticipated GMR approaches 0.80 or 1.25, progressively more subjects are needed to demonstrate that the confidence interval remains entirely within the equivalence interval.

Planning From Historical Pharmacokinetic Data

The most important input for sample-size planning is often the estimate of within-subject variability. This can come from:

• Previous bioequivalence studies
• Reference-product studies
• Clinical pharmacology studies
• Published literature
• Internal development data

Historical estimates should be interpreted carefully. If the variability estimate is too optimistic, the resulting sample size may be insufficient.

A conservative approach may therefore evaluate several plausible CV assumptions.

Sample Size Sensitivity Analysis

A useful planning exercise is to vary both the expected GMR and within-subject CV. For example:

The exact numbers depend on the design and statistical procedure, but the direction of the relationship is consistent.

What Happens if the Study Is Underpowered?

Suppose a study is designed using a CV of 20%, but the true within-subject CV turns out to be 30%. The actual confidence interval will be wider than anticipated.

Consequently, the probability of successfully demonstrating bioequivalence will be lower than the planned power.

This is one of the principal reasons why reliable variability estimates are important in protocol development.

Rounding the Sample Size

Sample-size calculations should always be rounded upward. If a calculation produces:

the design should not use 31 subjects simply because 31 is the nearest integer. Instead:

For crossover studies, the final sample size may also need to be compatible with the planned sequence allocation.

For a two-sequence design, investigators often prefer balanced allocation, which may mean rounding the final target to an appropriate multiple of the number of sequences.

Balance Across Crossover Sequences

In a 2 × 2 crossover:

If 40 subjects are randomized, a balanced allocation might assign:

If the calculated sample size is 41, the protocol may choose to randomize 42 subjects so that the sequence allocation remains balanced.

Sample Size Planning in R

A simple exploratory calculation can be implemented in R using the PowerTOST package, which provides functions designed specifically for power and sample-size calculations in average bioequivalence studies.

install.packages("PowerTOST")
library(PowerTOST)

CV <- 0.25
theta0 <- 0.95
theta1 <- 0.80
theta2 <- 1.25
target_power <- 0.90

sampleN.TOST(
  CV = CV,
  theta0 = theta0,
  theta1 = theta1,
  theta2 = theta2,
  targetpower = target_power,
  design = "2x2"
)

The important inputs are:

• CV: anticipated within-subject coefficient of variation.
• theta0: anticipated GMR.
• theta1: lower equivalence limit.
• theta2: upper equivalence limit.
• targetpower: desired probability of demonstrating bioequivalence.
• design: the crossover design being evaluated.

The exact output depends on the package version and the statistical assumptions used by the selected design.

Exploring the Effect of CV in R

A useful sensitivity analysis is to evaluate several plausible CV assumptions:

library(PowerTOST)

CV.values <- c(0.15, 0.20, 0.25, 0.30, 0.35)

results <- lapply(
  CV.values,
  function(cv) {
    sampleN.TOST(
      CV = cv,
      theta0 = 0.95,
      theta1 = 0.80,
      theta2 = 1.25,
      targetpower = 0.90,
      design = "2x2"
    )
  }
)

results

This type of sensitivity analysis can show how strongly the enrollment requirement depends on the assumed intra-subject variability.

Planning for More Than One Pharmacokinetic Endpoint

Many bioequivalence studies evaluate multiple PK parameters. For example, both AUC and C_max may be important.

If the same study must establish bioequivalence for both endpoints, the study size should be sufficient for the endpoint with the more demanding requirement, subject to the applicable regulatory framework.

For example:

C_max may therefore drive the sample-size requirement even if AUC is less variable.

What About T_max?

T_max is fundamentally different from AUC and C_max. It is a time-to-peak measure and is commonly summarized using descriptive statistics and nonparametric methods rather than the standard log-transformed ANOVA framework used for AUC and C_max.

Consequently, a conventional ABE sample-size calculation for log(AUC) should not automatically be interpreted as a sample-size calculation for T_max.

Sequence, Period, and Treatment Effects

A standard 2 × 2 crossover model may include terms such as:

where the model components represent overall mean, subject, period, treatment, and residual variation according to the specific analysis parameterization.

The primary treatment comparison is the estimated difference between test and reference on the log scale.

The crossover design allows subject-specific variation to be separated from the within-subject residual variation that is particularly relevant to the precision of the treatment comparison.

Carryover and Washout

Crossover studies require sufficient washout between periods to minimize the possibility that drug exposure from the first treatment affects the second period.

Inadequate washout can undermine the assumptions of the crossover design.

First-Period Data and Incomplete Subjects

Because crossover studies rely on within-subject comparisons, complete information across periods is particularly valuable.

If a participant contributes only one evaluable period, the information available for the primary treatment comparison may be substantially reduced.

The protocol and statistical analysis plan should therefore define how incomplete subjects will be handled and how the sample-size target relates to the primary analysis population.

Common Sample Size Mistakes

1. Using between-subject variability instead of intra-subject variability. The standard crossover bioequivalence calculation is driven by within-subject variability.
2. Assuming GMR = 1.00 without justification. A GMR farther from 1.00 requires more information.
3. Using a 95% confidence interval automatically. Conventional average bioequivalence is generally evaluated using a 90% confidence interval corresponding to two one-sided 5% tests.
4. Thinking that an observed GMR between 80% and 125% proves bioequivalence. The confidence interval, not merely the point estimate, must satisfy the equivalence criterion.
5. Ignoring the crossover design. A 2 × 2 crossover, replicate crossover, and parallel design do not have the same sample-size calculation.
6. Ignoring high variability. Highly variable drugs may require substantially larger studies or specialized designs.
7. Using an optimistic CV estimate. Underestimating variability can result in inadequate power.
8. Ignoring incomplete crossover data. Subjects who fail to provide complete PK information may not contribute the same information as fully evaluable subjects.
9. Failing to maintain sequence balance. The final randomized sample may need to be rounded to preserve balanced treatment sequences.
10. Using a generic clinical-trial sample-size formula. Bioequivalence sample size requires a method appropriate to the log-transformed PK endpoint and equivalence framework.

A Practical Bioequivalence Sample Size Workflow

A useful workflow is:

1. Define the bioequivalence objective and estimand.
2. Specify the PK endpoint, such as AUC or C_max.
3. Specify the study design, such as a 2 × 2 crossover.
4. Obtain a scientifically defensible estimate of within-subject CV.
5. Specify the anticipated GMR.
6. Specify the applicable equivalence limits.
7. Specify the desired statistical power.
8. Use the appropriate confidence-interval and TOST framework.
9. Calculate the number of evaluable subjects.
10. Assess sensitivity of the calculation to CV and expected GMR.
11. Inflate for anticipated non-evaluable subjects and incomplete crossover data.
12. Round the final randomized sample appropriately for sequence balance.
13. Separately account for screening failures if necessary.
14. Document the statistical software, version, assumptions, and calculation.

What a Protocol Should State

A bioequivalence protocol should make the sample-size assumptions transparent. At minimum, the statistical section should identify:

• Primary pharmacokinetic endpoint
• Bioequivalence estimand
• Study design and treatment sequences
• Expected geometric mean ratio
• Within-subject coefficient of variation
• Lower and upper equivalence limits
• Target statistical power
• Type I error framework
• Confidence interval methodology
• Required number of evaluable subjects
• Expected non-evaluable rate
• Planned number randomized
• Sequence allocation
• Sample-size software and version
• Source of the variability estimate

This documentation allows reviewers to reconstruct the calculation and evaluate whether the study is appropriately designed.

Example Protocol Language

A protocol might describe the sample-size methodology along the following lines:

Why Bioequivalence Sample Size Is Different From Superiority Trials

The Central Mathematical Relationships

The key relationships can be summarized compactly.

First, convert the coefficient of variation to log-scale variance:

Then express the anticipated treatment difference on the log scale:

The conventional equivalence boundaries become:

The statistical objective is to obtain sufficient precision that the confidence interval for the treatment difference remains within those limits with the desired probability.

Summary

Sample-size planning for bioequivalence studies is a specialized application of equivalence testing and crossover-study methodology. Unlike a conventional superiority trial, the goal is not to demonstrate that one treatment is different from another. The objective is to demonstrate that the test and reference products are sufficiently similar according to prespecified equivalence limits.

For many conventional pharmacokinetic endpoints, AUC and C_max are analyzed after logarithmic transformation. The treatment effect is therefore expressed as a difference in mean log exposure, which corresponds to a geometric mean ratio after exponentiation.

The conventional bioequivalence limits of 80% and 125% correspond to:

or, on the log scale:

The major drivers of sample size are the within-subject coefficient of variation, the anticipated GMR, the equivalence limits, the desired power, and the crossover design.

Higher within-subject variability requires more participants. An expected GMR farther from 1.00 also requires more participants because the anticipated treatment ratio is closer to an equivalence boundary.

The standard 2 × 2 crossover design is particularly efficient because each subject receives both test and reference treatments, allowing the comparison to be based largely on within-subject information.

The conventional average bioequivalence framework uses two one-sided tests, which are equivalent to evaluating a two-sided 90% confidence interval and requiring the entire interval to fall within the prespecified equivalence limits.

After determining the required number of evaluable subjects, the study should be inflated for expected non-evaluable subjects, incomplete crossover data, and other operational losses. The final randomized sample may also need to be rounded to preserve balanced allocation across treatment sequences.

References

Food and Drug Administration. Bioavailability Studies Submitted in NDAs or INDs — General Considerations. U.S. Food and Drug Administration.

Food and Drug Administration. Statistical Approaches to Establishing Bioequivalence. U.S. Food and Drug Administration.

European Medicines Agency. Guideline on the Investigation of Bioequivalence. European Medicines Agency.

Chow, S.-C. & Liu, J.-P. (2008). Design and Analysis of Bioavailability and Bioequivalence Studies, 3rd ed. CRC Press.

Diletti, E., Hauschke, D. & Steinijans, V.W. (1991). Sample size determination for bioequivalence assessment by means of confidence intervals. International Journal of Clinical Pharmacology, Therapy and Toxicology, 29, 1–8.

Julious, S.A. (2004). Sample Sizes for Clinical Trials. Chapman & Hall/CRC.

Potvin, D., DiLiberti, C.E., Hauck, W.W., Parr, A.F., Schuirmann, D.J. & Sethuraman, V.S. (2008). Sequential design approaches for bioequivalence studies with crossover designs. Pharmaceutical Statistics, 7(4), 245–262.

Hauschke, D., Steinijans, V.W. & Pigeot, I. (2007). Bioequivalence Studies in Drug Development. Wiley.