Sample Size for Equivalence Trials

the most favorable location is often near:

while power decreases as the true difference approaches either boundary.

Sample Size Sensitivity Analysis

A robust sample-size assessment should examine how the required sample changes under different assumptions. For example:

This type of sensitivity analysis can reveal which planning assumptions dominate the study size.

R Implementation for the Worked Example

The approximate continuous-endpoint calculation can be implemented directly in R.

alpha <- 0.05
power <- 0.90

sigma <- 10
margin <- 5
delta <- 0

z_alpha <- qnorm(1 - alpha)
z_beta  <- qnorm(power)

n_per_group <-
  2 * sigma^2 *
  (z_alpha + z_beta)^2 /
  (margin - abs(delta))^2

ceiling(n_per_group)

The resulting sample size is approximately:

# 69

Thus approximately 69 evaluable patients are required per group.

Inflating the Sample Size for Dropout in R

dropout <- 0.10

n_randomized_per_group <-
  ceiling(n_per_group / (1 - dropout))

n_randomized_total <-
  2 * n_randomized_per_group

n_randomized_per_group
n_randomized_total

The result is:

# 77 patients per group
# 154 patients total

Writing a Reusable R Function

A reusable function makes it easier to perform sensitivity analyses.

equiv_n <- function(
  margin,
  sigma,
  delta = 0,
  alpha = 0.05,
  power = 0.90,
  dropout = 0
) {

  effective_margin <-
    margin - abs(delta)

  if (effective_margin <= 0) {
    stop(
      "Expected difference must lie inside the equivalence region."
    )
  }

  z_alpha <- qnorm(1 - alpha)
  z_beta  <- qnorm(power)

  n_evaluable <-
    2 * sigma^2 *
    (z_alpha + z_beta)^2 /
    effective_margin^2

  n_randomized <-
    ceiling(
      n_evaluable / (1 - dropout)
    )

  list(
    n_evaluable_per_group =
      ceiling(n_evaluable),

    n_randomized_per_group =
      n_randomized,

    n_randomized_total =
      2 * n_randomized
  )
}

For the worked example:

equiv_n(
  margin = 5,
  sigma = 10,
  delta = 0,
  alpha = 0.05,
  power = 0.90,
  dropout = 0.10
)

This returns approximately:

$n_evaluable_per_group
# 69

$n_randomized_per_group
# 77

$n_randomized_total
# 154

Performing a Margin Sensitivity Analysis

The effect of the equivalence margin can be examined by evaluating several candidate margins.

margins <- c(3, 4, 5, 6, 7)

sample_sizes <- sapply(
  margins,
  function(m) {

    ceiling(
      2 * sigma^2 *
      (qnorm(1 - alpha) + qnorm(power))^2 /
      m^2
    )
  }
)

data.frame(
  margin = margins,
  n_per_group = sample_sizes
)

This produces a simple sensitivity table showing how dramatically the required sample can change as the margin changes.

Evaluating Different Power Levels

powers <- c(0.80, 0.85, 0.90, 0.95)

sample_sizes <- sapply(
  powers,
  function(pow) {

    ceiling(
      2 * sigma^2 *
      (
        qnorm(1 - alpha) +
        qnorm(pow)
      )^2 /
      margin^2
    )
  }
)

data.frame(
  power = powers,
  n_per_group = sample_sizes
)

This allows the design team to see the cost of increasing the target power.

Evaluating Different Standard Deviations

sds <- c(8, 9, 10, 11, 12)

sample_sizes <- sapply(
  sds,
  function(s) {

    ceiling(
      2 * s^2 *
      (
        qnorm(1 - alpha) +
        qnorm(power)
      )^2 /
      margin^2
    )
  }
)

data.frame(
  standard_deviation = sds,
  n_per_group = sample_sizes
)

This is particularly useful when historical studies provide a range rather than a single reliable estimate of variability.

Exact vs. Approximate Sample Size Calculations

The formula presented for the continuous endpoint is an approximate normal-theory calculation. For many planning situations, this is a useful starting point. However, exact sample-size calculations may account for:

• Finite-sample t distributions
• Unknown variance
• Unequal variances
• Unequal allocation
• Non-normal endpoints
• Binary outcomes
• Ratio-scale outcomes
• Specific analysis models

When the final analysis uses a particular statistical model, the sample-size calculation should be aligned with that analysis.

Repeated Measures and Covariate Adjustment

Some equivalence trials analyze repeated measurements or adjust for baseline covariates. Covariate adjustment can improve precision when the covariate explains a substantial proportion of outcome variability.

For example, if an ANCOVA model reduces the residual variance relative to an unadjusted analysis, the required sample size may decrease.

However, this should not be assumed automatically. The planning calculation should reflect the anticipated residual variability under the prespecified analysis.

Clustered or Multicenter Designs

If participants are clustered within centers or other units, observations may not be independent. The effective sample size can be reduced by within-cluster correlation. A design effect may be approximated by:

where:

• \(m\) is the average cluster size.
• \(\rho\) is the intracluster correlation coefficient.

A rough inflation is then:

The appropriate approach depends on the actual analysis model and trial structure.

Equivalence Margin Selection

The most important non-statistical decision in an equivalence study is often the choice of the margin. A defensible process should consider:

• Clinical relevance
• Historical treatment effects
• Measurement variability
• Patient perspective
• Risk-benefit considerations
• Regulatory expectations
• The intended interpretation of the endpoint

The margin should be established before the study begins.

Preserving the Comparator Effect

In some equivalence settings, the scientific justification for the margin depends on historical evidence concerning the comparator. This is particularly important when the goal is to establish that a new treatment retains clinically important properties of an established treatment.

The equivalence margin should therefore be justified in relation to the clinical effect that would be considered acceptable to lose.

Sample Size and Study Credibility

A study can be statistically valid but clinically unconvincing if its equivalence margin is poorly justified. Conversely, a clinically reasonable margin may require a large sample.

This creates an important design tradeoff:

Common Sample-Size Mistakes

1. Using a superiority sample-size formula. Equivalence has a different hypothesis structure and requires an equivalence margin.
2. Defining equivalence as "p > 0.05." A nonsignificant superiority test does not establish equivalence.
3. Choosing the margin for convenience. The equivalence margin should be clinically justified.
4. Using the wrong confidence level. A 5% TOST framework corresponds to a 90% two-sided confidence interval.
5. Ignoring the expected treatment difference. If the expected difference is close to an equivalence boundary, the required sample size can increase substantially.
6. Underestimating variability. Because sample size is proportional to \(\sigma^2\), modest changes in variability can materially affect the required N.
7. Forgetting dropout inflation. The calculated number of evaluable patients is not necessarily the number that should be randomized.
8. Ignoring the analysis scale. Ratio-based endpoints often require calculations on the log scale.
9. Using a continuous-endpoint formula for a binary endpoint. Binary endpoints require their own variance and effect-measure assumptions.
10. Failing to match the sample-size calculation to the primary analysis. The design, estimand, endpoint, and analysis model should be internally consistent.

A Practical Equivalence Trial Workflow

What Should Be Reported in the Protocol?

The sample-size section should allow another statistician to reproduce the calculation. At minimum, document:

• Primary endpoint
• Effect measure
• Equivalence margins
• Clinical rationale for the margins
• Expected treatment difference
• Standard deviation or event-rate assumptions
• Type I error
• Target power
• Allocation ratio
• Required evaluable sample size
• Expected dropout rate
• Final randomized sample size
• Sample-size calculation method
• Software used
• Relevant assumptions

Worked Example Summary

The Most Important Concept

The most important idea in equivalence-trial sample-size planning is that the study must be precise enough to rule out clinically important differences.

The calculation therefore depends heavily on the relationship between:

• The equivalence margin
• The expected treatment difference
• The variability of the endpoint
• The desired power
• The sample size

For a continuous endpoint with a symmetric margin and expected difference of zero, the central relationship is:

This immediately shows why:

• Higher variability requires more patients.
• Higher power requires more patients.
• A narrower equivalence margin requires substantially more patients.
• A treatment difference closer to an equivalence boundary requires more patients.
• Dropout requires additional randomized patients.

References

Schuirmann, D.J. (1987). A comparison of the two one-sided tests procedure and the power approach for assessing the equivalence of average bioavailability. Journal of Pharmacokinetics and Biopharmaceutics, 15, 657–680.

Westlake, W.J. (1976). Symmetrical confidence intervals for bioequivalence trials. Biometrics, 32, 741–744.

Chow, S.-C., Shao, J. & Wang, H. Sample Size Calculations in Clinical Research. Chapman & Hall/CRC.

Chow, S.-C. & Liu, J.-P. Design and Analysis of Bioavailability and Bioequivalence Studies. Marcel Dekker.

Blackwelder, W.C. (1982). Proving the null hypothesis in clinical trials. Controlled Clinical Trials, 3, 345–353.

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.

Hauschke, D., Pigeot, I. & Tamhane, A.C. Statistical Principles for Clinical Trials. Wiley.