Population Bioequivalence Calculator

Methodology

Population bioequivalence (PBE) extends ordinary average bioequivalence by considering both the difference in population means and the variability of the test and reference formulations. For a 2×2 crossover design, Chow, Shao & Wang formulate the PBE criterion through the quantity λ.

PBE corresponds to the alternative λ < 0. Thus, the sample-size calculation is intended for parameter settings in which the assumed value of λ is negative.

Approximate sample-size formula

For equal sample sizes in the two crossover sequences, with n subjects per sequence, Chow, Shao & Wang give the following large-sample approximation:

Here, z_q is the q-th standard-normal quantile. The scaling coefficient is

The calculator rounds the resulting value of n upward to the next whole subject because the formula gives the required number of subjects per sequence. The total evaluable sample size is therefore 2n.

Parameter definitions

• δ: population mean difference between test and reference formulations.
• σ_TT²: total variance for the test formulation, combining between- and within-subject components.
• σ_TR²: total variance for the reference formulation.
• σ_BT² and σ_BR²: between-subject variances for test and reference.
• ρ: correlation between the subject-level test and reference random effects.
• σ_1,1: standard deviation associated with the paired test-reference contrast under a = b = 1.
• σ₀: FDA-specified constant used in the PBE criterion.
• θ_U: upper PBE limit.

Worked validation example

The published TrialSize implementation of the Chow, Shao & Wang procedure gives a worked PBE example with α = 0.05, β = 0.20 (80% power), σ_1,1 = 0.20, σ_TT = σ_TR = √0.17, σ_BT = σ_BR = 0.40, ρ = 0.75, θ_PBE = 1.74, δ = 0, and λ = −0.2966. The published result is n = 12 per sequence.

Using the same inputs, the implementation evaluates the approximation at approximately 11.73 before rounding, giving 12 subjects per sequence, or 24 total evaluable subjects. This reproduces the published worked-example result.

Important interpretation

This is an approximate planning formula for the balanced 2×2 crossover PBE procedure described by Chow, Shao & Wang. It is not the same calculation as the commonly used average-bioequivalence 90% confidence-interval/TOST approach. The authors specifically developed the method for PBE under 2×2, 2×3, and 2×4 crossover designs.

References