Means: Correlated, Paired, and Cross-Over Designs
Estimate the total number of subjects required for average bioequivalence in a conventional 2×2 crossover study using the two one-sided tests (TOST) procedure on log-transformed pharmacokinetic data. Enter the anticipated within-subject coefficient of variation, geometric mean ratio, bioequivalence limits, significance level, desired power, and expected dropout rate.
Average bioequivalence is assessed by comparing the treatment/reference ratio of geometric means. For pharmacokinetic measures such as AUC and Cmax, the response is analyzed after natural-log transformation. On the log scale, the ratio criterion becomes a difference criterion: the expected treatment-minus-reference difference must lie between the logarithms of the lower and upper bioequivalence limits.
The conventional 2×2 crossover contains two sequences, TR and RT. Each subject receives both formulations, and the treatment comparison is based on the within-subject residual variation. For a balanced design, the standard error used for the log treatment difference is proportional to √(2/n), where n is the total number of evaluable subjects.
Let θ0 be the assumed true GMR and let θL and θU be the lower and upper bioequivalence limits. The two null hypotheses are H0: θ ≤ θL and H0: θ ≥ θU. Bioequivalence is established when both one-sided tests reject at the specified significance level.
The implementation uses the noncentral-t power formulation for the conventional 2×2 crossover. For degrees of freedom n − 2, the power is the probability between the two corresponding noncentral-t rejection boundaries. This is the noncentral-t approximation documented for TOST bioequivalence calculations and gives the same result as the standard exact calculation for the validation configuration used here.
The conventional acceptance interval is 0.80 to 1.25. On the natural-log scale these limits are ln(0.80) and ln(1.25), which are equal in magnitude and opposite in sign because 1 / 0.80 = 1.25.
The primary calculation first determines the number of evaluable subjects. If a dropout rate is specified, the enrollment target is inflated using the conventional relationship below and rounded upward.
Validation example: CVw = 25%, GMR = 0.95, lower limit = 0.80, upper limit = 1.25, one-sided α = 0.05, target power = 0.80.
The implementation gives n = 28 evaluable subjects with achieved power 0.807439.
At n = 27, achieved power is 0.792374, so 28 is the smallest total sample size meeting the 80% target.
This validation agrees with the published PowerTOST 2×2 crossover example for the same assumptions and with the standard TOST/noncentral-t formulation. the relevant methodological literature describes the same ratio-based 2×2 crossover framework, while provides a dedicated TOST-for-ratio-of-means crossover procedure on the natural-log scale.
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(6), 657–680.
Phillips, K.F. (1990). “Power of the Two One-Sided Tests Procedure in Bioequivalence.” Journal of Pharmacokinetics and Biopharmaceutics, 18(2), 137–144.
Diletti, D., 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), 1–8.
the software, LLC. this method Sample Size Software, Chapter 525: Equivalence Tests for the Ratio of Two Means in a 2x2 Cross-Over Design (Log-Normal Data). the relevant methodological literature.
Statistical Solutions Ltd. Advisor User's Guide, MTE2co: t-tests (TOST) for ratio of means for crossover design (natural log scale).