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This calculator uses the 2 x 2 crossover design recommended by FDA guidelines. The test of average bioequivalence is a the same as testing the equivalence for means. If another design is used, the two sample crossover calculator for hypotheses on means can be used.

Letting \(y_{ijk}\) be the PK response of interest (in log transfored or in original units) of the \(i^{th}\) subject in the \(k^{th}\) sequence in the \(j^{th}\) dosing period, the model used to test average bioequivalence is given by:

\[ y_{ijk} = \mu + F_l + P_j + Q_k + S_{ikl} + \epsilon_{ijk} \]
where \(\mu\) is the overall mean, \(P_j\) is the fixed period effect, \(Q_k\) is the fixed sequence effect, \(F_l\) is the fixed effect of the \(l^{th}\) formulation (\(l = T, R\)), and \(S_{ikl}\) is the random effect of the \(i^{th}\) subject in the \(k^{th}\) sequence using the \(l^{th}\) formulation (see Chow *et at.* and the FDA guidance for a more detailed specification of the model).

To test average bioequivalence, we test the hypothesis:

\[ H_0: \delta \le \delta_L \text{ or } \delta \ge \delta_U \text{ versus } H_1: \delta_L \lt \delta \lt \delta_U \]where \(\delta = F_{T} - F_{R}\) and \(\delta_L\) and \(\delta_U\) are bioequivalence limits.

To perform this calculation, we need to estimate the standard deviation of the intersubject comparison: \(\sigma_{1,1}\). The value can be estimated by:

\[ \hat{\sigma}_{1,1} = \frac{1}{2(n - 1)} \sum^2_{k=1} \sum^n_i=1 (y_{i1k} - y_{i2k} - \bar{y}_{\cdot1k} + \bar{y}_{\cdot2k})^2 \]To input multiple values, seperate them by a comma.

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- Solve For
- The unknown you are interested in solving for.
- N
- The sample size used to test the hypothesis.
- Alpha
- The \(\alpha\) (Type I error rate) level of the hypothesis test.
- Power
- The power (1 - Type II error rate) of the hypothesis test.
- \(\delta\)
- The difference of the fixed effects.
- Standard Deviation (\(\sigma_{1,1}\))
- The standard deviation for the inter-subject comparison.
- Bioequivalence Margin
- The difference between the upper (lower) bioequivalence limit and \(\delta\) (
*i.e.*\(\delta_U - \delta\)). Note: for this calculator, we assume \(\delta_U - \delta = \delta - \delta_L\) - Known Standard Deviation
- If the standard deviation is known or estimated (usually it is estimated).

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Known Standard Deviation

Unknown Standard Deviation

N | Alpha | Power | \(\delta\) | Standard Deviation | Margin |
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- Chow, S., Shao, J., & Wang, H. (2003),
*Sample size calculations in clinical research,*New York: Marcel Dekker. - FDA (2001). Guidance for Industry on Statistical Approaches to Establishing Bioequivalence. Center for Drug Evaluation and Research, Food and Drug Administration, Rockville, MD.