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Population Bioequivalence

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).

Population bioequivalence (PBE) is tested using the following hypothesis:

\[ H_0: \lambda \ge 0 \text{ versus } H_1: \lambda < 0 \]

where

\[ \lambda = \delta^2 + \sigma^2_{TT} - \sigma^2_{TR} - \theta_{PBE} \max\{ \sigma^2_0, \sigma^2_{TR} \} \]

\(\delta = F_T - F_R \), \(\sigma^2_{TT}\) is the total variance for the test formulation, \(\sigma^2_{TR}\) is the total variance for the reference formulation, \(\theta_{PBE}\) and \(\sigma_0\) are a constants specified by the FDA.

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

\[ \hat{\sigma}_{1,1} = \frac{1}{2(n - 1)} \sum^2_{k=1} \sum^n_i=1 (y_{i1k} - y_{i2k} - \bar{y}_{\cdot1k} + \bar{y}_{\cdot2k})^2 \]

as well as the between subject variances within each formulation (\(\sigma_{BT}\) and \(\sigma_{BR}\)) and the intersubject correlation (\(\rho\)).

Power Calculation Parameters

To input multiple values, seperate them by a comma.

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Power Calculation Explanation

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.
\(\lambda\)
The effect size defined above.
\(\rho\)
The intersubject correlation.
\(\sigma_{1,1}\)
The standard deviation for the inter-subject comparison.
\(\sigma_{TT}\)
The total standard deviation for test formulation.
\(\sigma_{TR}\)
The total standard deviation for reference formulation.
\(\sigma_{BT}\)
The intersubject standard deviation for test formulation.
\(\sigma_{BR}\)
The intersubject standard deviation for reference formulation.
\(\theta_{PBE}\)
The bioequivalence limit (a constant set by the FDA; see their 2001 guidance).

Calculation Results

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N Alpha Power \(\delta\) \(\lambda\) \(\rho\) \(\sigma_{1,1}\) \(\sigma_{TT}\) \(\sigma_{TR}\) \(\sigma_{BT}\) \(\sigma_{BR}\) \(\theta_{PBE}\)
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Power Graph

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References