{{ message }}

# 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.

{{ this.errors }}

## 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

No calculation has been generated yet.

N Alpha Power $$\delta$$ $$\lambda$$ $$\rho$$ $$\sigma_{1,1}$$ $$\sigma_{TT}$$ $$\sigma_{TR}$$ $$\sigma_{BT}$$ $$\sigma_{BR}$$ $$\theta_{PBE}$$
{{ val }}