Bioequivalence and Bioavailability Forum

Dear Detlew,

❝ I think all the confusion comes from that sigma_w, sigma_d, sigma_delta values including their estimates which are used by all the authors cited within this thread in a different meaning.

Quite possible.

❝ I'm not able to figure out who is who, what is what. Sorry.

F**ing terminology.

❝ The only thing I'm convinced of is that your formula (2) above is correct.

❝ If you write the confidence interval for the BE decision as

❝ PE(T-R) +- SD_(d)*tval(0.95, df)

Exactly.

❝ The rest of your algebra is straight forward.

❝ And correct if you ask me .

THX. Now three people agree. I even didn’t trust my rusty algebra and asked Maxima for help:

❝ BTW: the formula (2) is not the error term in the 2×2×2 crossover.

How would you call it? We use \((3)\) in PowerTOST’s BE_CI.R line 30:

sqrt(mse*ades$bkni*nc)

where for design = "2x2" ades$bkni is 0.5 and nc is sum(1/n[1]+1/n[2]).
If we agree that \(\small{\widehat{\sigma}_\textrm{w}=\sqrt{MSE}}\) * we end up with \((2)\):

      For the special case of a two-treatment, two-period crossover study in which \(\small{n_\textrm{1}}\) subjects receive the test formulation in period one and the reference formulation in period two, while \(\small{n_\textrm{2}}\) subjects receive the reference formulation in period one and the reference formulation in period two, the unbiased estimator is given by $$\small{Est.=\frac{\left(\bar{X}_\textrm{T1}+\bar{X}_\textrm{T2} \right)}{2}-\frac{\left(\bar{X}_\textrm{R1}+\bar{X}_\textrm{R2} \right)}{2}}$$ where
      \(\small{\bar{X}_\textrm{T1}=}\) the observed mean of the \(\small{n_\textrm{1}}\) observations of the test formulation in period one.
      \(\small{\bar{X}_\textrm{T2}=}\) the observed mean of the \(\small{n_\textrm{2}}\) observations of the test formulation in period two.
      \(\small{\bar{X}_\textrm{R1}=}\) the observed mean of the \(\small{n_\textrm{2}}\) observations of the reference formulation in period one.
      \(\small{\bar{X}_\textrm{R2}=}\) the observed mean of the \(\small{n_\textrm{1}}\) observations of the reference formulation in period two.
The standard error of this estimator is $$\small{SE=s\sqrt{\frac{1}{2}\left (\frac{1}{n_\textrm{1}}+\frac{1}{n_\textrm{2}} \right)}}$$ where […] \(\small{s}\) is the square root of the “error” mean square from the crossover analysis of variance, based on \(\small{\nu}\) degrees of freedom.

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• Schuirmann DJ. A comparison of the Two One-Sided Tests Procedure and the Power Approach for Assessing the Equivalence of Average Bioavailability. J Pharmacokin Biopharm. 1987; 15(6): 657–80. doi:10.1007/BF01068419.