Bioequivalence and Bioavailability Forum

Dear Detlew,

❝ It is explicitly stated that way (< and not ≤) in

❝

❝ Hauschke, Steinijans, Pigeot

❝ "Bioequivalence Studies in Drug Development"

❝ Wiley, Chichester, 2007, page 90

That’s what I wrote above. See also on top of page 89. The Nulls are given including the boundaries and the alternatives excluding them.

❝ ... but I have 2 other minority reports for you (proof/evidence by authority: "Well, Lieschen Mueller says it's true, so it must be." )

❝ Westlake, W.J.

❝ "Symmetrical Confidence Intervals for Bioequivalence Trials"

❝ Biometrics, 32, p 741-744 (1976)

❝

❝ stating the confidence interval inclusion rule explicitly with ≤ and

❝

❝ Diletti et.al.

❝ "Sample size determination for bioequivalence assessment by means of confidence intervals"

❝ Int. J. Clin. Pharm., Ther. and Tox., Vol.30, Supl. 1, p. S51-58 (192)

❝

❝ stating the two one-sided tests explicitly as

❝ t1=(mT-mR-ln(Θ1))/(sD*sqrt(2/n)) ≥ t(1-α,df)


❝ t2=(mT-mR-ln(Θ2))/(sD*sqrt(2/n)) ≤ -t(1-α,df)

Yep, but before (p. S52):

H₀: ln µ_T/µ_R≤ln θ₁ or ln µ_T/µ_R≥ln θ₂ (bioinequivalence)
H₁: ln θ₁<ln µ_T/µ_R<ln θ₂ (bioequivalence)

Drives me nuts.

What about Mr Schuirmann (1987)…

H₀₁: µ_T–µ_R≤θ₁
H₁₁: µ_T–µ_R>θ₁

and

H₀₂: µ_T–µ_R≥θ₁
H₁₂: µ_T–µ_R<θ₁

Bioequivalent if

t₁=≥t_1-α(ν) and t₂=≥t_1-α(ν)

Continuing with

“The two one-sided tests procedure turns out to be operationally identical to the procedure of declaring equivalence only if the ordinary 1-2α confidence (not 1-α) confidence interval for µ_T–µ_R is completely contained in the equivalence interval [θ₁, θ₂].”

Completely contained?

Kem Phillips (1990)

H₀: µ_T–µ_R<θ_L or µ_T–µ_R>θ_U
H₁: θ_L≤µ_T–µ_R≤θ_U

continuing with

“H₀ is rejected in favor of bioequivalence if T_L and -T_U equal or exceed t_1-α,ν […]”

And so on and so forth in many papers…

❝ Other papers state the interval inclusion rule as

❝ I ⊂ (Θ1,Θ2)

Oh yes. I use it sometimes myself as well.

❝ That time the "Theory of sets" was dealt with I have skipped school .

When I was in school from one year to the next everything was given as sets. Was fashionable for a while. Didn’t bother me too much because I’ve spent many schooldays in one of the many Viennese coffee houses anyhow.

❝ This is only an incomplete selection of findings which led to my uncertainness. As stated above: Using real numbers (not rounded) it will not make much a difference how we implement it,…

Agree. In my home-brew BE software I didn’t round at all, but tested for θ_L ≤ 90% CI ≤ θ_U. Ever since I’m using commercial software I’m in limbo. I validated Phoenix/WinNonlin with data sets from the literature (and even a very small one manually). But none of them “scratched at the edge”. The manual isn’t helpful:

If the interval (CI_Lower, CI_Upper) is contained within LowerBound and UpperBound, average bioequivalence has been shown.

Contained? Meaning ⊂ or ⊆?

❝ … thus we can't empirical test it via simulations.

Not sure what you mean here.