## I'm seeking to understand the math behind our current regulation [General Sta­tis­tics]

» Hi Victor, I tried to reconstruct your original post as good as I could. Since it was broken before the first “$$\mathcal{A}$$”, I guess you used an UTF-16 character whereas the forum is coded in UTF-8.

Hi Helmut, Thanks for helping me out :)

Edit: after a quick experiment (click here to see screenshot), it seems that the “$$\mathcal{A}$$” I used was a UTF-8 character after all? ⊙.☉

» Please don’t link to large images breaking the layout of the posting area and forcing us to scroll our viewport. THX.

Noted, and thanks for downscaling my original image :)

» I think that your approach has same flaws.
• You shouldn’t transform the profiles but the PK metrics AUC and Cmax.

I see; I thought it would make sense for Tmax to also be transformed because of googling stuff like this: coupled with the fact that the population distribution that is being analyzed looks a lot like a Log-normal distribution; so I thought normalizing Tmax just made sense, since almost all distributions studied in undergraduate (e.g. F-distribution used by ANOVA) are ultimately transformations of one or more standard normals. With that said, is the above stuff that I googled, wrong?

»
• The Null hypothesis is bioinequivalence, i.e.,
» $$H_0:\mu_T/\mu_R\notin \left [ \theta_1,\theta_2 \right ]\:vs\:H_1:\theta_1<\mu_T/\mu_R<\theta_2$$ where $$[\theta_1,\theta_2]$$ are the limits of the acceptance range. Testing for a statistically significant difference is futile (i.e., asking whether treatments are equal). We are interested in a clinically relevant difference $$\Delta$$. With the common 20% we get back-transformed $$\theta_1=1-\Delta,\:\theta_2=1/(1-\Delta)$$ or 80–125%.

Thanks for enlightening me that I can now restate the current standard's hypothesis in a "more familiar (undergraduate-level)" form:
$$H_0: ln(\mu_T) - ln(\mu_R) \notin \left [ ln(\theta_1), ln(\theta_2) \right ]\:vs\:H_1: ln(\theta_1) < ln(\mu_T) - ln(\mu_R) < ln(\theta_2)$$

I now realize that I was actually using the old standard's hypothesis (whose null tested for bioequivalence, instead of the current standard's null for bioinequivalence), which had problems with their α & β (highlighted in red below, cropped from this paper), thus rendering my initial question pointless, because I was analyzing an old problem; i.e. before Hauck and Anderson's 1984 paper. »
• Nominal $$\alpha$$ is fixed by the regulatory agency (generally at 0.05). With low sample sizes and/or high variability the actual $$\alpha$$ can be substantially lower.
• Since you have to pass both AUC and Cmax (each tested at $$\alpha$$ 0.05) the intersection-union tests keep the familywise error rate at ≤0.05.

With that said, regarding the old standard's hypothesis (whose null tested for bioequivalence), I was originally curious (although it may be a meaningless problem now, but I'm still curious) on how they bounded the family-wise error rate (FWER) if α=5% for each hypothesis test, since the probability of committing one or more type I errors when performing three hypothesis tests = 1 - (1-α)^3 = 1 - (1-0.05)^3 = 14.26% (if those three hypothesis tests were actually independent).

The same question more importantly applied to β, since in the old standard's hypothesis (whose null tested for bioequivalence), "the consumer’s risk is defined as the probability (β) of accepting a formulation which is bioinequivalent, i.e. accepting H0 when H0 is false (Type II error)." (as quoted from page 212 of the same paper).

Do you know how FDA bounded the "global" α & β before 1984? Because I am curious on "what kind of secret math technique" was happening behind-the-scenes that allowed 12 random-samples to be considered "good enough by the FDA"; i.e.
• How to calculate the probability of committing one or more type I errors when performing three hypothesis tests, when the null was tested for bioequivalence (before 1984)?
• How to calculate the probability of committing one or more type II errors when performing three hypothesis tests, when the null was tested for bioequivalence (before 1984)?

ଘ(੭*ˊᵕˋ)੭* ̀ˋ Ing. Helmut Schütz 