Bioequivalence and Bioavailability Forum

Dear Dave,

❝ [...] The FDA asked for: [...]

❝

❝ a 95% upper bound for (uT - uR)² / (sig_wr)² - theta < 0

❝

❝ The FDA wants you to use this formula to calculate theta:

❝

❝ theta = [ln(1.25)]² / sig_w₀², where sig_w₀ = 0.25.

❝ theta = 0.796689

❝

❝ Theta is a constant, and sig_wr is also a constant once you calculate what it is.

(emphasis by me)

This emphasized sentence was the reason why I wondered about Helmut's suggestion.

We do not know sigma_WR, but have an estimate of it from our data!
From Proc MIXED or Methods of moments or whatever. This estimate has accompanying an uncertainty which must IMHO be taken into account in the calculation of the 95% upper bound for the scaled ABE criterion.
BTW the linearized criterion is stated as
(µ_T - µ_R)² - theta×sigma²_WR < 0
(lineal linear except the super- and subscripts )

I recently found a paper from Tothfalusi et. al.¹⁾ in which the details of the calculations of the upper bound (bases on Hyslop's method invented for the IBE criterion, I think) are given to some extent.

Let me try to excerpt it:
"[...] The two terms of the criterion can be estimated by their respective expected values:
   Em=(mT-mR)2
   Ew= Theta*s2WR
where m_T and m_R are the observed overall means of the test and reference formulations, respectively and s²_WR the estimated within-subject variance for the reference product.

The confidence limits for the two terms in the rearranged BE criterion are:
   Cm=[Abs(mT-mR)+t(1-alpha,N-2)*SE]2
   Cw= Theta*(N-2)sWR2/chi2(alpha,N-2)

Here as already noted, SE =(s²_Int/N)^1/2.
(this is for 4 period design as indicated in the heading not excerpted D.L.)
t and chisq are the respective tabulated statistics [...]
(here begins the Hyslop method I think, D.L.)
The squared length of the onfidence intervals from their respective means are:
   Lm=(Cm-Em)²
   Lw=(Cw-Ew)²

Finally the confidence limit (CL) for the rearranged BE criterion is:
   CL=Em-Ew+(Lm+Lw)½
[...]" (EoE end of excerpt)

Seems easy enough to implement (of course in R ) . Provided it is that easy.
My only doubt here is that I don't know, what they really mean with s²_Int in their formula for replicate design.
But this does not matter. Let your software calculate the SE for µ_T-µ_R for your design and use it in C_m.

¹⁾ Tothfalusi, L., L. Endrenyi, K.K. Midha, M.J. Rawson and J.W. Hubbard
Evaluation of the bioequivalence of highly-variable Drug products
Pharm. Res. Vol.18, No. 6 (2001), 728-733

Hope this helps someone.
I myself haven't used this up to now.
Anybody out there who has an example or knows one, preferably elaborated?

❝ [...] I'd be interested to see some SAS code from someone who's done the job properly and knows what they're doing.

                        (emphasis by me)
Bible, New testament, Luke 23:34