Bioequivalence and Bioavailability Forum

Dear Detlew,

sorry for the confusing post.

Summary of CIs of the 50331 sim’s passing the unrounded criterion:
     Unrd.lo              Unrd.hi               Rd.lo                Rd.hi
Min.   :1.03984217   Min.   :1.15101402   Min.   :1.03980000   Min.   :1.15100000
1st Qu.:1.13630237   1st Qu.:1.23050864   1st Qu.:1.13630000   1st Qu.:1.23050000
Median :1.15065573   Median :1.23972299   Median :1.15070000   Median :1.23970000
Mean   :1.15006990   Mean   :1.23640877   Mean   :1.15006972   Mean   :1.23640889
3rd Qu.:1.16470993   3rd Qu.:1.24556820   3rd Qu.:1.16470000   3rd Qu.:1.24560000
Max.   :1.22939131   Max.   :1.24999989   Max.   :1.22940000   Max.   :1.25000000
Summary of CIs of the 50503 sim’s passing the rounded criterion:
     Unrd.lo              Unrd.hi               Rd.lo                Rd.hi
Min.   :1.03984217   Min.   :1.15101402   Min.   :1.0398000   Min.   :1.15100000
1st Qu.:1.13631513   1st Qu.:1.23056103   1st Qu.:1.1363000   1st Qu.:1.23060000
Median :1.15069540   Median :1.23976572   Median :1.1507000   Median :1.23980000
Mean   :1.15011128   Mean   :1.23645515   Mean   :1.1501111   Mean   :1.23645518
3rd Qu.:1.16475001   3rd Qu.:1.24561372   3rd Qu.:1.1647500   3rd Qu.:1.24560000
Max.   :1.22939131   Max.   :1.25004974   Max.   :1.2294000   Max.   :1.25000000

❝ ❝     Unrounded.lo     Unrounded.hi  Rd.lo  Rd.hi

❝ ❝ 1.14406215516423 1.25000008857710 1.1441 1.2500

❝ ❝ …

❝ ❝ 1.16009942086582 1.25004974319996 1.1601 1.2500

❝

❝ I'm not certain if I understand your numbers given. How do the above correspond to below? Numbers from cases which were judged BE if rounded and not BE if not rounded?

❝ ❝     Unrounded.lo         Unrounded.hi         Rounded.lo         Rounded.hi

❝ ❝ Min.   :1.11998223   Min.   :1.25000009   Min.   :1.12000000   Min.   :1.25

❝ ❝ 1st Qu.:1.14730961   1st Qu.:1.25001184   1st Qu.:1.14730000   1st Qu.:1.25

❝ ❝ Median :1.16346327   Median :1.25002528   Median :1.16345000   Median :1.25

❝ ❝ Mean   :1.16222013   Mean   :1.25002554   Mean   :1.16221977   Mean   :1.25

❝ ❝ 3rd Qu.:1.17608605   3rd Qu.:1.25003838   3rd Qu.:1.17610000   3rd Qu.:1.25

❝ ❝ Max.   :1.20299871   Max.   :1.25004974   Max.   :1.20300000   Max.   :1.25

The list of the the 172 studies failing the unrounded but passing the rounded criterion above were ordered by the unrounded upper CL. So the lowest was 1.2500000885771 and the highest 1.25004974319996. This matches the summary above.

❝ ❝ Mr X will tell me “Nice simulations proving the patient’s risk is not maintained.”

❝ Augmented with the reply “If and only if one uses your (assuming Mister X to be a regulator) f*#*g rule of rounding the CI's.”

I would try to use a different wording – likely after consulting our capt’n.

❝ BTW: My original question was more concerned with empirical alpha>0.05 significant without rounding. I wouldn't expect such cases to be real. Otherwise the theory behind our BE statistics is wrong.

I wouldn’t say simulations disprove theory here. The convergence is slow (see the plot for Method B). Significant results might be pure chance.

❝ BTW2: There is a question that bothers me, every time I think about it:

❝ assuming BE if

❝    0.8 ≤ lCL and uCL ≤ 1.25 (I)

❝ or better

❝    0.8 < lCL and uCL < 1.25  (II).

Ouch, that hurts! Wellek (2003), Patterson & Jones (2006), Hauschke et al. (2007), Chow and Liu (2009):
$$\begin{matrix}
H_0:\mu_\textrm{T}-\mu_\textrm{R}\,{\color{Red}\leq}\,\theta_\textrm{L}\;\textrm{or}\;\mu_\textrm{T}\,{\color{Red}\geq}\,\theta_\textrm{U}\\
H_\textrm{a}:\theta_\textrm{L}\,{\color{Green}<}\,\mu_\textrm{T}-\mu_\textrm{R}\,{\color{Green}<}\theta_\textrm{U}
\end{matrix}$$Minority report^*:$$-\theta_\textrm{A}\,{\color{Red}\leq}\,\mu_\textrm{T}-\mu_\textrm{R}\,{\color{Red}\leq}\,\,\theta_\textrm{A}$$

❝ At least in formulating the bioequivalence alternative hypothesis it is always written:

❝    Θ1< µT/µR < Θ2

Yep, based on the above.

❝ and the corresponding two one-sided t-statistics have to be t_l < -t(1-α,df) and t_u > t(1-α,df). Does this transform really to (I) for the confidence interval inclusion rule?

No. Transforms definitely into (II).

❝ The EMA guidance is here clear: "To be inside the acceptance interval the lower bound should be ≥ 80.00% when rounded to two decimal places and the upper bound should be ≤ 125.00% when rounded to two decimal places." But the regulatory point of view is not necessarily the scientific one as we noticed more than once.

Wonderful. You discovered a flaw in the GL! According to the model BE should not be [0.8, 1.25] (borders inclusive) but ]0.8, 1.25[ (borders exclusive).

❝ In case of no rounding this doesn't make much difference since lCL=0.8 and uCL=1.25 (without rounding) are obtained with probability of nearly zero. But in case of rounding ...

¡Fantástico!
             unrounded         rnd. (commerc.)        rnd. (R)   
         [80,125]  ]80,125[  [80,125]  ]80,125[  [80,125]  ]80,125[
 79.900     –         –         –         –         –         –
 79.995     –         –         +         –         +         –
 79.999     –         –         +         –         +         –
 80.000     +         –         +         –         +         –
 80.001     +         +         +         –         +         –
 80.005     +         +         +         +         +         –
 80.010     +         +         +         +         +         +
124.900     +         +         +         +         +         +
124.995     +         +         +         –         +         –
124.999     +         +         +         –         +         –
125.000     +         –         +         –         +         –
125.001     –         –         +         –         +         –
125.005     –         –         –         –         +         –
125.010     –         –         –         –         –         –

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• Verbeeck R, Musuamba FT. The Revised 2010 EMA Guideline for the Investigation of Bioequivalence for Immediate Release Oral Formulations with Systemic Action. J Pharm Pharmaceut Sci. 2012; 15(3): 376–8. online

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P.S.: Another goodie from the FDA (see this post; downscaling the AR for NTIDs). Have a close look at this line of code:
theta=((log(1.11111))/0.1)**2;