## What do you want to achieve? [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. Please don’t link to large images breaking the layout of the posting area and forcing us to scroll our viewport. THX.

I think that your approach has same flaws.

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

I think that your approach has same flaws.

- You shouldn’t transform the profiles but the PK metrics AUC and C
_{max}.

- 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%.

*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 C_{max}(each tested at \(\alpha\) 0.05) the intersection-union tests keep the familywise error rate at ≤0.05.

- For given design, sample size, variability, and point estimate calculation of \(\alpha\) and \(\beta\) is straightforward. R-code for the package
`PowerTOST`

at the end.

- t
_{max}follows a discrete distribution and hence, should be assessed by a nonparametric test.

`library(PowerTOST)`

design <- "2x2x2" # for others, see known.designs()

n <- 24

CV <- 0.25

PE <- 0.95

alpha <- power.TOST(CV = CV, n = n, theta0 = 1.25, design = design)

beta <- 1 - power.TOST(CV = CV, n = n, theta0 = PE, design = design)

cat("alpha =", alpha,

"\nbeta =", beta, "\n"

Gives

`alpha = 0.04999527 `

beta = 0.2608845

—

Helmut Schütz

The quality of responses received is directly proportional to the quality of the question asked. 🚮

Science Quotes

*Dif-tor heh smusma*🖖Helmut Schütz

The quality of responses received is directly proportional to the quality of the question asked. 🚮

Science Quotes

### Complete thread:

- What is the largest α (Alpha) & β (Beta) allowed by FDA? victor 2019-11-16 21:57
- What do you want to achieve?Helmut 2019-11-17 01:26
- I'm seeking to understand the math behind our current regulation victor 2019-11-17 10:53
- Some answers Helmut 2019-11-17 14:35
- Wow! Amazing answers! victor 2019-11-18 08:26
- More answers Helmut 2019-11-18 15:09
- Wow! More amazing answers! victor 2019-11-18 20:16
- Books & intersection-union Helmut 2019-11-19 12:01
- My progress on IUT so far victor 2019-11-22 01:28
- Update: Counterexamples victor 2019-11-23 09:05

- My progress on IUT so far victor 2019-11-22 01:28

- Books & intersection-union Helmut 2019-11-19 12:01

- Wow! More amazing answers! victor 2019-11-18 20:16

- More answers Helmut 2019-11-18 15:09

- Wow! Amazing answers! victor 2019-11-18 08:26

- Some answers Helmut 2019-11-17 14:35

- I'm seeking to understand the math behind our current regulation victor 2019-11-17 10:53

- What do you want to achieve?Helmut 2019-11-17 01:26