Bioequivalence and Bioavailability Forum

Hi Imph,

❝ Thank you so much. Your response has been of great help.

Welcome.

❝ – By consulting Dileti's paper, I noticed that there is no formula for the calculation of the number of subjects. Is there a way or a reference to get the formula please.

You can only calculate power for a given sample size, i.e., you start with an assumption and then increase (or decrease) n to obtain at least the target power. See there for the details.

Step by step:

library(PowerTOST)
CV     <- 0.25
theta0 <- 0.95
target <- 0.80
n      <- 12                    # minimum acc. to the GLs
power  <- power.TOST(CV = CV, theta0 = theta0, n = n)
iter   <- 1
res    <- data.frame(iter = iter, n = n, power = power)
if (res$power[iter] < target) { # upwards
  repeat {
    power <- power.TOST(CV = CV, theta0 = theta0, n = n)
    res[iter, ] <- c(iter, n, power)
    if (power >= target) {
      break
    } else {
      iter <- iter + 1
      n    <- n + 2
    }
  }
} else {                        # downwards
  repeat {
    power <- power.TOST(CV = CV, theta0 = theta0, n = n)
    res[iter, ] <- c(iter, n, power)
    if (power < target) {
      res <- res[-nrow(res), ]
      break
    } else {
      iter <- iter + 1
      n    <- n - 2
    }
  }
}
print(res, row.names = FALSE)

 iter  n     power
    1 12 0.3137351
    2 14 0.4141013
    3 16 0.5041795
    4 18 0.5801284
    5 20 0.6430574
    6 22 0.6953401
    7 24 0.7391155
    8 26 0.7760553
    9 28 0.8074395

In PowerTOST’s sample size functions you can show the iterations by setting the argument details to TRUE (by default only the final result is shown):

sampleN.TOST(CV = 0.25, theta0 = 0.95, targetpower = 0.80, details = TRUE)

+++++++++++ Equivalence test - TOST +++++++++++
            Sample size estimation
-----------------------------------------------
Study design: 2x2 crossover
Design characteristics:
df = n-2, design const. = 2, step = 2

log-transformed data (multiplicative model)

alpha = 0.05, target power = 0.8
BE margins = 0.8 ... 1.25
True ratio = 0.95,  CV = 0.25

Sample size search (ntotal)
 n     power
26   0.776055
28   0.807439

Exact power calculation with
Owen's Q functions.

❝ – […] in the calculation of the degrees of freedom, for example (2n-2), where "n" has to be estimated. I would like to know on what basis this "n" is estimated.

See above.

Dear Imph,

beside to all what Helmut has told you you may find in the /doc directory of the package a PDF file "BE_power_sample_size_excerpt.pdf" which has some small further informations about the sample size estimation.

You may obtain an approximation of n by using the large sample approximation.
See P. Zhang
“A simple formula for sample size calculation in equivalence studies.”
J. Biopharm. Stat. 2003 Aug;13(3):529-38
This approximate number is in many cases relatively close to the exact value, and is used in PowerTOST as starting value of the iterative search of the sample size.

Hi Willem,

❝ Could you elaborate on your note of the use of PowerTOST by FDA? It would provide me an additional argument for using PowerTOST for formal sample size estimation in our company.

Maybe this post helps.

About the validation of PowerTOST with examples:

• Parallel design
  
• 2×2×2 crossover design
  
• Reference-scaling

See this document¹ about the acceptability of Base R. R is updated every couple of months with documented changes² and maintaining a bug-tracking system.³ I recommend to use always the latest release.

The authors of PowerTOST tried to do their best to provide reliable and valid results. The package’s NEWS-file documents the development of the package, bug-fixes, and introduction of new methods. Issues are tracked at Git­Hub (as of today none is still open). So far the package had ≈93,000 downloads. Therefore, it is extremely unlikely that bugs were not detected given its large user base.

However, the ultimate responsibility of validating any software (yes, of SAS as well…) lies in the hands of the user.

• The software programs […] should be suitable and validated for the intended use. Whether standard, off-the-shelf software is purchased or bespoke software is developed, developer, vendor and/or service provider qualification and/or validation certificates may be provided but it is the user’s responsibility to ensure that the software is validated for its intended use and that it was developed in a controlled manner in accordance with a QA system. […] There should be SOPs in place for usage of each software program that is used to perform activities of a BE study. […] Software programs used […] should be specified in writing. […] These requirements apply to all systems used in clinical BE studies, e.g. […] for statistical analysis […].⁴
  
• The credibility of the numerical results of the analysis depends on the quality and validity of the methods and software (both internally and externally written) used […] for processing the data statistically. Data management activities should therefore be based on thorough and effective standard operating procedures. The computer software used for […] statistical analysis should be reliable, and documentation of appropriate software testing procedures should be available.⁵
  
• FDA does not require use of any specific software for statistical analyses […]. However, the software package(s) used for statistical analyses should be fully documented in the submission, including version and build identification.⁶

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1. The R Foundation for Statistical Computing. A Guidance Document for the Use of R in Regulated Clinical Trial Environments. Vienna. October 18, 2021. Online.
   
2. The R Foundation. R News. 2022-04-22. Online.
   
3. Bugzilla. R bug tracking system. Online.
   
4. WHO. Guidance for organizations performing in vivo bioequivalence studies. Geneva. May 2016. Technical Report Series No. 996, Annex 9. Section 4.4. Online.
   
5. FDA, CDER/CBER. E9 Statistical Principles for Clinical Trials. Rockville. September 1998. Section V.H. Download.
   
6. FDA. Statistical Software Clarifying Statement. May 6, 2015. Download.