Bioequivalence and Bioavailability Forum

Stephen Senn in Statistical issues in drug development: "Power: That which statisticians are always calculating but never have."
Dear All!

As discussed frequently here in the forum the CV used in the formulas / software for sample size estimation in BE studies are usually not "carved in stone" like Moses ten commandments (to cite the famous admin of this forum ) but known to us only with less or more imprecision as estimated CV with some degrees of freedom from one or more (pilot) studies done in the past or taken from the literature.

Helmut has described in his famous lectures how to combine such CV's and how to get an one-sided upper limit of the variability. With that upper limit one can then do a sensitivity analysis of the power / sample size.

Just discovered a paper and a book written by the well-known S.A. Julious

S. A. Julious, R. J. Owen
"Sample size calculations for clinical studies allowing for uncertainty about the variance"
Pharmaceutical Statistics, Vol 5(1), 29 - 37 (2006)

Steven A. Julious
"Sample Sizes for Clinical Trials"
CRC press, Chapman/Hall, Boca Raton - London - New York
© 2010 by Taylor and Francis Group, LLC
ISBN 978-1-58488-739-3

The latter had in chapter 7 "Sample Size Calculations for Bioequivalence Trials" - 7.2.6 or 7.3.2 "Calculations Taking Account of the Imprecision of the Variance Used in the Sample Size Calculations" some handy formulas how to calculate (expected) power taking into account the uncertainty of the variance estimate. Covered are cross-over studies and parallel-group design.

These formulas, approximate of course although not so named in the book, rely on the non-central t-distribution and are not so hard to code in any statistical software which has that distribution at hand, f.i. in R.

Using the design helper functions of my super-duper code here now the additions for the Julious formulas:

# approximate "expected" power according to Joulious book
# taking into account the uncertainty of a se estimated with dfse degrees of freedom
# raw function: see the call in the next function
.exp.power.TOST <- function(alpha=0.05, ltheta1, ltheta2, diffm,
                            se, dfse, n, df, bk=2)
{
  tval   <- qt(1 - alpha, df, lower.tail = TRUE)
  d1 <- sqrt(n*(diffm-ltheta1)^2/(bk*se^2))
  d2 <- sqrt(n*(diffm-ltheta2)^2/(bk*se^2))
  # Julious formula(s), the rest of code is decoration
  pow <- pt(d1,dfse,tval) + pt(d2,dfse,tval) - 1

  return(pow)
 
}
# --------------------------------------------------------------------
# "Expected" power according to Julious
# See known.designs() for implemented study designs
# Only for log-transformed data
# uses helper functions for the various BE designs
exp.power.TOST <- function(alpha=0.05, theta1=0.8, theta2, diff=0.95,
                           CV, dfCV, n, design="2x2")
{
  # check if design is implemented
  d.no <- .design.no(design)
  if (is.na(d.no)) stop("Err: unknown design!")
 
  # design characteristics
  ades <- .design.props(d.no)
  dfe  <- parse(text=ades$df[1],srcfile=NULL) #degrees of freedom as expression
  bk   <- ades$bk                             #design const.
 
  if (missing(CV)) stop("Err: CV must be given!")
  if (missing(n))  stop("Err: number of subjects must be given!")

  if (missing(theta2)) theta2 <- 1/theta1
  ltheta1 <- log(theta1)
  ltheta2 <- log(theta2)
  ldiff   <- log(diff)
  se      <- sqrt(log(1.+CV^2))
  df <- eval(dfe)
  pow <- .exp.power.TOST(alpha, ltheta1, ltheta2, ldiff, se, dfCV, n, df, bk)

  return( pow )
}

Homework: Calculate the "expected" power with diff=0.95, CV=0.3, dfCV=10 (12 subjects pilot) and n=40 and compare it to the 'classical' solution using D. Labes Eierlegende Wollmilchsau or ElMaestro's Apfelstrudel .

The story will continue within a short time. (Oh no, that guy. There we go again!)