Bioequivalence and Bioavailability Forum

❝ My bad habit to use the terms acc. to guidelines (EMA 1401).

❝ The number of subjects to be included in the study should be based on an appropriate sample size calculation.

❝ Neverthless sample size estimation is correct.

❝ […] recently I was asked about response to insufficient sample size in performed study which demonstrated BE. It was study in 2x2 crossover design, no drop outs, sample size estimation performed and discussed in protocol. Only CV and GMR gone really much worse than expected.

❝ If we assumed the study results for another study, double sample size would be required for 80% power.

❝ When null hypothesis is false (formulations are BE) and we failed to reject bioinequivalence (probability beta = type II error = producer's risk) is for example 50%.

❝ One of two studies will fail by chance?

library(PowerTOST)
alpha  <- 0.05
theta0 <- 0.95
CV     <- 0.23
x      <- sampleN.TOST(alpha=alpha, CV=CV, theta0=theta0,
          targetpower=0.8, print=FALSE)
n      <- x[["Sample size"]]
pwr1   <- x[["Achieved power"]]
n
# [1] 24
rho    <- seq(0, 1, 0.005) # correlation
pwr2   <- vector()
for (j in seq_along(rho)) {
  pwr2[j] <- power.2TOST(CV=rep(CV, 2), theta0=rep(theta0, 2),
                         n=n, rho=rho[j])
}
round(pwr1, 4)   # one study
# [1] 0.8067
round(pwr1^2, 4) # intuitive for two studies
# [1] 0.6507
round(range(pwr2), 4)
# [1] 0.6562 0.8066
plot(rho, pwr2, type="l", ylab="power of 2 TOSTs", las=1)


❝ With such under powered study there will be low reproducibility of results (by the other (supporting) studies)?

❝ If someone got luck to win with power 50% or 33% ... it should be still no problem for regulatory or is there a problem that if someone want to remake the whole study (don't know why) it would be hard acc. to needed luck.

❝

❝ (I am just thinking and writing.)

❝ It is non sense when I got the idea to forget the power/producer's risk at all and design the study only for getting the 90% CI in 0.8000-1.2500.

The number of subjects in a clinical trial should always be large enough to provide a reliable answer to the questions addressed.
Using the usual method for determining the appropriate sample size, the following items should be specified: […] the probability of erroneously failing to reject the null hypothesis (the type II error) […]

❝ Without required power (not good statistical practice - I am just only playing with the numbers when I am using algebra for LL:

library(PowerTOST)
# functions to find the highest CV which will pass BE for
# given PE and n:
opt1 <- function(x) CI.BE(CV=x, pe=theta0, n=n)[["lower"]]-crit
opt2 <- function(x) CI.BE(CV=x, pe=theta0, n=n)[["upper"]]-crit
alpha  <- 0.05
theta0 <- 0.95
n      <- rep(12, 2)
if (theta0 <= 1) {
  crit <- 0.80
  CV   <- uniroot(opt1, interval=c(0.01, 5), tol=1e-8)$root
} else {
  crit <- 1.25
  CV   <- uniroot(opt2, interval=c(0.01, 5), tol=1e-8)$root
}
CV
# [1] 0.3573667
round(100*CI.BE(CV=CV, pe=theta0, n=n), 2)
# lower  upper
# 80.00 112.81
power.TOST(alpha=alpha, theta0=theta0, CV=CV, n=n)
# [1] 0.3534668

...
CV
# [1] 0.3020968
round(100*CI.BE(CV=CV, pe=theta0, n=n), 2)
# lower  upper
# 80.00 101.25
power.TOST(alpha=alpha, theta0=theta0, CV=CV, n=n)
# [1] 0.503441

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