Bioequivalence and Bioavailability Forum

Hi zizou,

❝ 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.

Apart from the sloppy terminology this section is unfortunate. According to members of the EWP-PK drafting group (now PKWP) the GL should offer a kind of a “cook-book”. So why the heck the perfect recipe –

The number of subjects required is determined by

1. the error variance associated with the primary characteristic to be studied as estimated from a pilot experiment, from previous studies or from published data,
   
2. the significance level desired,
   
3. the expected deviation from the reference product compatible with bioequivalence (delta) and
   
4. the required power.

which was part of all [sic] previous versions – was replaced by the laconic elastic clause* “appropriate”?

❝ […] 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.

Still I hold that if BE was demonstrated the sample size was sufficient indeed. The former implies the latter. Only assumptions were “disproved”.
The assessor could write a letter (not a deficiency letter!) essentially saying “You were lucky this time. Granted. In the future please pay more attention to the sample size, i.e., that assumptions will not be overly optimistic. A copy of this letter will be send to the responsible IEC. Thank you.”

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

Yes, that’s the purpose of power. Plan another study properly.

❝ 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?

On the long run, yes. Also by chance, you can have another study passing as well. We discussed the behavior of power in multiple studies in another thread.
Intuitively p = ∏p_i (e.g., 0.64 for two studies with 0.8). Benjamin Lang coded the function power.2TOST() in PowerTOST. Type help(power.2TOST) for references. His original idea was to provide power for testing two correlated PK-metrics (hence the argument rho in the function) in the same study. Maybe we can misuse it for two studies?

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)


If studies are independent (ρ = 0) we arrive approximately at the intuitive result. If ρ = 1 combined power would be ~ the single studies’ power. The latter is practically impossible (first of all we would have to repeat the study in the same subjects). The “truth” may lie somewhere in between but don’t ask me where.

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

Tricky. I cannot imagine that any BE-study ever was repeated. The closest would be the combination of a pilot and a pivotal study in a larger sample size. If you consider a pilot study supportive, its (generally low) power is not relevant anyhow. IMHO, it is only the pivotal study which counts.

❝ 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.)

Indeed. But I also can’t imagine why one being already aware of a lucky strike (low power in the first study) would gamble yet another time.
The guy in the Armani suit? ©2010 by ElMaestro

❝ 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.

True. Remember that in the old NfG/GL power was mentioned. For the initiates “appropriate” is backed by ICH-E9 Section 3.5:

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:

Confirmed your results.

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

and

...
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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