Power with a Danish twist [Power / Sample Size]

posted by Helmut Homepage – Vienna, Austria, 2017-02-07 15:23  – Posting: # 17031
Views: 19,677

Hi ElMaestro,

» […] DKMA are currently to the best of my knowledge only enforcing the requirement in relation to substitution.

Seems so. Acceptance limits for AUC and Cmax of antiepileptics apart from levetiracetam and benzodiazepines are 90.00-111.11%. Now we could ask what is the maximum deviation of the PE from 100% for a given sample size (based on CV and target power). Example valproic acid, CV for Cmax 10% (AUC 5–7%), target power 80%, θ0 0.975 (as suggested by the FDA; tighter release spec’s for NTIDs):

library(PowerTOST)
dk <- function(CV, n, lower, upper, theta0) { # Danish power
        p0 <- power.TOST(CV=CV, n=n, theta0=theta0,
                         theta1=lower, theta2=upper)
        p1 <- power.TOST(CV=CV, n=n, theta0=theta0,
                         theta1=lower, theta2=1)
        p2 <- power.TOST(CV=CV, n=n, theta0=theta0,
                         theta1=1, theta2=upper)
        p0-p1-p2
      }
of <- function(x) { # objective function
        (dk(CV=CV, n=n, lower=lower, upper=upper, theta0=x)-target)^2
      }
lower  <- 0.9 # NTID-range
upper  <- 1/lower
CV     <- 0.1
target <- 0.8
theta0 <- 0.975
n      <- sampleN.TOST(CV=0.1, theta0=theta0, theta1=lower, theta2=upper,
                       targetpower=target, print=FALSE)[["Sample size"]]
theta0.dk <- optimize(of, interval=c(lower, 1), tol=1e-12)$minimum
power  <- dk(CV=CV, n=n, lower=lower, upper=upper, theta0=theta0.dk)
GMR    <- seq(lower, upper, length.out=101)
pwr    <- vector()
for (j in seq_along(GMR)) {
  pwr[j] <- dk(CV=CV, n=n, lower=lower, upper=upper, theta0=GMR[j])
}
plot(GMR, pwr, log="x", ylim=c(0, 1), type="l", lwd=2, col="blue",
     las=1, ylab="power with a Danish twist", cex.main=1,
     main=paste0("theta0 = ", theta0, " (n = ", n, ")"))
grid()
abline(h=target)
abline(v=c(theta0, 1/theta0), lty=3, col="red")
abline(v=c(theta0.dk, 1/theta0.dk), lty=2, col="blue")
arrows(theta0, 0, theta0.dk, 0, length=0.1, angle=25,
       code=2, lwd=2, col="red")
arrows(1/theta0, 0, 1/theta0.dk, 0, length=0.1, angle=25,
       code=2, lwd=2, col="red")
BE     <- sprintf("%s %.4f%s%.4f", "BE-limits:", lower, "\u2026", upper)
op     <- par(no.readonly=TRUE)
par(family="mono")
if (round(power, 6) >= target) {
  legend("topright", bg="white", cex=0.9, x.intersp=0,
         legend=c(BE, sprintf("%s %.4f%s%.4f",
                              "GMR-range:", theta0.dk, "\u2026", 1/theta0.dk)))
} else {
  legend("topright", bg="white", cex=0.9, x.intersp=0,
         legend=c(BE, sprintf("%s %.4f", "Achievable power:", power)))
}
par(op)


That’s tough. The “conventional” sample size would be 22. To fulfill the Danish requirement the GMR must not be outside 0.9817…1.0186 without compromising power. Wow. if we want higher power we soon discover that the GMR must be exactly 1.
Side effect: If we originally assumed a GMR of 0.95 (n 44), the GMR range has to be even tighter with 0.9834…1.0169 since Danish power decreases with the sample size (see the second plot in this post).

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