Bioequivalence and Bioavailability Forum

Dear Detlew,

» Side effect 2:
» The Danish are ultra-conservative!

Confirmed. Try this one:

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 <- 0.9
n      <- 22 # fixed
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)
pwr.1  <- dk(CV=CV, n=n, lower=lower, upper=upper, theta0=theta0)
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)
pwr    <- sprintf("%s %.4g", "Power (theta0):", pwr.1)
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),
                  pwr))
} else {
  legend("topright", bg="white", cex=0.9, x.intersp=0,
         legend=c(BE, sprintf("%s %.4f", "Achievable power:", power), pwr))
}
par(op)


It is courageous to assume theta0 0.9 for a NTID (sampleN.TOST() would tell you to get lost). If you believe in 0.95, n would be 44. To get 80% power the GMR-range is 0.9834…1.0169. Power for 0.95 is only 0.2333. Would it be ethical to perform the study?

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