Bioequivalence and Bioavailability Forum

Hi Brus,

❝ I am trying to adapt your example of power calculation with plot replesentation, but for the calculation of the sample size. My objetive is to plot a graphicala representation with ratio in X axis and sample size in Y axis and setting the CV and power as a fixed value.

I suggest this goody first. May sound picky but we can’t get the exact desired power, only one which is at least that high.*

Try this one:

library(PowerTOST)
design <- "2x2x2" # any in known.designs()
target <- 0.80    # target (desired) power
x.min  <- 0.85    # minimum T/R-ratio (> 0.8)
x.res  <- 250     # 'resolution' of the x-axis
y.min  <- 0.15    # minimum CV
y.max  <- 0.30    # maximum CV
y.res  <- 0.05    # step size of CVs
# T/R-ratios equally spaced in log-scale
theta0 <- exp(seq(log(x.min), log(1 / x.min), length.out = x.res))
CV     <- seq(y.min, y.max, y.res)
res    <- data.frame(CV = rep(CV, each = length(theta0)),
                     target = target, theta0 = theta0,
                     power = NA_real_, n = NA_integer_)
for (j in 1:nrow(res)) {
  tmp <- sampleN.TOST(CV = res$CV[j], theta0 = res$theta0[j],
                      targetpower = target, design = design,
                      print = FALSE) # no output to the console
  if (tmp[["Sample size"]] >= 12) {  # full throttle
    res[j, 4:5]  <- tmp[8:7]
  } else {
    res$n[j]     <- 12               # acc. to the guidelines
    res$power[j] <- power.TOST(CV = res$CV[j], theta0 = res$theta0[j],
                               design = design, n = res$n[j])
  }
}
# theta0 on log-axis to demonstrate the symmetry
plot(res$theta0, res$n, type = "n", log = "x", las = 1,
     ylim = c(12, max(res$n)), font.main = 1,
     main = paste(design, "design\n",
                  sprintf("target power \u2265 %.f%%", 100 * target)),
     xlab = expression(theta[0]), ylab = expression(italic(n)))
grid()
clr <- colorRampPalette(c("blue", "red"))(length(CV))
for (j in seq_along(CV)) {
  lines(res$theta0[res$CV == CV[j]],
        res$n[res$CV == CV[j]], lwd = 2, type = "s", col = clr[j],
        lend = 2, ljoin = 1)
}
legend("top", bg = "white", inset = 0.02, box.lty = 0,
       legend = sprintf("%.f%%", 100 * CV),
       title = expression(italic(CV)), lwd = 2, col = clr,
       seg.len = 2.5, ncol = 2)

The sample size is a staircase function and due to the symmetry in \(\small{\log_{e}}\)-scale we require the same sample size for any given \(\small{\theta_0}\) and \(1/\small{\theta_0}\) (see also there).

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• OK, if we would be able to dose fractional subjects, then yes. I can’t. Amazingly some software packages report sumfink like \(\small{n=23.45}\) and leave the rounding up to the user. Crazy.

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Diving deeper into the matter: Power is a hypersurface depending on \(\small{\theta_0}\), \(\small{n}\), and \(\small{CV}\). We can only show spatial projections at certain \(\small{CV}\text{-}\)values (the panels) with \(\small{\log_{e}\theta_0}\) (x-axis), \(\small{n}\) (y-axis), and power (z-axis).
\(\small{n=12-292}\) and \(\small{\theta_0=0.80-1.25}\) to demonstrate that at the limits of the acceptance range power (which is there the Type I Error) is ≤0.05.

The intersections with the horizontal plane (the target power) show the U-shaped dependency of \(\small{n}\) on \(\small{\theta_0}\) as in the plot above. The slices at certain \(\small{\theta_0}\)- and \(\small{n}\)-values give the plots like in the very old post above.

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Don’t try to imagine four-dimensional objects. Even a simple hypercube (tesseract) may fry you brain.

CC0 2007 Jason Hise @ Wikimedia Commons