Bioequivalence and Bioavailability Forum

Dear Detlew & Osama,

❝ ❝ ... how could this type I error inflation happen?

❝ May be Helmut has a better (paedogogical) explanation because he is an exceptionally gifted preacher .

Only you say so. I’ll try.

Let’s start with two power curves (first -script at the end). The 2-sequence 4-period replicate study is designed for ABE with conventional limits, 80% power. Since we are wary, we assume a \(\small{\theta_0}\) (T/R-ratio) of 0.90 and estimate a sample size of 40. One power curve for the conventional limits as planned (──) and one for the widened limits (──).

The probability of a type I error (i.e., the consumer risk) to be treated with a formulation which is not BE (true \(\small{\theta_0}\) outside the limits) is ≤5%. You see that the blue power curve intersects 0.05 at \(\small{\theta_0=0.80}\) and \(\small{\theta_0=1.25}\). That means also the chance of falsely passing BE is 5%. The same is applicable to South Africa’s approach where we could use pre-specified widened limits independent from the observed CV. Then we have a TIE of 0.05 as well (the red power curve intersects 0.05 at \(\small{\theta_0=0.75}\) and \(\small{\theta_0=1.\dot{3}}\)).

Imagine an even more extreme case than the one Detlew mentioned. We observe in the study a CV_wR of 30.01% although the true one is 30%. That means we will use the widened limits of 75.00–133.33% instead of the correct 80.00–125.00%. With the same \(\small{\theta_0=0.80}\) and \(\small{\theta_0=1.25}\) we jump to the red power curve and therefore, have a much higher chance of passing (~40% instead of 5%).

Now for the tricky part (sorry): In the previous posts we estimated an inflation of the TIE of ~20%. Why not 40%? Say, the true CV_wR is 30% and acc. to the convention the drug is considered to be not highly variable. Hence, we would have to apply the conventional limits. There is a ~50% chance that in the actual study we will observe a CV_wR >30% and misclassify the drug/formulation as highly variable and apply the widened limits. But there is also a ~50% chance that we observe a CV_wR ≤30% and use the conventional limits. Both together gives the ~20% inflation (actually it is more complicated*).

What if we estimate the sample size for the GCC’s approach? With 28 it will be lower (second -script).

Since with a CV of 30% we have to use the conventional limits (power at \(\small{\theta_0=0.80}\) and \(\small{\theta_0=1.25}\) will be ~0.16) and we felt into the trap of an inflated type I error. Note that the TIE depends also on the sample size. Hence, it will be smaller than with 40 subjects.

If you think about iteratively adjusted α like for the reference-scaling methods (third -script) – examples for the 2x2x4 design (sample sizes estimated for the GCC’s approach):

CV = 0.150, n = 12 (power = 0.8496)
TIE = 0.050148: inflated (α-adjustment necessary).
 2 iterations: adjusted α = 0.049854 (90.029% CI), TIE = 0.05 (power = 0.8493).
CV = 0.200, n = 18 (power = 0.8015)
TIE = 0.050910: inflated (α-adjustment necessary).
 7 iterations: adjusted α = 0.049082 (90.184% CI), TIE = 0.05 (power = 0.7988).
CV = 0.250, n = 28 (power = 0.8220)
TIE = 0.072558: inflated (α-adjustment necessary).
 5 iterations: adjusted α = 0.032075 (93.585% CI), TIE = 0.05 (power = 0.7644).
CV = 0.300, n = 28 (power = 0.8079)
TIE = 0.164829: inflated (α-adjustment necessary).
 8 iterations: adjusted α = 0.009071 (98.186% CI), TIE = 0.05 (power = 0.5859).
CV = 0.301, n = 28 (power = 0.8085)
TIE = 0.022390: not inflated (no α-adjustment necessary).

However, we could face a substantial loss in power (for CV 30% and the adjusted α of ~0.0091 it would drop from 81% to 59%).

---

1. Power-curves for fixed limits

library(PowerTOST)
design <- "2x2x4" # any replicate - see known.designs()
CV     <- 0.30
target <- 0.80
theta0 <- 0.90
n      <- sampleN.TOST(CV = CV, theta0 = theta0, targetpower = target,
                       design = design, details = FALSE,
                       print = FALSE)[["Sample size"]]
theta0 <- c(theta0, seq(0.75*0.96, 1, length.out = 60))
theta0 <- sort(unique(c(theta0, 0.8, 1, 1.25, 1/theta0)))
res    <- data.frame(theta0 = theta0)
for (j in seq_along(theta0)) {
  res$pwr1[j] <- power.TOST(CV = CV, n = n, theta0 = theta0[j],
                            theta1 = 0.80, design = design)
  res$pwr2[j] <- power.TOST(CV = CV, n = n, theta0 = theta0[j],
                            theta1 = 0.75, design = design)
}
if (is.null(attr(dev.list(), "names")))
  windows(width = 4.5, height = 3.3, record = TRUE)
op  <- par(no.readonly = TRUE)
par(mar = c(4, 4, 0, 0) + 0.1, cex.axis = 0.8)
plot(log(theta0), res$pwr1, type = "n", axes = FALSE, yaxs = "i",
     xlim = log(c(0.75, 1/0.75)), ylim = c(0, 1.04),
     xlab = expression(theta[0]), ylab = "power")
axis(2, las = 1)
axis(2, at = 0.05, las = 1)
axis(1, at = log(c(0.75, seq(0.8, 1.2, 0.1), 1.25, 1/0.75)),
     labels = sprintf("%.2f", c(0.75, seq(0.8, 1.2, 0.1), 1.25, 1/0.75)))
abline(h = pretty(c(0, 1)), lty = 3, col = "lightgrey")
abline(h = 0.05, lty = 2)
abline(v = log(c(0.9, 1, 1.1, 1.2)), lty = 3, col = "lightgrey")
abline(v = log(c(0.75, 0.8, 1.25, 1/0.75)), lty = 2)
box()
lines(log(theta0), res$pwr1, lwd = 3, col = "blue")
lines(log(theta0), res$pwr2, lwd = 3, col = "red")
par(op)

2. Power curves for sample sizes acc. to the GCC’s approach

library(PowerTOST)
design <- "2x2x4" # only "2x2x4", "2x2x3", "2x3x3" implemented!
CV     <- 0.30
target <- 0.80
theta0 <- 0.90
reg    <- reg_const("user", r_const = log(1/0.75)/CV2se(0.3),
                    CVswitch = 0.3, CVcap = 0.3, pe_const = TRUE)
n      <- sampleN.scABEL(CV = CV, theta0 = theta0, targetpower = target,
                         design = design, regulator = reg, details = FALSE,
                         print = FALSE)[["Sample size"]]
theta0 <- c(theta0, seq(0.75*0.96, 1, length.out = 60))
theta0 <- sort(unique(c(theta0, 0.8, 1, 1.25, 1/theta0)))
res    <- data.frame(theta0 = theta0)
for (j in seq_along(theta0)) {
  res$pwr[j] <- power.scABEL(CV = CV, n = n, theta0 = theta0[j],
                             theta1 = 0.80, design = design,
                             regulator = reg)
}
if (is.null(attr(dev.list(), "names")))
  windows(width = 4.5, height = 3.3, record = TRUE)
op  <- par(no.readonly = TRUE)
par(mar = c(4, 4, 0, 0) + 0.1, cex.axis = 0.8)
plot(log(theta0), res$pwr, type = "n", axes = FALSE, yaxs = "i",
     xlim = log(c(0.75, 1/0.75)), ylim = c(0, 1.04),
     xlab = expression(theta[0]), ylab = "power")
axis(2, las = 1)
axis(2, at = 0.05, las = 1)
axis(1, at = log(c(0.75, seq(0.8, 1.2, 0.1), 1.25, 1/0.75)),
     labels = sprintf("%.2f", c(0.75, seq(0.8, 1.2, 0.1), 1.25, 1/0.75)))
abline(h = pretty(c(0, 1)), lty = 3, col = "lightgrey")
abline(h = 0.05, lty = 2)
abline(v = log(c(0.9, 1, 1.1, 1.2)), lty = 3, col = "lightgrey")
abline(v = log(c(0.8, 1.25)), lty = 2)
if (CV > 0.3) abline(v = log(c(0.75, 1/0.75)), lty = 2)
box()
lines(log(theta0), res$pwr, lwd = 3, col = "blue")
par(op)

3. Iteratively adjusted α for the GCC’s approach

library(PowerTOST)
opt <- function(x) {
  if (sdsims) {
    power.scABEL.sds(alpha = x, CV = CV, n = n, theta0 = U, design = design,
                     regulator = reg, nsims = 1e6, setseed = setseed,
                     progress = FALSE) - alpha
  } else {
    power.scABEL(alpha = x, CV = CV, n = n, theta0 = U, design = design,
                 regulator = reg, nsims = 1e6, setseed = setseed) - alpha
  }
}
design  <- "2x2x4" # only "2x2x4", "2x2x3", "2x3x3" implemented!
alpha   <- 0.05    # nominal level of the test
CV      <- 0.30    # can be a 2-element vector: CV[1] for T, CV[2] for R
theta0  <- 0.90    # assumed T/R-ratio
target  <- 0.80    # target (desired) power
setseed <- TRUE    # for reproducibility
sdsims  <- FALSE   # set to TRUE for partial replicate (much slower)
reg     <- reg_const("user", r_const = log(1/0.75)/CV2se(0.3),
                     CVswitch = 0.3, CVcap = 0.3, pe_const = TRUE)
# power and sample size for the GCC’s approach
x       <- sampleN.scABEL(CV = CV, theta0 = theta0, design = design,
                          targetpower = target, regulator = reg,
                          details = FALSE, print = FALSE)
power   <- x[["Achieved power"]]
n       <- x[["Sample size"]]
U       <- scABEL(CV = CV, regulator = reg)[["upper"]]
power   <- power.TOST(CV = CV, n = n, theta0 = theta0, design = design)
if (!sdsims) {     # simulate underlying statistics
  TIE <- power.scABEL(CV = CV, n = n, theta0 = U, design = design,
                      regulator = reg, nsims = 1e6)
} else {           # simulate subject data
  TIE <- power.scABEL.sds(CV = CV, n = n, theta0 = U, design = design,
                          regulator = reg, nsims = 1e6, progress = FALSE)
}
txt <- paste0("CV = ", sprintf("%.3f", CV), ", n = ", n,
              " (power = ", sprintf("%.4f)", power))
if (TIE <= alpha) {
  txt <- paste0(txt, "\nTIE = ", sprintf("%.6f", TIE), ": not inflated ",
                "(no \u03B1-adjustment necessary).\n")
} else {
  txt <- paste0(txt, "\nTIE = ", sprintf("%.6f", TIE), ": inflated ",
                "(\u03B1-adjustment necessary).")
  x   <- uniroot(opt, interval = c(0, alpha), tol = 1e-8)
  alpha.adj <- x$root
  TIE.adj   <- alpha + x$f.root
  power.adj <- power.TOST(alpha = alpha.adj, CV = CV, n = n, theta0 = theta0,
                          design = design)
  txt <- paste0(txt, "\n", sprintf("%2i", x$iter), " iterations: ",
                "adjusted \u03B1 = ", sprintf("%.6f", alpha.adj), " (",
                sprintf("%.3f%%", 100*(1-2*alpha.adj)), " CI), TIE = ",
                TIE.adj, " (power = ", sprintf("%.4f)", power.adj), ".\n")
}
cat(txt)

---

• \(\small{\theta_0}\) follows a lognormal distribution and \(\small{CV_\textrm{wR}}\) a \(\small{\chi^2}\) distribution. Both distributions are not symmetric but skewed to the right:
  
  Hence, at a true \(\small{\theta_0}\) of 1.25 and a true \(\small{CV_\textrm{wR}}\) of 30% in a particular study the chance of a classify the drug falsely as highly variable (based on the observed \(\small{CV_\textrm{wR}}\)) and proceed with scaling is slightly higher than 50%.

---

Edit: We implemented regulator = "GCC" in version 1.5-3 (2021-01-18) of PowerTOST. Example:

library(PowerTOST)
design  <- "2x2x4
CV      <- 0.30
theta0  <- 0.90
target  <- 0.80
n       <- sampleN.scABEL(CV = CV, theta0 = theta0, design = design,
                          targetpower = target, regulator = "GCC",
                          details = FALSE, print = FALSE)[["Sample size"]]
scABEL.ad(CV = CV, theta0 = theta0, design = design, n = n,
          regulator = "GCC", details = TRUE)

+++++++++++ scaled (widened) ABEL ++++++++++++
         iteratively adjusted alpha
   (simulations based on ANOVA evaluation)
----------------------------------------------
Study design: 2x2x4 (4 period full replicate)
log-transformed data (multiplicative model)
1,000,000 studies in each iteration simulated.

CVwR 0.3, CVwT 0.3, n(i) 14|14 (N 28)
Nominal alpha                 : 0.05
True ratio                    : 0.9000
Regulatory settings           : GCC (ABE)
Switching CVwR                : 0.3
BE limits                     : 0.8000 ... 1.2500
PE constraints                : 0.8000 ... 1.2500
Empiric TIE for alpha 0.0500  : 0.16483 (rel. change of risk: +230%)
Power for theta0 0.9000       : 0.808
Iteratively adjusted alpha    : 0.00907
Empiric TIE for adjusted alpha: 0.05000
Power for theta0 0.9000       : 0.586 (rel. impact: -27.5%)

Runtime    : 6.6 seconds
Simulations: 8,100,000 (7 iterations)

The impact on power is massive. Which sample size would we need to maintain the target power?

sampleN.scABEL.ad(CV = CV, theta0 = theta0, design = design,
                  targetpower = target, regulator = "GCC",
                  details = TRUE)

+++++++++++ scaled (widened) ABEL ++++++++++++
            Sample size estimation
        for iteratively adjusted alpha
   (simulations based on ANOVA evaluation)
----------------------------------------------
Study design: 2x2x4 (4 period full replicate)
log-transformed data (multiplicative model)
1,000,000 studies in each iteration simulated.

Assumed CVwR 0.3, CVwT 0.3
Nominal alpha      : 0.05
True ratio         : 0.9000
Target power       : 0.8
Regulatory settings: GCC (ABE)
Switching CVwR     : 0.3
BE limits          : 0.8000 ... 1.2500
PE constraints     : 0.8000 ... 1.2500

n  28, nomin. alpha: 0.05000 (power 0.8079), TIE: 0.1648

Sample size search and iteratively adjusting alpha
n  28,   adj. alpha: 0.00907 (power 0.5859), rel. impact on power: -27.48%
n  48,   adj. alpha: 0.00343 (power 0.7237)
n  46,   adj. alpha: 0.00376 (power 0.7136)
n  48,   adj. alpha: 0.00343 (power 0.7237)
n  50,   adj. alpha: 0.00313 (power 0.7330)
n  52,   adj. alpha: 0.00283 (power 0.7402)
n  54,   adj. alpha: 0.00258 (power 0.7490)
n  56,   adj. alpha: 0.00233 (power 0.7554)
n  58,   adj. alpha: 0.00215 (power 0.7641)
n  60,   adj. alpha: 0.00198 (power 0.7703)
n  62,   adj. alpha: 0.00180 (power 0.7789)
n  64,   adj. alpha: 0.00164 (power 0.7851)
n  66,   adj. alpha: 0.00152 (power 0.7909)
n  68,   adj. alpha: 0.00138 (power 0.7958)
n  70,   adj. alpha: 0.00126 (power 0.8010), TIE: 0.05000
Compared to nominal alpha's sample size increase of 150.0% (~study costs).

Runtime    : 96.4 seconds
Simulations: 120,700,000

Oops! Since the TIE depends on the sample size itself (see the plots in this post), we have to adjust more.