Bioequivalence and Bioavailability Forum

Hi Laura,

❝ If there is a compound that we already know is highly variable, do you think that making a 2x2 pilot is robust enough to later, using the pilot results, make a 2x4 pivotal or would it be more reliable to make a 2x4 pilot?

The latter. In a 2×2 pilot you get only the within-subject CV_w (pooled from the – unknowns – of T and R, i.e., CV_wT and CV_wR). See this article why that is not a good idea for planning a replicate design. In the sample size estimation of the pivotal study you would have to assume CV_wT = CV_wR. That’s both ethically and economically questionable. See also this article.

Let’s explore a hypothetical example (-script at the end). Say, you have three pilot studies.

1. A 2×2 with CV_w 40%. You have to assume CV_wT = CV_wR in planning the pivotal.
   
2. A 2x4 replicate with a variance ratio of the first’s ≈0.67, i.e., CV_wT < CV_wR. It is not uncommon that CV_wT < CV_wR. Then you will get an incentive in planning the pivotal, i.e., require a smaller sample size compared to CV_wT = CV_wR.
   
3. Another 2x4 replicate with a variance ratio of the first’s 1.5, i.e., CV_wT > CV_wR. Rare but possible. You will need a larger sample size than in the two other cases.

You design the pivotals assuming a T/R-ratio of 0.9 (recommended for HVDs) and target ≥ 80% power for the EMA’s Average Bio­equi­va­lence with Expanding Limits (ABEL). Note that these are the defaults in the reference-scaling functions of PowerTOST and therefore, don’t have to be specified.

pivotal studies based on pilots
   pilot     CVw    CVwT    CVwR s2.ratio      L       U  n   power
     2×2 0.40000 0.40000 0.40000    1.00  74.62% 134.02% 30 0.80656
 1st 2×4 0.40000 0.35507 0.44153   ~0.67  72.56% 137.81% 24 0.81029
 2nd 2×4 0.40000 0.44153 0.35507    1.50  76.96% 129.94% 42 0.81378

L and U are the expanded limits in ABEL based on CV_wR:$$\small{
\eqalign{s_\text{wR}&=\sqrt{\log_e(CV_\text{wR}^2+1)}\\
\left\{L,U\right\}&=100\exp(\mp0.76\cdot s_\text{wR})}}$$n are the estimated sample sizes based on CV_wT, CV_wR, the T/R-ratio, target power, and the design.
The confidence interval depends on the pooled variance of T and R$$\small{\eqalign{s_\text{w}^2&=\log_e(CV_\text{w}^2+1)\\
&=\log_e(0.4^2+1)\approx0.14842\ldots\textsf{,}}}$$which is identical in all our cases.

power of pivotals compared to planned based on 2×2 with 30 subjects
 pivotal   power power.30
       2 0.81029  0.87752
       3 0.81378  0.69853

If you plan the pivotal based on the 2×2 with 30 subjects:

• If actually CV_wT < CV_wR, you gain power (≈88% instead of ≈81% because you can expand more than assumed and have 30 subjects instead of the required 24) but waste money.
  
• If actually CV_wT > CV_wR, your study will be underpowered (≈70% instead of ≈81% because you can expand less than assumed and have only 30 subjects instead of the required 42).

Quoting Section 3.5 of ICH M9:

The number of subjects in a clinical trial should always be large enough to provide a reliable answer to the questions addressed.

Statistics are not exactly one of the strengths of ethics committees, but I [sic] would not accept a protocol for ABEL based on the results of a 2×2 pilot study.

A final hint: If you don’t have your own replicate design pilot (preferrable anyway) but the results of another study (report, publication), you can back-calculate CV_wR from the upper confidence limit. For our examples:

      U   CVwR
 134.02 0.4000
 137.81 0.4415
 129.94 0.3551

Hope that helps.

---

library(PowerTOST)
CVw       <- 0.4
pilots    <- c("2×2", "1st 2×4", "2nd 2×4")
ratios    <- c(1, 2 / 3, 3 / 2)
CV        <- data.frame(T = NA_real_, R = NA_real_)
for (j in 1:3) {
  CV[j, 1:2] <- CVp2CV(CV = CVw, ratio = ratios[j])
}
pivotals  <- data.frame(pilot = pilots, CVw = CVw, CVwT =  CV[, "T"], CVwR =  CV[, "R"],
                        s2.ratio = c(sprintf("%5.2f ", ratios[1]),
                                     sprintf("~%.2f ", ratios[2]),
                                     sprintf("%5.2f ", ratios[3])),
                        L = NA_real_, U = NA_real_, n = NA_integer_, power = NA_real_)
for (j in 1:3) {
  pivotals[j, 2:4]  <- sprintf("%.5f", c(CVw = CVw, CV[j, 1:2]))
  pivotals[j, 6:7]  <- sprintf("%.2f%%", 100 * scABEL(CV = CV[j, "R"]))
  # using the defaults: theta0 = 0.9 and targetpower = 0.8
  tmp               <- sampleN.scABEL(CV = as.numeric(CV[j, 1:2]), design = "2x2x4",
                                      details = FALSE, print = FALSE)
  pivotals$n[j]     <- tmp[["Sample size"]]
  pivotals$power[j] <- tmp[["Achieved power"]]
}
comp      <- data.frame(pivotal = 2:3, power = pivotals$power[2:3], power.30 = NA_real_)
for (j in 1:2) {
  comp$power.30[j] <- power.scABEL(CV = as.numeric(CV[j + 1, 1:2]), design = "2x2x4",
                                   n = pivotals$n[1])
}
t         <- c("pivotal studies based on pilots\n",
               paste("\npower of pivotals compared to planned based on 2×2 with",
                     n = pivotals$n[1], "subjects\n"))
cat(t[1]); print(pivotals, row.names = FALSE); cat(t[2]); print(comp, row.names = FALSE)

# calculate CVwR from the upper expanded confidence limit U
# (it has to be 1.2500 < U < 1.4319)
back      <- data.frame(U = c(134.02, 137.81, 129.94), CVwR = NA_real_)
for (j in 1:3) {
  back$CVwR[j] <- sprintf("%.4f", CVwRfromU(U = back$U[j] / 100))
}
print(back, row.names = FALSE)