Bioequivalence and Bioavailability Forum

Hi AB661,

❝ - Pilot BE Study with full replicate design. Calculate sample size for pivotal two way Cross over study design based on this study.

It’s always better to perform a pilot study than to trust in data from the literature. Even more, the sample size you mentioned is pretty small and hence, the estimates (T/R-ratio, CV) are unreliable. Note that any replicate design can be assessed for conventional (unscaled) ABE as well. I would state in the protocol that you aim at reference-scaling by ABEL for AUC. If CV_wR ≤ 30%, the study will be assessed by ABE (90% CI within 80.0 – 125.0%). C_max is not relevant in the sample size estimation because only the point estimate has to lie within 80.0 – 125.0%. An -script for 2×2×2 and 4-period full replicate designs at the end.

Examples:

• Say, the CV of AUC is 30% and thus, you must not scale.
  previous data
 metric design theta0   CV  N
   Cmax  2x2x2   1.11 0.43 14
    AUC  2x2x2   1.11 0.30 14
planned pivotal studies
 metric design method theta0   CV     L     U  n     power
   Cmax  2x2x2     PE   1.11 0.43 0.800 1.250 20 0.8131388
   Cmax  2x2x4     PE   1.11 0.43 0.800 1.250 10 0.8131388
    AUC  2x2x2    ABE   1.11 0.30 0.800 1.250 78 0.8049580
    AUC  2x2x4    ABE   1.11 0.30 0.800 1.250 40 0.8157494
  
• If it is 33%, ABEL is acceptable – of course only if the study is performed in a replicate design.
  previous data
 metric design theta0   CV  N
   Cmax  2x2x2   1.11 0.43 14
    AUC  2x2x2   1.11 0.33 14
planned pivotal studies
 metric design method theta0   CV     L     U  n     power
   Cmax  2x2x2     PE   1.11 0.43 0.800 1.250 20 0.8131388
   Cmax  2x2x4     PE   1.11 0.43 0.800 1.250 10 0.8131388
    AUC  2x2x2    ABE   1.11 0.33 0.800 1.250 92 0.8001078
    AUC  2x2x4   ABEL   1.11 0.33 0.783 1.277 36 0.8096300

---

library(PowerTOST)            # see https://cran.r-project.org/package=PowerTOST
# data of a previous study; will also be used in planning pivotal ones
prev   <- data.frame(metric = c("Cmax", "AUC"), design = "2x2x2",
                     theta0 = c(1.11, 1.11), CV = c(0.43, 0.30), N = c(14, 14))
design <- c("2x2x2", "2x2x4") # 2×2×2 and 4-period full replicates (TRTR|RTRT, TRRT|RTTR, TTRR|RRTT)
                              # "2x2x3" for 3-period full replicates (TRT|RTR, TTR|RRT)
                              # "2x3x3" for partial replicate (TRR|RTR|RRT)
target <- 0.8                 # target (desired) power
# planned pivotal studies
piv1   <- cbind(metric = prev[, 1], design = design[1], method = "ABE",
                prev[, 3:4], L = 0.8, U = 1.25, n = NA_integer_, power = NA_real_)
piv2   <- cbind(metric = prev[, 1], design = design[2], piv1[, 3:9])
for (j in 1:nrow(prev)) {
  if (prev$metric[j] == "Cmax") {
    # alpha = 0.5 means that only the T/R-ratio has
    # to lie within 80–125% (without observing its CI)
    for (k in seq_along(design)) {
      tmp <- sampleN.TOST(alpha = 0.5, CV = prev$CV[j], theta0 = prev$theta0[j],
                          design = design[k], targetpower = target, print = FALSE)
      ifelse (k == 1, piv1[j, 8:9] <- tmp[7:8], piv2[j, 8:9] <- tmp[7:8])
    }
  } else {
    # ABE, 90% CI of the T/R-ratio within 80–125%
    for (k in seq_along(design)) {
      tmp <- sampleN.TOST(CV = prev$CV[j], theta0 = prev$theta0[j],
                          design = design[k], targetpower = target, print = FALSE)
      ifelse (k == 1, piv1[j, 8:9] <- tmp[7:8], piv2[j, 8:9] <- tmp[7:8])
    }
    if (prev$CV[j] > 0.3) {
      # ABEL, 90% CI of the T/R-ratio within expanded limits {L, U}; for Health
      # Canada intra-subject contrasts are used (instead of an ANOVA)
      tmp          <- sampleN.scABEL(CV = prev$CV[j], theta0 = prev$theta0[j],
                                     design = piv2$design[j], targetpower = target,
                                     regulator = "HC", print = FALSE, details = FALSE)
      piv2[j, 3]   <- "ABEL"
      piv2[j, 6:7] <- scABEL(CV = prev$CV[j], regulator = "HC") # L and U
      piv2[j, 8:9] <- tmp[8:9]
    }
  }
}
pivotals   <- rbind(piv1, piv2)
pivotals   <- pivotals[order(pivotals$metric, pivotals$design, method = "radix",
                             decreasing = c(TRUE, FALSE)), ] # cosmetics
pivotals$L <- sprintf("%.3f", pivotals$L)
pivotals$U <- sprintf("%.3f", pivotals$U)
pivotals$method[pivotals$metric =="Cmax"] <- "PE"
t1         <- "previous data\n"; t2 <- "planned pivotal studies\n"
cat(t1); print(prev, row.names = FALSE); cat(t2); print(pivotals, row.names = FALSE)