Reference-scaling: only simulations possible [Two-Stage / GS Designs]

posted by Helmut Homepage – Vienna, Austria, 2019-10-20 13:58  – Posting: # 20708
Views: 431

Hi mittyri,

» sad story

“So sad!” (© Mr Trump)

» BTW: in your lecture you suggested to adjust the CIs for ABEL due to TIE inflation. Is that also dead?
» Does it mean that you cannot prove anything using sims??

I hope not. For all reference-scaling methods we need already simulations to estimate the sample size. It would be strange to allow them here but not for the TIE. RSABE/ABEL is tricky anyhow. The model is based on (population) parameters $$-\theta_s\leq\tfrac{\mu_T-\mu_R}{\sigma_{wR}}\leq+\theta_s$$ which are unknown. We have only their estimates. Exactly this misspecification (apply scaling although the drug is not highly variable) leads to the inflated TIE. As I wrote above, Andreas Brandt said that

[simulation-based] methods might [sic] be acceptable if no alternative which shows analytically control of the TIE is available”.

Clearly the case here. In the current implementation reference-scaling is a framework with two (RSABE) and three (ABEL) decisions. No way to solve that analytically (given, at least the GMR-restriction could be implemented by setting α 0.5).
Our method2 follows the ‘spirit’ of the GL, i.e., we assume that \(s_{wR}=\sigma_{wR}\). Molins et al.2 proposed to assume the worst, i.e., regardless of \(s_{wR}\) adjust α as if \(CV_{wR}=0.30\). Conservative but it has a substantial negative impact on power (esp. for really high variability where an inflated TIE is unlikely). For examples see the RSABE vignette of the working version of the next release of PowerTOST and R-code at the end.

What is better? Rely on the ad hoc solutions (which are not perfect) or follow the book, ignore the inflation and put the patients in jeopardy?


  1. Labes D, Schütz H. Inflation of Type I Error in the Evaluation of Scaled Average Bioequivalence, and a Method for its Control. Pharm Res. 2016: 33(11); 2805–14. doi:10.1007/s11095-016-2006-1.
  2. Molins E, Cobo E, Ocaña J. Two-Stage Designs Versus European Scaled Average Designs in Bioequivalence Studies for Highly Variable Drugs: Which to Choose? Stat Med. 2017: 36(30); 4777–88. doi:10.1002/sim.7452.

library(PowerTOST)
CV  <- c(0.35, 0.80) # in-/outside region of inflation
des <- "2x2x4"
n   <- pwr <- numeric()
for (j in seq_along(CV)) {
  x      <- sampleN.scABEL(CV = CV[j], design = des,
                           print = FALSE, details = FALSE)
  n[j]   <- x[["Sample size"]]
  pwr[j] <- x[["Achieved power"]]
}
res <- data.frame(CV = rep(CV, each = 2), n = rep(n, each = 2),
                  TIE = NA, power = rep(pwr, each = 2),
                  method = rep(c("Labes and Schütz", "Molins et al."), 2),
                  alpha.adj = NA, TIE.adj = NA, pwr.adj = NA,
                  stringsAsFactors = FALSE)
for (j in 1:nrow(res)) {
  if (res$method[j] == "Labes and Schütz") {
    x <- scABEL.ad(CV = res$CV[j], design = des, n = res$n[j], print = FALSE)
    res$pwr.adj[j] <- x$pwr.adj
  } else {
    x <- scABEL.ad(CV = 0.30, design = des, n = res$n[j], print = FALSE)
    res$pwr.adj[j] <- power.scABEL(alpha = x$alpha.adj, CV = res$CV[j],
                                   design = des, n = res$n[j])
  }
  res$TIE[j]       <- x$TIE.unadj
  res$alpha.adj[j] <- x$alpha.adj
  res$TIE.adj[j]   <- x$TIE.adj
}
print(res, row.names = FALSE)

  CV  n      TIE   power           method alpha.adj TIE.adj pwr.adj
0.35 34 0.065566 0.81184 Labes and Schütz  0.036299    0.05 0.77281
0.35 34 0.081626 0.81184    Molins et al.  0.028572    0.05 0.74046
0.80 50 0.049600 0.81235 Labes and Schütz        NA      NA      NA
0.80 50 0.082115 0.81235    Molins et al.  0.028201    0.05 0.73198


Cheers,
Helmut Schütz
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