Bioequivalence and Bioavailability Forum

Hi ElMaestro,

❝ […] When I assessed my very first BE-dossier I asked for post-hoc power and justification and I felt divine. Today I am not very proud of it.

Was a good while ago, right?

I’m guilty as well. Mea culpa, mea maxima culpa.
Starting in the early 1980s I assessed studies solely by nonparametric (!) methods. Yep, about 300 of them. No dead people lying in the street. Only in the annex of the report I gave the ANOVA (the CV and yes, bloody post hoc power). When working in a CRO it is difficult to convince the client that sum­fink is crap (“everybody reports it, we are always right and pay you,…”). My last report giving post hoc power dates back to the late 1990s. Fuck, more than twenty years ago. I’m getting old.
Was a long way to go.

❝ Now I confessed.

So did I.

---

Johannes Kepler has spent years and decades arithmetically and geometrically proving the Divine Harmony of the Spheres postulated by Pythagoras. And yet, he has discarded the work of his half life, thousands of pages of records and calculations, at one go, when he finally realized that the planetary orbits are not circles but ellipses. […] This is the way true and scientific knowledge gain works.
Leo Lukas: Der Kepler-Komplex (May 2018)

Hi Valiveti,

❝ ❝ ❝ […] 48 subjects considering ~80% power, ISCV: 23%.

❝ ❝ ❝ The study was bioequivalent and 90% CI is within 80-125 for both Cmax & AUCt.

❝ ❝ ❝ Power of Cmax and AUCt are 92% & 66% respectively.

❝ ❝ In this order? Can you give us their CVs and GMRs?

❝ CV- Cmax: 27.731; AUCt: 22.678

❝ GMR-Cmax: 97.76; AUCt: 87.99

Sorry, I forgot to ask for the sample sizes. Hence, I estimated them – which leaves me confused. 

library(PowerTOST)
guess.GMR.0 <- function(GMR) { # function to estimate the assumed GMR
  sampleN.TOST(alpha=alpha, CV=CV.0/100, theta0=GMR, theta1=theta1,
               theta2=theta2, targetpower=pwr.0/100, design=design,
               print=FALSE)[["Sample size"]]-n.0
}
guess.n.fin <- function(n) { # function to estimate the final sample size
  if (PK == "Cmax") {
    CV       <- CV.1/100
    GMR      <- GMR.1/100
    post.hoc <- post.hoc.1/100 }
  if (PK == "AUC") {
    CV       <- CV.2/100
    GMR      <- GMR.2/100
    post.hoc <- post.hoc.2/100 }
  round(power.TOST(alpha=alpha, CV=CV, theta0=GMR, theta1=theta1,
                   theta2=theta2, n=n, design=design)-post.hoc, 2)
}
assess <- function(CI, theta1=0.80, theta2=1.25) { # assess for BE
  if (CI[["lower"]] > 100*theta2 | CI[["upper"]] < 100*theta1) {
    res <- "bioinequivalence proven"
  } else {
    if (CI[["lower"]] >= 100*theta1 & CI[["upper"]] <= 100*theta2) {
      res <- "bioequivalence demonstrated"
    } else {
      res <- "failed to demonstrated BE (inconclusive)"
    }
  }
}
# design: CV and power in percent
design     <- "2x2x2"
theta1     <- 0.80 # lower BE limit
theta2     <- 1.25 # upper BE limit
CV.0       <- 23   # assumed CV
pwr.0      <- 80   # target (desired) power
n.0        <- 48   # initial sample size
alpha      <- 0.05 # for 90% CI
CI.pct     <- 100*(1-2*alpha)
GMR.0      <- round(uniroot(guess.GMR.0, interval=c(theta1+1e-8, 1))$root, 4)
pwr.ach    <- power.TOST(alpha=alpha, CV=CV.0/100, theta0=GMR.0,
                         theta1=theta1, theta2=theta2, n=n.0, design=design)
# study: CV, GMR, and post hoc power in percent
PK         <- "Cmax"
CV.1       <- 27.731
GMR.1      <- 97.76
post.hoc.1 <- 92
n.fin.1    <- round(uniroot(guess.n.fin, interval=c(12, n.0))$root, 0)
ph.est.1   <- power.TOST(alpha=alpha, CV=CV.1/100, theta0=GMR.1/100,
                         theta1=theta1, theta2=theta2, n=n.fin.1,
                         design=design)
CI.1       <- round(100*CI.BE(alpha=alpha, pe=GMR.1/100, CV=CV.1/100,
                              n=n.fin.1, design=design), 2)
concl.1    <- assess(CI=CI.1, theta1=theta1, theta2=theta2)
PK         <- "AUC"
CV.2       <- 22.678
GMR.2      <- 87.99
post.hoc.2 <- 66
n.fin.2    <- round(uniroot(guess.n.fin, interval=c(12, n.0))$root, 0)
ph.est.2   <- power.TOST(alpha=alpha, CV=CV.2/100, theta0=GMR.2/100,
                         theta1=theta1, theta2=theta2, n=n.fin.2,
                         design=design)
CI.2       <- round(100*CI.BE(alpha=alpha, pe=GMR.2/100, CV=CV.2/100,
                              n=n.fin.2, design=design), 2)
concl.2    <- assess(CI=CI.2, theta1=theta1, theta2=theta2)
cat(design, "design", "\nBE limits:",
    sprintf("%5.2f%% \u2013 %6.2f%%", 100*theta1, 100*theta2),
    paste0("\nAssumed: CV = ", CV.0, "%, GMR = ", 100*GMR.0,
    "%, target power = ", pwr.0, "%"), "\nPlanned: n =", n.0,
    sprintf("(achieved power = %.2f%%)", 100*pwr.ach),
    paste0("\n", paste0(rep("\u2500", 48), collapse=""), "\nStudy"),
    "\nCmax   :",
    sprintf("%s %.2f%%, %s %.2f%%", "CV =", CV.1, "GMR =", GMR.1),
    "\n         n =", n.fin.1,
    sprintf("(post hoc power = %.2f%% ~ %.0f%%)", 100*ph.est.1, post.hoc.1),
    sprintf("%s%.2f%% CI:", "\n         ", CI.pct),
    sprintf("%6.2f%%", CI.1[["lower"]]), "\u2013",
    sprintf("%6.2f%%", CI.1[["upper"]]),
    "\n        ", concl.1,
    "\nAUC    :",
    sprintf("%s %.2f%%, %s %.2f%%", "CV =", CV.2, "GMR =", GMR.2),
    "\n         n =", n.fin.2,
    sprintf("(post hoc power = %.2f%% ~ %.0f%%)", 100*ph.est.2, post.hoc.2),
    sprintf("%s%.2f%% CI:", "\n         ", CI.pct),
    sprintf("%6.2f%%", CI.2[["lower"]]), "\u2013",
    sprintf("%6.2f%%", CI.2[["upper"]]),
    "\n        ", concl.2, "\n")

I got (my confidence intervals might not be exactly yours if sequences were unbalanced, i.e., n₁ ≠ n₂):

2x2x2 design
BE limits: 80.00% – 125.00%
Assumed: CV = 23%, GMR = 90%, target power = 80%
Planned: n = 48 (achieved power = 80.48%)
────────────────────────────────────────────────
Study
Cmax   : CV = 27.73%, GMR = 97.76%
         n = 38 (post hoc power = 92.04% ~ 92%)
         90.00% CI:  87.98% – 108.63%
         bioequivalence demonstrated
AUC    : CV = 22.68%, GMR = 87.99%
         n = 48 (post hoc power = 65.80% ~ 66%)
         90.00% CI:  81.49% –  95.01%
         bioequivalence demonstrated

Is this correct? If yes, why did you have much less eligible subjects in the data set of C_max than in the one of AUC?
Generally it is the other way ’round (n_AUC ≤ n_Cmax): If the AUC can’t be reliably estimated (missing samples in the late phase and/or the extrapolated AUC >20% of AUC_0–∞) its value is not reported whereas the one of C_max still is.

Hi Valiveti,

you logged in to the forum last Monday – so I assume that you have read my previous post. Would you mind answering my questions above?
By definition a forum isn’t a one-way street.