Bioequivalence and Bioavailability Forum • adjusting α

adjusting α [RSABE / ABEL]

posted by Helmut – Vienna, Austria, 2013-08-11 18:19 (4696 d 23:51 ago) – Posting: # 11262
Views: 30,170

Hi all!

slow code to estimate sample sizes for adjusted α.

library(PowerTOST)

CV    <- 0.30    # intra-subject CV of reference

reg   <- "EMA"   # "EMA" for scABEL or "FDA" for RSABE

des   <- "2x2x4" # full: "2x2x4"; partial: "2x3x3"

target<- 0.80    # target power

TR    <- 0.90    # expected T/R ratio

alpha <- 0.05    # target alpha

print <- FALSE   # TRUE fo intermediate results

nsims <- 1e6

if (reg == "EMA") { # scABEL (rounded switching cond. acc. to GL)

  ifelse (CV <= 0.5, GMR <- exp(0.76*CV2se(CV)),

                     GMR <- exp(0.76*CV2se(0.5)))

  if (CV <= 0.3) GMR <- 1.25

    } else {        # RSABE (exact switching cond.: 0.8925742…)

  ifelse (CV > 0.3, GMR <- exp(log(1.25)/0.25*CV2se(CV)),

                    GMR <- 1.25)

}

if (reg == "EMA") { # first estimate (unadjusted alpha)

  res <- sampleN.scABEL(alpha=alpha, theta0=TR, CV=CV,

                        targetpower=target, design=des,

                        print=FALSE, details=FALSE)

  } else {

  res <- sampleN.RSABE(alpha=alpha, theta0=TR, CV=CV,

                       targetpower=target, design=des,

                       print=FALSE, details=FALSE)

}

n1   <- res[["Sample size"]]

pwr  <- res[["Achieved power"]]

infl <-NULL

if (reg == "EMA") {

  pBE <- power.scABEL(alpha=alpha, CV=CV, theta0=GMR,

                      n=n1, design=des, nsims=nsims)

  } else {

  pBE <- power.RSABE(alpha=alpha, CV=CV, theta0=GMR,

                     n=n1, design=des, nsims=nsims)

}

if (pBE > alpha) infl <- "alpha-inflation!"

cat("regulator:", reg, " design:", des,

  "\nexpected CV:", CV, "expected PE:", TR,

  " target power:", target, " achieved power:", pwr,

  "\nn:", n1, " target alpha:", alpha, " pBE at extended limit:",

  round(pBE, 5), infl, "\n")

n.adj <- n1      # start of search: sample size with unadjusted alpha

pwr   <- 0.5     # that’s rather low!

prec  <- 1e-8    # precision of bisection algo

iter1 <- 0

while (pwr < target) {

  infl  <- NULL

  iter1 <- iter1 + 1

  start <- alpha   # unadjusted alpha

  stop  <- 0.05    # target alpha

  nom   <- c(0.001, start)           # from conservative to unadjusted alpha

  lower <- min(nom); upper <- max(nom)

  delta <- upper - lower             # interval

  iter2 <- 0

  while (abs(delta) > prec) {        # until precision reached

    iter2  <- iter2 + 1

    adj <- (lower+upper)/2           # bisection

    if (reg == "EMA") {

      pBE <- power.scABEL(alpha=adj, theta0=GMR, CV=CV,

                          n=n.adj, design=des, nsims=nsims)

      } else {

      pBE <- power.RSABE(alpha=adj, theta0=GMR, CV=CV,

                         n=n.adj, design=des, nsims=nsims)

    }

    if (abs(pBE - stop) <= prec) break # precision reached

    if (pBE > stop) upper <- adj       # move limit downwards

    if(pBE < stop) lower <- adj        # move limit upwards

    delta <- upper - lower             # new interval

  }

  if (reg == "EMA") {                  # current alpha and power

    pwr <- power.scABEL(alpha=adj, theta0=TR, CV=CV,

                        n=n.adj, design=des, nsims=nsims)

    pBE <- power.scABEL(alpha=adj, theta0=GMR, CV=CV,

                        n=n.adj, design=des, nsims=nsims)

    } else {

    pwr <- power.RSABE(alpha=adj, theta0=TR, CV=CV,

                       n=n.adj, design=des, nsims=nsims)

    pBE <- power.RSABE(alpha=adj, theta0=GMR, CV=CV,

                       n=n.adj, design=des, nsims=nsims)

  }

  if (print) {

    cat("iteration:", iter1, " n:", n.adj, " power:", pwr,

        " adj. alpha:", adj, "\n")

    if (.Platform$OS.type == "windows") flush.console()

  }

  if (pBE > alpha) infl <- "alpha-inflation!"

  if (!is.null(infl) | pwr < target) { # increase sample size if inflation or insufficient power

    ifelse (des == "2x2x4", n.adj <- n.adj+2,

                            n.adj <- n.adj+3)

  }

}

if (n.adj > n1) {

cat("regulator:", reg, " design:", des,

    "\nexpected CV:", CV, "expected PE:", TR,

    " target power:", target, " achieved power:", pwr,

    "\nn:", n.adj, " target alpha:", alpha, " adj. alpha:",

    round(adj, 5), " pBE at extended limit:", round(pBE, 5), infl,

    "\nSample size penalty due to adjusted alpha of",

    round(adj, 5), "( ~", round(100*(1-2*adj), 2),

    "% confidence interval):\n", n.adj-n1, "(",

    round(100*(n.adj-n1)/n1, 1), "% )\n")

  } else {

cat("regulator:", reg, " design:", des,

    "\nexpected CV:", CV, "expected PE:", TR,

    " target power:", target, " achieved power:", pwr,

    "\nn:", n.adj, " target alpha:", alpha, " adj. alpha:",

    round(adj, 5), " pBE at extended limit:", round(pBE, 5), infl,

    "\nNo sample size penalty due to adjusted alpha of", round(adj, 5),

    "(~", round(100*(1-2*adj), 2), "% confidence interval)\n")

}

I set the expected ratio to 0.9 (more realistic than 0.95 for HVDs/HVDPs). Observations: Sample size penality due to α_adj is higher in the fully replicated design. Generally no more penalty for CV_WR > ~40%.
Bonus question: α_adj depends on CV_WR observed in the study, which regulators probably don’t like. In a deficiency letter on Potvin’s method C The Netherland’s MEB stated:

“Adapting the confidence intervals based upon power is not acceptable and also not in accordance with the EMA guideline. Confidence intervals should be selected a priori, without evaluation of the power.”

(my emphases)
Let’s say in study planning we expect a CV_WR of 40% (T/R 0.9, 90% power). In a fully replicate design we would estimate the sample size with 42 – but α will be inflated to 0.05555. So we adjust α to 0.04377 (91.25% CI), which would need 44 subjects. In the study CV_WR turned out to be 35%, which would call for a more conservative α_adj of 0.03812 (92.38% CI). If we would use the α_adj from study planning, we would face an inflation to 0.05681. Maybe it is acceptable to regulators to apply a lower α (wider CI) than in study planning, but what if it is the other way round? If it is not acceptable to use a higher α (narrower CI) we would loose power.

—
Dif-tor heh smusma 🖖🏼 Довге життя Україна!
Helmut Schütz

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Complete thread:

‘alpha’ of scaled ABE? d_labes 2013-03-15 15:56 [RSABE / ABEL]
- ‘alpha’ of scaled ABE? Helmut 2013-03-15 17:27
  - ’alpha’ of scaled ABE: design d_labes 2013-03-16 20:10
    - N=24, 48 d_labes 2013-03-18 08:11
      - adaptive design without adjustment martin 2013-03-25 20:47
        
        α adjustment? Helmut 2013-03-26 15:21
        
        iteratively adjusted alpha martin 2013-03-26 16:19
        
        throw away our sample size tables? Helmut 2013-03-26 18:02
        
        if real martin 2013-03-26 19:44
        
        regulators don’t care? Helmut 2013-03-26 21:18
        
        adjusting αHelmut 2013-08-11 16:19
        
        missing letter apapanas 2024-10-31 06:02
        
        PowerTOST: sampleN.scABEL.ad() Helmut 2024-10-31 08:41
        
        PowerTOST: sampleN.scABEL.ad() apapanas 2024-11-08 10:48
        
        Molins et al. (2017) Helmut 2024-11-08 14:34
        
        target α in sim’s? Helmut 2013-03-27 15:38
        
        target α in sim’s? martin 2013-03-27 16:36
        
        If it is real: results Helmut 2013-03-28 23:20
        
        having alpha martin 2013-03-28 15:08
        
        fixed swr Helmut 2013-03-28 23:34
        
        fixed swr martin 2013-03-29 15:07
        
        Yes but no but yes but no but… Helmut 2013-03-29 16:10