Bioequivalence and Bioavailability Forum

d_labes
★★★

Berlin, Germany,
2013-03-15 16:56
(4417 d 09:38 ago)

Posting: # 10202
Views: 31,070

‘alpha’ of scaled ABE? [RSABE / ABEL]

Dear All!

We had already from time to time the question which combination of CV_wR and GMR apply to find alpha ≤ 0.05 from power calculations. No one was really sure. Except the statement that power calculations at the conventional BE limits 1.25 (or 0.8) as for the ABE test will not be appropriate.

A rather natural choice IMHO would be to calculate the power at the border(s) of the widened BE acceptance limits.
Thus I employed power.scABEL() and power.RSABE() from R-package PowerTOST.

Here the results for each method on its own:
GMR=1.25 if CV<=0.3 or GMR=exp(0.8925742*CV2se(CV)) if CV>0.3 for the FDA RSABE
GMR=1.25 if CV<=0.3 or GMR=exp(0.7601283*CV2se(CV)) if CV>0.3 for the EMA scABEL and cap on if CV>0.5

    CV      GMR pBE_ABEL     GMR2 pBE_RSABE

 0.200 1.250000 0.050220 1.250000  0.050153

 0.225 1.250000 0.050632 1.250000  0.051969

 0.250 1.250000 0.053143 1.250000  0.061009

 0.275 1.250000 0.061006 1.250000  0.082647

 0.300 1.250000 0.076460 1.250000  0.116120 ┐

 0.302 1.251782 0.075453 1.301734  0.048548 ┘ FDA discontinuity

 0.325 1.272352 0.067721 1.326887  0.047741

 0.350 1.294853 0.064379 1.354484  0.046156

 0.375 1.317485 0.062291 1.382326  0.043941

 0.400 1.340231 0.059506 1.410391  0.041041

 0.425 1.363072 0.055282 1.438658  0.037799

 0.450 1.385991 0.049795 1.467104  0.034401

 0.475 1.408971 0.043420 1.495709  0.031150

 0.500 1.431997 0.036960 1.524452  0.028097

 0.525 1.431997 0.040644 1.553312  0.025383

 0.550 1.431997 0.043412 1.582270  0.022875

 0.575 1.431997 0.045423 1.611308  0.020699

 0.600 1.431997 0.046647 1.640406  0.018660

 0.625 1.431997 0.047379 1.669548  0.016948

 0.650 1.431997 0.047754 1.698717  0.015457

 0.675 1.431997 0.047698 1.727897  0.014135

 0.700 1.431997 0.047192 1.757073  0.012861

 0.725 1.431997 0.046282 1.786230  0.011799

 0.750 1.431997 0.045020 1.815356  0.010865

 0.775 1.431997 0.043338 1.844438  0.010057

 0.800 1.431997 0.041320 1.873464  0.009425

Interesting! At least for me :yes:

.

Both methods start for low CV's with pBE ~ 0.05. Classical ABE evaluation controls alpha≤0.05 :cool:

.
Approaching CV=0.3 the probability of (falsely?) accepting BE raises, for the FDA method up to 0.116. This is understandable since a increasing part of our simulated studies will have sample CV's >0.3 which are then evaluated with widened limits or downscaled linear scaled ABE criterion.
After/at CV=0.3 pBE for the FDA method drops sharply to or below 0.05 due to the known discontinuity in the widened BE limits and becomes with rising CV lower and lower (dominated by the point estimator criterion? conservative?). For the EMA method the pBE stays above 0.05 up to CV=0.45. The cap at CV≥0.5 then prevents the decrease for higher CV's seen in the FDA method. pBE first rises slightly with increasing CV. This behaviour (power increases with increasing CV) we have already seen here. But then a decrease is seen (due to the point estimator criterion?).

If we really estimating here the type I error then both methods are cases of an alpha inflation!? The FDA method in the vicinity of CV≤0.3 and the EMA method also but additionally up to CV=0.45. And the 'alpha inflation' is much more pronounced as that what the EMA lead to question Potvin method C.

—
Regards,

Detlew

Helmut
★★★

Vienna, Austria,
2013-03-15 18:27
(4417 d 08:07 ago)

@ d_labes
Posting: # 10203
Views: 24,563

‘alpha’ of scaled ABE?

Post reply

Dear Detlew!

❝ A rather natural choice IMHO would be to calculate the power at the border(s) of the widened BE acceptance limits.

Agree.

❝ GMR=1.25 if CV<=0.3 or GMR=exp(0.8925742*CV2se(CV)) if CV>0.3 for the FDA RSABE

❝ GMR=1.25 if CV<=0.3 or GMR=exp(0.7601283*CV2se(CV)) if CV>0.3 for the EMA scABEL and cap on if CV>0.5

I would prefer a GMR of 0.80 since PowerTOST employs simulations – remember the “rounding thread”? ;-)

</nitpicking>

Which designs / sample sizes have you used? I’m struggling reproducing your numbers.

❝ Interesting! At least for me :yes: .

For me as well. I will ask the two Lászlós and maybe Panos Macheras.

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

The quality of responses received is directly proportional to the quality of the question asked. 🚮
Science Quotes

d_labes
★★★

Berlin, Germany,
2013-03-16 21:10
(4416 d 05:24 ago)

@ Helmut
Posting: # 10208
Views: 24,219

’alpha’ of scaled ABE: design

Post reply

Dear Helmut!

❝ <nitpicking>

I would prefer a GMR of 0.80 since PowerTOST employs simulations – remember the “rounding thread”? ;-)

</nitpicking>

No rounding at all employed. Even 0.760... not.

❝ Which designs / sample sizes have you used? I’m struggling reproducing your numbers.

No explicit design written down in the code. That means default -> "2x3x3". n=48 if I remember correctly.

—
Regards,

Detlew

d_labes
★★★

Berlin, Germany,
2013-03-18 09:11
(4414 d 17:23 ago)

(edited on 2013-03-18 09:52)
@ d_labes
Posting: # 10216
Views: 24,074

N=24, 48

Post reply

Dear Helmut!

❝ No explicit design written down in the code. That means default -> "2x3x3". n=48 if I remember correctly.

Must correct me: N=24 in the calculations above. nsims=1E6.

N=48 comes here:

    CV      GMR pBE_ABEL     GMR2 pBE_RSABE

 0.200 1.250000 0.050071 1.250000  0.050197

 0.225 1.250000 0.050103 1.250000  0.050458

 0.250 1.250000 0.050896 1.250000  0.055809

 0.275 1.250000 0.057014 1.250000  0.084654

 0.300 1.250000 0.076539 1.250000  0.147126

 0.302 1.251782 0.075132 1.301734  0.045037

 0.325 1.272352 0.066421 1.326887  0.044747

 0.350 1.294853 0.063700 1.354484  0.040058

 0.375 1.317485 0.061786 1.382326  0.033091

 0.400 1.340231 0.058847 1.410391  0.026019

 0.425 1.363072 0.054528 1.438658  0.019674

 0.450 1.385991 0.048974 1.467104  0.014709

 0.475 1.408971 0.042496 1.495709  0.010821

 0.500 1.431997 0.035153 1.524452  0.008119

 0.525 1.431997 0.041243 1.553312  0.006059

 0.550 1.431997 0.045210 1.582270  0.004553

 0.575 1.431997 0.047477 1.611308  0.003430

 0.600 1.431997 0.048748 1.640406  0.002631

 0.625 1.431997 0.049389 1.669548  0.002035

 0.650 1.431997 0.049721 1.698717  0.001592

 0.675 1.431997 0.049877 1.727897  0.001291

 0.700 1.431997 0.049954 1.757073  0.001048

 0.725 1.431997 0.049987 1.786230  0.000879

 0.750 1.431997 0.049998 1.815356  0.000727

 0.775 1.431997 0.050014 1.844438  0.000621

 0.800 1.431997 0.050007 1.873464  0.000538

—
Regards,

Detlew

martin ★★ Austria, 2013-03-25 21:47 (4407 d 04:47 ago) @ d_labes Posting: # 10279 Views: 24,001	adaptive design without adjustment Post reply
	dear all! this behavior may be due to an adaptive choice of a statistical method in face of the data without any "adjustment" for this adaptation. this reminds me on the "flowchart" approach many CROs are using which should be avoided. best regards martin PS.: no idea at the moment how to account for this. PPS.: why flowcharts should not be used link.

Helmut
★★★

Vienna, Austria,
2013-03-26 16:21
(4406 d 10:13 ago)

@ martin
Posting: # 10286
Views: 24,500

α adjustment?

Post reply

Dear all!

❝ this behavior may be due to an adaptive choice of a statistical method in face of the data without any "adjustment" for this adaptation.

Yep. The acceptance range is data-driven (random variable dependent on CV_WR).

❝ PS.: no idea at the moment how to account for this.

Adjust α? Quick shot:

library(PowerTOST)

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

CV    <- 0.300   # intra-subject CV of reference

n     <- 24      # total sample size

                 # in case of imbalanced sequences use a vector:

                 # e.g., c(n1,n2,n3)

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

print <- FALSE   # intermediate results

if (reg == "EMA") { # scABEL

  ifelse (CV <= 0.5, GMR <- exp(0.7601283*CV2se(CV)),

    GMR <- exp(0.7601283*CV2se(0.5)))

  if (CV <= 0.3) GMR <- 1.25

    } else {        # RSABE

  ifelse (CV > 0.3, GMR <- exp(0.8925742*CV2se(CV)), GMR <- 1.25)

}

nsims <- 1e6

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

x     <- 0.05     # target alpha

sig   <- binom.test(x*nsims, nsims,

           alternative='less')$conf.int[2] # significance limit of sim’s

nom   <- c(0.001, x)                # from conservative to target alpha

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

delta <- upper - lower              # interval

ptm   <- proc.time()                # start timer

iter  <- 0

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

  iter  <- iter + 1

  x.new <- (lower+upper)/2          # bisection

  if (reg == "EMA") {

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

                        design=des, nsims=nsims)

    } else {

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

                       design=des, nsims=nsims)

  }

  if (print == TRUE) cat(iter, x.new, pBE, "\n")

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

  if (pBE > sig) upper <- x.new     # move limit downwards

  if (pBE < sig) lower <- x.new     # move limit upwards

  delta <- upper - lower            # new interval

}

if (print) cat("run-time:", round((proc.time()[3]-ptm[3]),1),

  "seconds  iterations:", iter, "\n")

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

  "\ntarget alpha:", x, " adjusted alpha:", x.new, " pBE:", pBE, "\n")



  CV   adj alpha pBE_scABEL  adj alpha pBE_RSABE

0.200  0.050000  0.050220    0.050000  0.050153

0.225  0.049730  0.050360    0.048467  0.050359

0.250  0.047370  0.050359    0.040790  0.050360

0.275  0.040833  0.050360    0.028606  0.050359

0.300  0.031124  0.050359    0.017999  0.050360

0.302  0.031612  0.050359    0.050000  0.048548

0.325  0.035545  0.050360    0.050000  0.047741

0.350  0.037636  0.050360    0.050000  0.046156

0.375  0.039190  0.050360    0.050000  0.043941

0.400  0.041417  0.050360    0.050000  0.041041

0.425  0.045221  0.050360    0.050000  0.037799

0.450  0.050000  0.049795    0.050000  0.034401

0.475  0.050000  0.043420    0.050000  0.031150

0.500  0.050000  0.036960    0.050000  0.028097

0.525  0.050000  0.040644    0.050000  0.025383

0.550  0.050000  0.043412    0.050000  0.022875

0.575  0.050000  0.045423    0.050000  0.020699

0.600  0.050000  0.046647    0.050000  0.018660

0.625  0.050000  0.047379    0.050000  0.016948

0.650  0.050000  0.047754    0.050000  0.015457

0.675  0.050000  0.047698    0.050000  0.014135

0.700  0.050000  0.047192    0.050000  0.012861

0.725  0.050000  0.046282    0.050000  0.011799

0.750  0.050000  0.045020    0.050000  0.010865

0.775  0.050000  0.043338    0.050000  0.010057

0.800  0.050000  0.041320    0.050000  0.009425

Take my code with a grain of salt. Cases where no adjustment is necessary agree with Detlew’s results.

Note: Any result <0.05035995 is not significantly >0.05 in 10⁶ simulations. However, if you want to get a more conservative estimate replace

sig <- binom.test(x*nsims, nsims, alternative='less')$conf.int[2]

sig <- 0.05

Edit: Much faster code in this post.

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

The quality of responses received is directly proportional to the quality of the question asked. 🚮
Science Quotes

martin ★★ Austria, 2013-03-26 17:19 (4406 d 09:15 ago) @ Helmut Posting: # 10287 Views: 23,952	iteratively adjusted alpha Post reply
	dear helmut! iteratively adjusted alpha – super idea! you may find of interest that two weirdos published a paper about iterative alpha adjustment a few years ago (paper) soley based on simulation results (not enough skilled to provide evidence via mathematical proof). best regards martin

Helmut
★★★

Vienna, Austria,
2013-03-26 19:02
(4406 d 07:32 ago)

@ martin
Posting: # 10289
Views: 24,031

throw away our sample size tables?

Post reply

Hi Martin!

❝ iteratively adjusted alpha – super idea!

If the inflation of the consumer’s risk is real (based on our assumption to estimate pBE with a GMR at the borders of the scaled acceptance range) regulators should revise their methods. Would increase the sample size and IMHO requires a multi-step procedure.

Estimate the sample size for unadjusted α
Estimate adjusted α for this n
Estimate the sample size for adjusted α
Repeat steps 2–3 until you are in limbo

❝ […] two weirdos published a paper […]

Not their only crime. ;-)

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

The quality of responses received is directly proportional to the quality of the question asked. 🚮
Science Quotes

martin ★★ Austria, 2013-03-26 20:44 (4406 d 05:50 ago) @ Helmut Posting: # 10290 Views: 23,889	if real Post reply
	dear Helmut! totally agree: If the inflation of the type-I-error is real = serious risk to public health (to quote a famous colleague of mine). best regards martin PS.: IMHO - a-priori specification of the method should control the type-I-error (i.e. CV from historical data or more stringent the corresponding lower confidence limit should be greater than 30% to allow scaling).

Helmut
★★★

Vienna, Austria,
2013-03-26 22:18
(4406 d 04:16 ago)

@ martin
Posting: # 10291
Views: 24,027

regulators don’t care?

Post reply

Hi Martin!

❝ totally agree: If the inflation of the type-I-error is real = serious risk to public health (to quote a famous colleague of mine).

Is this the same guy who is (in)famous for publicly calling some approaches “bullshit”?

BTW, α-inflation is not unknown. See the figures and table 1 of the two Lászlós.¹

❝ PS.: IMHO - a-priori specification of the method should control the type-I-error (i.e. CV from historical data or more stringent the corresponding lower confidence limit should be greater than 30% to allow scaling).

I would also say so. Already noted in the past:

Besides, it is not clear at present what kind of regulatory policy will be followed when several σ_WR estimates (i.e. results of previous submissions) are available to a drug regulatory agency. If regulators use all available data, then they can get an improved estimate for σ_WR, and possibly can draw a different conclusion from the sponsor.²

But likely it is wishful thinking to expect a list of acceptance ranges or CVs of reference formulations.

Endrényi L, Tóthfalusi L. Regulatory Conditions for the Determination of Bioequivalence of Highly Variable Drugs. J Pharm Pharmaceut Sci. 2009;12(1):138–49. free online
Tóthfalusi L, Endrényi L, García Arieta A. Evaluation of Bioequivalence for Highly Variable Drugs with Scaled Average Bioequivalence. Clin Pharmacokinet. 2009;48(11):725–43. doi:10.2165/11318040-000000000-00000

Edit: EMA’s data set I (full replicate, n₁ 36, n₂ 37, CV_WR 0.4696) needs no α-adjustment, pBE 0.012061. If the two outliers are excluded (n₁ 34, n₂ 37, CV_WR 0.3216) we require an adjusted α of 0.03149 in order to maintain the consumer’s risk. The 93.70% CIs of 106.03–126.16% (Method A) and 106.10–126.24% (Method B) lie still within the scaled acceptance range of 78.79–126.93%, but it is a close shave…

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

The quality of responses received is directly proportional to the quality of the question asked. 🚮
Science Quotes

Helmut
★★★

Vienna, Austria,
2013-08-11 18:19
(4268 d 09:15 ago)

@ Helmut
Posting: # 11262
Views: 22,836

adjusting α

Post reply

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

The quality of responses received is directly proportional to the quality of the question asked. 🚮
Science Quotes

apapanas ☆ Greece, 2024-10-31 07:02 (169 d 19:32 ago) @ Helmut Posting: # 24251 Views: 2,959	missing letter Post reply
	Noticed that a letter is missing below, which would not let me run the code ❝ `} else {` ❝ `pBE <- power.RSABE(alpha=adj, theta0=GMR, CV=CV,` ❝ `n=n.adj, esign=des, nsims=nsims)` ❝ `}`

Helmut
★★★

Vienna, Austria,
2024-10-31 09:41
(169 d 16:53 ago)

@ apapanas
Posting: # 24252
Views: 2,980

PowerTOST: sampleN.scABEL.ad()

Post reply

Hi Antigoni,

❝ Noticed that a letter is missing below, which would not let me run the code :-)

Oops! I corrected it.

However, much easier with the function sampleN.scABEL.ad(), which was implemented in version 1.3-3 (2016-01-15) of PowerTOST.

library(PowerTOST) CV <- 0.4 target <- 0.9 design <- "2x2x4" sampleN.scABEL(CV = CV, targetpower = target, design = design, details = TRUE) # theta0 = 0.9 is the default in the HVD reference-scaling functions +++++++++++ scaled (widened) ABEL +++++++++++ Sample size estimation (simulation based on ANOVA evaluation) --------------------------------------------- Study design: 2x2x4 (4 period full replicate) log-transformed data (multiplicative model) 1e+05 studies for each step simulated. alpha = 0.05, target power = 0.9 CVw(T) = 0.4; CVw(R) = 0.4 True ratio = 0.9 ABE limits / PE constraint = 0.8 ... 1.25 EMA regulatory settings - CVswitch = 0.3 - cap on scABEL if CVw(R) > 0.5 - regulatory constant = 0.76 - pe constraint applied Sample size search n power 36 0.8630 38 0.8782 40 0.8924 42 0.9032 sampleN.scABEL.ad(CV = CV, targetpower = target, design = design, details = TRUE) # using the values of above +++++++++++ scaled (widened) ABEL ++++++++++++ Sample size estimation for iteratively adjusted alpha (simulations based on ANOVA evaluation) ---------------------------------------------- Study design: 2x2x4 (4 period full replicate) log-transformed data (multiplicative model) 1,000,000 studies in each iteration simulated. Assumed CVwR 0.4, CVwT 0.4 Nominal alpha : 0.05 True ratio : 0.9000 Target power : 0.9 Regulatory settings: EMA (ABEL) Switching CVwR : 0.3 Regulatory constant: 0.76 Expanded limits : 0.7462 ... 1.3402 Upper scaling cap : CVwR > 0.5 PE constraints : 0.8000 ... 1.2500 n 42, nomin. alpha: 0.05000 (power 0.9032), TIE: 0.0557 Sample size search and iteratively adjusting alpha n 42, adj. alpha: 0.04256 (power 0.8914), rel. impact on power: -1.31% n 44, adj. alpha: 0.04326 (power 0.9052), TIE: 0.05000 Compared to nominal alpha's sample size increase of 4.8% (~study costs). Runtime : 12.7 seconds Simulations: 16,400,000

To get only the sample size use

sampleN.scABEL.ad(CV = CV, targetpower = target, design = design, print = FALSE)[["Sample size"]]

I leave it to you to loop over the target power, CV_wR, and designs to get the plots… :-D

For examples and more elaborated [image]

-scripts see this article.

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

The quality of responses received is directly proportional to the quality of the question asked. 🚮
Science Quotes

apapanas ☆ Greece, 2024-11-08 11:48 (161 d 14:46 ago) @ Helmut Posting: # 24264 Views: 2,099	PowerTOST: sampleN.scABEL.ad() Post reply
	Many thanks dear Helmut!

Helmut
★★★

Vienna, Austria,
2024-11-08 15:34
(161 d 11:00 ago)

@ apapanas
Posting: # 24265
Views: 2,052

Molins et al. (2017)

Post reply

Hi Antigoni,

❝ Many thanks dear Helmut!

Welcome.
See also the function sample.size.adj() in this article. It implements our method^1,2 and a more conservative one.^3,4
Of course, such a conservatism comes with a price. The sample sizes can be prohibitively large.

method CVwT CVwR Target power Sample size Achieved power EMA (2010) 0.4 0.4 0.9 42 0.90325 Labes and Schütz (2016) 0.4 0.4 0.9 44 0.90523 Molins et al. (2017) 0.4 0.4 0.9 78 0.90246 method alpha TIE EMA (2010) 0.05000 0.05572 Labes and Schütz (2016) 0.04326 0.05000 Molins et al. (2017) 0.03061 0.05000

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.
Assuming that the observed $\small{CV_\text{wR}=}$ the true $\small{CV_\text{wR}}$.
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.
Assuming the ‘worst case’ $\small{CV_\text{wR}=30\%}$ – irrespective of the observed $\small{CV_\text{wR}}$ because the maximum inflation of the Type I Error is always at the switching $\small{CV_\text{wR}=30\%}$ regardless the design and sample size.

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

The quality of responses received is directly proportional to the quality of the question asked. 🚮
Science Quotes

Helmut
★★★

Vienna, Austria,
2013-03-27 16:38
(4405 d 09:56 ago)

@ martin
Posting: # 10297
Views: 23,972

target α in sim’s?

Post reply

Hi Martin,

a technical question (see the note at the end of my previous post). Though any result <0.05035995 is not statistically significant >0.05 shouldn’t I set the target (sig) in the algo to 0.05?

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

The quality of responses received is directly proportional to the quality of the question asked. 🚮
Science Quotes

martin ★★ Austria, 2013-03-27 17:36 (4405 d 08:58 ago) @ Helmut Posting: # 10298 Views: 23,885	target α in sim’s? Post reply
	Hi Helmut! IMHO: yes as this is a constant and should not dependent on the number of simulation runs. however, differences should be rather small by using 1E6 runs. best regards martin Ps.: ... and yes its the same guy who is famous for publicly calling some approaches “bullshit”

Helmut
★★★

Vienna, Austria,
2013-03-29 00:20
(4404 d 02:14 ago)

@ martin
Posting: # 10305
Views: 24,148

If it is real: results

Post reply

Hi Martin!

❝ IMHO: yes as this is a constant and should not dependent on the number of simulation runs.

Oh yes, sure. Sorry for my stupidity.

❝ Ps.: ... and yes its the same guy who is famous for publicly calling some approaches “bullshit” :-D

I always thought he should work on his social skills.

Here the results of my sim’s for target α 0.05. Partial and full replicate designs, n=24/48.

[image]

The FDA’s method would lead to a large α-inflation (as presented by Detlew above) for CV_WR ≤30%. This could be corrected by adjusting α. However, since scaling is only allowed for CV_WR >30% nothing has to be done. Like the conventional TOST the test gets more conservative with increasing CV (and n).

[image]

The EMA’s method is another cup of tea. α-inflation is seen up to 45%! The test is closer to the nominal level than the FDA’s.
Interesting the behavior of the partial replicate with n=24. :confused:

Code (10–25 iterations to reach convergence, runtime on my machine 15–45 seconds depending on design, n, CV):

require(PowerTOST)

reg   <- "EMA"   # "EMA" for ABEL or "FDA" for scABE

CV    <- 0.300   # intra-subject CV of reference

n     <- 24      # total sample size

                 # in case of imbalanced sequences use a vector:

                 # e.g., c(n1,n2,n3)

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

                 # for others see known.designs()

print <- TRUE    # intermediate results

if (reg == "EMA") { # scABEL

  ifelse (CV <= 0.5, GMR <- exp(0.7601283*CV2se(CV)),

                     GMR <- exp(0.7601283*CV2se(0.5)))

  if (CV <= 0.3) GMR <- 1.25

  } else {          # RSABE

  ifelse (CV > 0.3, GMR <- exp(0.8925742*CV2se(CV)),

                    GMR <- 1.25)

}

nsims <- 1e6

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

x     <- 0.05    # target alpha

nom   <- c(0.001, x)                # from conservative to target alpha

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

delta <- upper - lower              # interval

ptm   <- proc.time()                # start timer

iter  <- 0

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

  iter  <- iter + 1

  x.new <- (lower+upper)/2          # bisection

  if (reg == "EMA") {

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

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

    } else {

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

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

  }

  if (print) {                      # show progress

    if (iter == 1) cat(" i  adj. alpha       pBE\n")

    cat(format(iter, digits=2, width=2),

        format(x.new, digits=6, width=11, nsmall=7),

        format(pBE, digits=6, width=9, nsmall=6), "\n")

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

  }

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

  if (pBE > x) upper <- x.new       # move upper limit downwards

  if (pBE < x) lower <- x.new       # move lower limit upwards

  delta <- upper - lower            # new interval

}

if (print) cat("run-time:", round((proc.time()[3]-ptm[3]), 1),

               "seconds  iterations:", iter, "\n")

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

    "\ntarget alpha:", x, " adjusted alpha:", x.new,

    " pBE:", pBE, "(empiric alpha)\n")

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

The quality of responses received is directly proportional to the quality of the question asked. 🚮
Science Quotes

martin ★★ Austria, 2013-03-28 16:08 (4404 d 10:26 ago) @ Helmut Posting: # 10304 Views: 23,760	having alpha Post reply
	Hi Helmut & simulants ! What do you think about a kind of adaptive group-sequential design with one interim analysis (i.e. estimate CV) allowing to make a decision regarding average BE or scaled BE in the final analysis? This may solve the problem. best regards martin

Helmut
★★★

Vienna, Austria,
2013-03-29 00:34
(4404 d 02:00 ago)

@ martin
Posting: # 10306
Views: 24,051

fixed swr

Post reply

Hi Martin!

❝ What do you think about a kind of adaptive group-sequential design with one interim analysis (i.e. estimate CV) allowing to make a decision regarding average BE or scaled BE in the final analysis? This may solve the problem.

Nope. Actually the methods are (hidden) sequential.
Most clear in the FDA’s guidance:

Determine s_WR, the within-subject standard deviation (SD) of the reference product, for the pharmacokinetic (PK) parameters AUC and C_max.

If s_WR < 0.294,^1,2 use the two one-sided tests procedure to determine bioequivalence (BE) for the individual PK parameter(s)
If s_WR ≥ 0.294,² use the reference-scaled procedure to determine BE for the individual PK parameter(s)

However, once scaling is justified (s_WR ≥ 0.294), the extent of scaling is still data-driven (dependent on s_WR from the study). IMHO the problem would only vanish if regulators publish a fixed value of s_W for each reference which has to be used independent from the findings in the study.

s_WR 0.294 ~ CV_WR 0.300469…
We see the worst inflation at CV_WR 0.3. Maybe it would be better to use ABE if s_WR ≤ 0.294 and RSABE if s_WR > 0.294.

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

The quality of responses received is directly proportional to the quality of the question asked. 🚮
Science Quotes

martin ★★ Austria, 2013-03-29 16:07 (4403 d 10:27 ago) @ Helmut Posting: # 10311 Views: 23,735	fixed swr Post reply
	Hi Helmut ! Yep; your are totally right (should have done my homework first ). what about defining the extent of scaling after interim analysis of the first stage data (CV estimated in 1st stage) within a group-sequential adaptive design? Drawback: BUT what if the CV at interim is smaller than 0.3 ... best regards martin

Helmut
★★★

Vienna, Austria,
2013-03-29 17:10
(4403 d 09:24 ago)

@ martin
Posting: # 10312
Views: 24,023

Yes but no but yes but no but…

Post reply

Hi Martin!

❝ what about defining the extent of scaling after interim analysis of the first stage data (CV estimated in 1st stage) within a group-sequential adaptive design?

Theoretically OK. ;-)

❝ Drawback: BUT what if the CV at interim is smaller than 0.3 ... :confused:

… or it is larger than 0.3 in stage 1 and in the pooled data set (n↑ = more precise estimate) you find one which is ≤0.3? If the inflation is real (still not sure) we have a problem – only with EMA’s method – in the CV-range 0.3–0.5. IMHO there are two options:

Iteratively adjust α.
Use a fixed α of ~0.030 for partial replicates and ~0.025 for full replicates if CV_WR ≤0.5 (see plots above). This should cover the entire range of designs and sample sizes of up to 96. The maximum adjustment (at 0.3) is almost independent from sample size (see plot below). Conservative for any CV >0.3…

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

The quality of responses received is directly proportional to the quality of the question asked. 🚮
Science Quotes

‘alpha’ of scaled ABE? [RSABE / ABEL]

‘alpha’ of scaled ABE?

’alpha’ of scaled ABE: design

N=24, 48

adaptive design without adjustment

α adjustment?

iteratively adjusted alpha

throw away our sample size tables?

if real

regulators don’t care?

adjusting α

missing letter

PowerTOST: sampleN.scABEL.ad()

PowerTOST: sampleN.scABEL.ad()

Molins et al. (2017)

target α in sim’s?

target α in sim’s?

If it is real: results

having alpha

fixed swr

fixed swr

Yes but no but yes but no but…