Bioequivalence and Bioavailability Forum • Determination of sample size including the factor of σD

junhao
☆

China,
2016-07-08 06:34
(3216 d 08:10 ago)

Posting: # 16480
Views: 7,925

Determination of sample size including the factor of σD [Power / Sample Size]

Hello everybody,

In Table 1, Appendix C of FDA 2001 guidance Statistical Approaches to Establishing Bioequivalence, the sample size is calculated for different σ_Ds. I was wondering how this is done? Nothing can I find in the two references listed with the talbe.

One further question, when power.RSABE() of R package TOST is used in simulation that gives sample size for RSABE design, does it account for σ_D, or the σ_D is assumed to be zero?

In a previous post, A reference is mentioned:

Scott D Patterson and Byron Jones
REPLICATE DESIGNS AND AVERAGE, INDIVIDUAL, AND POPULATION BIOEQUIVALENCE
GSK BDS Technical Report 2002 – 01

A link was given below but I failed opening it :lookaround:

. Can somebody help to try if this link is still correct?
http://biometrics.com/wp-content/uploads/2009/06/tr2002-01.pdf

Thank you,

Junhao//

d_labes
★★★

Berlin, Germany,
2016-07-13 15:49
(3210 d 22:56 ago)

@ junhao
Posting: # 16486
Views: 6,575

Sample size taking into account SbF

Post reply

Dear Junhao,

❝ In Table 1, Appendix C of FDA 2001 guidance Statistical Approaches to Establishing Bioequivalence, the sample size is calculated for different σ_Ds. ...

Don't invest too much in understanding or verifying this table.
It shows sample sizes for the so-called "individual bioequivalence" in which the SbF (subject by formulation) interaction, your σ_D, plays a prominent role. "Individual bioequivalence" is nowadays history only.

❝ One further question, when power.RSABE() of R package TOST is used in simulation that gives sample size for RSABE design, does it account for σ_D, or the σ_D is assumed to be zero?

The code of PowerTOST is open source, means you can inspect it, either in the source code tar-ball or even without that, just typing

library(PowerTOST)

power.RSABE

At the very beginning of the code there is an expression s2D <- 0. Guess what it means :-D

❝ In a previous post, A reference is mentioned:

❝ ...

❝ A link was given below but I failed opening it :lookaround: . Can somebody help to try if this link is still correct?

Unfortunately this link doesn't function since some time ago. Further I couldn't find the reference at other places of the Inder-nett. Drop me a mail and I can send you a backup copy of this technical report. BTW: It has two parts.

Hope this helps.

—
Regards,

Detlew

junhao
☆

China,
2016-07-14 07:54
(3210 d 06:51 ago)

@ d_labes
Posting: # 16489
Views: 6,504

Sample size taking into account SbF

Post reply

Dear Detlew,

Thank you for your reply!

❝ Don't invest too much in understanding or verifying this table.

❝ It shows sample sizes for the so-called "individual bioequivalence" in which the SbF (subject by formulation) interaction, your σ_D, plays a prominent role. "Individual bioequivalence" is nowadays history only.

But when we assume different between-subject variation for T and R, σ_D is never gonna be 0, and I right? I am not interested in the old-fashion "Individual Bioequivalence" either, but in the RSABE context, we are assuming same between-subject variation for T and R so that σ_D could be 0?

❝ The code of PowerTOST is open source, means you can inspect it, either in the source code tar-ball or even without that, just typing

❝ library(PowerTOST)

❝ power.RSABE

❝ At the very beginning of the code there is an expression s2D <- 0. Guess what it means :-D .

I will try it out. Thank you for the information.

❝ Unfortunately this link doesn't function since some time ago. Further I couldn't find the reference at other places of the Inder-nett. Drop me a mail and I can send you a backup copy of this technical report. BTW: It has two parts.

Thank you and I would appreciate your help!

Junhao//

d_labes
★★★

Berlin, Germany,
2016-07-19 11:18
(3205 d 03:26 ago)

@ junhao
Posting: # 16492
Views: 6,557

Sample size taking into account SbF

Post reply

Dear Junhao,

❝ ❝ Don't invest too much in understanding or verifying this table.

❝ ❝ It shows sample sizes for the so-called "individual bioequivalence" in which the SbF (subject by formulation) interaction, your σ_D, plays a prominent role ...

Here I have to correct myself. Table 1 in the Appendix of the mentioned FDA guidance actually shows sample sizes for ABE.

Here some code to reproduce this table:

library(PowerTOST)

# function to calculate the 'pooled' CV if the individual component of the variability are given 

# Wang/Chow 2002

# "ON Statistical Power for Average Bioequivalence Testing Under Replicated Crossover Designs"

# J. BIOPHARM. STAT. Vol. 12, No. 3, pp. 295–309, 2002

CVm <- function(sD, swT, swR=swT, design){

  if (design=="2x2"){

    s2diff <- (sD^2 + swT^2 + swR^2)/2

  }

  else if (design=="2x2x4"){

    s2diff <- sD^2 + (swT^2 + swR^2)/2

  } else stop("Design not implemented.")

  mse2CV(s2diff)

}



# gimme the numbers

delta <- 0.05 # this is measured in the log domain, corresponds to theta0 = 0.9512294

# assuming swT=swR as in Table 1

swT   <- 0.5

sD    <- 0.15

targetpower <- 0.8



design <- "2x2"

CV <- CVm(sD, swT, design=design)

sampleN.TOST(CV=CV, theta0=exp(-delta), targetpower=targetpower, design=design)



design <- "2x2x4"

CV <- CVm(sD, swT, design=design)

sampleN.TOST(CV=CV, theta0=exp(-delta), targetpower=targetpower, design=design)

Note that we need a 'pooled' variability for ABE only. Which we can get from a non-replicated design also.
Only in case of a full replicate design we may have informations about the components. But here it depends on the statistical model if a σ_D plays a role.
In case of a partial replicate design you can't get reasonable estimates of the variability components in all cases. Here a model with a SbF is overspecified and may lead to estimation problems like non-convergence of the REML procedure.

See f.i.
Endrenyi et al.
"Properties of the estimated variance component for subject-by-formulation interaction in studies of individual bioequivalence"
Statist. Med. 2000; 19:2867–2878
to find out that σ_D=0 is a reasonable assumption.

❝ But when we assume different between-subject variation for T and R, σ_D is never gonna be 0, and I right?

Correct. If your statistical model of the between-subject variability incorporates a SbF interaction. Otherwise this component is absorbed in the estimates of the within-subject variabilities σ_wT and σ_wR, respectively.

❝ ... but in the RSABE context, we are assuming same between-subject variation for T and R so that σ_D could be 0?

Also correct. But beside the CI component of T-R of RSABE, where we again have to use a pooled variability, there is no influence of σ_D.

—
Regards,

Detlew

Helmut
★★★

Vienna, Austria,
2016-07-19 17:07
(3204 d 21:37 ago)

@ d_labes
Posting: # 16493
Views: 6,402

Appendix C, Table I

Post reply

Dear Detlew,

hijacking your code:

library(PowerTOST) CVm <- function(sD, swT, swR=swT, design){ if (design == "2x2") { s2diff <- (sD^2 + swT^2 + swR^2)/2 } else if (design == "2x2x4") { s2diff <- sD^2 + (swT^2 + swR^2)/2 } else stop("Design not implemented.") mse2CV(s2diff) } delta <- 0.05 targetpower <- c(0.80, 0.90) design <- c("2x2", "2x2x4") swT <- c(0.15, 0.23, 0.30, 0.50) sD <- c(0.01, 0.10, 0.15) method <- "exact" n <- matrix(data=NA, ncol=length(targetpower)*2+2, nrow=length(swT)*length(sD)) dimnames(n) <- list(NULL, c("swT", "sD", sort(paste0(sprintf("%i%%:", 100*targetpower), rep(c("2P", "4P"), each=length(targetpower)))))) n[, 1] <- rep(swT, each=length(sD)) n[, 2] <- rep(sD, length(swT)) r <- 0 for (j in seq_along(swT)) { for (k in seq_along(sD)) { r <- r + 1 c <- 2 for (l in seq_along(targetpower)) { for (m in seq_along(design)) { c <- c + 1 CV <- CVm(sD[k], swT[j], design=design[m]) n[r, c] <- sampleN.TOST(CV=CV, theta0=exp(-delta), targetpower=targetpower[l], design=design[m], method=method, print=FALSE)[["Sample size"]] } } } } print(as.data.frame(n), row.names=FALSE)

Gives:

swT sD 80%:2P 80%:4P 90%:2P 90%:4P 0.15 0.01 12 6 16 8 0.15 0.10 14 8 18 10 0.15 0.15 16 12 22 14 0.23 0.01 24 12 32 16 0.23 0.10 28 14 36 20 0.23 0.15 30 18 40 24 0.30 0.01 40 20 54 28 0.30 0.10 42 22 56 30 0.30 0.15 46 26 60 34 0.50 0.01 108 54 146 74 0.50 0.10 110 56 148 76 0.50 0.15 114 60 152 80

Equal to, lower / higher than the reference (Appendix C, Table I):

Δ = 0.05 80% Power 90% Power ─────────────────── σWT σD 2P 4P 2P 4P ────────────────────────────── 0.15 0.01 12 6 16 8 0.10 14 10 18 12 0.15 16 12 22 16 ────────────────────────────── 0.23 0.01 24 12 32 16 0.10 26 16 36 20 0.15 30 18 38 24 ────────────────────────────── 0.30 0.01 40 20 54 28 0.10 42 24 56 30 0.15 44 26 60 34 ────────────────────────────── 0.50 0.01 108 54 144 72 0.10 110 58 148 76 0.15 112 60 150 80

—
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, 2016-07-20 13:27 (3204 d 01:18 ago) @ Helmut Posting: # 16497 Views: 6,265	Appendix C, Table I Post reply
	Dear Helmut, power testing and nitpicking, as always . What's your resumé? — Regards, Detlew

Helmut
★★★

Vienna, Austria,
2016-07-20 17:45
(3203 d 21:00 ago)

@ d_labes
Posting: # 16500
Views: 6,258

I don’t care

Post reply

Dear Detlew,

❝ power testing and nitpicking, as always :-D .

❝ What's your resumé?

My personal resumé as requested.

Since we have no clue how the FDA arrived at their numbers (as usual), who cares? PowerTOST is the gold standard!
BTW, method="nct" gives the same results as method="exact". Sample sizes with method="shifted" are (with a few exceptions) larger.

—
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