Bioequivalence and Bioavailability Forum • Narrower Acceptance Range [0.90/1.11]

Helmut
★★★

Vienna, Austria,
2006-05-26 18:02
(6912 d 07:59 ago)

Posting: # 139
Views: 9,881

Narrower Acceptance Range [0.90/1.11] [Power / Sample Size]

Dear all,

since at least the Danish Authority requires a narrower acceptance range for some drugs in the following you may find a table giving sample sizes for the acceptance range of 0.90-1.11 (cave: the upper limit is 1/0.9=1.1111..., except for Canada, where the upper limit is set to 1.12).

Sample sizes were calculated for an expected deviation of -5% from the reference (80% and 90% power) according to approximations (which can easily implemented e.g. in a speadsheet for any combination of power and deviation) given in
[1] Hauschke D, Steinijans VW, Diletti E and M Burke
Sample Size Determination for Bioequivalence Assessment Using a Multiplicative Model
J Pharmacokin Biopharm 20/5, 557-561, 1992
and
[2] S-C Chow and H Wang
On Sample Size Calculation in Bioequivalence Trials
J Pharmacokin Pharmacodyn 28/2, 155-169, 2001
Errata: J Pharmacokin Pharmacodyn 29/2, 101-102, 2002

Approximate sample sizes differ slightly from exact values given by
[3] Diletti E, Hauschke D and VW Steinijans
Sample size determination: Extended tables for the multiplicative model
and bioequivalence ranges of 0.9 to 1.11 and 0.7 to 1.43
Int J Clin Pharm Ther Toxicol 30/Suppl.1, S59-62, 1992 and
[4] B Oloffson
StudySize2.01
CreoStat HB, 2006


       +-----------+-----------+-----------+-----------+

       |    [1]    |     [2]   |     [3]   |     [4]   |

+------+-----------+-----------+-----------+-----------+

|  CV% |  80%  90% |  80%  90% |  80%  90% |  80%  90% |

+------+-----------+-----------+-----------+-----------+

|  5.0 |  12    16 |  12    16 |  14    18 |  14    18 |

|  6.0 |  18    24 |  18    22 |   .     . |  18    24 |

|  7.0 |  22    30 |  22    30 |   .     . |  24    32 |

|  7.5 |  26    36 |  26    34 |  26    36 |  26    36 |

|  8.0 |  30    40 |  28    40 |   .     . |  30    40 |

|  9.0 |  36    50 |  36    50 |   .     . |  36    50 |

| 10.0 |  44    60 |  44    60 |  44    60 |  44    60 |

| 11.0 |  54    74 |  52    72 |   .     . |  54    72 |

| 12.0 |  62    86 |  62    86 |   .     . |  62    86 |

| 12.5 |  68    94 |  68    94 |  68    94 |  68    94 |

| 13.0 |  74   102 |  74   100 |   .     . |  74   100 |

| 14.0 |  84   116 |  84   116 |   .     . |  84   116 |

| 15.0 |  98   134 |  96   134 |  96   132 |  96   132 |

| 17.5 | 132   182 | 132   182 | 130   180 | 130   180 |

| 20.0 | 172   236 | 170   236 | 168   232 | 168   232 |

| 22.5 | 216   298 | 216   298 |   .     . | 212   292 |

| 25.0 | 266   368 | 266   368 |   .     . | 258   358 |

+------+-----------+-----------+-----------+-----------+

If using approximations you should allow for a safety margin; sample sizes are sometimes a little larger (overlined) than exact ones, except for small CVs (underlined) where the sample size is underestimated.

Hopefully your narrow therapeutic index drug shows low variability - which is often the case; otherwise it will be very difficult to demonstrate BE in a conventional 2x2 cross-over study...

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Helmut Schütz

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Helmut
★★★

Vienna, Austria,
2008-04-04 18:13
(6233 d 07:48 ago)

@ Helmut
Posting: # 1764
Views: 7,428

Table/R-code for AR [0.90/1.11]

Post reply

Dear all,

I updated the sample size table (expected deviation of -5% from the reference, 80% and 90% power) according to exact results given in Table 5.3,^[1] which are matched by results obtained by R-code given in this post with modifications^[2] and David Dubin's FARTSSIE^[3] (see this post).

       +-----------+-----------+-----------+

       |    [1]    |    [2]    |    [3]    |

+------+-----------+-----------+-----------+

|  CV% |  80%  90% |  80%  90% |  80%  90% |

+------+-----------+-----------+-----------+

|  5.0 |  14    18 |  14    18 |  14    18 |

|  6.0 |   .     . |  18    24 |  18    24 |

|  7.0 |   .     . |  24    32 |  24    32 |

|  7.5 |  26    36 |  26    36 |  26    36 |

|  8.0 |   .     . |  30    40 |  30    40 |

|  9.0 |   .     . |  36    50 |  36    50 |

| 10.0 |  44    60 |  44    60 |  44    60 |

| 11.0 |   .     . |  54    72 |  54    72 |

| 12.0 |   .     . |  62    86 |  62    86 |

| 12.5 |  68    94 |  68    94 |  68    94 |

| 13.0 |   .     . |  74   100 |  74   100 |

| 14.0 |   .     . |  84   116 |  84   116 |

| 15.0 |  96   132 |  96   132 |  96   132 |

| 17.5 | 130   180 | 130   180 | 130   180 |

| 20.0 | 168   232 | 168   232 | 168   232 |

| 22.5 | 212   292 | 212   292 | 212   292 |

| 25.0 | 258   358 | 258   358 | 258   358 |

+------+-----------+-----------+-----------+

The code (download link) was tested in versions 2.6.2, 2.5.1, 1.9.1, and 1.9.0.

#######################################################################

# Sample size calculation for a standard RT/TR                        #

# 2x2x2 cross-over design (multiplicative model)                      #

#######################################################################

# R-code based on SAS-code by

# (1) B. Jones and M.G. Kenward

#     Design and Analysis of Cross-Over Trials

#     Chapman & Hall/CRC, Boca Raton (2nd Edition 2000)

# (2) S. Patterson and B. Jones

#     Bioequivalence and Statistics in Clinical Pharmacology

#     Chapman & Hall/CRC, Boca Raton (2006)

# /*** WARNING : PROGRAM OFFERED FOR USE WITHOUT ANY GUARANTEES    ***/

# /*** NO LIABILITY IS ACCEPTED FOR ANY LOSS RESULTING FROM USE OF ***/

# /*** THIS SET OF SAS INTRUCTIONS                                 ***/

# Modification of degrees of freedom according to a

# personal message by D. Hauschke (E-mail 2006-01-05)

# Tested in R-versions 2.6.2 / 2.5.1 / 1.9.1 / 1.9.0

# 2008-04-04

# Helmut Schuetz

# BEBAC - Consultancy services for Bioequivalence

# and Bioavailability Studies

# 1070 Vienna, Austria

# ----------------------------------------------

a           <- 0.05      # alpha

theta1      <- 0.90      # lower acceptance limit

theta2      <- 1/theta1  # upper acceptance limit

ratio       <- 0.95      # expected ratio T/R

assign("target",

  c(0.8,0.9))            # target power

assign("CV",

  c(.05,.06,.07,.075,.08,

    .09,.1,.11,.12,.125,

    .13,.14,.15,.175,.2,

    .225,.25))           # intra-subject coefficients of variation

limit       <- 500       # upper sample size limit for search

# ----------------------------------------------

# Do not modify the code below this line unless

# you know what you are doing!

# ----------------------------------------------

for (i in CV)            # CV loop

  {

  if(i==CV[1]){

    title=paste(

      paste("   Sample size estimation for a standard\n",

        "  RT/TR 2x2x2 cross-over design (multi-\n",

        "  plicative model).\n",

        "  Expected ratio T/R =",

        format(round(ratio*100,2),nsmall=2,width=4),"%.\n\n"),

      paste(paste(format(round(target*100,2),nsmall=2,width=17),

        "%",sep="",collapse=""),"\n"),

      paste("  CV",

        paste(c(rep("   spl.size (pwr.)",

        length(target))),collapse=""),"\n"),

      paste("----",

        paste(c(rep(paste(rep("-",18),collapse=""),

        length(target))),collapse=""),"\n") )

    cat(title)

    }

  cat(paste(format(i*100,nsmall=1,width=4),"% ",sep=""))

  sigmaW    <- sqrt(log(1+i^2))

  s         <- sqrt(2)*sigmaW

  for (j in target)      # power loop

    {

    n       <- 6         # start value of sample size search

    repeat{

      df    <- n-2

      t1    <- qt(1-a,df)

      t2    <- -t1

      nc1   <- (sqrt(n))*((log(ratio)-log(theta1))/s)

      nc2   <- (sqrt(n))*((log(ratio)-log(theta2))/s)

      prob1 <- pt(t1,df,nc1)

      prob2 <- pt(t2,df,nc2)

      power <- prob2-prob1

      ppct  <- power*100

      if(power >= j | n > limit) break

      n     <- n+2       # increment for even sample size

      }

    if(n <= limit){

      cat(paste(format(n,width=5)," (",

        format(round(ppct,5),nsmall=5,width=8),"%) ",

        sep=""))

      } else

      cat(paste(">",format(limit,width=4)," (",

        format(round(ppct,5),nsmall=5,width=8),"%) ",

        sep=""))

    }

  cat("\n")

  }

You can modify the code to your needs:
a (one-sided) alpha - needed in TOST / 90% CI (to obtain sample sizes for a 95% CI as needed for NTIDs by ANVISA, set a <- 0.025)
theta1 lower acceptance limit, where theta2=1/theta1
target set of target powers; if you want to get additionally e.g., 70% power, use: assign("target",c(0.7,0.8,0.9))
CV set of CVs; each value gives one line of output
limit upper limit of sample size; useful if set to a number lower than 500 to get the power obtained with a maximum feasible sample size

[1] D Hauschke, VW Steinijans and I Pigeot
Bioequivalence Studies in Drug Development: Methods and Applications
Wiley, New York, pp 118 (2007)
[3] D Dubins
FARTSSIE v1.4
Toronto, Ontario, Canada (2008)
download link

—
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