Encoding:	UTF-8
Version:	1.1.3
Date:	2022-05-02
Title:	Average Bioequivalence with Expanding Limits (ABEL)
Author:	Helmut Schütz [aut, cre], Michael Tomashevskiy [ctb], Detlew Labes [ctb]
Maintainer:	Helmut Schütz <helmut.schuetz@bebac.at>
Depends:	R (≥ 3.5.0)
Imports:	readxl (≥ 1.0.0), PowerTOST (≥ 1.5.3), lmerTest, nlme, pbkrtest, graphics, grDevices
Suggests:	knitr, rmarkdown, testthat, devtools
Description:	Performs comparative bioavailability calculations for Average Bioequivalence with Expanding Limits (ABEL). Implemented are 'Method A' / 'Method B' and the detection of outliers. If the design allows, assessment of the empiric Type I Error and iteratively adjusting alpha to control the consumer risk. Average Bioequivalence - optionally with a tighter (narrow therapeutic index drugs) or wider acceptance range (South Africa: Cmax) - is implemented as well.
License:	GPL (≥ 3)
LazyData:	true
VignetteBuilder:	knitr
URL:	https://github.com/Helmut01/replicateBE
BugReports:	https://github.com/Helmut01/replicateBE/issues
NeedsCompilation:	no
Packaged:	2022-05-02 19:06:36 UTC; HS
Repository:	CRAN
Date/Publication:	2022-05-02 22:52:03 UTC

Comparative BA-calculation for Average Bioequivalence

Description

This function performs the required calculations for the BE decision via conventional (unscaled) Average Bioequivalence based on ANOVA as recommended in the EMA’s guideline.

Usage

ABE(alpha = 0.05, path.in, path.out = tempdir(), file, set = "",
    ext, na = ".", sep = ",", dec = ".", logtrans = TRUE,
    print = TRUE, details = FALSE, verbose = FALSE, ask = FALSE,
    data = NULL, theta1, theta2)

Arguments

Details

The model for the treatment comparison is
lm(log(PK) ~ sequence + subject%in%sequence + period + treatment, data = data)
where all effects are fixed.

Tested designs

• 4-period 2-sequence full replicates
  ⁠TRTR | RTRT⁠
⁠TRRT | RTTR⁠
⁠TTRR | RRTT⁠

• 2-period 4-sequence replicate
  ⁠TR | RT | TT | RR ⁠ (Balaam’s design)

• 4-period 4-sequence full replicates
  ⁠TRTR | RTRT | TRRT | RTTR⁠
⁠TRRT | RTTR | TTRR | RRTT⁠

• 3-period 2-sequence full replicates
  ⁠TRT | RTR⁠
⁠TRR | RTT⁠

• 3-period (partial) replicates
  ⁠TRR | RTR | RRT⁠
⁠TRR | RTR ⁠ (extra-reference design)

Data structure

• Columns must have the headers subject, period, sequence, treatment, PK, and/or logPK.
  Any order of columns is acceptable.
  Uppercase and mixed case headers will be internally converted to lowercase headers.

  • subject must be integer numbers or (any combination of) alphanumerics
    ⁠[A-Z, a-z, -, _, #, 0-9]⁠

  • period must be integer numbers.

  • sequence must be contained in the tested designs (numbers or e.g., ⁠ABAB⁠ are not acceptable).

  • The Test treatment must be coded T and the Reference R.

Value

Prints results to a file if argument print = TRUE (default).
If argument print = FALSE, returns a data frame with the elements:

Warning

Files may contain a commentary header. If reading from a CSV-file, each line of the commentary header must start with ⁠"# "⁠ (hashmark space = ⁠ASCII 35 ASCII 32⁠). If reading from an Excel-file all lines preceding the column headers are treated as a comment.

Clarification

The ‘ASCII line chart’ in the result file gives the confidence limits with filled black squares and the point estimate as a white rhombus. If a confidence limit exceeds the drawing range, it is shown as a triangle. The BE limits and 100% are given with single vertical lines. The ‘resolution’ is approximatelly 0.5% and therefore, not all symbols might be shown. The CI and PE take presedence over the limits.

Disclaimer

Program offered for Use without any Guarantees and Absolutely No Warranty. No Liability is accepted for any Loss and Risk to Public Health Resulting from Use of this R-Code.

Note

The EMA’s model assumes equal [sic!] intra-subject variances of test and reference (like in 2×2×2 trials) – even if proven false in one of the full replicate designs (were both CV_wT and CV_wR can be estimated). Hence, amongst biostatisticians it is called the ‘crippled model’ because the replicative nature of the study is ignored.
Conventional unscaled ABE has to be employed for C_max (if widening of the acceptance range is clinically not justifiable), AUC_0–t, AUC_0–72 (immediate release products) and C_max,ss, C_τ,ss, _partialAUC (if widening of the acceptance range is clinically not justifiable), and AUC_0–t, AUC_0–∞, AUC_0–τ (modified release products).

Author(s)

Helmut Schütz

References

European Medicines Agency, Committee for Medicinal Products for Human Use. Guideline on the Investigation of Bioequivalence. CPMP/EWP/QWP/1401/98 Rev. 1/ Corr **. London. 20 January 2010. online

European Medicines Agency, Committee for Medicinal Products for Human Use. Guideline on the pharmacokinetic and clinical evaluation of modified release dosage forms. EMA/CPMP/EWP/280/96 Corr1. London. 20 November 2014. online

See Also

Examples

# Importing from a CSV-file, using most of the defaults: variable
# separator comma, decimal separator period, print to file.
# Note: You must adapt the path-variables. The example reads from
# the data provided by the library. Write-permissions must be granted
# for 'path.out' in order to save the result file. Here the deafault
# (R's temporary folder) is used. If you don't know where it is,
# type tempdir() in the console.
path.in <- paste0(find.package("replicateBE"), "/extdata/")
ABE(path.in = path.in, file = "DS", set = "02", ext = "csv")
# Should result in:
#   BE-limits          :  80.00% ... 125.00%
#   Confidence interval:  97.32% ... 107.46%  pass
#   Point estimate     : 102.26%
# Generate the data.frame of results (7-digits precision) and show
# in the console. Use an internal dataset.
x <- ABE(details = TRUE, print = FALSE, data = rds02)
print(x, row.names = FALSE)

# Assuming a NTID and assess BE with narrower limits for one
# of the internal datasets.
ABE(data = rds02, theta1 = 0.90)
# Should result in:
#   BE-limits          :  90.00% ... 111.11%
#   Confidence interval:  97.32% ... 107.46%  pass
#   Point estimate     : 102.26%

Reference Dataset for TR|RT|TT|RR Replicate Designs

Description

Dataset for Balaam’s design obtained by simulations to be evaluated by method.A(), method.B().

Usage

rds27

Format

• Reference Dataset 27 (rds27)
  312 subjects. Balanced (78 subjects in each of the four sequences) and incomplete (T of subject 111 missing in period 2 of sequence RT). No outliers.
  A data frame with 624 observations on the following 5 variables:

  rds27

  subject	a factor with 312 levels: 1, 2, ..., 18
  period	a factor with 2 levels: 1, 2
  sequence	a factor with 4 levels: TR, RT, TT, RR
  treatment	a factor with 2 levels: T, R
  PK	a numeric vector of pharmacokinetic responses acceptable for reference-scaling (generally Cmax)

Details

Note

In software sequences and treatments are ranked in lexical order. Hence, executing str() or summary() will show sequence as "RR", "RT", "TR", "TT" and treatment as "R", "T". In BE – by convention – sequences are ordered with T first. The package follows this convention.

Author(s)

Helmut Schütz (R-code for simulations by Detlew Labes)

Source

Examples

str(rds27)
row <- c(1:2, 157:158, 313:314, 469:470)
rds27[row, ]
summary(rds27[2:5])

Reference Dataset for TRR|RTR (extra-reference) Designs

Description

Dataset simulated to be evaluated by method.A(), method.B().

Usage

rds22

Format

• Reference dataset 22
  Simulated with CV_wT = CV_wR = 45%, CV_bT = CV_bR = 100% GMR 0.90. 42 subjects.
  Balanced (21 subjects in each of the sequences) and complete (no missing data). No outliers.
  A data frame with 126 observations on the following 5 variables:

  rds22

  subject	a factor with 42 levels: 1, 2, ..., 42
  period	a factor with 3 levels: 1, 2, 3
  sequence	a factor with 2 levels: TRR, RTR
  treatment	a factor with 2 levels: T, R
  PK	a numeric vector of pharmacokinetic responses acceptable for reference-scaling (generally Cmax)
  logPK	a numeric vector of the natural logarithms of PK

Details

Note

In software sequences and treatments are ranked in lexical order. Hence, executing str() or summary() will show sequence as "RTR", "TRR" and treatment as "R", "T". In BE – by convention – sequences are ordered with T first. The package follows this convention.
This partial replicate design is also known as the ‘extra-reference design’. Since the Test is not administered in all periods, lacking period effects must be assumed. In the presence of true period effects the treatment comparison will be biased. Hence, this design is not recommended.

Author(s)

Helmut Schütz (R-code for simulations by Detlew Labes)

Source

Examples

str(rds22)
rds22[61:66, ]
summary(rds22[2:5])

Reference Datasets for TRR|RTR|RRT (partial) Replicate Designs

Description

Datasets from the public domain or simulated to be evaluated by method.A(), method.B(), or ABE().

Format

• Reference Dataset 02
  24 subjects.
  Balanced (eight subjects in each of the three sequences) and complete (no missing data). No outliers.
  A data frame with 72 observations on the following 6 variables:

  rds02

  subject	a factor with 24 levels: 1, 2, ..., 24
  period	a factor with 3 levels: 1, 2, 3
  sequence	a factor with 3 levels: TRR, RTR, RRT
  treatment	a factor with 2 levels: T, R
  PK	a numeric vector of pharmacokinetic responses acceptable for reference-scaling (generally Cmax)
  logPK	a numeric vector of the natural logarithms of PK

  In the source evaluated by SAS v9.1 for ABEL. Reported results:

  SAS Proc GLM

  CVwR	11.2%
  PE	102.26% (Method A and B)
  90% CI	97.32% – 107.46% (Method A and B)

• Reference Dataset 04
  Data set of Table II given by Patterson & Jones. 51 subjects.
  Balanced (17 subjects in each of the three sequences) and complete. No outliers.
  A data frame with 153 observations on the following 5 variables:

  rds04

  subject	a factor with 51 levels: 1, 2, ..., 56
  period	a factor with 3 levels: 1, 2, 3
  sequence	a factor with 3 levels: TRR, RTR, RRT
  treatment	a factor with 2 levels: T, R
  PK	a numeric vector of pharmacokinetic responses (here Cmax)

  In the source evaluated by SAS with the FDA’s mixed effects model (termed ‘Method C’ by the EMA; not compatible with the guideline). Reported results:

  SAS Proc MIXED

  CVwR	61%
  PE	137%
  90% CI	119% – 159%

• Reference Dataset 07
  Simulated with CV_wT = CV_wR = 35%, GMR 0.90. 360 subjects.
  Balanced (120 subjects in each of the three sequences) and complete. No outliers.
  A data frame with 1,080 observations on the following 5 variables:

  rds07

  subject	a factor with 360 levels: 1, 2, ..., 360
  period	a factor with 3 levels: 1, 2, 3
  sequence	a factor with 3 levels: TRR, RTR, RRT
  treatment	a factor with 2 levels: T, R
  PK	a numeric vector of pharmacokinetic responses (generally Cmax)

• Reference Dataset 30
  Simulated with heteroscedasticity (CV_wT = 14%, CV_wR = 28%, CV_bT = 28%, CV_bR = 56%), GMR = 0.90. 12 subjects. 14 subjects.
  Imbalanced (six subjects in sequence TRR, five in RTR, and three RRT) and incomplete (two missings in sequences TRR and RTR and three in sequence RRT). Missings / period: 0/1, 0/2, 7/3. No outliers.
  A data frame with 35 observations on the following 5 variables:

  rds30

  subject	a factor with 14 levels: 1, 2, ..., 39
  period	a factor with 3 levels: 1, 2, 3
  sequence	a factor with 3 levels: TRR, RTR, RRT
  treatment	a factor with 2 levels: T, R
  PK	a numeric vector of pharmacokinetic responses (generally Cmax)

Details

Note

In software sequences and treatments are ranked in lexical order. Hence, executing str() or summary() will show sequence as "RRT", "RTR", "TRR" and treatment as "R", "T". In BE – by convention – sequences are ordered with T first. The package follows this convention.

Author(s)

Helmut Schütz (R-code for simulations by Detlew Labes)

Source

References

European Medicines Agency. London, 21 September 2016. Annex I, Annex III.

Patterson SD, Jones B. Viewpoint: observations on scaled average bioequivalence. Pharm Stat. 2012; 11(1): 1–7. doi:10.1002/pst.498

Examples

str(rds02)
row <- c(10:12, 1:3, 16:18)
rds02[row, ]
summary(rds02[2:6])

Reference Dataset for TRR|RTT Replicate Designs

Description

Dataset from the public domain to be evaluated by method.A(), method.B(), or ABE().

Usage

rds10

Format

• Reference Dataset 10
  18 subjects.
  Balanced (nine subjects in both sequences) and complete. No outliers.
  A data frame with 54 observations on the following 5 variables:

  rds10

  subject	a factor with 18 levels: 1, 2, ..., 18
  period	a factor with 3 levels: 1, 2, 3
  sequence	a factor with 2 levels: TRR, RTT
  treatment	a factor with 2 levels: T, R
  PK	a numeric vector of pharmacokinetic responses (here AUC)

Details

Note

In software sequences and treatments are ranked in lexical order. Hence, executing str() or summary() will show sequence as "RTT", "TRR" and treatment as "R", "T". In BE – by convention – sequences are ordered with T first. The package follows this convention.
In analogy to the EMA’s Q&A: Uncertain estimate of CVwR since less than twelve subjects in sequence TRR.

Source

References

Chow SC, Liu JP. Design and Analysis of Bioavailability and Bioequivalence Studies. Boca Raton: CRC Press; 3^rd edition 2009. p275.

Examples

str(rds10)
row <- c(1:3, 28:30)
rds10[row, ]
summary(rds10[2:5])

Reference Datasets for TRRT|RTTR Replicate Designs

Description

Datasets from the public domain to be evaluated by method.A(), method.B(), or ABE().

Format

• Reference Dataset 05
  26 subjects.
  Balanced (13 subjects in both sequences) and complete. No outliers.
  A data frame with 104 observations on the following 5 variables:

  rds05

  subject	a factor with 26 levels: 1, 2, ..., 26
  period	a factor with 4 levels: 1, 2, 3, 4
  sequence	a factor with 2 levels: TRRT, RTTR
  treatment	a factor with 2 levels: T, R
  PK	a numeric vector of pharmacokinetic responses (here Cmax)

  In the source evaluated by SAS with the FDA’s mixed effects model (termed ‘Method C’ by the EMA; not compatible with the guideline). Reported results:

  SAS Proc Mixed

  CVwR	5.47%
  CVwT	6.75%
  PE	107.90%
  90% CI	103.66% – 112.2%

• Reference Dataset 11
  37 subjects.
  Unbalanced (18 subjects in sequence TRRT and 19 subjects in RTTR) and complete. No outliers.
  A data frame with 148 observations on the following 5 variables

  rds11

  subject	a factor with 37 levels: 1, 2, ..., 37
  period	a factor with 4 levels: 1, 2, 3, 4
  sequence	a factor with 2 levels: TRRT, RTTR
  treatment	a factor with 2 levels: T, R
  PK	a numeric vector of pharmacokinetic responses (here Cmax)

  In the source evaluated by SAS with the FDA’s mixed effects model (termed ‘Method C’ by the EMA; not compatible with the guideline). Reported results:

  SAS Proc MIXED

  PE	90.0%
  90% CI	79.6% – 101.7%

• Reference Dataset 16
  38 subjects.
  Unbalanced (18 subjects in sequence TRRT and 20 in RTTR) and complete. No outliers.
  A data frame with 152 observations on the following 5 variables:

  rds16

  subject	a factor with 38 levels: 1, 2, ..., 38
  period	a factor with 4 levels: 1, 2, 3, 4
  sequence	a factor with 2 levels: TRRT, RTTR
  treatment	a factor with 2 levels: T, R
  PK	a numeric vector of pharmacokinetic responses (here Cmax)

Details

Note

In software sequences and treatments are ranked in lexical order. Hence, executing str() or summary() will show sequence as "RTTR", "TRRT" and treatment as "R", "T". In BE – by convention – sequences are ordered with T first. The package follows this convention.

Source

References

Shumaker RC, Metzler CM. The Phenytoin Trial is a Case Study of ‘Individual’ Bioequivalence. Drug Inf J. 1998; 32(4): 1063–72. doi:10.1177/009286159803200426

Hauschke D, Steinijans VW, Pigeot I. Bioequivalence Studies in Drug Development. Chichester: John Wiley; 2007. p216.

U.S. Food and Drug Administration, Center for Drug Evaluation and Research. Bioequivalence Studies. Rockville, 1997. bioequivalence study files (archived 2017-07-23)

Examples

str(rds05)
summary(rds05[2:5])
head(rds11, 8)

Reference Dataset for TRRT|RTTR|TTRR|RRTT Designs

Description

Dataset from the public domain to be evaluated by method.A() and/or method.B().

Format

• Reference Dataset 24
  40 subjects (one completely missing).
  Unbalanced (nine subjects in sequence TRRT and ten in each of the other three) and complete. Two outliers (subject 3 in sequence RTTR and subject 30 in sequence TTRR).
  A data frame with 160 observations on the following 5 variables:

  rds24

  subject	a factor with 40 levels: 1, 2, ..., 932
  period	a factor with 4 levels: 1, 2, 3, 4
  sequence	a factor with 4 levels: TRRT, RTTR, TTRR, RRTT
  treatment	a factor with 2 levels: T, R
  PK	a numeric vector of pharmacokinetic responses acceptable for reference-scaling (here Cmax)

Details

Note

In software sequences and treatments are ranked in lexical order. Hence, executing str() or summary() will show sequence as "RRTT", "RTTR", "TRRT", "TTRR" and treatment as "R", "T". In BE – by convention – sequences are ordered with T first. The package follows this convention.

Source

References

U.S. Food and Drug Administration, Center for Drug Evaluation and Research. Bioequivalence Studies. Rockville, 1997. bioequivalence study files (archived 2017-07-23)

Examples

str(rds24)
row <- c(13:16, 9:12, 1:4, 5:8)
rds24[row, ]
summary(rds24[2:5])

Reference Datasets for TRT|RTR Replicate Designs

Description

Datasets from the public domain and edited to be evaluated by method.A() and/or method.B().

Format

• Reference dataset 03
  Based on rds01. Removed all data of period 4. 77 subjects.
  Unbalanced (39 subjects in sequence TRT and 38 in RTR) and incomplete (six missings in sequence TRT and two in RTR). Missings / period: 0/1, 1/2, 7/3. Two outliers (subjects 45 and 52) in sequence RTR.
  A data frame with 223 observations on the following 6 variables:

  rds03

  subject	a factor with 77 levels: 1, 2, ..., 78
  period	a factor with 3 levels: 1, 2, 3
  sequence	a factor with 2 levels: TRT, RTR
  treatment	a factor with 2 levels: T, R
  PK	a numeric vector of pharmacokinetic responses acceptable for reference-scaling (generally Cmax)

• Reference dataset 17
  Based on rds03. 19 subjects.
  Unbalanced (seven subjects in sequence TRT and twelve in RTR) and incomplete (one missing in sequence TRT). Missings / period: 0/1, 0/2, 1/3. One outlier (subject 18) in sequence RTR.
  A data frame with 56 observations on the following 6 variables:

  rds17

  subject	a factor with 19 levels: 1, 2, ..., 22
  period	a factor with 3 levels: 1, 2, 3
  sequence	a factor with 2 levels: TRT, RTR
  treatment	a factor with 2 levels: T, R
  PK	a numeric vector of pharmacokinetic responses acceptable for reference-scaling (generally Cmax)

Details

Note

In software sequences and treatments are ranked in lexical order. Hence, executing str() or summary() will show sequence as "RTR", "TRT" and treatment as "R", "T". In BE – by convention – sequences are ordered with T first. The package follows this convention.

Author(s)

Helmut Schütz

Source

Examples

head(rds03, 6)
summary(rds03[2:5])

Reference Datasets for TRTR|RTRT Designs

Description

Datasets from the public domain, edited, or obtained by simulations to be evaluated by method.A() and/or method.B().

Format

• Reference dataset 01
  77 subjects.
  Unbalanced (39 subjects in sequence TRTR and 38 in RTRT) and incomplete (seven missings in sequence TRTR and three in sequence RTRT). Missings / period: 0/1, 1/2, 7/3, 2/4. Two outliers (subjects 45 and 52) in sequence RTRT.
  A data frame with 298 observations on the following 6 variables:

  rds01

  subject	a factor with 77 levels: 1, 2, ..., 78
  period	a factor with 4 levels: 1, 2, 3, 4
  sequence	a factor with 2 levels: TRTR, RTRT
  treatment	a factor with 2 levels: T, R
  PK	a numeric vector of pharmacokinetic responses acceptable for reference-scaling (generally Cmax)
  logPK	a numeric vector of the natural logarithms of PK

  In the source evaluated by SAS v9.1 for ABEL. Reported results:

  SAS Proc GLM

  CVwR	47.0%
  PE	115.66% (Method A)
  	115.73% (Method B)
  90% CI	107.11% – 124.89% (Method A)
  	107.17% – 124.97% (Method B)

• Reference dataset 06
  Based on rds01. 77 subjects. Responses of T and R switched.
  Unbalanced (39 subjects in sequence TRTR and 38 in RTRT) and incomplete (seven missings in sequence TRTR and three in sequence RTRT). Missings / period: 0/1, 1/2, 7/3, 2/4. No outliers.
  A data frame with 298 observations on the following 6 variables:

  rds06

  subject	a factor with 77 levels: 1, 2, ..., 78
  period	a factor with 4 levels: 1, 2, 3, 4
  sequence	a factor with 2 levels: TRTR, RTRT
  treatment	a factor with 2 levels: T, R
  PK	a numeric vector of pharmacokinetic responses acceptable for reference-scaling (generally Cmax)

• Reference dataset 08
  Simulated with slight heteroscedasticity (CV_wT = 70%, CV_wR = 80%), CV_bT = CV_bR = 150%, GMR = 0.85. 222 subjects.
  Balanced (222 subjects in both sequences) and complete. No outliers.
  The extreme sample size results from high variability, an assumed true GMR 0.85, and target power 90%.
  A data frame with 888 observations on the following 5 variables:

  rds08

  subject	a factor with 222 levels: 1, 2, ..., 222
  period	a factor with 4 levels: 1, 2, 3, 4
  sequence	a factor with 2 levels: TRTR, RTRT
  treatment	a factor with 2 levels: T, R
  PK	a numeric vector of pharmacokinetic responses acceptable for reference-scaling (generally Cmax)

• Reference dataset 09
  Based on rds08. Wide numeric range (data of last 37 subjects multiplied by 1,000,000). 222 subjects.
  Balanced (222 subjects in both sequences) and complete. No outliers.
  A data frame with 888 observations on the following 5 variables:

  rds09

  subject	a factor with 222 levels: 1, 2, ..., 222
  period	a factor with 4 levels: 1, 2, 3, 4
  sequence	a factor with 2 levels: TRTR, RTRT
  treatment	a factor with 2 levels: T, R
  PK	a numeric vector of pharmacokinetic responses acceptable for reference-scaling (generally Cmax)

• Reference dataset 12
  Simulated with extreme intra- and intersubject variability, GMR = 1.6487. 77 subjects.
  Unbalanced (39 subjects in sequence TRTR and 38 in RTRT) and incomplete (seven missings in sequence TRTR and three in sequence RTRT). Missings / period: 0/1, 1/2, 7/3, 2/4. No outliers.
  A data frame with 298 observations on the following 6 variables:

  rds12

  subject	a factor with 77 levels: 1, 2, ..., 78
  period	a factor with 4 levels: 1, 2, 3, 4
  sequence	a factor with 2 levels: TRTR, RTRT
  treatment	a factor with 2 levels: T, R
  PK	a numeric vector of pharmacokinetic responses acceptable for reference-scaling (generally Cmax)

• Reference dataset 13
  Based on rds08. Highly incomplete (approx. 50% of period 4 data deleted). 222 subjects.
  Balanced (111 subjects in both sequences) and incomplete (56 missings in both sequences). Missings / period: 0/0, 0/0, 0/0, 112/4. No outliers.
  A data frame with 776 observations on the following 5 variables:

  rds13

  subject	a factor with 222 levels: 1, 2, ..., 222
  period	a factor with 4 levels: 1, 2, 3, 4
  sequence	a factor with 2 levels: TRTR, RTRT
  treatment	a factor with 2 levels: T, R
  PK	a numeric vector of pharmacokinetic responses acceptable for reference-scaling (generally Cmax)

• Reference dataset 14
  Simulated with high variability, GMR = 1. Dropouts as a hazard function growing with period. 77 subjects.
  Unbalanced (39 subjects in sequence TRTR and 38 in RTRT) and incomplete (18 missings in sequence TRTR and 17 in sequence RTRT). Missings / period: 0/1, 4/2, 12/3, 19/4. No outliers.
  A data frame with 273 observations on the following 6 variables:

  rds14

  subject	a factor with 77 levels: 1, 2, ..., 78
  period	a factor with 4 levels: 1, 2, 3, 4
  sequence	a factor with 2 levels: TRTR, RTRT
  treatment	a factor with 2 levels: T, R
  PK	a numeric vector of pharmacokinetic responses acceptable for reference-scaling (generally Cmax)

• Reference dataset 15
  Based on ref08. Highly incomplete (approx. 50% of period 4 data coded as missing 'NA'). 222 subjects.
  Balanced (111 subjects in both sequences) and incomplete (56 missings in both sequences). Missings / period: 0/1, 0/2, 0/3, 112/4. No outliers.
  A data frame with 888 observations (112 NA) on the following 5 variables

  rds15

  subject	a factor with 222 levels: 1, 2, ..., 222
  period	a factor with 4 levels: 1, 2, 3, 4
  sequence	a factor with 2 levels: TRTR, RTRT
  treatment	a factor with 2 levels: T, R
  PK	a numeric vector of pharmacokinetic responses acceptable for reference-scaling (generally Cmax)

• Reference dataset 18
  Data set based on rds14. Removed T data of subjects 63–78. 77 subjects.
  Unbalanced (39 subjects in sequence TRTR and 38 in RTRT) and incomplete (32 missings in sequence TRTR and 31 in sequence RTRT). Missings / period: 8/1, 12/2, 18/3, 25/4. No outliers.
  A data frame with 245 observations on the following 6 variables:

  rds18

  subject	a factor with 77 levels: 1, 2, ..., 78
  period	a factor with 4 levels: 1, 2, 3, 4
  sequence	a factor with 2 levels: TRTR, RTRT
  treatment	a factor with 2 levels: T, R
  PK	a numeric vector of pharmacokinetic responses acceptable for reference-scaling (generally Cmax)

• Reference dataset 19
  Data set based on rds18. Removed data of subjects 63–78. 61 subjects.
  Unbalanced (31 subjects in sequence TRTR and 30 in RTRT) and incomplete (14 missings in both sequences). Missings / period: 0/1, 4/2, 9/3, 15/4. Two outliers (subjects 18 and 51 in sequence RTRT).
  A data frame with 216 observations on the following 6 variables:

  rds19

  subject	a factor with 61 levels: 1, 2, ..., 62
  period	a factor with 4 levels: 1, 2, 3, 4
  sequence	a factor with 2 levels: TRTR, RTRT
  treatment	a factor with 2 levels: T, R
  PK	a numeric vector of pharmacokinetic responses acceptable for reference-scaling (generally Cmax)

• Reference dataset 20
  Data set based on rds19. Extreme outlier of R (subject 1) introduced: original value ×100). 61 subjects.
  Unbalanced (31 subjects in sequence TRTR and 30 in RTRT) and incomplete (14 missings in both sequences). Missings / period: 0/1, 4/2, 9/3, 15/4. Two outliers (subjects 1 and 51 in sequence RTRT).
  A data frame with 216 observations on the following 6 variables:

  rds20

  subject	a factor with 61 levels: 1, 2, ..., 62
  period	a factor with 4 levels: 1, 2, 3, 4
  sequence	a factor with 2 levels: TRTR, RTRT
  treatment	a factor with 2 levels: T, R
  PK	a numeric vector of pharmacokinetic responses acceptable for reference-scaling (generally Cmax)

• Reference dataset 21
  Based on ds01. 77 subjects. One extreme result of subjects 45 & 52 set to NA.
  Unbalanced (39 subjects in sequence TRTR and 38 in RTRT) and incomplete (seven missings in sequence TRTR and five in sequence RTRT). Missings / period: 1/1, 1/2, 8/3, 2/4. No outliers.
  A data frame with 298 observations (2 NA) on the following 6 variables:

  rds21

  subject	a factor with 61 levels: 1, 2, ..., 62
  period	a factor with 4 levels: 1, 2, 3, 4
  sequence	a factor with 2 levels: TRTR, RTRT
  treatment	a factor with 2 levels: T, R
  PK	a numeric vector of pharmacokinetic responses acceptable for reference-scaling (generally Cmax)

• Reference dataset 25
  Simulated with heteroscedasticity (CV_wT = 50%, CV_wR = 80%), CV_bT = CV_bR = 130%, GMR = 0.85. 70 subjects.
  Balanced (70 subjects in both sequences) and complete. No outliers.
  A data frame with 280 observations on the following 5 variables:

  rds25

  subject	a factor with 70 levels: 1, 2, ..., 70
  period	a factor with 4 levels: 1, 2, 3, 4
  sequence	a factor with 2 levels: TRTR, RTRT
  treatment	a factor with 2 levels: T, R
  PK	a numeric vector of pharmacokinetic responses acceptable for reference-scaling (generally Cmax)

• Reference dataset 26
  54 subjects.
  Balanced (27 subjects in both sequences) and incomplete (two missings in both sequences). Missings / period: 0/1, 0/2, 2/3, 2/4. One outlier (subject 49) in sequence RTRT.
  A data frame with 216 observations on the following 5 variables:

  rds26

  subject	a factor with 54 levels: 1, 2, ..., 57
  period	a factor with 4 levels: 1, 2, 3, 4
  sequence	a factor with 2 levels: TRTR, RTRT
  treatment	a factor with 2 levels: T, R
  PK	a numeric vector of pharmacokinetic responses acceptable for reference-scaling (here Cmax)

  In the source evaluated by SAS for ABEL. Reported results (Method A):

  SAS Proc GLM

  CVwR	60.25%
  PE	151.3%
  90% CI	133.5% – 171.4%

• Reference dataset 29
  Simulated with heteroscedasticity (CV_wT = 14%, CV_wR = 28%, CV_bT = 28%, CV_bR = 56%), GMR = 0.90. 12 subjects.
  Imbalanced (five subjects in sequence TRTR and seven in sequence RTRT) and incomplete (three missings in sequence TRTR and four in sequence RTRT). Missings / period: 0/1, 1/2, 2/3, 4/4. One outlier (subject 11) in sequence RTRT.
  A data frame with 41 observations on the following 5 variables:

  rds29

  subject	a factor with 12 levels: 1, 2, ..., 20
  period	a factor with 4 levels: 1, 2, 3, 4
  sequence	a factor with 2 levels: TRTR, RTRT
  treatment	a factor with 2 levels: T, R
  PK	a numeric vector of pharmacokinetic responses acceptable for reference-scaling (generally Cmax)

Details

Note

In software sequences and treatments are ranked in lexical order. Hence, executing str() or summary() will show sequence as "RTRT", "TRTR" and treatment as "R", "T". In BE – by convention – sequences are ordered with T first. The package follows this convention.

Author(s)

Helmut Schütz (R-code for simulations by Detlew Labes), Michael Tomashevskiy (simulations in Phoenix NLME)

Source

References

European Medicines Agency. London, 21 September 2016. Annex I, Annex II.

Patterson SD, Jones B. Bioequivalence and Statistics in Clinical Pharmacology. Boca Raton: CRC Press; 2^nd edition 2016. p105–6.

Examples

str(rds01)
summary(rds01[2:6])

Reference Dataset for TRTR|RTRT|TRRT|RTTR Designs

Description

Dataset from the public domain to be evaluated by method.A() and/or method.B().

Format

• Reference Dataset 23
  22 subjects.
  Unbalanced (four subjects in sequence RTRT and six in each of the other three) and complete. Two outliers (subjects 8 and 17) in sequence TRTR.
  A data frame with 88 observations on the following 5 variables:

  rds23

  subject	a factor with 22 levels: 1, 2, ..., 27
  period	a factor with 4 levels: 1, 2, 3, 4
  sequence	a factor with 4 levels: TRTR, RTRT, TRRT, RTTR
  treatment	a factor with 2 levels: T, R
  PK	a numeric vector of pharmacokinetic responses acceptable for reference-scaling (here Cmax)

Details

Note

In software sequences and treatments are ranked in lexical order. Hence, executing str() or summary() will show sequence as "RTRT", "RTTR", "TRRT", "TRTR" and treatment as "R", "T". In BE – by convention – sequences are ordered with T first. The package follows this convention.

Source

References

U.S. Food and Drug Administration, Center for Drug Evaluation and Research. Bioequivalence Studies. Rockville, 1997. bioequivalence study files (archived 2017-07-23)

Examples

str(rds23)
row <- c(25:28, 5:8, 9:12, 1:4)
rds23[row, ]
summary(rds23[2:5])

Reference Datasets for TTRR|RRTT Designs

Description

Dataset obtained by simulations to be evaluated by method.A() and/or method.B().

Format

• Reference Dataset 28
  64 subjects. Balanced (64 subjects in both sequences) and complete. No outliers.
  A data frame with 256 observations on the following 5 variables:

  rds28

  subject	a factor with 64 levels: 1, 2, ..., 64
  period	a factor with 4 levels: 1, 2, 3, 4
  sequence	a factor with 2 levels: TTRR, RRTT
  treatment	a factor with 2 levels: T, R
  PK	a numeric vector of pharmacokinetic responses acceptable for reference-scaling (generally Cmax)

Details

Note

In software sequences and treatments are ranked in lexical order. Hence, executing str() or summary() will show sequence as "RRTT", "TTRR" and treatment as "R", "T". In BE – by convention – sequences are ordered with T first. The package follows this convention.

Author(s)

Helmut Schütz (R-code for simulations by Detlew Labes)

Source

Examples

str(rds28)
summary(rds28[1:5])

Comparative BA-calculation for Average Bioequivalence with Expanding Limits by the EMA's 'Method A'

Description

This function performs the required calculations for the mixed (or aggregate) BE decision via Average Bioequivalence with Expanding Limits (ABEL) based on ANOVA (‘Method A’) as recommended in Annex I.

Usage

method.A(alpha = 0.05, path.in, path.out = tempdir(), file, set = "",
         ext, na = ".", sep = ",", dec = ".", logtrans = TRUE,
         regulator = "EMA", ola = FALSE, print = TRUE, details = FALSE,
         adjust = FALSE, verbose = FALSE, ask = FALSE,
         plot.bxp = FALSE, fence = 2, data = NULL)

Arguments

Details

The model for the estimation of CVwR is
lm(log(PK) ~ sequence + subject%in%sequence + period, data = data[data$treatment == "R", ])
where all effects are fixed.
The model for the treatment comparison is
lm(log(PK) ~ sequence + subject%in%sequence + period + treatment, data = data)
where all effects are fixed.

Tested designs

• 4-period 2-sequence full replicates
  ⁠TRTR | RTRT⁠
⁠TRRT | RTTR⁠
⁠TTRR | RRTT⁠

• 2-period 4-sequence replicate
  ⁠TR | RT | TT | RR ⁠ (Balaam’s design)

• 4-period 4-sequence full replicates
  ⁠TRTR | RTRT | TRRT | RTTR⁠
⁠TRRT | RTTR | TTRR | RRTT⁠

• 3-period 2-sequence full replicates
  ⁠TRT | RTR⁠
⁠TRR | RTT⁠

• 3-period (partial) replicates
  ⁠TRR | RTR | RRT⁠
⁠TRR | RTR ⁠ (extra-reference design)

Data structure

• Columns must have the headers subject, period, sequence, treatment, PK, and/or logPK.
  Any order of columns is acceptable.
  Uppercase and mixed case headers will be internally converted to lowercase headers.

  • subject must be integers or (any combination of) alphanumerics
    ⁠[A-Z, a-z, -, _, #, 0-9]⁠

  • period must be integer numbers.

  • sequence must be contained in the tested designs (numbers or e.g., ⁠ABAB⁠ are not acceptable).

  • The Test treatment must be coded T and the Reference R.

Value

Prints results to a file if argument print = TRUE (default).
If argument print = FALSE, returns a data frame with the elements:

• If reference-scaling is applicable (i.e., CVwR(%) >30%):

  L(%)	lower expanded limit of the acceptance range (AR)
  U(%)	upper expanded limit of the acceptance range (AR)

• If reference-scaling is not applicable (i.e., CVwR(%) ≤30%):

  BE.lo(%)	lower limit of the conventional AR (⁠ 80⁠)
  BE.hi(%)	upper limit of the conventional AR (⁠125⁠)

If ola = TRUE and at least one studentized outlier was detected:

• If reference-scaling is applicable (i.e., CVwR(%) >30):

  L.rec(%)	recalculated lower expanded limit of the AR
  U.rec(%)	recalculated upper expanded limit of the AR

• If reference-scaling is not applicable (i.e., CVwR(%) ≤30):

  BE.rec.lo(%)	lower limit of the conventional AR (⁠ 80⁠)
  BE.rec.hi(%)	upper limit of the conventional AR (⁠125⁠)

Warning

Files may contain a commentary header. If reading from a CSV-file, each line of the commentary header must start with ⁠"# "⁠ (hashmark space = ⁠ASCII 35 ASCII 32⁠). If reading from an Excel-file all lines preceding the column headers are treated as a comment.

Clarification

The ‘ASCII line chart’ in the result file gives the confidence limits with filled black squares and the point estimate as a white rhombus. If a confidence limit exceeds the maximum possible expansion limit, it is shown as a triangle. Expanded limits are given as double vertical lines. Unscaled limits, the GMR restriction, and 100% are given with single vertical lines. The ‘resolution’ is approximatelly 0.5% and therefore, not all symbols might be shown. The CI and PE take presedence over the limits and the expanded limits over unscaled ones.

Disclaimer

Program offered for Use without any Guarantees and Absolutely No Warranty. No Liability is accepted for any Loss and Risk to Public Health Resulting from Use of this R-Code.

Note

The EMA’s model specified as ‘Method B’ in Annex I assumes equal [sic] intra-subject variances of test and reference (like in 2×2×2 trials) – even if proven false in one of the full replicate designs (were both CV_wT and CV_wR can be estimated). Hence, amongst biostatisticians it is called the ‘crippled model’ because the replicative nature of the study is ignored.
The half-width of the CI in log-scale allows a comparison of methods (B vs A) where a higher value might point towards a more conservative decision. In the provided reference datasets – with one exception – the conclusion of BE (based on the mixed CI and GMR criteria) agrees between ‘Method A’ and ‘Method B’. However, for the highly incomplete dataset 14 ‘Method A’ was liberal (passing by ANOVA but failing by the mixed effects model).

Reference-scaling is acceptable for C_max (immediate release products) and C_max,ss, C_τ,ss, and _partialAUC (modified release products). However, quoting the BE guideline:
The applicant should justify that the calculated intra-subject variability is a reliable estimate and that it is not the result of outliers.
Quoting the Q&A on the Revised EMA Bioequivalence Guideline:
... a study could be acceptable if the bioequivalence requirements are met both including the outlier subject (using the scaled average bioequivalence approach and the within-subject CV with this subject) and after exclusion of the outlier (using the within-subject CV without this subject).
An outlier test is not an expectation of the medicines agencies but outliers could be shown by a box plot. This would allow the medicines agencies to compare the data between them.

The EMA’s method of reference-scaling for highly variable drugs / drug products is currently recommended in other jurisdictions as well (e.g., the WHO; ASEAN States, Australia, Belarus, Brazil, Chile, Egypt, the Eurasian Economic Union, the East African Community, New Zealand, the Russian Federation).

In a pilot phase the WHO accepted reference-scaling for AUC (4-period full replicate studies are mandatory in order to assess the variability associated with each product). It was an open issue how this assessment should be done. In Population Bioequivalence (PBE) and Individual Bioequivalence (IBE) the s_wT/s_wR ratio was assessed and similar variability was concluded for a ratio within 0.667–1.500. However, the power of comparing variabilities in a study designed to demonstrate ABE is low. This was one of the reasons why PBE and IBE were not implemented in regulatory practice. An alternative approach is given in the FDA’s draft ANDA guidance. Variabilities are considered comparable if the upper confidence limit of σ_wT/σ_wR is less than or equal to 2.5.
In 2021 the requirement of comparing variabilities was lifted.

Author(s)

Helmut Schütz, Michael Tomashevskiy, Detlew Labes

References

European Medicines Agency, Committee for Medicinal Products for Human Use. Guideline on the Investigation of Bioequivalence. CPMP/EWP/QWP/1401/98 Rev. 1/ Corr **. London. 20 January 2010. online

European Generic and Biosimilar Medicines Association. 3^rd EGA Symposium on Bioequivalence. Questions and Answers on the Revised EMA Bioequivalence Guideline. London. 1 June 2010. online

European Medicines Agency, Committee for Medicinal Products for Human Use. Questions & Answers: positions on specific questions addressed to the Pharmacokinetics Working Party (PKWP). EMA/618604/2008 Rev. 13. London. 19 November 2015. online

European Medicines Agency. Clinical pharmacology and pharmacokinetics: questions and answers. 3.1 Which statistical method for the analysis of a bioequivalence study does the Agency recommend? Annex I. EMA/582648/2016. London. 21 September 2016. online

Executive Board of the Health Ministers’ Council for GCC States. The GCC Guidelines for Bioequivalence. Version 3.0. May 2021. online

Health Canada. Guidance Document. Conduct and Analysis of Comparative Bioavailability Studies. Ottawa. 2018/06/08. online

European Medicines Agency, Committee for Medicinal Products for Human Use. Guideline on the pharmacokinetic and clinical evaluation of modified release dosage forms. EMA/CPMP/EWP/280/96 Corr1. London. 20 November 2014. online

World Health Organization, Prequalification Team: medicines. Guidance Document: Application of reference-scaled criteria for AUC in bioequivalence studies conducted for submission to PQTm. Geneva. 22 November 2018. online

World Health Organization. Application of reference-scaled criteria for AUC in bioequivalence studies conducted for submission to PQT/MED. Geneva. 02 July 2021. online

U.S. Food and Drug Administration, Center for Drug Evaluation and Research. Draft Guidance for Industry. Bioequivalence Studies with Pharmacokinetic Endpoints for Drugs Submitted Under an ANDA. August 2021. download

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

See Also

Examples

# Importing from a CSV-file, using most of the defaults: variable
# separator colon, decimal separator period, no outlier-analyis,
# print to file.
# Note: You must adapt the path-variables. The example reads from
# the data provided by the library. Write-permissions must be granted
# for 'path.out' in order to save the result file. Here the default
# (R's temporary folder) is used. If you don't know where it is,
# type tempdir() in the console.

path.in  <- paste0(find.package("replicateBE"), "/extdata/")
method.A(path.in = path.in, file = "DS", set = "01", ext = "csv")
# Should result in:
#   CVwT               :  35.16%
#   swT                :   0.34138
#   CVwR               :  46.96% (reference-scaling applicable)
#   swR                :   0.44645
#   Expanded limits    :  71.23% ... 140.40% [100exp(±0.760·swR)]
#   swT / swR          :   0.7647 (similar variabilities of T and R)
#   sw-ratio (upper CL):   0.9324 (comparable variabilities of T and R)
#   Confidence interval: 107.11% ... 124.89%  pass
#   Point estimate     : 115.66%              pass
#   Mixed (CI & PE)    :                      pass
#
# Internal reference dataset 01 used and results to R's
# temporary folder. Additional outlier-analyis.
method.A(ola = TRUE, data = rds01)
# Should give the same as above. Additionally:
#   Outlier fence      :  2×IQR of studentized residuals.
#   Recalculation due to presence of 2 outliers (subj. 45|52)
#   CVwR (outl. excl.) :  32.16% (reference-scaling applicable)
#   swR (recalculated) :   0.31374
#   Expanded limits    :  78.79% ... 126.93% [100exp(±0.760·swR)]
#   swT / swR (recalc.):   1.0881 (similar variabilities of T and R)
#   sw-ratio (upper CL):   1.3282 (comparable variabilities of T and R)
#   Confidence interval: pass
#   Point estimate     : pass
#   Mixed (CI & PE)    : pass
# Same dataset. Show information about outliers and the ANOVA-table.
method.A(ola = TRUE, print = FALSE, verbose = TRUE, data = rds01)
# Generate the data.frame of results (full precision) and show it
# in the console
x <- method.A(ola = TRUE, details = TRUE, print = FALSE, data = rds01)
print(x, row.names = FALSE)
#
# Assess the Type I Error and iteratively adjust alpha if necessary.
# Not run: due to timing policy of CRAN for examples

method.A(adjust = TRUE, data = rds01)
# Should give in the result file:
#   Assessment of the empiric Type I Error (TIE); 1,000,000 studies simulated.
#     TIE not > nominal 0.05; consumer risk is controlled.
#
# Same with recalculation based on outliers, iteratively adjust alpha
# if necessary

method.A(ola = TRUE, adjust = TRUE, data = rds01)
# Should give in the result file:
#   Assessment of the empiric Type I Error (TIE) based on original CVwR;
#   1,000,000 studies simulated.
#     TIE not > nominal 0.05; consumer risk is controlled.
#   Assessment of the empiric Type I Error (TIE) based on recalculated CVwR;
#   1,000,000 studies in each of the 8 iterations simulated.
#     TIE for alpha 0.050000         : 0.07018
#     TIE for adjusted alpha 0.033416: 0.05000
#
# Repeat the evaluation with the adjusted alpha.

method.A(alpha = 0.033416, ola = TRUE, adjust = TRUE, data = rds01)
# Should give in the result file:
#   alpha              :   0.033416 (93.3168% CI)
#   Confidence interval: 106.16% ... 126.00%  pass
#   Point estimate     : 115.66%              pass
#   Mixed (CI & PE)    :                      pass
#   Assessment based on recalculated CVwR 32.16%
#   Confidence interval: pass
#   Point estimate     : pass
#   Mixed (CI & PE)    : pass
#   Assessment of the empiric Type I Error (TIE) based on original CVwR;
#   1,000,000 studies simulated.
#     TIE not > nominal 0.05; consumer risk is controlled.
#   Assessment of empiric Type I Error (TIE) based on recalculated CVwR;
#   1,000,000 studies in each of the 8 iterations simulated.
#     TIE for alpha 0.033416         : 0.05000
#     TIE not > nominal 0.05; consumer risk is controlled.

Comparative BA-calculation for Average Bioequivalence with Expanding Limits by the EMA's 'Method B'

Description

This function performs the required calculations for the mixed (or aggregate) BE decision via Average Bioequivalence with Expanding Limits (ABEL) based on a linear mixed effects model with subjects as a random effect (‘Method B’) as specified in Annex I.

Usage

method.B(alpha = 0.05, path.in, path.out = tempdir(), file, set = "",
         ext, na = ".", sep = ",", dec = ".", logtrans = TRUE,
         regulator = "EMA", ola = FALSE, print = TRUE, details = FALSE,
         verbose = FALSE, ask = FALSE, plot.bxp = FALSE, fence = 2,
         data = NULL, option = 2)

Arguments

Details

The model for the estimation of CVwR is
lm(log(PK) ~ sequence + subject%in%sequence + period, data = data[data$treatment == "R", ])
where all effects are fixed.
The model for the treatment comparison is with the default option=2
lme(log(PK) ~ sequence + period + treatment, random = ~1|subject, data = data)
and with option=1, option=3
lmer(log(PK) ~ sequence + period + treatment + (1|subject), data = data)
where sequence, period, and treatment are fixed effects and subject(sequence) is a random effect.

Tested designs

• 4-period 2-sequence full replicates
  ⁠TRTR | RTRT⁠
⁠TRRT | RTTR⁠
⁠TTRR | RRTT⁠

• 2-period 4-sequence replicate
  ⁠TR | RT | TT | RR ⁠ (Balaam’s design)

• 4-period 4-sequence full replicates
  ⁠TRTR | RTRT | TRRT | RTTR⁠
⁠TRRT | RTTR | TTRR | RRTT⁠

• 3-period 2-sequence full replicates
  ⁠TRT | RTR⁠
⁠TRR | RTT⁠

• 3-period (partial) replicates
  ⁠TRR | RTR | RRT⁠
⁠TRR | RTR ⁠ (extra-reference design)

Data structure

• Columns must have the headers subject, period, sequence, treatment, PK, and/or logPK.
  Any order of columns is acceptable.
  Uppercase and mixed case headers will be internally converted to lowercase headers.

  • subject must be integer numbers or (any combination of) alphanumerics
    ⁠[A-Z, a-z, -, _, #, 0-9]⁠

  • period must be integer numbers.

  • sequence must be contained in the tested designs (numbers or e.g., ABAB are not acceptable).

  • The Test treatment must be coded T and the Reference R.

Value

Prints results to a file if argument print = TRUE (default).
If argument print = FALSE, returns a data.frame with the elements:

• If reference-scaling is applicable (i.e., CVwR(%) >30):

  L(%)	lower expanded limit of the acceptance range (AR)
  U(%)	upper expanded limit of the acceptance range (AR)

• If reference-scaling is not applicable (i.e., ≤30):

  BE.lo(%)	lower limit of the conventional AR (⁠ 80⁠)
  BE.hi(%)	upper limit of the conventional AR (⁠125⁠)

If ola = TRUE and at least one studentized outlier was detected:

• If reference-scaling is applicable (i.e., CVwR.rec(%) >30):

  L.rec(%)	recalculated lower expanded limit of the AR
  U.rec(%)	recalculated upper expanded limit of the AR

• If reference-scaling is not applicable (i.e., CVwR.rec(%) ≤30):

  BE.rec.lo(%)	lower limit of the conventional AR (⁠ 80⁠)
  BE.rec.hi(%)	upper limit of the conventional AR (⁠125⁠)

Warning

Files may contain a commentary header. If reading from a CSV-file, each line of the commentary header must start with ⁠"# "⁠ (hashmark space = ⁠ASCII 35 ASCII 32⁠). If reading from an Excel-file all lines preceding the column headers are treated as a comment.

Clarification

The ‘ASCII line chart’ in the result file gives the confidence limits with filled black squares and the point estimate as a white rhombus. If a confidence limit exceeds the maximum possible expansion limit, it is shown as a triangle. Expanded limits are given as double vertical lines. Unscaled limits, the GMR restriction, and 100% are given with single vertical lines. The ‘resolution’ is approximatelly 0.5% and therefore, not all symbols might be shown. The CI and PE take presedence over the limits and the expanded limits over unscaled ones.

Disclaimer

Program offered for Use without any Guarantees and Absolutely No Warranty. No Liability is accepted for any Loss and Risk to Public Health Resulting from Use of this R-Code.

Note

The EMA’s model specified as ‘Method B’ in Annex I assumes equal [sic] intra-subject variances of test and reference (like in 2×2×2 trials) – even if proven false in one of the full replicate designs (were both CV_wT and CV_wR can be estimated). Hence, amongst biostatisticians it is called the “crippled model” because the replicative nature of the study is ignored.
The method for calculating the degrees of freedom is not specified in the SAS code provided by the EMA in Annex I. Hence, the default in PROC MIXED, namely DDFM=CONTAIN is applied.
For incomplete data (i.e., missing periods) Satterthwaite’s approximation of the degrees of freedom (option = 1) or Kenward-Roger (option = 3) might be better choices – if stated as such in the statistical analysis plan.
The half-width of the confidence interval in log-scale allows a comparison of methods (B v.s. A) or options (2 v.s. 1). A higher value might point towards a more conservative decision. Quoting the Q&A-document:
A simple linear mixed model, which assumes identical within-subject variability (Method B), may be acceptable as long as results obtained with the two methods do not lead to different regulatory decisions. However, in borderline cases [...] additional analysis using Method A might be required.
In the provided reference datasets – with one exception – the conclusion of BE (based on the mixed CI and GMR criteria) agrees between ‘Method A’ and ‘Method B’. However, for the highly incomplete dataset 14 ‘Method A’ was liberal (passing by ANOVA but failing by the mixed effects model).

Reference-scaling is acceptable for C_max (immediate release products) and C_max,ss, C_τ,ss, and _partialAUC (modified release products). However, quoting the BE guideline:
The applicant should justify that the calculated intra-subject variability is a reliable estimate and that it is not the result of outliers.
Quoting the Q&A on the Revised EMA Bioequivalence Guideline:
... a study could be acceptable if the bioequivalence requirements are met both including the outlier subject (using the scaled average bioequivalence approach and the within-subject CV with this subject) and after exclusion of the outlier (using the within-subject CV without this subject).
An outlier test is not an expectation of the medicines agencies but outliers could be shown by a box plot. This would allow the medicines agencies to compare the data between them.

The EMA’s method of reference-scaling for highly variable drugs / drug products is currently recommended in other jurisdictions as well (e.g., the WHO; ASEAN States, Australia, Belarus, Brazil, Chile, Egypt, the Eurasian Economic Union, the East African Community, New Zealand, the Russian Federation).

Health Canada’s variant of ABEL (upper cap of scaling ~57.4% limiting the expansion at 67.7–150.0%) is only approximate because a mixed-effects model would be required.

In a pilot phase the WHO accepted reference-scaling for AUC (4-period full replicate studies are mandatory in order to assess the variability associated with each product). It was an open issue how this assessment should be done. In Population Bioequivalence (PBE) and Individual Bioequivalence (IBE) the s_wT/s_wR ratio was assessed and similar variability was concluded for a ratio within 0.667–1.500. However, the power of comparing variabilities in a study designed to demonstrate ABE is low. This was one of the reasons why PBE and IBE were not implemented in regulatory practice. An alternative approach is given in the FDA’s draft ANDA guidance. Variabilities are considered comparable if the upper confidence limit of σ_wT/σ_wR is less than or equal to 2.5.
In 2021 the requirement of comparing variabilities was lifted by the WHO.

Author(s)

Helmut Schütz, Michael Tomashevskiy, Detlew Labes

References

European Medicines Agency, Committee for Medicinal Products for Human Use. Guideline on the Investigation of Bioequivalence. CPMP/EWP/QWP/1401/98 Rev. 1/ Corr **. London. 20 January 2010. online

European Generic and Biosimilar Medicines Association. 3^rd EGA Symposium on Bioequivalence. Questions and Answers on the Revised EMA Bioequivalence Guideline. London. 1 June 2010. online

European Medicines Agency, Committee for Medicinal Products for Human Use. Questions & Answers: positions on specific questions addressed to the Pharmacokinetics Working Party (PKWP). EMA/618604/2008 Rev. 13. London. 19 November 2015. online

European Medicines Agency. Clinical pharmacology and pharmacokinetics: questions and answers. 3.1 Which statistical method for the analysis of a bioequivalence study does the Agency recommend? Annex I. EMA/582648/2016. London. 21 September 2016. online

Executive Board of the Health Ministers’ Council for GCC States. The GCC Guidelines for Bioequivalence. Version 3.0. May 2021. online

Health Canada. Guidance Document. Conduct and Analysis of Comparative Bioavailability Studies. Ottawa. 2018/06/08. online

European Medicines Agency, Committee for Medicinal Products for Human Use. Guideline on the pharmacokinetic and clinical evaluation of modified release dosage forms. EMA/CPMP/EWP/280/96 Corr1. London. 20 November 2014. online

World Health Organization, Prequalification Team: medicines. Guidance Document: Application of reference-scaled criteria for AUC in bioequivalence studies conducted for submission to PQTm. Geneva. 22 November 2018. online

World Health Organization. Application of reference-scaled criteria for AUC in bioequivalence studies conducted for submission to PQT/MED. Geneva. 02 July 2021. online

U.S. Food and Drug Administration, Center for Drug Evaluation and Research. Draft Guidance for Industry. Bioequivalence Studies with Pharmacokinetic Endpoints for Drugs Submitted Under an ANDA. August 2021. download

See Also

Examples

# Importing from a CSV-file, using most of the defaults: variable
# separator colon, decimal separator period, no outlier-analyis,
# print to file.
# Note: You must adapt the path-variables. The example reads from
# the data provided by the library. Write-permissions must be granted
# for 'path.out' in order to save the result file. Here the default
# (R's temporary folder) is used. If you don't know where it is,
# type tempdir() in the console.
path.in <- paste0(find.package("replicateBE"), "/extdata/")
method.B(path.in = path.in, file = "DS", set = "01", ext = "csv")
# Should result in:
#   CVwT               :  35.16%
#   swT                :   0.34138
#   CVwR               :  46.96% (reference-scaling applicable)
#   swR                :   0.44645
#   Expanded limits    :  71.23% ... 140.40% [100exp(±0.760·swR)]
#   swT / swR          :   0.7647 (similar variabilities of T and R)
#   sw-ratio (upper CL):   0.9324 (comparable variabilities of T and R)
#   Confidence interval: 107.17% ... 124.97%  pass
#   Point estimate     : 115.73%              pass
#   Mixed (CI & PE)    :                      pass
#
# Internal reference dataset 01 used and results to R's temporary
# folder. Additional outlier-analyis and box plot saved as PNG.
method.B(ola = TRUE, plot.bxp = TRUE, data = rds01)
# Should give the same as above. Additionally:
#   Recalculation due to presence of 2 outliers (subj. 45|52)
#   CVwR (outl. excl.) :  32.16% (reference-scaling applicable)
#   swR  (recalc.)     :   0.31374
#   Expanded limits    :  78.79% ... 126.93% [100exp(±0.760·swR)]
#   swT / swR (recalc.):   1.0881 (similar variabilities of T and R)
#   sw-ratio (upper CL):   1.3282 (comparable variabilities of T and R)
#   Confidence interval: pass
#   Point estimate     : pass
#   Mixed (CI & PE)    : pass
#
# Same dataset. Show information about outliers and the model-table.
method.B(ola = TRUE, print = FALSE, verbose = TRUE, data = rds01)
# data.frame of results (full precision) shown in the console.
x <- method.B(ola = TRUE, print = FALSE, details = TRUE, data = rds01)
print(x, row.names = FALSE)
# Compare Method B with Method A for all reference datasets.

ds <- substr(grep("rds", unname(unlist(data(package = "replicateBE"))),
                  value = TRUE), start = 1, stop = 5)
for (i in seq_along(ds)) {
  A <- method.A(print=FALSE, details=TRUE, data=eval(parse(text=ds[i])))$BE
  B <- method.B(print=FALSE, details=TRUE, data=eval(parse(text=ds[i])))$BE
  r <- paste0("A ", A, ", B ", B, " - ")
  cat(paste0(ds[i], ":"), r)
  if (A == B) {
    cat("Methods A and B agree.\n")
  } else {
    if (A == "fail" & B == "pass") {
      cat("Method A is conservative.\n")
    } else {
      cat("Method B is conservative.\n")
    }
  }
}
# should give
#   rds01: A pass, B pass - Methods A and B agree.
#   ...
#   rds14: A pass, B fail - Method B is conservative.
#   ...

# Health Canada: Only the PE of Cmax has to lie within 80.0-125.0%
# (i.e., no CI is required). With alpha = 0.5 the CI is practically
# supressed (zero width) and ignored in the assessment.
x    <- method.B(alpha = 0.5, regulator = "HC", option = 1,
                 data = rds03, print = FALSE, details = TRUE)[19:20]
x[1] <- round(x[1], 1) # only one decimal place for HC
print(x, row.names = FALSE)
# Should result in:
# PE(%)  GMR
# 124.5 pass

Reference Datasets

Description

Datasets of replicate designs from the public domain, edited, or obtained by simulations to be evaluated by method.A(), method.B(), or ABE().

Details

In full replicate designs both R and T are administered twice (in 3-period designs to ½ of the subjects).
Balaam’s design is a mixture of a conventional crossover (½ of the subjects) and a replicate design (¼ of the subjects receive either R or T twice).
In partial replicate designs only R is administered twice.

Author(s)

Helmut Schütz (R-code for simulations by Detlew Labes), Michael Tomashevskiy (simulations in Phoenix NLME)

Source

References

European Medicines Agency. London, 21 September 2016. Annex II, Annex III.

Patterson SD, Jones B. Viewpoint: observations on scaled average bioequivalence. Pharm Stat. 2012; 11(1): 1–7. doi:10.1002/pst.498

Shumaker RC, Metzler CM. The Phenytoin Trial is a Case Study of ‘Individual’ Bioequivalence. Drug Inf J. 1998; 32(4): 1063–72. doi:10.1177/009286159803200426

Chow SC, Liu JP. Design and Analysis of Bioavailability and Bioequivalence Studies. Boca Raton: CRC Press; 3^rd edition 2009. p275.

Hauschke D, Steinijans VW, Pigeot I. Bioequivalence Studies in Drug Development. Chichester: John Wiley; 2007. p216.

Patterson SD, Jones B. Bioequivalence and Statistics in Clinical Pharmacology. Boca Raton: CRC Press; 2^nd edition 2016. p105–6.

U.S. Food and Drug Administration, Center for Drug Evaluation and Research. Bioequivalence Studies. Rockville, 1997. bioequivalence study files (archived 2017-07-23)

See Also

4-period full replicates
TRTR.RTRT, TRRT.RTTR, TTRR.RRTT, TRTR.RTRT.TRRT.RTTR, TRRT.RTTR.TTRR.RRTT
2-period replicate (Balaam’s design)
TR.RT.TT.RR
3-period full replicates
TRT.RTR, TRR.RTT
3-period partial replicates
TRR.RTR.RRT, TRR.RTR

Examples

# show structure of all data sets
ds <- substr(grep("rds", unname(unlist(data(package = "replicateBE"))),
                  value = TRUE), start = 1, stop = 5)
for (i in seq_along(ds)) {
  cat(ds[i], "\n")
  str(eval(parse(text = ds[i])))
}