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Comparative BA-calculation for the EMA's Average Bioequivalence with Expanding Limits (ABEL)

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replicateBE

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Version 1.1.3.9000 built 2022-05-03 with R 4.2.0 (development version not on CRAN).

Comparative BA-calculation for the EMA’s Average Bioequivalence with Expanding Limits (ABEL)

Introduction

The package provides data sets (internal .rda and in CSV-format in /extdata/) supporting users in a black-box performance qualification (PQ) of their software installations. Users can analyze own data imported from CSV- and Excel-files (in xlsx or the legacy xls format). The methods given by the EMA for reference-scaling of HVD(P)s, i.e., Average Bioequivalence with Expanding Limits (ABEL)1,2 are implemented.
Potential influence of outliers on the variability of the reference can be assessed by box plots of studentized and standardized residuals as suggested at a joint EGA/EMA workshop.3
Health Canada’s approach4 requiring a mixed-effects model is approximated by intra-subject contrasts.
Direct widening of the acceptance range as recommended by the Gulf Cooperation Council5 (Bahrain, Kuwait, Oman, Qatar, Saudi Arabia, United Arab Emirates) is provided as well.
In full replicate designs the variability of test and reference treatments can be assessed by swT/swR and the upper confidence limit of σwT/σwR. This was required in a pilot phase by the WHO but lifted in 2021; reference-scaling of AUC is acceptable if the protocol is submitted to the PQT/MED.6

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Methods

Estimation of CVwR (and CVwT in full replicate designs)

Called internally by functions method.A() and method.B(). A linear model of log-transformed pharmacokinetic (PK) responses and effects
    sequence, subject(sequence), period
where all effects are fixed (i.e., by an ANOVA). Estimated by the function lm() of package stats.

modCVwR <- lm(log(PK) ~ sequence + subject %in% sequence + period,
                        data = data[data$treatment == "R", ])
modCVwT <- lm(log(PK) ~ sequence + subject %in% sequence + period,
                        data = data[data$treatment == "T", ])

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Method A

Called by function method.A(). A linear model of log-transformed PK responses and effects
    sequence, subject(sequence), period, treatment
where all effects are fixed (e.g., by an ANOVA). Estimated by the function lm() of package stats.

modA <- lm(log(PK) ~ sequence + subject %in% sequence + period + treatment,
                     data = data)

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Method B

Called by function method.B(). A linear model of log-transformed PK responses and effects
    sequence, subject(sequence), period, treatment
where subject(sequence) is a random effect and all others are fixed.
Three options are provided:

  • Estimation by the function lmer() of package lmerTest. method.B(..., option = 1) employs Satterthwaite’s approximation of the degrees of freedom equivalent to SAS’ DDFM=SATTERTHWAITE, Phoenix WinNonlin’s Degrees of Freedom Satterthwaite, and Stata’s dfm=Satterthwaite. Note that this is the only available approximation in SPSS.
modB <- lmer(log(PK) ~ sequence + period + treatment + (1|subject),
                       data = data)
  • Estimation by the function lme() of package nlme. method.B(..., option = 2) employs degrees of freedom equivalent to SAS’ DDFM=CONTAIN, Phoenix WinNonlin’s Degrees of Freedom Residual, STATISTICA’s GLM containment, and Stata’s dfm=anova. Implicitly preferred according to the EMA’s Q&A document and hence, the default of the function.
modB <- lme(log(PK) ~ sequence +  period + treatment, random = ~1|subject,
                      data = data)
  • Estimation by the function lmer() of package lmerTest. method.B(..., option = 3) employs the Kenward-Roger approximation equivalent to Stata’s dfm=Kenward Roger (EIM) and SAS’ DDFM=KENWARDROGER(FIRSTORDER) i.e., based on the expected information matrix. Note that SAS with DDFM=KENWARDROGER and JMP calculate Satterthwaite’s [sic] degrees of freedom and apply the Kackar-Harville correction, i.e., based on the observed information matrix.
modB <- lmer(log(PK) ~ sequence + period + treatment + (1|subject),
                       data = data)

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Average Bioequivalence

Called by function ABE(). The model is identical to Method A. Conventional BE limits (80.00 – 125.00%) are employed by default. Tighter limits (90.00 – 111.11%) for narrow therapeutic index drugs (EMA and others) or wider limits (75.00 – 133.33%) for Cmax according to the guideline of South Africa7 can be specified.

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Tested designs

Four period (full) replicates

TRTR | RTRT
TRRT | RTTR
TTRR | RRTT
TRTR | RTRT | TRRT | RTTR
TRRT | RTTR | TTRR | RRTT

Three period (full) replicates

TRT | RTR
TRR | RTT

Two period (full) replicate

TR | RT | TT | RR (Balaam’s design; not recommended due to poor power characteristics)

Three period (partial) replicates

TRR | RTR | RRT
TRR | RTR (Extra-reference design; biased in the presence of period effects, not recommended)

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Cross-validation

Details about the reference datasets:

help("data", package = "replicateBE")
?replicateBE::data

Results of the 30 reference datasets agree with ones obtained in SAS (v9.4), Phoenix WinNonlin (v6.4 – v8.3.4.295), STATISTICA (v13), SPSS (v22.0), Stata (v15.0), and JMP (v10.0.2).8

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Examples

  • Evaluation of the internal reference dataset 019 by Method A.
library(replicateBE) # attach the package
res <- method.A(verbose = TRUE, details = TRUE,
                print = FALSE, data = rds01)
# 
# Data set DS01: Method A by lm() 
# ─────────────────────────────────── 
# Type III Analysis of Variance Table
# 
# Response: log(PK)
#                   Df   Sum Sq  Mean Sq  F value     Pr(>F)
# sequence           1   0.0077 0.007652  0.00268  0.9588496
# period             3   0.6984 0.232784  1.45494  0.2278285
# treatment          1   1.7681 1.768098 11.05095  0.0010405
# sequence:subject  75 214.1296 2.855061 17.84467 < 2.22e-16
# Residuals        217  34.7190 0.159995                    
# 
# treatment T – R:
#   Estimate Std. Error    t value   Pr(>|t|) 
# 0.14547400 0.04650870 3.12788000 0.00200215 
# 217 Degrees of Freedom
cols <- c(12, 17:21)           # extract relevant columns
# cosmetics: 2 decimal places acc. to the GL
tmp  <- data.frame(as.list(sprintf("%.2f", res[cols])))
names(tmp) <- names(res)[cols]
tmp  <- cbind(tmp, res[22:24]) # pass|fail
print(tmp, row.names = FALSE)
#  CVwR(%)  L(%)   U(%) CL.lo(%) CL.hi(%)  PE(%)   CI  GMR   BE
#    46.96 71.23 140.40   107.11   124.89 115.66 pass pass pass

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  • The same dataset evaluated by Method B, Satterthwaite approximation of degrees of freedom.
res  <- method.B(option = 1, verbose = TRUE, details = TRUE,
                 print = FALSE, data = rds01)
# 
# Data set DS01: Method B (option = 1) by lmer() 
# ────────────────────────────────────────────── 
# Response: log(PK)
# Type III Analysis of Variance Table with Satterthwaite's method
#             Sum Sq  Mean Sq NumDF    DenDF F value    Pr(>F)
# sequence  0.001917 0.001917     1  74.7208 0.01198 0.9131536
# period    0.398078 0.132693     3 217.1188 0.82881 0.4792840
# treatment 1.579332 1.579332     1 216.9386 9.86464 0.0019197
# 
# treatment T – R:
#   Estimate Std. Error    t value   Pr(>|t|) 
#  0.1460900  0.0465130  3.1408000  0.0019197 
# 216.939 Degrees of Freedom (equivalent to SAS’ DDFM=SATTERTHWAITE)
cols <- c(12, 17:21)
tmp  <- data.frame(as.list(sprintf("%.2f", res[cols])))
names(tmp) <- names(res)[cols]
tmp  <- cbind(tmp, res[22:24])
print(tmp, row.names = FALSE)
#  CVwR(%)  L(%)   U(%) CL.lo(%) CL.hi(%)  PE(%)   CI  GMR   BE
#    46.96 71.23 140.40   107.17   124.97 115.73 pass pass pass

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  • The same dataset evaluated by Method B, Kenward-Roger approximation of degrees of freedom. Outlier assessment, recalculation of CVwR after exclusion of outliers, new expanded limits.
res  <- method.B(option = 3, ola = TRUE, verbose = TRUE,
                 details = TRUE, print = FALSE, data = rds01)

# 
# Outlier analysis
#  (externally) studentized residuals
#  Limits (2×IQR whiskers): -1.717435, 1.877877
#  Outliers:
#  subject sequence  stud.res
#       45     RTRT -6.656940
#       52     RTRT  3.453122
# 
#  standarized (internally studentized) residuals
#  Limits (2×IQR whiskers): -1.69433, 1.845333
#  Outliers:
#  subject sequence stand.res
#       45     RTRT -5.246293
#       52     RTRT  3.214663
# 
# Data set DS01: Method B (option = 3) by lmer() 
# ────────────────────────────────────────────── 
# Response: log(PK)
# Type III Analysis of Variance Table with Kenward-Roger's method
#             Sum Sq  Mean Sq NumDF    DenDF F value    Pr(>F)
# sequence  0.001917 0.001917     1  74.9899 0.01198 0.9131528
# period    0.398065 0.132688     3 217.3875 0.82878 0.4792976
# treatment 1.579280 1.579280     1 217.2079 9.86432 0.0019197
# 
# treatment T – R:
#   Estimate Std. Error    t value   Pr(>|t|) 
#  0.1460900  0.0465140  3.1408000  0.0019197 
# 217.208 Degrees of Freedom (equivalent to Stata’s dfm=Kenward Roger EIM)
cols <- c(27, 31:32, 19:21)
tmp  <- data.frame(as.list(sprintf("%.2f", res[cols])))
names(tmp) <- names(res)[cols]
tmp  <- cbind(tmp, res[22:24])
print(tmp, row.names = FALSE)
#  CVwR.rec(%) L.rec(%) U.rec(%) CL.lo(%) CL.hi(%)  PE(%)   CI  GMR   BE
#        32.16    78.79   126.93   107.17   124.97 115.73 pass pass pass

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  • The same dataset evaluated according to the conditions of the GCC (if CVwR > 30% widened limits 75.00 – 133.33%, GMR-constraint 80.00 – 125.00%).
res <- method.A(regulator = "GCC", details = TRUE,
                print = FALSE, data = rds01)
cols <- c(12, 17:21)
tmp  <- data.frame(as.list(sprintf("%.2f", res[cols])))
names(tmp) <- names(res)[cols]
tmp  <- cbind(tmp, res[22:24])
print(tmp, row.names = FALSE)
#  CVwR(%)  L(%)   U(%) CL.lo(%) CL.hi(%)  PE(%)   CI  GMR   BE
#    46.96 75.00 133.33   107.11   124.89 115.66 pass pass pass

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  • The same dataset evaluated according to the conditions of South Africa (if CVwR > 30% fixed limits 75.00 – 133.33%).
res <- ABE(theta1 = 0.75, details = TRUE,
           print = FALSE, data = rds01)
tmp <- data.frame(as.list(sprintf("%.2f", res[12:17])))
names(tmp) <- names(res)[12:17]
tmp <- cbind(tmp, res[18])
print(tmp, row.names = FALSE)
#  CVwR(%) BE.lo(%) BE.hi(%) CL.lo(%) CL.hi(%)  PE(%)   BE
#    46.96    75.00   133.33   107.11   124.89 115.66 pass

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  • Evaluation of the internal reference dataset 05.10 Tighter limits for the NTID phenytoin.
res <- ABE(theta1 = 0.90, details = TRUE,
           print = FALSE, data = rds05)
cols <- c(13:17)
tmp  <- data.frame(as.list(sprintf("%.2f", res[cols])))
names(tmp) <- names(res)[cols]
tmp  <- cbind(tmp, res[18])
print(tmp, row.names = FALSE)
#  BE.lo(%) BE.hi(%) CL.lo(%) CL.hi(%)  PE(%)   BE
#     90.00   111.11   103.82   112.04 107.85 fail

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Installation

The package requires R ≥3.5.0. However, for the Kenward-Roger approximation method.B(..., option = 3) R ≥3.6.0 is required.

  • Install the released version from CRAN:
install.packages("replicateBE", repos = "https://cloud.r-project.org/")
  • To use the development version, please install the released version from CRAN first to get its dependencies right (readxl ≥1.0.0, PowerTOST ≥1.5.3, lmerTest, nlme, pbkrtest).

    You need tools for building R packages from sources on your machine. For Windows users:

    • Download Rtools from CRAN and follow the suggestions of the installer.
    • Install devtools and build the development version by:
install.packages("devtools", repos = "https://cloud.r-project.org/")
devtools::install_github("Helmut01/replicateBE")

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Session Information

Inspect this information for reproducibility. Of particular importance are the versions of R and the packages used to create this workflow. It is considered good practice to record this information with every analysis.

options(width = 66)
print(sessionInfo(), locale = FALSE)
# R version 4.2.0 (2022-04-22 ucrt)
# Platform: x86_64-w64-mingw32/x64 (64-bit)
# Running under: Windows 10 x64 (build 22000)
# 
# Matrix products: default
# 
# attached base packages:
# [1] stats     graphics  grDevices utils     datasets  methods  
# [7] base     
# 
# other attached packages:
# [1] replicateBE_1.1.3.9000
# 
# loaded via a namespace (and not attached):
#  [1] tidyselect_1.1.2    xfun_0.30           purrr_0.3.4        
#  [4] splines_4.2.0       lmerTest_3.1-3      lattice_0.20-45    
#  [7] colorspace_2.0-3    vctrs_0.4.1         generics_0.1.2     
# [10] htmltools_0.5.2     yaml_2.3.5          utf8_1.2.2         
# [13] rlang_1.0.2         pillar_1.7.0        nloptr_2.0.0       
# [16] glue_1.6.2          PowerTOST_1.5-4     readxl_1.4.0       
# [19] lifecycle_1.0.1     stringr_1.4.0       munsell_0.5.0      
# [22] gtable_0.3.0        cellranger_1.1.0    mvtnorm_1.1-3      
# [25] evaluate_0.15       knitr_1.39          fastmap_1.1.0      
# [28] parallel_4.2.0      pbkrtest_0.5.1      fansi_1.0.3        
# [31] highr_0.9           broom_0.8.0         Rcpp_1.0.8.3       
# [34] backports_1.4.1     scales_1.2.0        lme4_1.1-29        
# [37] TeachingDemos_2.12  ggplot2_3.3.5       digest_0.6.29      
# [40] stringi_1.7.6       dplyr_1.0.8         numDeriv_2016.8-1.1
# [43] grid_4.2.0          cli_3.3.0           tools_4.2.0        
# [46] magrittr_2.0.3      tibble_3.1.6        tidyr_1.2.0        
# [49] crayon_1.5.1        pkgconfig_2.0.3     MASS_7.3-57        
# [52] ellipsis_0.3.2      Matrix_1.4-1        minqa_1.2.4        
# [55] rmarkdown_2.14      rstudioapi_0.13     cubature_2.0.4.4   
# [58] R6_2.5.1            boot_1.3-28         nlme_3.1-157       
# [61] compiler_4.2.0

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Contributors

Helmut Schütz (author) ORCID iD
Michael Tomashevskiy (contributor)
Detlew Labes (contributor) ORCID iD

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Disclaimer

Package 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.

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  1. EMA. EMA/582648/2016. Annex I. London. 21 September 2016. Online
  2. EMA, CHMP. CPMP/EWP/QWP/1401/98 Rev. 1/ Corr **. London. 20 January 2010. Online.
  3. EGA. Revised EMA Bioequivalence Guideline. Questions & Answers. London. June 2010. Online
  4. Health Canada. Guidance Document. Conduct and Analysis of Comparative Bioavailability Studies. Ottawa. 2018/06/08. Online.
  5. Executive Board of the Health Ministers’ Council for GCC States. The GCC Guidelines for Bioequivalence. Version 3.0. May 2021. Online.
  6. WHO. Application of reference-scaled criteria for AUC in bioequivalence studies conducted for submission to PQT/MED. Geneva. 02 July 2021. Online.
  7. MCC. Registration of Medicines. Biostudies. Pretoria. June 2015. Online.
  8. Schütz H, Tomashevskiy M, Labes D, Shitova A, González-de la Parra M, Fuglsang A. Reference Datasets for Studies in a Replicate Design Intended for Average Bioequivalence with Expanding Limits. AAPS J. 2020; 22(2): Article 44. doi:10.1208/s12248-020-0427-6.
  9. EMA. EMA/582648/2016. Annex II. London. 21  September 2016. Online.
10. 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.

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Comparative BA-calculation for the EMA's Average Bioequivalence with Expanding Limits (ABEL)

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