replicateBE

• Comparative BA-calculation for the EMA’s Average Bioequivalence with Expanding Limits (ABEL)
• Introduction
  • Methods
    • Estimation of CV_wR (and CV_wT in full replicate designs)
    • Method A
    • Method B
    • Average Bioequivalence
  • Tested designs
    • Four period (full) replicates
    • Three period (full) replicates
    • Two period (full) replicate
    • Three period (partial) replicates
  • Cross-validation
• Examples
• Installation
• Disclaimer

# Version 1.0.15.9999 built 2020-10-12 with R 4.0.3 
# (development version not on CRAN).

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

Introduction

The library provides data sets (internal .rda and in CSV-format in /extdata/) which support users in a black-box performance qualification (PQ) of their software installations. Users can perform analysis of their own data imported from CSV- and Excel-files. The methods given by the EMA in Annex I for reference-scaling according to the EMA’s Guideline on the Investigation of Bioequivalence 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.
In full replicate designs the variability of test and reference treatments can be assessed by s_wT/s_wR and the upper confidence limit of σ_wT/σ_wR (required for the WHO’s approach for reference-scaling of AUC).

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Methods

Estimation of CV_wR (and CV_wT 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., ANOVA). Estimated by the function lm() of library 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 (i.e., ANOVA). Estimated by the function lm() of library 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

1. Estimated by the function lme() of library nlme. 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)

1. Estimated by the function lmer() of library lmerTest. Employs Satterthwaite’s approximation of the degrees of freedom method.B(..., option = 1) 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)

1. Estimated by the function lmer() of library lmerTest. Employs the Kenward-Roger approximation method.B(..., option = 3) 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 Sattertwaite’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) or wider limits (75.00 – 133.33%) for C_max according to the guidelines of the Gulf Cooperation Council (Bahrain, Kuwait, Oman, Qatar, Saudi Arabia, United Arab Emirates) and South Africa 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 (9.4), Phoenix WinNonlin (6.4 – 8.1), STATISTICA (13), SPSS (22.0), Stata (15.0), and JMP (10.0.2).

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Examples

• Evaluation of the internal reference dataset 01 of Annex II by Method A.

library(replicateBE) # attach the library
res <- method.A(verbose = TRUE, details = TRUE, print = FALSE,
                data = rds01)
# 
# Data set DS01: Method A by lm() 
# ------------------------------- 
# Analysis of Variance Table
# 
# Response: log(PK)
#                   Df   Sum Sq  Mean Sq  F value     Pr(>F)
# sequence           1   0.0077 0.007652  0.04783  0.8270958
# 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, 15:19)           # extract relevant columns
tmp  <- round(res[cols], 2)    # 2 decimal places acc. to the GL
tmp  <- cbind(tmp, res[20:22]) # pass|fail
print(tmp, row.names = FALSE)
#  CVwR(%)  L(%)  U(%) CL.lo(%) CL.hi(%)  PE(%)   CI  GMR   BE
#    46.96 71.23 140.4   107.11   124.89 115.66 pass pass pass

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• Same dataset evaluated by Method B, Kenward-Roger approximation of degrees of freedom. Outlier assessment, recalculation of CV_wR 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(25, 28:29, 17:19)    # extract relevant columns
tmp  <- round(res[cols], 2)    # 2 decimal places acc. to the GL
tmp  <- cbind(tmp, res[30:32]) # pass|fail
print(tmp, row.names = FALSE)
#  CVwR.rec(%) L.rec(%) U.rec(%) CL.lo(%) CL.hi(%)  PE(%) CI.rec GMR.rec BE.rec
#        32.16    78.79   126.93   107.17   124.97 115.73   pass    pass   pass

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• Evaluation of the internal reference dataset 05 of Shumaker and Metzler by ABE, tighter limits for the narrow therapeutic index drug phenytoin.

res <- ABE(verbose = TRUE, theta1 = 0.90, details = TRUE,
           print = FALSE, data = rds05)
# 
# Data set DS05: ABE by lm() 
# -------------------------- 
# Analysis of Variance Table
# 
# Response: log(PK)
#                  Df   Sum Sq   Mean Sq  F value     Pr(>F)
# sequence          1 0.092438 0.0924383  6.81025  0.0109629
# period            3 0.069183 0.0230609  1.69898  0.1746008
# treatment         1 0.148552 0.1485523 10.94435  0.0014517
# sequence:subject 24 2.526550 0.1052729  7.75581 4.0383e-12
# Residuals        74 1.004433 0.0135734                    
# 
# treatment T – R:
#   Estimate Std. Error    t value   Pr(>|t|) 
# 0.07558800 0.02284850 3.30822000 0.00145167 
# 74 Degrees of Freedom
cols <- c(13:17)            # extract relevant columns
tmp  <- round(res[cols], 2) # 2 decimal places acc. to the GL
tmp  <- cbind(tmp, res[18]) # pass|fail
print(tmp, row.names=FALSE)
#  BE.lo(%) BE.hi(%) CL.lo(%) CL.hi(%)  PE(%)   BE
#        90   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 ≥1.3.3, 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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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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