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

Version 1.0.15 built 2020-07-24 with R 4.0.2.

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}*/

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", ])
```

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`

.

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

- 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.

- 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.

- 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.

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.

`TRTR | RTRT`

`TRRT | RTTR`

`TTRR | RRTT`

`TRTR | RTRT | TRRT | RTTR`

(confounded effects, *not recommended*)

`TRRT | RTTR | TTRR | RRTT`

(confounded effects, *not recommended*)

`TRT | RTR`

`TRR | RTT`

`TR | RT | TT | RR`

(Balaam’s design; *not recommended* due to poor power characteristics)

`TRR | RTR | RRT`

`TRR | RTR`

(Extra-reference design; biased in the presence of period effects, *not recommended*)

Details about the reference datasets:

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).

- 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 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
```

- Same dataset evaluated by Method B, Kenward-Roger approximation of degrees of freedom. Outlier assessment, recalculation of
*CV*after exclusion of outliers, new expanded limits._{wR}

```
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 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
```

- 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 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
```

The package requires R ≥ 3.5.0; for the Kenward-Roger approximation `method.B(..., option = 3)`

R ≥ 3.6.0 is required.

- Install the released version from CRAN:

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")
```

*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.*