Defaults

Parameter	Argument	Purpose	Default
$\small{\alpha}$	`alpha`	Nominal level of the test	`0.05`
$\small{\pi}$	`targetpower`	Minimum desiredpower	`0.80`
$\small{\theta_0}$	`theta0`	‘True’ or assumed T/R ratio	`0.90`
$\small{\theta_1}$	`theta1`	Lower BE limit and PE constraint	`0.80`
$\small{\theta_2}$	`theta2`	Upper BE limit and PE constraint	`1.25`
CV	`CV`	CV	none
design	`design`	Planned replicate design	`"2x3x3"`
regulator	`regulator`	‘target’ jurisdiction	`"EMA"`
nsims	`nsims`	Number of simulations	see below
nstart	`nstart`	Start if a previous run failed	none
imax	`imax`	Maximum number of iterations	`100`
print	`print`	Show information in the console?	`TRUE`
details	`details`	Show details of the sample size search?	`FALSE`
setseed	`setseed`	Issue a fixed seed of the random number generator?	`TRUE`

Argumentstargetpower,theta0,theta1,theta2, andCV have to begiven as fractions, not in percent.
TheCV is thewithin (intra-) subject coefficient ofvariation. If one value is given, homoscedasticity (equal variances) isassumed and therefore,CV_wT =CV_wR. If two values are given (i.e.,CV = c(x, y)) heteroscedasticity (unequal variances) isassumed, whereCV[1] has to beCV_wT andCV[2]CV_wR.

If simulating for power (theta0 within the BE limits),nsims defaults to 100,000. If simulating for the empirictype I error (theta0 set to one of the BE limits),nsims defaults to one million.

Implemented Designs

#    design                        name   df#   "2x2x3"   2x2x3 replicate crossover 2n-3#   "2x2x4"   2x2x4 replicate crossover 3n-4#   "2x3x3"   partial replicate (2x3x3) 2n-3

The terminology of thedesign argument follows thispattern:treatments x sequences x periods.

Withfoo(..., details = FALSE, print = FALSE) resultsare given as a data frame ¹ with eleven columnsDesign,alpha,CVwT,CVwR,theta0,theta1,theta2,Sample size,Achieved power,Target power, andnlast. To accesse.g., the sample size use eitherfoo(...)[1, 8] orfoo(...)[["Sample size"]].We suggest to use the latter in your code for clarity.

The estimated sample size gives always thetotal number of subjects (not subject/sequence – like in someother software packages).

Conditions and Methods

Regulatory conditions and methods of evaluation are different.

#  regulator CVswitch   CVcap r_const      L      U pe_constr method#        EMA      0.3 0.50000 0.76000 0.6984 1.4319      TRUE  ANOVA#         HC      0.3 0.57382 0.76000 0.6667 1.5000      TRUE    ISC#        FDA      0.3    none 0.89257   none   none      TRUE    ISC

CVswitch is the lower cap of scaling,i.e., ifCV_wR is below this value reference-scaling is notacceptable. The upper cap of scalingCVcap differes betweenthe EMA and HC, whereas for the FDA scaling is unlimited. The regulatoryconstantr_const is used for calculating the expandedlimits (EMA, HC) and ‘implied limits’ (FDA) based ons_wR:$\small{\left [ L,U\right ]=100\cdot\exp (\mp 0.760\cdot s_{\textrm{wR}})}$
HereL andU give the maximum acceptableexpansion based on$\small{s_{\textrm{wR}}^{*}=\sqrt{\log_{e}(CV_\textrm{cap}^2)+1}}$.The point estimate constraintpe_constr [0.80, 1.25] isapplicable in all regulations. Evaluation has to be performed by ANOVA(EMA) or a mixed-effects model (HC, FDA). For the latter intra-subjectcontrasts are a sufficiently close approximation.

Sample Size

Highly Variable Drugs / Drug Products

Estimate the sample size for assumed intra-subjectCV 0.55(CV_wT = CV_wR). Employ thedefaults (theta0 = 0.90,targetpower = 0.80,design = "2x3x3",nsims = 1e5).

EMA

Average Bioequivalence with Expanding Limits (ABEL). Defaultregulator = "EMA".
Note that this approach is recommended in other jurisdictions as well(e.g., theWHO;ASEANStates, Australia, Brazil, Chile, the East African Community, Egypt, theEurasian Economic Union, New Zealand, the Russian Federation).

sampleN.scABEL(CV =0.55)## +++++++++++ scaled (widened) ABEL +++++++++++#             Sample size estimation#    (simulation based on ANOVA evaluation)# ---------------------------------------------# Study design: 2x3x3 (partial replicate)# log-transformed data (multiplicative model)# 1e+05 studies for each step simulated.## alpha  = 0.05, target power = 0.8# CVw(T) = 0.55; CVw(R) = 0.55# True ratio = 0.9# ABE limits / PE constraint = 0.8 ... 1.25# EMA regulatory settings# - CVswitch            = 0.3# - cap on scABEL if CVw(R) > 0.5# - regulatory constant = 0.76# - pe constraint applied### Sample size search#  n     power# 39   0.7807# 42   0.8085

Heterogenicity

Whilst in full replicate designs simulating via the ‘key’ statisticsclosely matches the ‘gold standard’ of subject simulations, this is lessso for unequal variances in the partial replicate design ifCV_wT >CV_wR. Let us keepCV_w 0.55 and split variances by a ratio of 1.5(i.e., T has a higher variance than R).

CV<-signif(CVp2CV(CV =0.55,ratio =1.5),4)sampleN.scABEL(CV = CV,details =FALSE)## +++++++++++ scaled (widened) ABEL +++++++++++#             Sample size estimation#    (simulation based on ANOVA evaluation)# ---------------------------------------------# Study design: 2x3x3 (partial replicate)# log-transformed data (multiplicative model)# 1e+05 studies for each step simulated.## alpha  = 0.05, target power = 0.8# CVw(T) = 0.6109; CVw(R) = 0.4852# True ratio = 0.9# ABE limits / PE constraint = 0.8 ... 1.25# Regulatory settings: EMA## Sample size#  n     power# 45   0.8114

Although the runtime will be longer, we recommend the functionsampleN.scABEL.sdsims() instead.

sampleN.scABEL.sdsims(CV = CV,details =FALSE)# Be patient. Simulating subject data may take a while!## +++++++++++ scaled (widened) ABEL +++++++++++#             Sample size estimation#    (simulation based on ANOVA evaluation)# ---------------------------------------------# Study design: 2x3x3 (partial replicate)# log-transformed data (multiplicative model)# 1e+05 studies for each step simulated.## alpha  = 0.05, target power = 0.8# CVw(T) = 0.6109; CVw(R) = 0.4852# True ratio = 0.9# ABE limits / PE constraint = 0.8 ... 1.25# Regulatory settings: EMA## Sample size#  n   power# 48   0.8161

The sample size is slightly larger.

Explore sample sizes for extreme heterogenicity (variance ratio 2.5)via the ‘key’ statistics and subject simulations (4- and 3-period fullreplicate and partial replicate designs).

CVp<-seq(0.40,0.70,0.05)CV<-signif(CVp2CV(CV = CVp,ratio =2.5),4)res<-data.frame(CVp = CVp,CVwT = CV[,1],CVwR = CV[,2],f4.key =NA,f4.ss =NA,# 4-period full replicatef3.key =NA,f3.ss =NA,# 3-period full replicatep3.key =NA,p3.ss =NA)# 3-period partial replicatefor (iinseq_along(CVp)) {  res$f4.key[i]<-sampleN.scABEL(CV = CV[i, ],design ="2x2x4",print =FALSE,details =FALSE)[["Sample size"]]  res$f4.ss[i]<-sampleN.scABEL.sdsims(CV = CV[i, ],design ="2x2x4",print =FALSE,details =FALSE,progress =FALSE)[["Sample size"]]  res$f3.key[i]<-sampleN.scABEL(CV = CV[i, ],design ="2x2x3",print =FALSE,details =FALSE)[["Sample size"]]  res$f3.ss[i]<-sampleN.scABEL.sdsims(CV = CV[i, ],design ="2x2x3",print =FALSE,details =FALSE,progress =FALSE)[["Sample size"]]  res$p3.key[i]<-sampleN.scABEL(CV = CV[i, ],design ="2x3x3",print =FALSE,details =FALSE)[["Sample size"]]  res$p3.ss[i]<-sampleN.scABEL.sdsims(CV = CV[i, ],design ="2x3x3",print =FALSE,details =FALSE,progress =FALSE)[["Sample size"]]}print(res,row.names =FALSE)#   CVp   CVwT   CVwR f4.key f4.ss f3.key f3.ss p3.key p3.ss#  0.40 0.4860 0.2975     62    62     90    90     81    87#  0.45 0.5490 0.3334     60    60     90    90     78    84#  0.50 0.6127 0.3688     54    54     82    82     69    75#  0.55 0.6773 0.4038     50    50     76    76     63    69#  0.60 0.7427 0.4383     46    46     72    72     60    66#  0.65 0.8090 0.4723     46    46     68    68     57    63#  0.70 0.8762 0.5059     46    46     70    70     57    63

As shown in the previous example, subject simulations are recommendedfor the partial replicate design. For full replicate designs simulationsvia the ‘key’ statistics give identical results and are recommended forspeed reasons. In this examplesampleN.scABEL() is 60timesfaster thansampleN.scABEL.sdsims().
However, ifCV_wT ≤CV_wR we getidentical results via the ‘key’ statistics.

Health Canada

Average Bioequivalence with Expanding Limits (ABEL). Defaults asabove butregulator = "HC".

sampleN.scABEL(CV =0.55,regulator ="HC")## +++++++++++ scaled (widened) ABEL +++++++++++#             Sample size estimation# (simulations based on intra-subject contrasts)# ----------------------------------------------# Study design:  2x3x3 (partial replicate)# log-transformed data (multiplicative model)# 1e+05 studies for each step simulated.## alpha  = 0.05, target power = 0.8# CVw(T) = 0.55; CVw(R) = 0.55# True ratio = 0.9# ABE limits / PE constraint = 0.8 ... 1.25# HC regulatory settings# - CVswitch            = 0.3# - cap on scABEL if CVw(R) > 0.57382# - regulatory constant = 0.76# - pe constraint applied### Sample size search#  n     power# 33   0.7539# 36   0.7864# 39   0.8142

Gulf Cooperation Council

Special case of ABEL: Conventional limits ifCV_wR≤30% and widened limits of 0.7500–1.3333 otherwise. No upper cap ofwidening. Defaults as above butregulator = "GCC". Only tocompare with previous studies because since in Version 3.1 of 10 August2022 the GCC implemented the EMA’s method.

sampleN.scABEL(CV =0.55,regulator ="GCC")## +++++++++++ scaled (widened) ABEL +++++++++++#             Sample size estimation#    (simulation based on ANOVA evaluation)# ---------------------------------------------# Study design: 2x3x3 (partial replicate)# log-transformed data (multiplicative model)# 1e+05 studies for each step simulated.## alpha  = 0.05, target power = 0.8# CVw(T) = 0.55; CVw(R) = 0.55# True ratio = 0.9# ABE limits / PE constraint = 0.8 ... 1.25# Widened limits = 0.75 ... 1.333333# GCC regulatory settings# - CVswitch            = 0.3# - cap on scABEL if CVw(R) > 0.3# - regulatory constant = 0.9799758# - pe constraint applied### Sample size search#  n     power# 72   0.7874# 75   0.8021

FDA

Apart from the FDA only required by China’s agency.

sampleN.RSABE(CV =0.55)## ++++++++ Reference scaled ABE crit. +++++++++#            Sample size estimation# ---------------------------------------------# Study design: 2x3x3 (partial replicate)# log-transformed data (multiplicative model)# 1e+05 studies for each step simulated.## alpha  = 0.05, target power = 0.8# CVw(T) = 0.55; CVw(R) = 0.55# True ratio = 0.9# ABE limits / PE constraints = 0.8 ... 1.25# FDA regulatory settings# - CVswitch            = 0.3# - regulatory constant = 0.8925742# - pe constraint applied### Sample size search#  n     power# 24   0.72002# 27   0.76591# 30   0.80034

Note the lower sample size compared to the other approaches (due tothe different regulatory constant and unlimited scaling).

Narrow Therapeutic Index Drugs (FDA, CDE)

Required by the FDA and the Chinese authority.
Assuming heteroscedasticity (CV_w 0.125,σ² ratio 2.5,i.e., T has a substantiallyhigher variability than R). Details of the sample size searchsuppressed. Assess additionally which one of the three components(scaled, ABE,s_wT/s_wR ratio)drives the sample size.

CV<-signif(CVp2CV(CV =0.125,ratio =2.5),4)n<-sampleN.NTID(CV = CV,details =FALSE)[["Sample size"]]## +++++++++++ FDA method for NTIDs ++++++++++++#            Sample size estimation# ---------------------------------------------# Study design:  2x2x4 (TRTR|RTRT)# log-transformed data (multiplicative model)# 1e+05 studies for each step simulated.## alpha  = 0.05, target power = 0.8# CVw(T) = 0.1497, CVw(R) = 0.09433# True ratio     = 0.975# ABE limits     = 0.8 ... 1.25# Regulatory settings: FDA## Sample size#  n     power# 38   0.816080suppressMessages(power.NTID(CV = CV,n = n,details =TRUE))#        p(BE)  p(BE-sABEc)    p(BE-ABE) p(BE-sratio)#      0.81608      0.93848      1.00000      0.85794

Thes_wT/s_wR component showsthe lowest power and hence, drives the sample size.
Compare that with homoscedasticity(CV_wT = CV_wR = 0.125):

CV<-0.125n<-sampleN.NTID(CV = CV,details =FALSE)[["Sample size"]]## +++++++++++ FDA method for NTIDs ++++++++++++#            Sample size estimation# ---------------------------------------------# Study design:  2x2x4 (TRTR|RTRT)# log-transformed data (multiplicative model)# 1e+05 studies for each step simulated.## alpha  = 0.05, target power = 0.8# CVw(T) = 0.125, CVw(R) = 0.125# True ratio     = 0.975# ABE limits     = 0.8 ... 1.25# Regulatory settings: FDA## Sample size#  n     power# 16   0.822780suppressMessages(power.NTID(CV = CV,n = n,details =TRUE))#        p(BE)  p(BE-sABEc)    p(BE-ABE) p(BE-sratio)#      0.82278      0.84869      1.00000      0.95128

Here the scaled ABE component shows the lowest power and drives thesample size, which is much lower than in the previous example.

Highly Variable Narrow Therapeutic Index Drugs (FDA, CDE)

Almost a contradiction in itself. Required for [dagibatran] (https://www.accessdata.fda.gov/drugsatfda_docs/psg/Dabigatran%20etexilate%20mesylate_oral%20capsule_NDA%20022512_RV05-17.pdf“Recommended Jun 2012; Revised Sep 2015, Jul 2017”),rivaroxaban, andedoxaban.
Assuming homoscedasticity (CV_wT =CV_wR = 0.30). Employ the defaults(theta0 = 0.95,targetpower = 0.80,design = "2x2x4",nsims = 1e5). Details of thesample size search suppressed.

sampleN.HVNTID(CV =0.30,details =FALSE)## +++++++++ FDA method for HV NTIDs ++++++++++++#            Sample size estimation# ----------------------------------------------# Study design: 2x2x4 (TRTR|RTRT)# log-transformed data (multiplicative model)# 1e+05 studies for each step simulated.## alpha  = 0.05, target power = 0.8# CVw(T) = 0.3, CVw(R) = 0.3# True ratio = 0.95# ABE limits = 0.8 ... 1.25## Sample size#  n     power# 22   0.829700

Assuming heteroscedasticity (CV_w 0.30,σ² ratio 2.5).

CV<-signif(CVp2CV(CV =0.125,ratio =2.5),4)sampleN.HVNTID(CV = CV,details =FALSE)## +++++++++ FDA method for HV NTIDs ++++++++++++#            Sample size estimation# ----------------------------------------------# Study design: 2x2x4 (TRTR|RTRT)# log-transformed data (multiplicative model)# 1e+05 studies for each step simulated.## alpha  = 0.05, target power = 0.8# CVw(T) = 0.1497, CVw(R) = 0.09433# True ratio = 0.95# ABE limits = 0.8 ... 1.25## Sample size#  n     power# 34   0.818800

In this case a substantially higher sample size is required since thevariability of T is higher than the one of R.

Type I Error

Contrary to average bioequivalence, where the Null-hypothesis isbased on fixed limits, in reference-scaling the Null is generated inface of the data (i.e, the limits are random variables).

Endrényi and Tóthfalusi (2009,² 2019³), Labes (2013⁴), Wonnemannetal. (2015⁵), Muñozet al. (2016⁶), Labes and Schütz(2016⁷),Tóthfalusi and Endrényi (2016,⁸ 2017⁹), Molinset al. (2017¹⁰), Dengand Zhou (2019¹¹) showed that under certain conditions(EMA, Health Canada:CV_wR ~0.22–0.45, FDA:CV_wR ≤0.30) the type I error will be substantiallyinflated.
Below the inflation region the study will be evaluated for ABE and thetype I error controlled by the TOST. Above the inflation region thetype I error is controlled by the PE restriction and for the EMA andHealth Canada additionally by the upper cap of scaling.

CV<-0.35res<-data.frame(n =NA,CV = CV,TIE =NA)res$n<-sampleN.scABEL(CV = CV,design ="2x2x4",print =FALSE,details =FALSE)[["Sample size"]]U<-scABEL(CV = CV)[["upper"]]res$TIE<-power.scABEL(CV = CV,n = res$n,theta0 = U,design ="2x2x4")print(res,row.names =FALSE)#   n   CV      TIE#  34 0.35 0.065566

With ~0.0656 the type I error is inflated (significantly larger thanthe nominal$\small{\alpha}$0.05).

$Fig. 1 Empiric type I error for the EMA’s ABEL, 4-period full replicate design. Pink plane at nominal \small{\alpha} 0.05. Contour lines enclose region of inflation.$

Fig. 1 Empiric type I error forthe EMA’s ABEL, 4-period full replicate design.
Pink plane atnominal$\small{\alpha}$ 0.05.Contour lines enclose region of inflation.

A substantially higher inflation of the type I error was reported forthe FDA’s model. However, Davitet al. (2012¹²) assessed the type Ierror not at the‘implied limits’ but with the‘desired consumer risk model’ if$\small{s_{\textrm{wR}}\geq s_0}$(CV_wR ≥~25.4%) at$\small{\exp\left ( \log_{e}(1.25)/s_0\sqrt{\log_{e}(CV_{\textrm{wR}}^2+1)} \right )}$. Somestatisticians call the latter ‘hocus-pocus’. However, even with thisapproach the type I error is still –although less – inflated.

res<-data.frame(CV =sort(c(seq(0.25,0.32,0.01),se2CV(0.25))),impl.L =NA,impl.U =NA,impl.TIE =NA,des.L =NA,des.U =NA,des.TIE =NA)for (iin1:nrow(res)) {  res[i,2:3]<-scABEL(CV = res$CV[i],regulator ="FDA")if (CV2se(res$CV[i])<=0.25) {    res[i,5:6]<-c(0.80,1.25)  }else {    res[i,5:6]<-exp(c(-1,+1)*(log(1.25)/0.25)*CV2se(res$CV[i]))  }  res[i,4]<-power.RSABE(CV = res$CV[i],theta0 = res[i,3],design ="2x2x4",n =32,nsims =1e6)  res[i,7]<-power.RSABE(CV = res$CV[i],theta0 = res[i,5],design ="2x2x4",n =32,nsims =1e6)}print(signif(res,4),row.names =FALSE)#     CV impl.L impl.U impl.TIE  des.L des.U des.TIE#  0.250 0.8000  1.250  0.06068 0.8000 1.250 0.06036#  0.254 0.8000  1.250  0.06396 0.8000 1.250 0.06357#  0.260 0.8000  1.250  0.07008 0.7959 1.256 0.05692#  0.270 0.8000  1.250  0.08352 0.7892 1.267 0.05047#  0.280 0.8000  1.250  0.10130 0.7825 1.278 0.04770#  0.290 0.8000  1.250  0.12290 0.7760 1.289 0.04644#  0.300 0.8000  1.250  0.14710 0.7695 1.300 0.04562#  0.310 0.7631  1.310  0.04515 0.7631 1.310 0.04466#  0.320 0.7568  1.321  0.04373 0.7568 1.321 0.04325

Fig. 2 Empiric type I error forthe FDA’s RSABE, 4-period full replicate design, n = 32.
Thick line‘implied limits’ (max. TIE 0.147 atCV_wR30%).
Thin line ‘desired consumer risk model’ (max. TIE 0.0636 atCV_wR 25.4%).

Various approaches were suggested to control the patient’s risk. Themethods of Labes and Schütz (2016) and Molinset al. (2017) areimplemented in the functionscABEL.ad(). The method ofTóthfalusi and Endrényi (2017) is implemented in the functionpower.RSABE2L.sds().

Iteratively adjusted α

If an inflated type I error is expected,$\small{\alpha}$ is adjusted based on theobservedCV_wR and the study should be evaluated witha wider confidence interval (Labes and Schütz 2016). Implementeddesigns:"2x3x3" (default),"2x2x3","2x2x4".
No adjustment is suggested if the study’s conditions(CV_wR, sample size, design) will not lead to aninflated type I error.

CV<-0.45n<-sampleN.scABEL(CV = CV,design ="2x2x4",print =FALSE,details =FALSE)[["Sample size"]]scABEL.ad(CV = CV,design ="2x2x4",n = n)# +++++++++++ scaled (widened) ABEL ++++++++++++#          iteratively adjusted alpha#    (simulations based on ANOVA evaluation)# ----------------------------------------------# Study design: 2x2x4 (4 period full replicate)# log-transformed data (multiplicative model)# 1,000,000 studies in each iteration simulated.## CVwR 0.45, CVwT 0.45, n(i) 14|14 (N 28)# Nominal alpha                 : 0.05# True ratio                    : 0.9000# Regulatory settings           : EMA (ABEL)# Switching CVwR                : 0.3# Regulatory constant           : 0.76# Expanded limits               : 0.7215 ... 1.3859# Upper scaling cap             : CVwR > 0.5# PE constraints                : 0.8000 ... 1.2500# Empiric TIE for alpha 0.0500  : 0.04889# Power for theta0 0.9000       : 0.811# TIE ≤ nominal alpha; no adjustment of alpha is required.

Inside the region of inflated type I errors.

CV<-0.35n<-sampleN.scABEL(CV = CV,design ="2x2x4",print =FALSE,details =FALSE)[["Sample size"]]scABEL.ad(CV = CV,design ="2x2x4",n = n)# +++++++++++ scaled (widened) ABEL ++++++++++++#          iteratively adjusted alpha#    (simulations based on ANOVA evaluation)# ----------------------------------------------# Study design: 2x2x4 (4 period full replicate)# log-transformed data (multiplicative model)# 1,000,000 studies in each iteration simulated.## CVwR 0.35, CVwT 0.35, n(i) 17|17 (N 34)# Nominal alpha                 : 0.05# True ratio                    : 0.9000# Regulatory settings           : EMA (ABEL)# Switching CVwR                : 0.3# Regulatory constant           : 0.76# Expanded limits               : 0.7723 ... 1.2948# Upper scaling cap             : CVwR > 0.5# PE constraints                : 0.8000 ... 1.2500# Empiric TIE for alpha 0.0500  : 0.06557# Power for theta0 0.9000       : 0.812# Iteratively adjusted alpha    : 0.03630# Empiric TIE for adjusted alpha: 0.05000# Power for theta0 0.9000       : 0.773

An adjusted$\small{\alpha}$ of0.0363 (i.e., the 92.74% CI) controls the patient’s risk.However, it leads to a slightly lower power (0.773 instead of0.812).

In order to counteract this loss in power, we can adjust the samplesize with the functionsampleN.scABEL.ad().

CV<-0.35sampleN.scABEL.ad(CV = CV,design ="2x2x4")## +++++++++++ scaled (widened) ABEL ++++++++++++#             Sample size estimation#         for iteratively adjusted alpha#    (simulations based on ANOVA evaluation)# ----------------------------------------------# Study design: 2x2x4 (4 period full replicate)# log-transformed data (multiplicative model)# 1,000,000 studies in each iteration simulated.## Assumed CVwR 0.35, CVwT 0.35# Nominal alpha      : 0.05# True ratio         : 0.9000# Target power       : 0.8# Regulatory settings: EMA (ABEL)# Switching CVwR     : 0.3# Regulatory constant: 0.76# Expanded limits    : 0.7723 ... 1.2948# Upper scaling cap  : CVwR > 0.5# PE constraints     : 0.8000 ... 1.2500# n  38,   adj. alpha: 0.03610 (power 0.8100), TIE: 0.05000

We have to increase the sample size to 38 in order to maintain power.Since the type I error depends to a minor degree on the sample size aswell, we have to adjust slightly more ($\small{\alpha}$ 0.0361 instead of 0.0363with 34 subjects).

Since the observedCV_wR is not the true –unknown – one, Molinset al. recommended to ‘assume the worst’and adjust forCV_wR 0.30 in all cases.

# CV = 0.35, n = 34, design = "2x2x4"#               method adj   alpha    TIE power#  EMA (nominal alpha)  no 0.05000 0.0656 0.812#     Labes and Schütz yes 0.03630 0.0500 0.773#        Molins et al. yes 0.02857 0.0500 0.740

Although Molin’s adjusted$\small{\alpha}$ controls the patient’srisk, it leads to a further loss in power.

Example with aCV_wR above the region of inflatedtype I errors (i.e., >0.45).

# CV = 0.8, n = 50, design = "2x2x4"#            method adj  alpha    TIE power#  Labes and Schütz  no 0.0500 0.0496 0.812#     Molins et al. yes 0.0282 0.0500 0.732

For high variability the negative impact on power is substantial.

‘Exact’ Procedure

Proposed by Tóthfalusi and Endrényi (2016). Example of the ‘ncTOST’method by the same authors (2017). Implemented designs:"2x3x3" (default),"2x2x3","2x2x4".

CV<-0.35n<-sampleN.scABEL(CV = CV,design ="2x2x4",print =FALSE,details =FALSE)[["Sample size"]]U<-scABEL(CV = CV)[["upper"]]# subject simulations and therefore, relatively slowpower.RSABE2L.sds(CV = CV,design ="2x2x4",theta0 = U,n = n,SABE_test ="exact",nsims =1e6,progress =FALSE)# [1] 0.048177

With ~0.0482 the patient’s risk is controlled. However, theregulatory acceptance is unclear.

Helpers

BE limits

Expanded Limits (EMA, Health Canada), widened limits (GCC)

CV<-c(0.30,0.40898,0.50,0.57382)res<-data.frame(CV = CV,EMA.L =NA,EMA.U =NA,EMA.cap ="",HC.L =NA,HC.U =NA,HC.cap ="",GCC.L =NA,GCC.U =NA,GCC.cap ="",stringsAsFactors =FALSE)# this line for R <4.0.0for (iinseq_along(CV)) {  res[i,2:3]<-sprintf("%.4f",scABEL(CV[i],regulator ="EMA"))  res[i,5:6]<-sprintf("%.3f",scABEL(CV[i],regulator ="HC"))  res[i,8:9]<-sprintf("%.3f",scABEL(CV[i],regulator ="GCC"))}res$EMA.cap[res$CV<=0.30]<- res$HC.cap[res$CV<=0.30]<-"lower"res$EMA.cap[res$CV>=0.50]<-"upper"res$HC.cap[res$CV>=0.57382]<-"upper"res$GCC.cap[res$CV<=0.30]<- res$GCC.cap[res$CV<=0.30]<-"lower"print(res,row.names =FALSE)#       CV  EMA.L  EMA.U EMA.cap  HC.L  HC.U HC.cap GCC.L GCC.U GCC.cap#  0.30000 0.8000 1.2500   lower 0.800 1.250  lower 0.800 1.250   lower#  0.40898 0.7416 1.3484         0.742 1.348        0.750 1.333#  0.50000 0.6984 1.4319   upper 0.698 1.432        0.750 1.333#  0.57382 0.6984 1.4319   upper 0.667 1.500  upper 0.750 1.333

For all agencies the lower cap for scaling is 30%. Whereas the uppercap for the EMA is at 50% (expanded limits 69.84–143.19%), for HealthCanada it is at ~57.4% (expanded limits 66.7–150.0%). The GCC had noupper cap (fixed widened limits 75.00–133.33%); since August 2022 theGCC uses the EMA’s method. ### FDA For the FDA there is no upper cap(scaling is unlimited).

‘Implied’ Limits

res<-data.frame(CV =c(0.25,se2CV(0.25),0.275,0.3,0.5,1.0),impl.L =NA,impl.U =NA,cap ="",stringsAsFactors =FALSE)# this line for R <4.0.0for (iin1:nrow(res)) {  res[i,2:3]<-sprintf("%.4f",scABEL(CV = res$CV[i],regulator ="FDA"))}res$cap[res$CV<=0.30]<-"lower"res$CV<-sprintf("%.3f", res$CV)print(res,row.names =FALSE)#     CV impl.L impl.U   cap#  0.250 0.8000 1.2500 lower#  0.254 0.8000 1.2500 lower#  0.275 0.8000 1.2500 lower#  0.300 0.8000 1.2500 lower#  0.500 0.6560 1.5245#  1.000 0.4756 2.1025

Limits of the ‘desired consumer risk model’

res<-data.frame(CV =c(0.25,se2CV(0.25),0.275,0.3,0.5,1.0),des.L =NA,des.U =NA,cap ="",stringsAsFactors =FALSE)# this line for R <4.0.0for (iin1:nrow(res)) {if (CV2se(res$CV[i])<=0.25) {    res[i,2:3]<-sprintf("%.4f",c(0.80,1.25))  }else {    res[i,2:3]<-sprintf("%.4f",exp(c(-1,+1)*(log(1.25)/0.25)*CV2se(res$CV[i])))  }}res$cap[res$CV<=0.30]<-"lower"res$CV<-sprintf("%.3f", res$CV)print(res,row.names =FALSE)#     CV  des.L  des.U   cap#  0.250 0.8000 1.2500 lower#  0.254 0.8000 1.2500 lower#  0.275 0.7858 1.2725 lower#  0.300 0.7695 1.2996 lower#  0.500 0.6560 1.5245#  1.000 0.4756 2.1025

Regulatory Settings

reg<-c("EMA","HC","GCC","FDA")for (iin1:4) {print(reg_const(regulator = reg[i]))cat("\n")}# EMA regulatory settings# - CVswitch            = 0.3# - cap on scABEL if CVw(R) > 0.5# - regulatory constant = 0.76# - pe constraint applied## HC regulatory settings# - CVswitch            = 0.3# - cap on scABEL if CVw(R) > 0.57382# - regulatory constant = 0.76# - pe constraint applied## GCC regulatory settings# - CVswitch            = 0.3# - cap on scABEL if CVw(R) > 0.3# - regulatory constant = 0.9799758# - pe constraint applied## FDA regulatory settings# - CVswitch            = 0.3# - no cap on scABEL# - regulatory constant = 0.8925742# - pe constraint applied

function	author(s)
`sampleN.scABEL`,`sampleN.RSABE`,`sampleN.NTID`,`sampleN.HVNTID`, `power.scABEL`,`power.RSABE2L.sdsims`,`scABEL`,`reg_const`	Detlew Labes
`power.scABEL.sdsims`	Detlew Labes, BenjaminLang
`sampleN.scABEL.ad`,`sampleN.scABEL.sdsims`,`sampleN.RSABE2L.sdsims`, `scABEL.ad`	Helmut Schütz

Movatterモバイル変換

Reference-Scaled AverageBioequivalence

Details and examples of other methods are accessible via the menu bar ontop of the page and in theonlinemanual of all functions.