library(PowerTOST) # attach the library
Parameter  Argument  Purpose  Default 

\(\small{\alpha}\)  alpha 
Nominal level of the test  0.05 
\(\small{\pi}\)  targetpower 
Minimum desired power  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 
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 
Arguments targetpower
, theta0
, theta1
, theta2
, and CV
have to be given as fractions, not in percent.
The CV is the within (intra) subject coefficient of variation. If one value is given, homoscedasticity (equal variances) is assumed and therefore, CV_{wT} = CV_{wR}. If two values are given (i.e., CV = c(x, y)
) heteroscedasticity (unequal variances) is assumed, where CV[1]
has to be CV_{wT} and CV[2]
CV_{wR}.
If simulating for power (theta0
within the BE limits), nsims
defaults to 100,000. If simulating for the empiric type I error (theta0
set to one of the BE limits), nsims
defaults to one million.
# design name df
# "2x2x3" 2x2x3 replicate crossover 2n3
# "2x2x4" 2x2x4 replicate crossover 3n4
# "2x4x4" 2x4x4 replicate crossover 3n4
# "2x3x3" partial replicate (2x3x3) 2n3
# "2x4x2" Balaam’s (2x4x2) n2
# "2x2x2r" Liu’s 2x2x2 repeated xover 3n2
The terminology of the design
argument follows this pattern: treatments x sequences x periods
.
With foo(..., details = FALSE, print = FALSE)
results are given as a data frame ^{1} with eleven columns Design
, alpha
, CVwT
, CVwR
, theta0
, theta1
, theta2
, Sample size
, Achieved power
, Target power
, and nlast
. To access e.g., the sample size use either foo(...)[1, 8]
or foo(...)[["Sample size"]]
. We suggest to use the latter in your code for clarity.
The estimated sample size gives always the total number of subjects (not subject/sequence – like in some other software packages).
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., if CV_{wR} is below this value referencescaling is not acceptable. The upper cap of scaling CVcap
differes between the EMA and HC, whereas for the FDA scaling is unlimited. The regulatory constant r_const
is used for calculating the expanded limits (EMA, HC) and ‘implied limits’ (FDA) based on s_{wR}: \(\small{\left [ L,U \right ]=100\cdot\exp (\mp 0.760\cdot s_{\textrm{wR}})}\)
Here L
and U
give the maximum acceptable expansion based on \(\small{s_{\textrm{wR}}^{*}=\sqrt{\log_{e}(CV_\textrm{cap}^2)+1}}\). The point estimate constraint pe_constr
[0.80, 1.25] is applicable in all regulations. Evaluation has to be performed by ANOVA (EMA) or a mixedeffects model (HC, FDA). For the latter intrasubject contrasts are a suffiently close approximation.
Estimate the sample size for assumed intrasubject CV 0.55 (CV_{wT} = CV_{wR}). Employ the defaults (theta0 = 0.90
, targetpower = 0.80
, design = "2x3x3"
, nsims = 1e5
).
Average Bioequivalence with Expanding Limits (ABEL). Default regulator = "EMA"
.
Note that this approach is recommended in other jurisdictions as well (e.g., the WHO; ASEAN States, Australia, Brazil, Chile, the East African Community, Egypt, the Eurasian 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)
# logtransformed 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
Whilst in full replicate designs simulating via the ‘key’ statistics closely matches the ‘gold standard’ of subject simulations, this is less so for unequal variances in the partial replicate design if CV_{wT} > CV_{wR}. Let us keep CV_{w} 0.55 and split variances by a ratio of 1.5 (i.e., T has a higher variance than R).
< signif(CVp2CV(CV = 0.55, ratio = 1.5), 4)
CV sampleN.scABEL(CV = CV, details = FALSE)
#
# +++++++++++ scaled (widened) ABEL +++++++++++
# Sample size estimation
# (simulation based on ANOVA evaluation)
# 
# Study design: 2x3x3 (partial replicate)
# logtransformed 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 function sampleN.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)
# logtransformed 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 3period full replicate and partial replicate designs).
< seq(0.40, 0.70, 0.05)
CVp < signif(CVp2CV(CV = CVp, ratio = 2.5), 4)
CV < data.frame(CVp = CVp, CVwT = CV[, 1], CVwR = CV[, 2],
res f4.key = NA, f4.ss = NA, # 4period full replicate
f3.key = NA, f3.ss = NA, # 3period full replicate
p3.key = NA, p3.ss = NA) # 3period partial replicate
for (i in seq_along(CVp)) {
$f4.key[i] < sampleN.scABEL(CV = CV[i, ], design = "2x2x4",
resprint = FALSE,
details = FALSE)[["Sample size"]]
$f4.ss[i] < sampleN.scABEL.sdsims(CV = CV[i, ], design = "2x2x4",
resprint = FALSE,
details = FALSE,
progress = FALSE)[["Sample size"]]
$f3.key[i] < sampleN.scABEL(CV = CV[i, ], design = "2x2x3",
resprint = FALSE,
details = FALSE)[["Sample size"]]
$f3.ss[i] < sampleN.scABEL.sdsims(CV = CV[i, ], design = "2x2x3",
resprint = FALSE,
details = FALSE,
progress = FALSE)[["Sample size"]]
$p3.key[i] < sampleN.scABEL(CV = CV[i, ], design = "2x3x3",
resprint = FALSE,
details = FALSE)[["Sample size"]]
$p3.ss[i] < sampleN.scABEL.sdsims(CV = CV[i, ], design = "2x3x3",
resprint = 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 recommended for the partial replicate design. For full replicate designs simulations via the ‘key’ statistics give identical results and are recommended for speed reasons. In this example sampleN.scABEL()
is 60times faster than sampleN.scABEL.sdsims()
.
However, if CV_{wT} ≤ CV_{wR} we get identical results via the ‘key’ statistics.
Average Bioequivalence with Expanding Limits (ABEL). Defaults as above but regulator = "HC"
.
sampleN.scABEL(CV = 0.55, regulator = "HC")
#
# +++++++++++ scaled (widened) ABEL +++++++++++
# Sample size estimation
# (simulations based on intrasubject contrasts)
# 
# Study design: 2x3x3 (partial replicate)
# logtransformed 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
Special case of ABEL: Conventional limits if CV_{wR} ≤30% and widened limits of 0.7500–1.3333 otherwise. No upper cap of widening. Defaults as above but regulator = "GCC"
.
sampleN.scABEL(CV = 0.55, regulator = "GCC")
#
# +++++++++++ scaled (widened) ABEL +++++++++++
# Sample size estimation
# (simulation based on ANOVA evaluation)
# 
# Study design: 2x3x3 (partial replicate)
# logtransformed 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
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)
# logtransformed 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 to the different regulatory constant and unlimited scaling).
Required by the FDA and the Chinese authority.
Assuming heteroscedasticity (CV_{w} 0.125, σ^{2} ratio 2.5, i.e., T has a substantially higher variability than R). Details of the sample size search suppressed. Assess additionally which one of the three components (scaled, ABE, s_{wT}/s_{wR} ratio) drives the sample size.
< signif(CVp2CV(CV = 0.125, ratio = 2.5), 4)
CV < sampleN.NTID(CV = CV, details = FALSE)[["Sample size"]]
n #
# +++++++++++ FDA method for NTIDs ++++++++++++
# Sample size estimation
# 
# Study design: 2x2x4 (TRTRRTRT)
# logtransformed 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 0.816080
suppressMessages(power.NTID(CV = CV, n = n, details = TRUE))
# p(BE) p(BEsABEc) p(BEABE) p(BEsratio)
# 0.81608 0.93848 1.00000 0.85794
The s_{wT}/s_{wR} component shows the lowest power and hence, drives the sample size.
Compare that with homoscedasticity (CV_{wT} = CV_{wR} = 0.125):
< 0.125
CV < sampleN.NTID(CV = CV, details = FALSE)[["Sample size"]]
n #
# +++++++++++ FDA method for NTIDs ++++++++++++
# Sample size estimation
# 
# Study design: 2x2x4 (TRTRRTRT)
# logtransformed 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 0.822780
suppressMessages(power.NTID(CV = CV, n = n, details = TRUE))
# p(BE) p(BEsABEc) p(BEABE) p(BEsratio)
# 0.82278 0.84869 1.00000 0.95128
Here the scaled ABE component shows the lowest power and drives the sample size, which is much lower than in the previous example.
Almost a contradiction in itself. Required for dagibatran, rivaroxaban, and edoxaban.
Assuming homoscedasticity (CV_{wT} = CV_{wR} = 0.30). Employ the defaults (theta0 = 0.95
, targetpower = 0.80
, design = "2x2x4"
, nsims = 1e5
). Details of the sample size search suppressed.
sampleN.HVNTID(CV = 0.30, details = FALSE)
#
# +++++++++ FDA method for HV NTIDs ++++++++++++
# Sample size estimation
# 
# Study design: 2x2x4 (TRTRRTRT)
# logtransformed 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, σ^{2} ratio 2.5).
< signif(CVp2CV(CV = 0.125, ratio = 2.5), 4)
CV sampleN.HVNTID(CV = CV, details = FALSE)
#
# +++++++++ FDA method for HV NTIDs ++++++++++++
# Sample size estimation
# 
# Study design: 2x2x4 (TRTRRTRT)
# logtransformed 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 the variability of T is higher than the one of R.
Power can by calculated by the counterparts of the respective sample size functions (instead the argument targetpower
use the argument n
and provide the observed theta0
), i.e.,
power.scABEL()
, power.RSABE()
, power.NTID()
, and power.HVNTID()
.
Contrary to average bioequivalence, where the Nullhypothesis is based on fixed limits, in referencescaling the Null is generated in face of the data (i.e, the limits are random variables).
Endrényi and Tóthfalusi (2009,^{2} 2019^{3}), Labes (2013^{4}), Wonnemann et al. (2015^{5}), Muñoz et al. (2016^{6}), Labes and Schütz (2016^{7}), Tóthfalusi and Endrényi (2016,^{8} 2017^{9}), Molins et al. (2017^{10}), Deng and Zhou (2019^{11}) 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 substantially inflated.
Below the inflation region the study will be evaluated for ABE and the type I error controlled by the TOST. Above the inflation region the type I error is controlled by the PE restriction and for the EMA and Health Canada additionally by the upper cap of scaling.
< 0.35
CV < data.frame(n = NA, CV = CV, TIE = NA)
res $n < sampleN.scABEL(CV = CV, design = "2x2x4", print = FALSE,
resdetails = FALSE)[["Sample size"]]
< scABEL(CV = CV)[["upper"]]
U $TIE < power.scABEL(CV = CV, n = res$n, theta0 = U, design = "2x2x4")
resprint(res, row.names = FALSE)
# n CV TIE
# 34 0.35 0.065566
With ~0.0656 the type I error is inflated (significantly larger than the nominal \(\small{\alpha}\) 0.05).
A substantially higher inflation of the type I error was reported for the FDA’s model. However, Davit et al. (2012^{12}) assessed the type I error 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 \(\small{\exp\left ( \log_{e}(1.25)/s_0 \sqrt{\log_{e}(CV_{\textrm{wR}}^2+1)} \right )}\). Some statisticians call the latter ‘hocuspocus’. However, even with this approach the type I error is still –although less – inflated.
< data.frame(CV = sort(c(seq(0.25, 0.32, 0.01), se2CV(0.25))),
res impl.L = NA, impl.U = NA, impl.TIE = NA,
des.L = NA, des.U = NA, des.TIE = NA)
for (i in 1:nrow(res)) {
2:3] < scABEL(CV = res$CV[i], regulator = "FDA")
res[i, if (CV2se(res$CV[i]) <= 0.25) {
5:6] < c(0.80, 1.25)
res[i, else {
} 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],
res[i, design = "2x2x4", n = 32, nsims = 1e6)
7] < power.RSABE(CV = res$CV[i], theta0 = res[i, 5],
res[i, 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
Various approaches were suggested to control the patient’s risk. The methods of Labes and Schütz (2016) and Molins et al. (2017) are implemented in the function scABEL.ad()
. The method of Tóthfalusi and Endrényi (2017) is implemented in the function power.RSABE2L.sds()
.
If an inflated type I error is expected, \(\small{\alpha}\) is adjusted based on the observed CV_{wR} and the study should be evaluated with a wider confidence interval (Labes and Schütz 2016). Implemented designs: "2x3x3"
(default), "2x2x3"
, "2x2x4"
.
No adjustment is suggested if the study’s conditions (CV_{wR}, sample size, design) will not lead to an inflated type I error.
< 0.45
CV < sampleN.scABEL(CV = CV, design = "2x2x4", print = FALSE,
n 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)
# logtransformed data (multiplicative model)
# 1,000,000 studies in each iteration simulated.
#
# CVwR 0.45, CVwT 0.45, n(i) 1414 (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.
< 0.35
CV < sampleN.scABEL(CV = CV, design = "2x2x4", print = FALSE,
n 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)
# logtransformed data (multiplicative model)
# 1,000,000 studies in each iteration simulated.
#
# CVwR 0.35, CVwT 0.35, n(i) 1717 (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 \(\small{\alpha}\) of 0.0363 (i.e., the 92.74% CI) controls the patient’s risk. However, it leads to a slightly lower power (0.773 instead of 0.812).
In order to counteract this loss in power, we can adjust the sample size with the function sampleN.scABEL.ad()
.
< 0.35
CV sampleN.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)
# logtransformed 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 as well, we have to adjust slightly more (\(\small{\alpha}\) 0.0361 instead of 0.0363 with 34 subjects).
Since the observed CV_{wR} is not the true – unknown – one, Molins et al. recommended to ‘assume the worst’ and adjust for CV_{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’s risk, it leads to a further loss in power.
Example with a CV_{wR} above the region of inflated type 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.
Proposed by Tóthfalusi and Endrényi (2016). Example of the ‘ncTOST’ method by the same authors (2017). Implemented designs: "2x3x3"
(default), "2x2x3"
, "2x2x4"
.
< 0.35
CV < sampleN.scABEL(CV = CV, design = "2x2x4", print = FALSE,
n details = FALSE)[["Sample size"]]
< scABEL(CV = CV)[["upper"]]
U # subject simulations and therefore, relatively slow
power.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, the regulatory acceptance is unclear.
< c(0.30, 0.40898, 0.50, 0.57382)
CV < data.frame(CV = CV, EMA.L = NA, EMA.U = NA, EMA.cap = "",
res HC.L = NA, HC.U = NA, HC.cap = "",
GCC.L = NA, GCC.U = NA, GCC.cap = "",
stringsAsFactors = FALSE) # this line for R <4.0.0
for (i in seq_along(CV)) {
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[i,
}$EMA.cap[res$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"
resprint(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 upper cap for the EMA is at 50% (expanded limits 69.84–143.19%), for Health Canada it is at ~57.4% (expanded limits 66.7–150.0%). The GCC has no upper cap (fixed widened limits 75.00–133.33%).
For the FDA there is no upper cap (scaling is unlimited).
< data.frame(CV = c(0.25, se2CV(0.25), 0.275, 0.3, 0.5, 1.0),
res impl.L = NA, impl.U = NA, cap = "",
stringsAsFactors = FALSE) # this line for R <4.0.0
for (i in 1:nrow(res)) {
2:3] < sprintf("%.4f", scABEL(CV = res$CV[i],
res[i, regulator = "FDA"))
}$cap[res$CV <= 0.30] < "lower"
res$CV < sprintf("%.3f", res$CV)
resprint(res, row.names = FALSE)
# CV impl.L impl.U cap
# 0.250 0.8000 1.2500 lower
# 0.254 0.8000 1.2500 lower
# 0.275 0.8000 1.2500 lower
# 0.300 0.8000 1.2500 lower
# 0.500 0.6560 1.5245
# 1.000 0.4756 2.1025
< data.frame(CV = c(0.25, se2CV(0.25), 0.275, 0.3, 0.5, 1.0),
res des.L = NA, des.U = NA, cap = "",
stringsAsFactors = FALSE) # this line for R <4.0.0
for (i in 1:nrow(res)) {
if (CV2se(res$CV[i]) <= 0.25) {
2:3] < sprintf("%.4f", c(0.80, 1.25))
res[i, else {
} 2:3] < sprintf("%.4f",
res[i, exp(c(1, +1)*(log(1.25)/0.25)*CV2se(res$CV[i])))
}
}$cap[res$CV <= 0.30] < "lower"
res$CV < sprintf("%.3f", res$CV)
resprint(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
< c("EMA", "HC", "GCC", "FDA")
reg for (i in 1:4) {
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
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