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Reference-Scaled Average Bioequivalence"
In PowerTOST: Power and Sample Size for (Bio)Equivalence Studies
Details and examples of other methods are accessible via the menu bar on top of the page and in the online manual of all functions.
library(PowerTOST) # attach the library
Defaults
| 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 | 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.
Implemented Designs
# design name df
# "2x2x3" 2x2x3 replicate crossover 2n-3
# "2x2x4" 2x2x4 replicate crossover 3n-4
# "2x4x4" 2x4x4 replicate crossover 3n-4
# "2x3x3" partial replicate (2x3x3) 2n-3
# "2x4x2" Balaam’s (2x4x2) n-2
# "2x2x2r" Liu’s 2x2x2 repeated x-over 3n-2
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 ^[R Documentation. Data Frames. 2022-02-08. R-manual.] 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).
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., if CV~wR~ is below this value reference-scaling 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 mixed-effects model (HC, FDA). For the latter intra-subject contrasts are a suffiently close approximation.
Sample Size
Highly Variable Drugs / Drug Products
Estimate the sample size for assumed intra-subject CV 0.55 (CV~wT~ = CV~wR~). Employ the defaults (theta0 = 0.90, targetpower = 0.80, design = "2x3x3", nsims = 1e5).
EMA
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)
# 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’ 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).
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 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)
# 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 full replicate 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 replicate
f3.key = NA, f3.ss = NA, # 3-period full replicate
p3.key = NA, p3.ss = NA) # 3-period partial replicate
for (i in seq_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 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.
Health Canada
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 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 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)
# 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 to the 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, 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.
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 0.816080
suppressMessages(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
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):
CV <- 0.125
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) = 0.125, 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(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 the sample 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, 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 (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, σ^2^ 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 the variability of T is higher than the one of R.
Power
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().
Type I Error
Contrary to average bioequivalence, where the Null-hypothesis is based on fixed limits, in reference-scaling the Null is generated in face of the data (i.e, the limits are random variables).
Endrényi and Tóthfalusi (2009,^[Endrényi L, Tóthfalusi L. Regulatory and study conditions for the determination of bioequivalence of highly variable drugs. J Pharm Sci. 2009: 12(1); 138--49. Open access.] 2019^[Endrényi L, Tóthfalusi L. Bioequivalence for highly variable drugs: regulatory agreements, disagreements, and harmonization. J Pharmacokin Pharmacodyn. 2019: 46(2); 117--26. doi:10.1007/s10928-019-09623-w.]), Labes (2013^[Labes D. ‘alpha’ of scaled ABE? BEBA Forum. Vienna, 2013. Open access.]), Wonnemann et al. (2015^[Wonnemann M, Frömke C, Koch A. Inflation of the Type I Error: Investigations on Regulatory Recommendations for Bioequivalence of Highly Variable Drugs. Pharm Res. 2015: 32(1); 135--43. doi:10.1007/s11095-014-1450-z.]), Muñoz et al. (2016^[Muñoz J, Alcaide D, Ocaña J. Consumer’s risk in the EMA and FDA regulatory approaches for bioequivalence in highly variable drugs. Stat Med. 2016: 35(12); 1933--43. doi:10.1002/sim.6834.]),
Labes and Schütz (2016^[Labes D, Schütz H. Inflation of Type I Error in the Evaluation of Scaled Average Bioequivalence, and a Method for its Control. Pharm Res. 2016: 33(11); 2805--14. doi:10.1007/s11095-016-2006-1.]), Tóthfalusi and Endrényi (2016,^[Tóthfalusi L, Endrényi L. An Exact Procedure for the Evaluation of Reference-Scaled Average Bioequivalence. AAPS J. 2016: 18(2); 476--89. doi:10.1208/s12248-016-9873-6.] 2017^[Tóthfalusi L, Endrényi L. Algorithms for Evaluating Reference Scaled Average Bioequivalence: Power, Bias, and Consumer Risk. Stat Med. 2017: 36(27); 4378--90. doi:10.1002/sim.7440.]), Molins et al. (2017^[Molins E, Cobo E, Ocaña J. Two-Stage Designs Versus European Scaled Average Designs in Bioequivalence Studies for Highly Variable Drugs: Which to Choose? Stat Med. 2017: 36(30); 4777--88. doi:10.1002/sim.7452.]), Deng and Zhou (2019^[Deng Y, Zhou X-H. Methods to control the empirical type I error rate in average bioequivalence tests for highly variable drugs. Stat Meth Med Res. 2019: 29(6). doi:10.1177/0962280219871589.]) 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.
CV <- 0.35
res <- 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 than the nominal $\small{\alpha}$ 0.05).
{width=450px}
A substantially higher inflation of the type I error was reported for the FDA’s model. However, Davit et al. (2012^[Davit BM, Chen ML, Conner DP, Haidar SH, Kim S, Lee CH, Lionberger RA, Makhlouf FT, Nwakama PE, Patel DT, Schuirmann DJ, Yu LX. Implementation of a Reference-Scaled Average Bioequivalence Approach for Highly Variable Generic Drug Products by the US Food and Drug Administration. AAPS J. 2012: 14(4); 915--24. doi:10.1208/s12248-012-9406-x.]) 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 ‘hocus-pocus’. However, even with this approach 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 (i in 1: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
{width=449px}
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().
Iteratively adjusted α
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.
CV <- 0.45
n <- 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 <- 0.35
n <- 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 $\small{\alpha}$ 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().
CV <- 0.35
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)
# 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 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.
‘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 <- 0.35
n <- sampleN.scABEL(CV = CV, design = "2x2x4", print = FALSE,
details = FALSE)[["Sample size"]]
U <- scABEL(CV = CV)[["upper"]]
# 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.
Helpers
BE limits
Expanded Limits (EMA, Health Canada), widened limits (GCC){#ExpLim}
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 <4.0.0
for (i in seq_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 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\%).
FDA
For the FDA there is no upper cap (scaling is unlimited).
‘Implied’ Limits{#ImpLim}
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 <4.0.0
for (i in 1: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$CV)
print(res, row.names = FALSE)
# CV impl.L impl.U cap
# 0.250 0.8000 1.2500 lower
# 0.254 0.8000 1.2500 lower
# 0.275 0.8000 1.2500 lower
# 0.300 0.8000 1.2500 lower
# 0.500 0.6560 1.5245
# 1.000 0.4756 2.1025
Limits of the ‘desired consumer risk model’{#DesCRM}
res <- data.frame(CV = c(0.25, se2CV(0.25), 0.275, 0.3, 0.5, 1.0),
des.L = NA, des.U = NA, cap = "",
stringsAsFactors = FALSE) # this line for R <4.0.0
for (i in 1:nrow(res)) {
if (CV2se(res$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 (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
Authors{#authors}
|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, Benjamin Lang|
|sampleN.scABEL.ad, sampleN.scABEL.sdsims, sampleN.RSABE2L.sdsims,
scABEL.ad|Helmut Schütz|
License{#license}
GPL-3 2022-02-19 Helmut Schütz
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PowerTOST documentation built on March 18, 2022, 5:47 p.m.
R Package Documentation
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Details and examples of other methods are accessible via the menu bar on top of the page and in the online manual of all functions.
library(PowerTOST) # attach the library
Defaults
| 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 | 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.
Implemented Designs
# design name df # "2x2x3" 2x2x3 replicate crossover 2n-3 # "2x2x4" 2x2x4 replicate crossover 3n-4 # "2x4x4" 2x4x4 replicate crossover 3n-4 # "2x3x3" partial replicate (2x3x3) 2n-3 # "2x4x2" Balaam’s (2x4x2) n-2 # "2x2x2r" Liu’s 2x2x2 repeated x-over 3n-2
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 ^[R Documentation. Data Frames. 2022-02-08. R-manual.] 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).
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., if CV~wR~ is below this value reference-scaling 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 mixed-effects model (HC, FDA). For the latter intra-subject contrasts are a suffiently close approximation.
Sample Size
Highly Variable Drugs / Drug Products
Estimate the sample size for assumed intra-subject CV 0.55 (CV~wT~ = CV~wR~). Employ the defaults (theta0 = 0.90, targetpower = 0.80, design = "2x3x3", nsims = 1e5).
EMA
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) # 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’ 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).
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 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) # 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 full replicate 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 replicate f3.key = NA, f3.ss = NA, # 3-period full replicate p3.key = NA, p3.ss = NA) # 3-period partial replicate for (i in seq_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 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.
Health Canada
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 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 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) # 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 to the 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, 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.
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 0.816080 suppressMessages(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
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):
CV <- 0.125 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) = 0.125, 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(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 the sample 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, 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 (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, σ^2^ 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 the variability of T is higher than the one of R.
Power
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().
Type I Error
Contrary to average bioequivalence, where the Null-hypothesis is based on fixed limits, in reference-scaling the Null is generated in face of the data (i.e, the limits are random variables).
Endrényi and Tóthfalusi (2009,^[Endrényi L, Tóthfalusi L. Regulatory and study conditions for the determination of bioequivalence of highly variable drugs. J Pharm Sci. 2009: 12(1); 138--49. Open access.] 2019^[Endrényi L, Tóthfalusi L. Bioequivalence for highly variable drugs: regulatory agreements, disagreements, and harmonization. J Pharmacokin Pharmacodyn. 2019: 46(2); 117--26. doi:10.1007/s10928-019-09623-w.]), Labes (2013^[Labes D. ‘alpha’ of scaled ABE? BEBA Forum. Vienna, 2013. Open access.]), Wonnemann et al. (2015^[Wonnemann M, Frömke C, Koch A. Inflation of the Type I Error: Investigations on Regulatory Recommendations for Bioequivalence of Highly Variable Drugs. Pharm Res. 2015: 32(1); 135--43. doi:10.1007/s11095-014-1450-z.]), Muñoz et al. (2016^[Muñoz J, Alcaide D, Ocaña J. Consumer’s risk in the EMA and FDA regulatory approaches for bioequivalence in highly variable drugs. Stat Med. 2016: 35(12); 1933--43. doi:10.1002/sim.6834.]), Labes and Schütz (2016^[Labes D, Schütz H. Inflation of Type I Error in the Evaluation of Scaled Average Bioequivalence, and a Method for its Control. Pharm Res. 2016: 33(11); 2805--14. doi:10.1007/s11095-016-2006-1.]), Tóthfalusi and Endrényi (2016,^[Tóthfalusi L, Endrényi L. An Exact Procedure for the Evaluation of Reference-Scaled Average Bioequivalence. AAPS J. 2016: 18(2); 476--89. doi:10.1208/s12248-016-9873-6.] 2017^[Tóthfalusi L, Endrényi L. Algorithms for Evaluating Reference Scaled Average Bioequivalence: Power, Bias, and Consumer Risk. Stat Med. 2017: 36(27); 4378--90. doi:10.1002/sim.7440.]), Molins et al. (2017^[Molins E, Cobo E, Ocaña J. Two-Stage Designs Versus European Scaled Average Designs in Bioequivalence Studies for Highly Variable Drugs: Which to Choose? Stat Med. 2017: 36(30); 4777--88. doi:10.1002/sim.7452.]), Deng and Zhou (2019^[Deng Y, Zhou X-H. Methods to control the empirical type I error rate in average bioequivalence tests for highly variable drugs. Stat Meth Med Res. 2019: 29(6). doi:10.1177/0962280219871589.]) 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.
CV <- 0.35 res <- 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 than the nominal $\small{\alpha}$ 0.05).
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A substantially higher inflation of the type I error was reported for the FDA’s model. However, Davit et al. (2012^[Davit BM, Chen ML, Conner DP, Haidar SH, Kim S, Lee CH, Lionberger RA, Makhlouf FT, Nwakama PE, Patel DT, Schuirmann DJ, Yu LX. Implementation of a Reference-Scaled Average Bioequivalence Approach for Highly Variable Generic Drug Products by the US Food and Drug Administration. AAPS J. 2012: 14(4); 915--24. doi:10.1208/s12248-012-9406-x.]) 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 ‘hocus-pocus’. However, even with this approach 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 (i in 1: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
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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().
Iteratively adjusted α
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.
CV <- 0.45 n <- 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 <- 0.35 n <- 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 $\small{\alpha}$ 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().
CV <- 0.35 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) # 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 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.
‘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 <- 0.35 n <- sampleN.scABEL(CV = CV, design = "2x2x4", print = FALSE, details = FALSE)[["Sample size"]] U <- scABEL(CV = CV)[["upper"]] # 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.
Helpers
BE limits
Expanded Limits (EMA, Health Canada), widened limits (GCC){#ExpLim}
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 <4.0.0 for (i in seq_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 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\%).
FDA
For the FDA there is no upper cap (scaling is unlimited).
‘Implied’ Limits{#ImpLim}
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 <4.0.0 for (i in 1: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$CV) print(res, row.names = FALSE) # CV impl.L impl.U cap # 0.250 0.8000 1.2500 lower # 0.254 0.8000 1.2500 lower # 0.275 0.8000 1.2500 lower # 0.300 0.8000 1.2500 lower # 0.500 0.6560 1.5245 # 1.000 0.4756 2.1025
Limits of the ‘desired consumer risk model’{#DesCRM}
res <- data.frame(CV = c(0.25, se2CV(0.25), 0.275, 0.3, 0.5, 1.0), des.L = NA, des.U = NA, cap = "", stringsAsFactors = FALSE) # this line for R <4.0.0 for (i in 1:nrow(res)) { if (CV2se(res$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 (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
Authors{#authors}
|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, Benjamin Lang|
|sampleN.scABEL.ad, sampleN.scABEL.sdsims, sampleN.RSABE2L.sdsims,scABEL.ad|Helmut Schütz|
License{#license}
GPL-3 2022-02-19 Helmut Schütz
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