PowerTOST (version 1.4-6)

scABEL.ad: Iteratively adjusted alpha for ABEL

Description

This function iteratively adjusts alpha for the BE decision via Average Bioequivalence with Expanding Limits (ABEL) based on simulations in order to maintain the consumer risk at the nominal level.

Usage

scABEL.ad(alpha = 0.05, theta0, theta1, theta2, CV,
          design = c("2x3x3", "2x2x4", "2x2x3"), regulator,
          n, alpha.pre = 0.05, imax = 100, tol, print = TRUE,
          details = FALSE, setseed = TRUE, nsims = 1e+06,
          sdsims = FALSE, progress)

Arguments

alpha

Type I Error (TIE) probability (nominal level of the test). Per convention commonly set to 0.05.

theta0

‘True’ or assumed bioavailability ratio. Defaults to 0.90 if not given explicitly.

theta1

Conventional lower ABE limit to be applied in the mixed procedure if CVwR==CVswitch. Also lower limit for the point estimate constraint. Defaults to 0.80 if not given explicitly.

theta2

Conventional upper ABE limit to be applied in the mixed procedure if CVwR==CVswitch. Also upper limit for the point estimate constraint. Defaults to 1.25 if not given explicitly.

CV

Intra-subject coefficient(s) of variation as ratio (not percent).

• If given as a scalar (length(CV)==1) the same CV of Test and Reference is assumed (homoscedasticity: CVwT==CVwR).

• If given as a vector (length(CV)==2) -- assuming heteroscedasticity -- the CV of Test must be given in the first element and the one of Reference in the second.

design

Design of the study. "2x3x3" is the partial replicate design (TRR|RTR|RRT). "2x2x3" is the 2-sequence 3-period full replicate design (TRT|RTR). "2x2x4" is the 2-sequence 4-period full replicate design (TRTR|RTRT). Defaults to "2x3x3".

regulator

Regulatory settings for the widening of the BE acceptance limits. Choose from "EMA" (default) or "HC". This argument may be given also in lower case.

n

Total sample size of the study or a vector of sample size / sequences. If n leads to an unbalanced design (i.e., is not a multiple of two in the full replicate designs or not a multiple of three in the partial replicate), the code tries to keep subjects / sequence as balanced as possible. In evaluating a particular unbalanced study always give n as a vector. If design="2x2x3" (TRT|RTR) the order of sample sizes is important. n[1] is for sequence TRT and n[2] is for sequence RTR. If n is missing, a sample size is estimated with target power 0.80 and pre-specified alpha if defined. Otherwise, alpha is used.

alpha.pre

Pre-specified alpha (optional). Must be <=alpha. ABEL will be performed at level alpha.pre and the TIE assessed at level alpha. Less powerful than adjusting alpha but an alternative in the critical region of maximum inflation of the TIE. In certain scenarios Bonferroni<U+2019>s 0.025 is not sufficient to preserve the Type I Error (e.g., the third example). Not recommended if CVwR >= 0.45 due to poor power characteristics.

imax

Maximum number of steps in sample size search. Defaults to 100.

tol

Desired accuracy (convergence tolerance). Defaults to 1E-6.

print

If TRUE (default), the function sends its results to the console.

details

If TRUE, the relative change of the consumer risk in percent is shown. Additionally information about the impact on power (for specified theta0 and target power 0.80), runtime, and number of simulations (iterations) are given. Defaults to FALSE.

setseed

Simulations are dependent on the starting point of the (pseudo) random number generator. To avoid differences in power for different runs set.seed(123456) is issued if setseed=TRUE (default).

nsims

Number of simulations to be performed to estimate the (empirical) TIE error and in each iteration of adjusting alpha. The default value 1,000,000 = 1E+6 should not be lowered.

sdsims

If FALSE (default) power is estimated by the respective statistics. Recommended for the full replicate designs for speed reasons. Set to TRUE if results of power.scABEL are expected to be inaccurate (small sample sizes and/or heteroscedasticity) and subject data (via power.scABEL.sdsims) should be simulated instead. Very time consuming (easily 100times slower)! If design="2x3x3" subject simulations are recommended -- even if homoscedasticity is assumed. Subject data simulations are not supported for regulator="HC".

progress

Set to TRUE if a progress bar should be displayed. Ignored if sdsims=FALSE.

Value

Sends results to the console if argument print=TRUE (default).

Returns a list with the input, adjusted alpha, and Type I Error (for nominal and adjusted alpha) if argument print=FALSE. If no adjustment is necessary, NAs will be returned for the respective variables (alpha.adj, TIE.adj, rel.change, pwr.adj, rel.loss).

Warning

See the Warning section of the function power.scABEL concerning the power value agreement to the one obtained by simulations via subject data.

Details

The simulations are done via the distributional properties of the statistical quantities necessary for assessing BE based on ABEL. Simulations for the TIE are performed at the upper (expanded) limit U of the acceptance range. Due to the symmetry around 1 results are valid for the lower (expanded) limit L as well. U at the EMA<U+2019>s and Health Canada<U+2019>s CVswitch and CVcap:

scABEL(CV=0.3, regulator="EMA")[["upper"]]; scABEL(CV=0.3, regulator="HC")[["upper"]]
[1] 1.25
[1] 1.25
scABEL(CV=0.5, regulator="EMA")[["upper"]]; scABEL(CV=0.57382, regulator="HC")[["upper"]]
[1] 1.43191
[1] 1.5

Simulated studies are evaluated by ANOVA (‘Method A’) as recommended in the EMA<U+2019>s Q&A-document and by intra-subject contrasts if regulator="HC". Health Canada requires a mixed-effects model which cannot be implemented in R. However, intra-subjects contrasts are a sufficiently close approximation. The Type I Error in ABEL depends only on CVwR and -- to a minor degree -- the sample size. Algorithm:

1. The TIE is assessed based on alpha (or alpha.pre) and compared to the nominal level of the test alpha.

2. If no inflation of the TIE is found, the algorithm stops.

3. Otherwise, alpha is iteratively adjusted (i.e., alpha.adj <alpha) until no more relevant inflation of the TIE is detected (i.e., abs(TIE - alpha) <= tol).

References

Wonnemann M, Fr<U+00F6>mke 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. 10.1007/s11095-014-1450-z

Mu<U+00F1>oz J, Alcaide D, Oca<U+00F1>a J. Consumer<U+2019>s risk in the EMA and FDA regulatory approaches for bioequivalence in highly variable drugs Stat Med. 2015;35(12):1933--43. 10.1002/sim.6834

Labes D, Sch<U+00FC>tz 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. 10.1007/s11095-016-2006-1

European Medicines Agency, Committee for Medicinal Products for Human Use. Guideline on the Investigation of Bioequivalence London, 20 January 2010. CPMP/EWP/QWP/1401/98 Rev. 1/ Corr **

European Medicines Agency, Committee for Medicinal Products for Human Use. Questions & Answers: positions on specific questions addressed to the Pharmacokinetics Working Party (PKWP) London, 19 November 2015. EMA/618604/2008 Rev. 13

Health Canada, Therapeutic Products Directorate. Policy on Bioequivalence Standards for Highly Variable Drug Products Ottawa, 18 April 2016. 16-104293-140

See Also

sampleN.scABEL.ad, power.scABEL, power.scABEL.sdsims, scABEL

Examples

# NOT RUN {
# Using all defaults:
# TRR|RTR|RRT, target power 80% for assumed ratio 0.90 (estimated sample size 54),
# EMA regulatory settings (ABE limits and PE constraint 0.80 - 1.25),
# 1E+6 simulated studies.
# Not run: due to timing policy of CRAN for examples
# }
# NOT RUN {
scABEL.ad(CV = 0.3)
# }
# NOT RUN {
# Should result in adjusted alpha 0.03389 (TIE 0.5000, TIE for nominal alpha 0.07189).
#
# As above but subject data simulations.
# }
# NOT RUN {
scABEL.ad(CV = 0.3, sdsims = TRUE)
# }
# NOT RUN {
# Should result in adjusted alpha 0.03336 (TIE 0.5000, TIE for nominal alpha 0.07237).
#
# TRT|RTR, heteroscedasticity, sample size 48 (unbalanced), subject data simulations.
# }
# NOT RUN {
scABEL.ad(CV = c(0.25, 0.3), design = "2x2x3", n = c(23, 25), sdsims = TRUE)
# }
# NOT RUN {
# Should result in adjusted alpha 0.02465 (TIE 0.5000, TIE for nominal alpha 0.09050).
#
# TRTR|RTRT, CV 0.35, sample size 33 (unbalanced).
# }
# NOT RUN {
scABEL.ad(CV = 0.35, design = "2x2x4", n = c(16, 17))
# }
# NOT RUN {
# Should result in adjusted alpha 0.03632 (TIE 0.5000, TIE for nominal alpha 0.06544).
# }