check.TSD: Checks Acceptability of Two-Stage Sequential Designs in...
In Helmut01/AdaptiveBE: Acceptability of Adaptive Bioequivalence Studies

Description Usage Arguments Details Note Author(s) References Examples

This function assesses the pre-specified alpha(s) for the BE decision in Adaptive Sequential Two-Stage Designs (TSDs) with sample size re-estimation based on simulations in order to control the consumer risk α at the nominal level.

check.TSD(Var1, PE1, n1, Var, PE, N, type = 1, usePE = FALSE, GMR,
          alpha0 = 0.05, alpha1 = 0.0294, alpha2 = 0.0294, theta1, theta2,
          target = 0.80, pmethod = c("shifted", "nct", "exact"),
          int.pwr = TRUE, min.n2 = 0, max.n = Inf, Nmax = Inf,
          fCrit = c("PE", "CI"), fClow = 0, nsims = 1e6, setseed = TRUE,
          tol = 1e-8, pa = FALSE, skip = TRUE, algo = 1, plot.it = FALSE,
          valid = FALSE, expl = 3, Xover = TRUE, stop1 = FALSE,
          KM = FALSE, KM.des = c("TSD", "TSD-1", "TSD-2"), CIs = FALSE)

`Var1`	Vector of observed variability in the first stage. If two elements are given: numeric value `"CV"` coefficient of variation (use e.g., 0.3 for 30%) or `"MSE"` (residual error from ANOVA) If four elements are given, the CV is calculated from the confidence interval by PowerTOST’s function `CVfromCI()`: lower confidence limit (use e.g., 0.81 for 81%) upper confidence limit (use e.g., 1.15 for 115%) alpha `"CI"`
`PE1`	Observed point estimate in the first stage. Must be given if a two-element vector of `Var1` is given. numeric value (use e.g., 0.96 for 96%) `"ratio"` or `"difflog"` (difference of LSMs in log-scale) In the case of a four-element vector of `Var1`, PE1 will be calculated as √(Var1[1]Var1[2])*.
`n1`	Sample size of the first stage of the study. In a parallel design with unequal group sizes may be given as a two-element vector where the first element is the group-size under Test and the second one under Reference. If `n1` leads to an unbalanced design (i.e., is not a multiple of two), the code tries to keep sequences (or groups) as balanced as possible.
`Var`	Vector of observed variability in the final (pooled) analysis. If two elements are given: numeric value `"CV"` coefficient of variation (use e.g., 0.3 for 30%) or `"MSE"` (residual error from ANOVA) If four elements are given, the CV is calculated from the confidence interval (taking the degrees of freedom of the stage-term into account): lower confidence limit (use e.g., 0.83 for 83%) upper confidence limit (use e.g., 1.13 for 113%) alpha `"CI"`
`PE`	Observed point estimate in the final (pooled) analysis. Must be given if a two-element vector of `Var` is given. numeric value (use e.g., 0.96 for 96%) `"ratio"` or `"difflog"` (difference of LSMs in log-scale) In the case of a four-element vector of `Var`, PE will be calculated as √(Var[1]Var[2])*.
`N`	Total sample size of the the study. If `N` leads to an unbalanced design (i.e., is not a multiple of two), the code tries to keep sequences (or groups) as balanced as possible.
`type`	‘Type’ of the Two-Stage Design (`1`, `2`, or `"MSDBE"`). For the definition see Schütz. Defaults to `1` if not given explicitly.
`usePE`	If `TRUE` the sample size estimation step is done with `PE1` and the MSE of stage 1 (i.e., fully adaptive). Defaults to `FALSE`, (i.e., the sample size is estimated with the fixed `GMR` and MSE of stage 1. NB: The power inspection steps in the Potvin methods are always done with the `GMR` argument and MSE (CV) of stage 1.
`GMR`	‘True’ or assumed bioavailability ratio for the sample size re-estimation. `0.95`: Potvin et al., Zheng et al., Fuglsang B and C/D (1), Fuglsang (parallel), Karalis & Macheras, Karalis, Xu et al. `0.90`: Montague D, Fuglsang C/D (2)
`alpha0`	Type I error (TIE) probability (α; nominal level of the test). Per convention commonly set to `0.05`.
`alpha1`	Specified adjusted α of the test in the interim. Defaults to `0.0294`. Common values: `0.0010`: Haybittle/Peto `0.0050`: O’Brien/Fleming `0.0100`: Zheng et al. `0.0248`: Xu et al. Method F (‘Type 2’) for CV 10–30% `0.0249`: Xu et al. Method E (‘Type 1’) for CV 10–30% `0.0254`: Xu et al. Method E (‘Type 1’) for CV 30–55% `0.0259`: Xu et al. Method F (‘Type 2’) for CV 30–55% `0.0269`: Fulgsang C/D (GMR 0.90, power 0.90) `0.0274`: Fuglsang C/D (GMR 0.95, power 0.90) `0.0280`: Montague D, Karalis TSD-1 `0.0284`: Fuglsang B (GMR 0.95, power 0.90) `0.0294`: Potvin B, Karalis & Macheras TSD, Karalis TSD-2, Fuglsang (parallel) `0.0304`: Kieser & Rauch
`alpha2`	Specified adjusted α of the test in the final (pooled) analysis. Defaults to `0.0294`. Common values: `0.0400`: Zheng et al. `0.0480`: O’Brien/Fleming `0.0490`: Haybittle/Peto `0.0269`: Fuglsang C/D (GMR 0.90, power 0.90) `0.0274`: Fuglsang C/D (GMR 0.95, power 0.90) `0.0280`: Montague D, Karalis TSD-1 `0.0284`: Fulgsang B (GMR 0.95, power 0.90) `0.0294`: Potvin B, Karalis & Macheras TSD, Karalis TSD-2, Fuglsang (parallel) `0.0304`: Kieser & Rauch `0.0349`: Xu et al. Method F (‘Type 2’) for CV 30–55% `0.0357`: Xu et al. Method E (‘Type 1’) for CV 30–55% `0.0363`: Xu et al. Method E (‘Type 1’) for CV 10–30% `0.0364`: Xu et al. Method F (‘Type 2’) for CV 10–30%
`theta1`	Lower acceptance limit for BE. Defaults to `0.80` if not given explicitly.
`theta2`	Upper acceptance limit for BE. Defaults to `1.25` if not given explicitly.
`target`	Power threshold in the first step of ‘Type 1’ designs and power to achieve in the sample size re-estimation step.
`pmethod`	Power calculation method; also to be used in the sample size re-estimation for stage 2. Implemented are `"nct"` (approximate calculations via the non-central t-distribution) and `"exact"` (exact calculations via Owen’s Q-functions), as well as `"shifted"` (approximation via the shifted central t-distribution, as used in most references). Defaults to `"nct"` as a reasonable compromise between speed and accuracy in the power and sample size estimation step. `"shifted"` is not state of the art and should be mainly used in validation. It is likely that in the study one of the better methods was used – which are available in current software; `"nct"` is available in e.g., FARTSSIE, PASS, SAS, and package `PowerTOST`, whereas `"exact"` is available in NQuery, SAS, and package `PowerTOST`. Caution: Adjusting α with `pmethod="exact"` can take hours.
`int.pwr`	If `TRUE` (the default) the interim power monitoring step in the stage 1 evaluation of ‘Type 1’ TSDs will be done as described in all [sic] references. Setting this argument to `FALSE` will omit this step (to satisfy some assessors of the Dutch MEB). Has no effect if `type=2` is choosen.
`min.n2`	Minimum sample size of stage 2. Defaults to 0 (as in all references). If the sample size estimation step gives `N < n1` the sample size for stage 2 will be set to `min.n2`, i.e., the total sample size to `n1+min.n2`.
`max.n`	If `max.n` is set to a finite value the re-estimated total sample size (`N`) is set to `min(max.n, N)`. Defaults to `Inf` which is equivalent to not constrain the re-estimated sample size. Caution: `max.n` here is not a futility criterion like `Nmax`.
`Nmax`	Futility criterion. If set to a finite value all simulated studies in which a sample size `>Nmax` is obtained will be regarded as failed in the interim. Mandatory if `KM=TRUE` (Karalis & Macheras recommend 150). Defaults to `Inf` (unconstrained total sample size).
`fCrit`	Futility criterion to use for the point estimate or confidence interval in the interim. Acceptable values are `"PE"` or `"CI"`. A suitable value must be given in the argument `fClow`.
`fClow`	Lower futility limit for `fCrit` in the interim. If the point estimate or confidence interval (as specified by `fCrit`) is outside `fClow` ... `1/fClow` the study is stopped in the interim with the result FAIL (not BE). May be missing. Defaults then to 0, i.e., no futility rule is applied. Values if `fCrit == "PE"`: `0.80`: Armitage, Karalis & Macheras, Karalis `0.85`: Charles Bon (AAPS Annual Meeting 2007) Values if `fCrit == "CI"`: `0.9250`: Detlew Labes (personal communication with Diane Potvin) `0.9374`: Xu et al. Method E (‘Type 1’) for CV 10–30% `0.9305`: Xu et al. Method E (‘Type 1’) for CV 30–55% `0.9492`: Xu et al. Method F (‘Type 2’) for CV 10–30% `0.9350`: Xu et al. Method F (‘Type 2’) for CV 30–55%
`nsims`	Number of simulations to be performed to estimate the (‘empiric’) TIE error and in optimizing adjusting α. The default value 1,000,000 = 1e+6 should not be lowered.
`setseed`	Simulations are dependent on the starting point of the (pseudo)random number generator.To avoid differences in power for different runs, `set.seed(1234567)` is issued if `TRUE` (default). Set to `FALSE` in order to assess robustness (i.e., a different seed is issued in every call of the function).
`tol`	Desired accuracy (convergence tolerance) of `uniroot()`. The objective function is \|TIE-alpha0\| ≤ tol. Defaults to 1e-8.
`pa`	Should results of the power analysis for adjusted α be shown? Defaults to `FALSE`.
`skip`	Should optimization of α be skipped if the TIE with the specified α already preserves the consumer risk (`alpha0`)? Defaults to `TRUE` for speed reasons.
`algo`	Defaults to `1`: Optimization by `uniroot()`. If set to `2` an attempt will be made to further improve the estimate by assessing a set of adjusted alphas close to the first estimated α. Easily doubles the computation time.
`plot.it`	Should a comparative plot of PE and CI int the final analysis be made? Defaults to `FALSE`. Only applicable if α was optimized (i.e., if the specified α inflates the TIE).
`valid`	Should one of the validation examples be assessed? Defaults to `FALSE`. If set to `TRUE` one of the built-in examples must be specified in the argument `expl`. Any of the other arguments can be additionally specified in order to assess their impact.
`expl`	Number of the validation examples (`1` to `14`). Potvin et al. Example 1, Method B (‘Type 1’), alpha1=alpha2=0.0294, GMR=0.95, target=0.80 Potvin et al. Example 1, Method C (‘Type 2’), alpha0=0.05, alpha1=alpha2=0.0294, GMR=0.95, target=0.80, stopped in the interim Potvin et al. Example 2, Method B (‘Type 1’), alpha1=alpha2=0.0294, GMR=0.95, target=0.80 Potvin et al. Example 2, Method C (‘Type 2’), alpha0=0.05, alpha1=alpha2=0.0294, GMR=0.95, target=0.80 Montague et al. Method D (‘Type 2’), alpha0=0.05, alpha1=alpha2=0.0280, GMR=0.90, target=0.80 Haybittle/Peto (‘Type 1’), alpha1=0.001, alpha2=0.049, GMR=0.95, target=0.80 Zheng et al. (MSDBE), alpha1=0.01, alpha2=0.04, GMR=0.95, target=0.80 Xu et al. Method E, high CV (‘Type 1’), alpha1=0.0254, alpha2=0.0357, GMR=0.95, target=0.80, n.max=180, futility on CI 0.9305...1/0.9305 Xu et al. Method F, high CV (‘Type 2’), alpha0=0.05, alpha1=0.0259, alpha2=0.0349, GMR=0.95, target=0.80, n.max=180, futility on CI 0.9350...1/0.9350 Fuglsang 2013, Method B (‘Type 1’), alpha1=alpha2=0.0284, GMR=0.95, target=0.90 Fuglsang 2013, Method C/D (‘Type 2’), alpha0=0.05, alpha1=alpha2=0.0269, GMR=0.90, target=0.90 Fuglsang 2014, parallel, Method C (‘Type 2’), alpha0=0.05, alpha1=alpha2=0.0294, GMR=0.95, target=0.80 (parallel design) Karalis/Macheras, TSD (‘Type 2’), alpha0=0.05, alpha1=alpha2=0.0294, target=0.80, futility on PE 0.80...1.25, Nmax=150 BEBAC internal, Method C (‘Type 2’), alpha0=0.05, alpha1=alpha2=0.0294, GMR=0.95, target=0.80, stopped in the interim
`Xover`	Defaults to `TRUE`.Set to `FALSE` if the study was performed in a parallel design.
`stop1`	Defaults to `FALSE` (i.e., the study proceeded to the second stage). Set to `TRUE` if the study stopped in the interim. Naturally in this case arguments `Var`, `PE`, and `N` are not required.
`KM`	Should a fully adaptive design according to Karalis & Macheras or Karalis be used? Defaults to `FALSE`. If set to `TRUE` the upper sample size for futility (`Nmax`) and the additional argument `KM.des` must be given.
`KM.des`	Design-specification of Karalis & Macheras or Karalis. Acceptable values: `TSD` ‘Type 2’ (Karalis & Macheras). `TSD-1` ‘Type 2’ (Karalis). `TSD-2` ‘Type 1’ (Karalis).
`CIs`	Defaults to `FALSE`. Set to `TRUE` to show the two-sided 95% CI of the empiric TIE(s).

The calculations follow in principle the simulations as described in the references.
The underlying subject data are assumed to be evaluated after log-transformation. Instead of – time-consuming – simulating subject data, the statistics PE1, PE, and MSE1, SS2 are simulated via their associated distributions (normal and χ²) as suggested by Zheng et al.

The code tries to reconstruct the applied method from the study’s conditions. In the strict sense applying a specified α is only justified if the study’s conditions were within the validated range (e.g., n1, CV1, GMR, target power, etc) given in the references.

Hoping to get in simulations an empiric Type I Error of exactly 0.05 (as – unoffically – demanded by the EMA’s Biostatistics Working Party) is futile. Already rounding the confidence interval in fixed sample designs to two digits according to the guideline (i.e., 79.995% upwards to 80.00% or 125.004% downwards to 125.00%) may transform into a TIE of up to 0.0508.
For more details see a description in the doc subdirectory of the package.
So far the author did not came across a TIE after optimization of >0.05001. The question whether this is acceptable to the BSWP remains open.

Program offered for Use without any Guarantees and Absolutely No Warranty.
No Liability is accepted for any Loss and Risk to Public Health Resulting from Use of this Code

Helmut Schütz

O’Brien PC, Fleming TR. A Multiple Testing Procedure for Clinical Trials.
Biometrics. 1979;35(3):549–56.

Armitage P. Interim Analysis in Clinical Trials.
Stat Med. 1991;10(6):925–37. doi: 10.1002/sim.4780100613

Potvin D, DiLiberti CE, Hauck WW, Parr AF, Schuirmann DJ, Smith RA. Sequential design approaches for bioequivalence studies with crossover designs.
Pharm Stat. 2008;7(4):245–62. doi: 10.1002/pst.294

Montague TH, Potvin D, DiLiberti CE, Hauck WW, Parr AF, Schuirmann DJ. Additional results for ‘Sequential design approaches for bioequivalence studies with crossover designs’.
Pharm Stat. 2012;11(1):8–13. doi: 10.1002/pst.483

Zheng Ch, Wang J, Zhao L. Testing bioequivalence for multiple formulations with power and sample size calculations.
Pharm Stat. 2012;11(4):334–41. doi: 10.1002/pst.1522

Fuglsang A. Sequential Bioequivalence Trial Designs with Increased Power and Controlled Type I Error Rates.
AAPS J. 2013;15(3):659–61. doi: 10.1208/s12248-013-9475-5

Karalis V, Macheras P. An Insight into the Properties of a Two-Stage Design in Bioequivalence Studies.
Pharm Res. 2013;30(7):1824–35. doi: 10.1007/s11095-013-1026-3

Karalis V. The role of the upper sample size limit in two-stage bioequivalence designs.
Int J Pharm. 2013;456(1):87–94. doi: 10.1016/j.ijpharm.2013.08.013

Fuglsang A. Futility Rules in Bioequivalence Trials with Sequential Designs.
AAPS J. 2014;16(1):79–82. doi: 10.1208/s12248-013-9540-0

Fuglsang A. Sequential Bioequivalence Approaches for Parallel Designs.
AAPS J. 2014;16(3):373–8. doi: 10.1208/s12248-014-9571-1

Zheng Ch, Zhao L, Wang J. Modifications of sequential designs in bioequivalence trials.
Pharm Stat. 2015;14(3):180–8. doi: 10.1002/pst.1672

Schütz H. Two-stage designs in bioequivalence trials.
Eur J Clin Pharmacol. 2015;71(3):271–81. doi: 10.1007/s00228-015-1806-2

Kieser M, Rauch G. Two-stage designs for cross-over bioequivalence trials.
Stat Med. 2015;34(16):2403–16. doi: 10.1002/sim.6487

Xu J, Audet C, DiLiberti CE, Hauck WW, Montague TH, Parr TH, Potvin D, Schuirmann DJ. Optimal adaptive sequential designs for crossover bioequivalence studies.
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# Potvin et al., Example 2, 'Method B'
# Not run: Due to timing policy of CRAN for examples.
## Not run: 
check.TSD(Var1=c(0.032634, "MSE"), PE1=c(0.08396, "difflog"), n1=12,
          Var=c(0.045896, "MSE"), PE=c(0.014439, "difflog"), N=20,
          pmethod="shifted", type=1)
# or simply using a built-in dataset
check.TSD(valid=TRUE, expl=3)
# Should give TIE 0.04307 and acceptance of the reported results.
#
# As above but variabilities and PEs calculated from CIs and alpha(s).
# Varibility as a 4-element vector, and PEs not given.
# Not run: Due to timing policy of CRAN for examples.
check.TSD(Var1=c(0.9293, 1.2728, 0.0294, "CI"), n1=12,
          Var=c(0.8845, 1.1638, 0.0294, "CI"), N=20,
          pmethod="shifted", type=1)
# Slightly different TIE (0.04300) since the precision of the input
# is limited.
#
# Potvin et al., Example 2, 'Method C'
check.TSD(Var1=c(0.032634, "MSE"), PE1=c(0.08396, "difflog"), n1=12,
          Var=c(0.045896, "MSE"), PE=c(0.014439, "difflog"), N=20,
          pmethod="shifted", type=2, plot.it=TRUE)
# or using the built-in dataset
check.TSD(valid=TRUE, expl=4)
# Should give TIE 0.05062 and optimized alpha 0.02858 (TIE 0.04992).
# Original and adjusted results agree; can accept the reported results.
#
# Xu et al., 'Method E' for high CVs
check.TSD(Var1=c(0.483, "CV"), PE1=c(0.943, "ratio"), n1=48,
          Var=c(0.429, "CV"), PE=c(1.01, "ratio"), N=104,
          pmethod="shifted", type=1, fCrit="CI", fClow=0.9305,
          alpha1=0.0254, alpha2=0.0357)
# or using the built-in dataset
check.TSD(valid=TRUE, expl=8)
## End(Not run)
# Should give TIE 0.04953 and acceptance of reported results.