power.tsd.fC: Power calculation of adaptive 2-stage BE studies (2x2...
In Power2Stage: Power and Sample-Size Distribution of 2-Stage Bioequivalence Studies

Description Usage Arguments Details Value Author(s) References See Also Examples

This function calculates the ‘empiric’ power of 2-stage BE studies according to Potvin et al. ‘method B/C’ via simulations. The Potvin methods are modified to include a futility criterion for the point estimate or for its 90%CI and to allow the sample size estimation step to be done with the point estimate (PE) and MSE of stage 1.

power.tsd.fC(method = c("B", "C", "B0"), alpha0 = 0.05, alpha = c(0.0294, 0.0294),
             n1, CV, GMR, targetpower = 0.8, pmethod = c("nct", "exact", "shifted"),
             usePE = FALSE, powerstep = TRUE, min.n2=0, max.n=Inf,
             fCrit=c("CI", "PE"), fClower, fCupper, theta0, theta1, theta2,
             npct = c(0.05, 0.5, 0.95), nsims, setseed = TRUE, details = FALSE)

`method`	Decision schemes according to Potvin et.al. (defaults to `"B"`). Montague’s ‘Method D’ can be obtained by choosing `"C"` but setting `alpha=c(0.028, 0.028)`. ‘Method E’ of Xu et al. can be obtained by choosing `"B"` and setting alphas, futility criterion `"CI"`, `max.n`, and `n1` according to the reference. ‘Method F’ can be obtained choosing `"C"` with the appropriate design setting according to the reference. `method="B0"` uses the decision scheme of Zheng et al. MSDBE (modified sequential design for BE studies) which differs from B in case of different alphas w.r.t. power monitoring and BE decision in case of power >= target power.
`alpha0`	Alpha value for the first step(s) in Potvin `"C"`, the power inspection and BE decision if power > targetpower. Defaults to 0.05. Only observed if `method="C"`
`alpha`	Vector (two elements) of the nominal alphas for the two stages. Defaults to Pocock’s setting `alpha=c(0.0294, 0.0294)`. Common values together with other arguments are: `rep(0.0294, 2)`: Potvin et al. ‘Method B’ `(fCrit="CI", fCupper=Inf)` `rep(0.0269, 2)`: Fulgsang ‘Method C/D’ `(method="C", GMR=0.9, targetpower=0.9, fCrit="CI", fCupper=Inf)` `rep(0.0274, 2)`: Fuglsang ‘Method C/D’ `(method="C", targetpower=0.9, fCrit="CI", fCupper=Inf)` `rep(0.0280, 2)`: Montague et al. ‘Method D’ `(method="C", GMR=0.9, fCrit="CI", fCupper=Inf)` `rep(0.0284, 2)`: Fulgsang ‘Method B’ `(GMR=0.9, targetpower=0.9, fCrit="CI", fCupper=Inf)` `rep(0.0304, 2)`: Kieser & Rauch `(fCrit="CI", fCupper=Inf)` `c(0.01, 0.04)`: Zheng et al. ‘MSDBE’ `(method="B0", fCrit="CI", fCupper=Inf)` `c(0.0249, 0.0357)`: Xu et al. ‘Method E’ for CV 10–30% `(fCrit="CI", fClower=0.9374, max.n=42)` `c(0.0254, 0.0363)`: Xu et al. ‘Method E’ for CV 30–55% `(fCrit="CI", fClower=0.9305, max.n=42)` `c(0.0248, 0.0364)`: Xu et al. ‘Method F’ for CV 10–30% `(method="C", fCrit="CI", fClower=0.9492, max.n=180)` `c(0.0259, 0.0349)`: Xu et al. ‘Method F’ for CV 30–55% `(method="C", fCrit="CI", fClower=0.9350, max.n=180)`
`n1`	Sample size of stage 1. For Xu’s methods the recommended sample size should be at least 18 (if CV 10–30%) or 48 (if CV 30–55%).
`CV`	Coefficient of variation of the intra-subject variability (use e.g., 0.3 for 30%).
`GMR`	Ratio T/R to be used in decision scheme (power calculations in stage 1 and sample size estimation for stage 2).
`targetpower`	Power threshold in the power monitoring steps and power to achieve in the sample size estimation step.
`pmethod`	Power calculation method, also to be used in the sample size estimation for stage 2. Implemented are `"nct"` (approximate calculations via non-central t-distribution, `"exact"` (exact calculations via Owen’s Q), and `"shifted"` (approximate calculation via shifted central t-distribution like in the paper of Potvin et al. Defaults to `"nct"` as a reasonable compromise between speed and accuracy in the sample size estimation step.
`usePE`	If `TRUE` the sample size estimation step is done with MSE and PE of stage 1. Defaults to `FALSE`, i.e., the sample size is estimated with anticipated (fixed) `GMR` given as argument and MSE of stage 1 (analogous to Potvin et. al.).
`powerstep`	If `TRUE` (the default) the interim power monitoring step in stage 1 evaluation of ‘method B’ will be done as described in Potvin et.al. Setting this argument to `FALSE` will omit this step. Has no effect if `method="C"` is choosen.
`min.n2`	Minimum sample size of stage 2. Defaults to zero. If the sample size estimation step gives `N < n1+min.n2` the sample size for stage 2 will be forced 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. Attention! `max.n` here is not a futility criterion like `Nmax` in other functions of this package.
`fCrit`	Futility criterion. If set to `"PE"` the study stops after stage 1 if not BE and if the point estimate (PE) of stage 1 evaluation is outside the range defined in the next two arguments `"fClower"` and `"fCupper"`. If set to `"CI"` the study stops after stage 1 if not BE and if the confidence interval of stage 1 evaluation is outside the range defined in the next two arguments. Defaults to `"PE"`. Futility criterion to use for `PE` or `CI`.
`fClower`	Lower futility limit for the `PE` or `CI` of stage 1. If the `PE` or `CI` is outside `fClower` ... `fCupper` the study is stopped in the interim with the result FAIL (not BE). May be missing. Defaults then to 0.8 if `fCrit="PE"` or 0.925 if `fCrit="CI"`.
`fCupper`	Upper futility limit for the `PE` or `CI` of stage 1. Will be set to `1/fClower` if missing.
`theta0`	Assumed ratio of geometric means (T/R) for simulations. If missing, defaults to `GMR`.
`theta1`	Lower bioequivalence limit. Defaults to 0.8.
`theta2`	Upper bioequivalence limit. Defaults to 1.25.
`npct`	Percentiles to be used for the presentation of the distribution of `n(total)=n1+n2`. Defaults to `c(0.05, 0.5, 0.95)` to obtain the 5% and 95% percentiles and the median.
`nsims`	Number of studies to simulate. If missing, `nsims` is set to 1E+05 = 100,000 or to 1E+06 = 1 Mio if estimating the empiric Type I Error (`'alpha'`), i.e., with `theta0` at the border or outside the acceptance range `theta1` ... `theta2`.
`setseed`	Simulations are dependent on the starting point of the (pseudo) random number generator. To avoid differences in power for different runs a `set.seed(1234567)` is issued if `setseed=TRUE`, the default. Set this argument to `FALSE` to view the variation in power between different runs.
`details`	If set to `TRUE` the function prints the results of time measurements of the simulation steps. Defaults to `FALSE`.

The calculations follow in principle the simulations as described in Potvin et al.
The underlying subject data are assumed to be evaluated after log-transformation. But instead of simulating subject data, the statistics pe1, mse1 and pe2, SS2 are simulated via their associated distributions (normal and χ² distributions).

Returns an object of class "pwrtsd" with all the input arguments and results as components.
The class "pwrtsd" has an S3 print method.
The results are in the components:

`pBE`	Fraction of studies found BE.
`pBE_s1`	Fraction of studies found BE in stage 1.
`pct_s2`	Percentage of studies continuing to stage 2.
`nmean`	Mean of n(total), aka average total sample size (ASN).
`nrange`	Range (min, max) of n(total).
`nperc`	Percentiles of the distribution of n(total).
`ntable`	Object of class `"table"` summarizing the discrete distribution of n(total) via its distinct values and counts of occurences of these values. This component is only given back if `usePE==FALSE` or `usePE==TRUE & fClower>0 & is.finite(fCupper)`, i.e., a futility range is used.

D. Labes

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. 2011; 11(1):8–13. doi: 10.1002/pst.483

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

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

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

Xu J, Audet C, DiLiberti CE, Hauck WW, Montague TH, Parr TH, Potvin D, Schuirmann DJ. Optimal adaptive sequential designs for crossover bioequivalence studies.
Pharm Stat. 2016;15(1):15–27. doi: 10.1002/pst.1721

power.tsd

# using all the defaults
power.tsd.fC(CV=0.25, n1=24)
# run-time ~1 sec
## Not run: 
# as above but storing the results
res <- power.tsd.fC(CV=0.25, n1=24)
# representation of the discrete distribution of n(total)
# via plot method of object with class "table" which creates a
# 'needle' plot
plot(res$ntable/sum(res$ntable), ylab="Density",
     xlab=expression("n"[total]), las=1,
     main=expression("Distribution of n"[total]))
## End(Not run)