power.tsd: Power calculation of adaptive 2-stage BE studies in a 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. via simulations. The Potvin methods are modified to include a futility criterion Nmax and to allow the sample size estimation step to be done with the point estimate (PE) and MSE (calculated from CV) of stage 1.

power.tsd(method = c("B", "C", "B0"), alpha0 = 0.05, alpha = c(0.0294, 0.0294), 
          n1, GMR, CV, targetpower = 0.8, pmethod = c("nct", "exact", "shifted"),
          usePE = FALSE, Nmax = Inf, min.n2 = 0, 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"`). Potvin’s ‘Method D’ can be obtained by choosing `"C"` but setting `alpha=c(0.028, 0.028)`. Montague’s ‘Method D’ can be obtained by choosing `"C"` but setting `alpha=c(0.028, 0.028)` and `GMR=0.9`. `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’ `rep(0.0269, 2)`: Fulgsang ‘Method C/D’ `(method="C", GMR=0.9, targetpower=0.9)` `rep(0.0274, 2)`: Fuglsang ‘Method C/D’ `(method="C", targetpower=0.9)` `rep(0.0280, 2)`: Montague et al. ‘Method D’ `(method="C", GMR=0.9)` `rep(0.0284, 2)`: Fulgsang ‘Method B’ `(GMR=0.9, targetpower=0.9)` `rep(0.0304, 2)`: Kieser & Rauch `c(0.01, 0.04)`: Zheng et al. ‘MSDBE’ `(method="B0")`
`n1`	Sample size of stage 1.
`GMR`	Ratio T/R to be used in decision scheme (power calculations in stage 1 and sample size estimation for stage 2).
`CV`	Coefficient of variation of the intra-subject variability (use e.g., 0.3 for 30%).
`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 `GMR` and MSE (calculated from `CV`) of stage 1 analogous to Potvin et. al. NB: The power inspection steps in the Potvin methods are always done with the `GMR` argument and MSE (`CV`) of stage 1.
`Nmax`	Futility criterion. If set to a finite value, all studies simulated in which a sample size `>Nmax` is obtained will be regarded as BE=FAIL. Set this argument to `Inf`, the default, to work without that futility criterion.
`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`.
`theta0`	True ratio of T/R for simulating. Defaults to the `GMR` argument if missing.
`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`	Vector of 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 otherwise if `is.finite(Nmax)`, i.e., a futility criterion 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. Controlling type I errors for two-stage bioequivalence study designs.
Clin Res Reg Aff. 2011; 28(4):100–5. doi: 10.3109/10601333.2011.631547

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

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

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

power.tsd.p for analogous calculations for 2-group parallel design.
power.tsd.fC for analogous calculations with futility check based on point estimate of stage 1.

# using all the defaults and 24 subjects in stage 1, CV of 25%
power.tsd(n1=24, CV=0.25)
# computation time ~ 1 sec
#
# as above, but save results for further use
res <- power.tsd(n1=24, CV=0.25)
## Not run: 
# 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]))
#
# If you prefer a histogram instead (IMHO, not the preferred plot):
# reconstruct the ntotal values from the ntable
ntot <- rep.int(as.integer(names(res$ntable)),
                times=as.integer(res$ntable))
# annotated histogram
hist(ntot, freq=FALSE, breaks=res$nrange[2]-res$nrange[1],
     xlab=expression("n"[total]), las=1,
     main=expression("Histogram of n"[total]))
abline(v=c(res$nmean, res$nperc[["50%"]]), lty=c(1, 3))
legend("topright", box.lty=0, legend=c("mean", "median"),
       lty=c(1, 3), cex=0.9)
## End(Not run)

TSD with 2x2 crossover 
Method B: alpha (s1/s2) = 0.0294 0.0294 
Target power in power monitoring and sample size est. = 0.8
Power calculation via non-central t approx. 
CV1 and GMR = 0.95 in sample size est. used
No futility criterion
BE acceptance range = 0.8 ... 1.25

CV = 0.25; n(stage 1) = 24; GMR = 0.95

1e+05 sims at theta0 = 0.95 (p(BE) = 'power').
p(BE)    = 0.84244
p(BE) s1 = 0.63203
Studies in stage 2 = 33.56%

Distribution of n(total)
- mean (range) = 29 (24 ... 86)
- percentiles
 5% 50% 95% 
 24  24  48