power.tsd.fC {Power2Stage}

Power calculation of adaptive 2-stage BE studies (2x2 crossover) with a futility criterion for the point estimate of T/R or its 90% CI

Description

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.

Usage

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)

Arguments

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.

Details

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).

Value

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.

Author(s)

D. Labes

References

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

See Also

power.tsd

Examples

# 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)