power.tsd.KM {Power2Stage}

Power calculation of adaptive 2-stage BE studies (2x2 crossover) (variant of power.2stage to obtain the results of Karalis / Macheras)

Description

This function calculates the ‘empiric’ power of 2-stage BE studies according to Potvin et al. via simulations. The Potvin methods are modified as described by Karalis & Macheras to include a futility criterion Nmax and to perform the power calculation steps and the sample size estimation step in the decision schemes with the MSE (calculated from CV) and the point estimate (PE) of T/R from stage 1.

Usage

power.tsd.KM(method = c("C", "B"), alpha0 = 0.05, alpha = c(0.0294, 0.0294),
             n1, CV, targetpower = 0.8, pmethod = c("nct", "exact"),
             Nmax = 150, 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.
Default is ⁠"C"⁠ aka TSD in the paper of Karalis & Macheras if setting ⁠alpha=c(0.0294, 0.0294)⁠.
TSD-1 of Karalis can be obtained by choosing ⁠"C"⁠ but setting ⁠alpha=c(0.028, 0.028)⁠.
TSD-2 of Karalis can be obtained by choosing ⁠"B"⁠ and setting ⁠alpha=c(0.0294, 0.0294)⁠.

alpha0

Alpha value for the first step(s) in Potvin C aka TSD of Karalis & Macheras or TSD-1 of Karalis, the power inspection and BE decision if power > targetpower.
Defaults to 0.05.

alpha

Vector (two elements) of the nominal alphas for the two stages.
Defaults to Pocock’s alpha setting ⁠alpha=c(0.0294, 0.0294)⁠ as in TSD of Karalis & Macheras.

n1

Sample size of stage 1.

CV

Coefficient of variation of the intra-subject variability (use e.g., 0.3 for 30%).

targetpower

Power threshold in the first step of Potvin ⁠"C"⁠ 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 and ⁠"exact"⁠ (exact calculations via Owen’s Q).
Defaults to ⁠"nct"⁠ as a reasonable compromise between speed and accuracy in the sample size estimation step.

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. Defaults to 150, as recommended by Karalis & Macheras.
Set this argument to ⁠Inf⁠, to work without that futility criterion.

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).

In contrast to Potvin et al. the power calculation steps as well as the sample size adaption step of the decision schemes are done using the MSE (calculated from CV) and the point estimate from stage 1.
This resembles the methods described in Karalis & Macheras and Karalis.

Value

Returns an object of class ⁠"pwrtsd"⁠ with all the input arguments and results as components.
The class ⁠"pwrtsd"⁠ has a 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).

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 ⁠is.finite(Nmax)⁠.

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

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

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

Examples

# using all the defaults
# but too low number of sims to complain with the CRAN policy:
# "check time only a few seconds per example"
# minimum number of sims should be 1E5 for power, 1E6 sims for 'alpha'
power.tsd.KM(n1=16, CV=0.2, nsims=1E4)
# ~3 sec if nsims=1E5