power.TOST {PowerTOST}

Power of the classical TOST procedure

Description

Calculates the exact or approximate power of the two-one-sided t-tests (TOST) procedure for various study designs used in BE studies.

Usage

power.TOST(alpha = 0.05, logscale = TRUE, theta1, theta2, theta0, CV, n, 
           design = "2x2", method="exact", robust=FALSE)

Arguments

alpha

Significance level (one-sided). Commonly set to 0.05.

logscale

Should the data used on log-transformed or on original scale? TRUE (default) or FALSE.

theta0

‘True’ or assumed T/R ratio or difference.
In case of logscale=TRUE it must be given as ratio T/R.
If logscale=FALSE, the difference in means. In this case, the difference may be expressed in two ways: relative to the same (underlying) reference mean, i.e. as (T-R)/R = T/R - 1; or as difference in means T-R. Note that in the former case the units of CV, theta1 and theta2 need also be given relative to the reference mean (specified as ratio).
Defaults to 0.95 if logscale=TRUE or to 0.05 if logscale=FALSE

theta1

Lower (bio-)equivalence limit.
In case of logscale=TRUE it is given as ratio.
If logscale=FALSE, the limit may be expressed in two ways: difference of means relative to the same (underlying) reference mean or in units of the difference of means. Note that in the former case the units of CV, theta0 and theta2 need also be given relative to the reference mean (specified as ratio).
Defaults to 0.8 if logscale=TRUE or to -0.2 if logscale=FALSE.

theta2

Upper (bio-)equivalence limit.
In case of logscale=TRUE it is given as ratio. If logscale=FALSE, the limit may be expressed in two ways: difference of means relative to the same (underlying) reference mean or in units of the difference of means. Note that in the former case the units of CV, theta0 and theta1 need also be given relative to the reference mean (specified as ratio).
If not given, theta2 will be calculated as 1/theta1 if logscale=TRUE or as -theta1 if logscale=FALSE.

CV

In case of logscale=TRUE the (geometric) coefficient of variation given as ratio.
If logscale=FALSE the argument refers to (residual) standard deviation of the response. In this case, standard deviation may be expressed two ways: relative to a reference mean (specified as ratio sigma/muR), i.e. again as a coefficient of variation; or untransformed, i.e. as standard deviation of the response. Note that in the former case the units of theta0, theta1 and theta2 need also be given relative to the reference mean (specified as ratio).

In case of cross-over studies this is the within-subject CV, in case of a parallel-group design the CV of the total variability.

n

Number of subjects under study.
Is total number if given as scalar, else number of subjects in the (sequence) groups. In the latter case the length of n vector has to be equal to the number of (sequence) groups.

design

Character string describing the study design.
See known.designs() for designs covered in this package.

method

Method for calculation of the power.
Defaults to "exact" in which case the calculation is done based on formulas with Owen’s Q. The calculation via Owen’s Q can also be choosen with method="owenq".
Another exact method via direct integration of the bivariate non-central t-distribution may be chosen with method="mvt". This may have somewhat lower precision compared to Owen’s Q and longer run-time.
Approximate calculations can be choosen via method="noncentral" or method="nct" for the approximation using the non-central t-distribution. With method="central" or method="shifted" the relative crude approximation via ‘shifted’ central t-distribution is chosen.
The strings for method may be abbreviated.

robust

Defaults to FALSE. With that value the usual degrees of freedom will be used.
Set to TRUE will use the degrees of freedom according to the ‘robust’ evaluation (aka Senn’s basic estimator). These degrees of freedom are calculated as n-seq.
See known.designs()$df2 for designs covered in this package.
Has only effect for higher-order crossover designs.

Details

The exact calculations of power are based on Owen’s Q-function or by direct integration of the bivariate non-central t-distribution via function pmvt of package mvtnorm.
Approximate power is implemented via the non-central t-distribution or the ‘shifted’ central t-distribution.

The formulas cover balanced and unbalanced studies w.r.t (sequence) groups.

In case of parallel group design and higher order crossover designs (replicate crossover or crossover with more than two treatments) the calculations are based on the assumption of equal variances for Test and Reference products under consideration.

The formulas for the paired means 'design' do not take a correlation parameter into account. They are solely based on the paired t-test (TOST of differences = zero).

Value

Value of power according to the input arguments.

Note

Of course it is highly recommended to use the default method="exact" :-).
There is no reason beside testing and for comparative purposes to use an approximation if the exact method is available.

If n is given as scalar (total sample size) and this number is not divisible by the number of (sequence) groups of the design an unbalanced design with small imbalance is assumed. A corresponding message is thrown showing the assumed numbers of subjects in (sequence) groups.
The function does not vectorize properly if design is a vector.
The function vectorizes properly if CV or theta0 are vectors.
Other vector input is not tested yet.

The former function power2.TOST() designd to handle unbalanced studies is defunct since power.TOST() handles balanced as well as unbalanced designs.

Author(s)

D. Labes, direct integration of bivariate non-central t-distribution by B. Lang

References

Phillips KF. Power of the Two One-Sided Tests Procedure in Bioequivalence. J Pharmacokin Biopharm. 1990;18(2):137–44. doi:10.1007/BF01063556

Diletti D, Hauschke D, Steinijans VW. Sample Size Determination for Bioequivalence Assessment by Means of Confidence Intervals. Int J Clin Pharmacol Ther Toxicol. 1991;29(1):1–8.

See Also

sampleN.TOST, known.designs

Examples

# power for the 2x2 cross-over design with 24 subjects and CV 25%
# using all the other default values
power.TOST(CV = 0.25, n = 24)
# should give: [1] 0.7391155
# nct approximation very good for this configuration
power.TOST(CV = 0.25, n = 24, method = "nct")
# gives also: [1] 0.7391155
# shifted-central-t  approximation 
power.TOST(CV = 0.25, n = 24, method = "shifted")
# gives:      [1] 0.7328894

# power for the 2x2 cross-over study with 24 subjects, CV 25%
# and 2 drop-outs in the same sequence group (unbalanced study)
power.TOST(CV=0.25, n=c(10,12))
# should give: [1] 0.6912935
# not the same compared to the balanced setting
power.TOST(CV=0.25, n=22)
# should give: [1] 0.6953401