Calculates the exact or approximate power of the two-one-sided t-tests (TOST) procedure for various study designs used in BE studies.
power.TOST(alpha = 0.05, logscale = TRUE, theta1, theta2, theta0, CV, n,
design = "2x2", method="exact", robust=FALSE)Value of power according to the input arguments.
Significance level (one-sided). Commonly set to 0.05.
Should the data used on log-transformed or on original scale? TRUE (default) or FALSE.
‘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
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.
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.
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.
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.
Character string describing the study design.
See known.designs() for designs covered in this package.
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.
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.
D. Labes, direct integration of bivariate non-central t-distribution by B. Lang
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).
Phillips KF. Power of the Two One-Sided Tests Procedure in Bioequivalence. J Pharmacokin Biopharm. 1990;18(2):137--44. tools:::Rd_expr_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.
sampleN.TOST, known.designs
# 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
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