Calculates the so-called expected, i.e., unconditional, power for a variety of study designs used in bioequivalence studies.
exppower.TOST(alpha = 0.05, logscale = TRUE, theta0, theta1, theta2,
CV, n, design = "2x2", robust = FALSE,
prior.type = c("CV", "theta0", "both"), prior.parm = list(),
method = c("exact", "approx"))Value of expected power according to the input.
Significance level (one-sided). Commonly set to 0.05.
Should the data be used on log-transformed or on original scale? TRUE (default) or FALSE.
Assumed ‘true’ (or ‘observed’ in case of prior.type != "CV") 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 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.
Defaults to FALSE. Set to TRUE will use the degrees of freedom according
to the ‘robust’ evaluation (aka Senn’s basic estimator). These df are calculated
as n-seq.
See known.designs()$df2 for designs covered in this package.
Specifies which parameter uncertainty should be accounted for. In case of
prior.type = "CV" (the default), only the uncertainty with respect to the
CV will be considered (i.e. the given treatment effect is assumed to be fix).
In case of prior.type = "theta0" only uncertainty with respect to the
treatment ratio/difference will be accounted for (i.e. the given CV is assumed
to be fix). In case of prior.type = "both" the power value will be
unconditional with respect to both the CV and theta0.
A list of parameters expressing the prior information about the
variability and/or treatment effect. Possible components are df,
SEM, m and design.
For prior.type = "CV" the degrees of freedom from the prior trial are
required. This information can be provided by specifying the single component
df or the combination consisting of m and design.
For prior.type = "theta0" the standard error of the treatment difference
from the prior trial is required. This information can be provided by specifying
the single component SEM or the combination consisting of
m and design.
For prior.type = "both" the degrees of freedom and the standard error of
the treatment difference are required. This information can be provided by
specifying the combination consisting of df and SEM or via the
combination m and design.
See 'Details' for a technical description on each component.
Defaults to method="exact".
In that case the expected power will be calculated as expected value of the
power with respect to the (prior) distribution of the respective parameter(s).
Set to method="approx" the expected power according to the
approximate formulas given in the book from Julious or in the Julious/Owen
paper will be calculated (using non-central t); this only affects
prior.type = "CV".
B. Lang (thanks to G. Nehmiz for the helpful discussions), D. Labes
This function calculates the so-called expected power taking into account that usually the parameters (CV and/or theta0) are not known but estimated from a prior study with some uncertainty. The expected power is an unconditional power and can therefore be seen as probability for success. See references for further details.
The prior.parm argument is a list that can supply any of the following
components:
dfError degrees of freedom from the prior trial (>4, maybe non-integer).
df = Inf is allowed and for method = "exact" the result will then
coincide with power.TOST(...).
Note: This corresponds to the df of both the CV and the difference of means.
SEMStandard error of the difference of means from the prior trial; must always be on additive scale (i.e., usually log-scale).
mNumber of subjects from prior trial. Specification is analogous to
the main argument n.
designStudy design of prior trial. Specification is analogous to the
main argument design.
For prior.parm, the combination consisting of df and SEM
requires a somewhat advanced knowledge of the prior trial (provided in the raw
output from for example the software SAS, or may be obtained via
emmeans of package emmeans.
However, it has the advantage that if there were
missing data the exact degrees of freedom and standard error of the difference
can be used, the former possibly being non-integer valued (e.g., if the
Kenward-Roger method was used).
Details on argument prior.type:
CVThe expectation is calculated with respect to the Inverse-gamma distribution.
theta0The expectation is calculated with respect to the conditional distribution theta0 | \(\sigma^2\) = s^2 of the posteriori distribution of (theta0, \(\sigma^2\)) from the prior trial.
bothThe expectation is calculated with respect to the posteriori
distribution of (theta0, \(\sigma^2\)) from the prior trial. Numerical calculation
of the two-dimensional integral is performed via hcubature.
Grieve AP. Confidence Intervals and Sample Sizes. Biometrics. 1991;47:1597--603. tools:::Rd_expr_doi("10.2307/2532411")
O’Hagan, Stevens, JW, Campell MJ. Assurance in Clinical Trial Design. Pharm Stat. 2005;4:187--201. tools:::Rd_expr_doi("10.1002/pst.175")
Julious SA, Owen RJ. Sample size calculations for clinical studies allowing for uncertainty in variance. Pharm Stat. 2006;5:29--37. tools:::Rd_expr_doi("10.1002/pst.197")
Julious SA. Sample sizes for Clinical Trials. Boca Raton: CRC Press / Chapman & Hall; 2010.
Bertsche A, Nehmitz G, Beyersmann J, Grieve AP. The predictive distribution of the residual variability in the linear-fixed effects model for clinical cross-over trials. Biom J. 2016;58(4):797--809. tools:::Rd_expr_doi("10.1002/bimj.201500245")
Box GEP, Tiao GC. Bayesian Inference in Statistical Analysis. Boston: Addison-Wesley; 1992.
Held L, Sabanes Bove D. Applied Statistical Inference. Likelihood and Bayes. Berlin, Heidelberg: Springer; 2014. tools:::Rd_expr_doi("10.1007/978-3-642-37887-4")
Senn S. Cross-over Trials in Clinical Research. Chichester: John Wiley & Sons; 2nd edition 2002.
Zierhut ML, Bycott P, Gibbs MA, Smith BP, Vicini P. Ignorance is not bliss: Statistical power is not probability of trial success. Clin Pharmacol Ther. 2015;99:356--9. tools:::Rd_expr_doi("10.1002/cpt.257")
expsampleN.TOST, power.TOST
# Expected power for a 2x2 crossover with 40 subjects
# CV 30% known from a pilot 2x2 study with 12 subjects
# using all the defaults for other parameters (theta0 carved in stone)
exppower.TOST(CV = 0.3, n = 40, prior.parm = list(df = 12-2))
# should give: [1] 0.7365519
# or equivalently
exppower.TOST(CV = 0.3, n = 40, prior.parm = list(m = 12, design = "2x2"))
# In contrast: Julious approximation
exppower.TOST(CV = 0.3, n = 40, prior.parm = list(df = 10), method = "approx")
# should give: [1] 0.7359771
# Compare this to the usual (conditional) power (CV known, "carved in stone")
power.TOST(CV = 0.3, n = 40)
# should give: [1] 0.8158453
# same as if setting df = Inf in function exppower.TOST()
exppower.TOST(CV = 0.3, n = 40, prior.parm = list(df = Inf))
# Expected power for a 2x2 crossover with 40 subjects
# CV 30% and theta0 = 0.95 known from a pilot 2x2 study with 12 subjects
# using uncertainty with respect to both CV and theta0
exppower.TOST(CV = 0.3, theta0 = 0.95, n = 40,
prior.parm = list(m = 12, design = "2x2"), prior.type = "both")
# should give [1] 0.5114685
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