power.RatioF {PowerTOST}

Power for equivalence of the ratio of two means with normality on original scale

Description

Calculates the power of the test of equivalence of the ratio of two means with normality on original scale.
This test is based on Fieller’s confidence (‘fiducial’) interval and Sasabuchi’s test (a TOST procedure as well).

Usage

power.RatioF(alpha = 0.025, theta1 = 0.8, theta2, theta0 = 0.95,
             CV, CVb, n, design = "2x2", setseed=TRUE)

Arguments

alpha

Type I error probability, aka significance level.
Defaults here to 0.025 because this function is intended for studies with clinical endpoints.

theta1

Lower bioequivalence limit. Typically 0.8 (default).

theta2

Upper bioequivalence limit. Typically 1.25.
Is set to 1/theta1 if missing.

theta0

‘True’ or assumed T/R ratio. Typically set to 0.95 for planning.

CV

Coefficient of variation as ratio. In case of design="parallel" this is the CV of the total variability, in case of design="2x2" the intra-subject CV (CVw in the reference).

CVb

CV of the between-subject variability. Only necessary for design="2x2".

n

Number of subjects to be planned.
n is for both designs implemented the total number of subjects.

design

A character string describing the study design.
design="parallel" or design="2x2" allowed for a two-parallel group design or a classical TR|RT crossover design.

setseed

If set to TRUE the dependence of the power from the state of the random number generator is avoided. With setseed = FALSE you may see the dependence from the state of the random number generator.

Details

The power is calculated exact using the bivariate non-central t-distribution via function pmvt of the package mvtnorm.
Due to the calculation method of the used package mvtnorm – randomized Quasi-Monte-Carlo – these probabilities are dependent from the state of the random number generator within the precision of the power. See argument setseed.

Value

Value of power according to the input.

Note

This function is intended for studies with clinical endpoints where the 95% confidence intervals are usually used for equivalence testing.
Therefore, alpha defaults here to 0.025 (see EMEA 2000).

The formulas given in the references rely on the assumption of equal variances in the two treatment groups for the parallel group design or on assuming equal within-subject and between-subject variabilities for the 2×2 crossover design.

Author(s)

D. Labes

References

Fieller EC. Some Problems in Interval Estimation. J Royal Stat Soc B. 1954;16(2):175–85. doi:10.1111/j.2517-6161.1954.tb00159.x

Sasabuchi S. A test of a multivariate normal mean with composite hypotheses determined by linear inequalities. Biometrika. 1980;67(2):429–39. doi:10.1093/biomet/67.2.429

Hauschke D, Kieser M, Diletti E, Burke M. Sample size determination for proving equivalence based on the ratio of two means for normally distributed data. Stat Med. 1999;18(1):93–105.

Hauschke D, Steinijans V, Pigeot I. Bioequivalence Studies in Drug Development. Chichester: Wiley; 2007. Chapter 10.

European Agency for the Evaluation of Medicinal Products, CPMP. Points to Consider on Switching between Superiority and Non-Inferiority. London, 27 July 2000. CPMP/EWP/482/99

See Also

sampleN.RatioF

Examples

# power for alpha=0.025, ratio0=0.95, theta1=0.8, theta2=1/theta1=1.25
# within-subject CV=0.2, between-subject CV=0.4 
# 2x2 crossover study, n=24
# using all the defaults:
power.RatioF(CV = 0.2, CVb = 0.4, n = 24)
# gives [1] 0.7315357