Density Underlying the TOST Power Calculation

Evaluates the integrand whose integral over \((0, \mathrm{upper})\) yields the power of the Schuirmann (1987) two one-sided tests procedure; see power_equivalence_md. Useful for plotting the integrand and for diagnostic work.

Usage

power_density_equivalence_md(
  power_sigma,
  alpha_level,
  theta1,
  theta2,
  diff,
  sigma,
  n,
  nu
)

Arguments

power_sigma

Numeric vector of \(\sigma\) values at which to evaluate the integrand.

alpha_level

Type I error rate for each of the two one-sided tests.

theta1

Lower limit of the equivalence interval on the appropriate scale (regular or log).

theta2

Upper limit of the equivalence interval on the appropriate scale (regular or log).

diff

True difference in treatment means (ratio on the log scale) on the appropriate scale.

sigma

\(\sqrt{\mathrm{error\ variance}}\).

n

Number of subjects per treatment.

nu

Degrees of freedom for sigma.

Value

A data.frame with one row per supplied power_sigma, and columns power_sigma and power_density.

Note

See the legacy MBESS package (Kelley, 2007a, 2007b) for additional details and discussion.

References

Kelley, K. (2007a). Confidence intervals for standardized effect sizes: Theory, application, and implementation. Journal of Statistical Software, 20(8), 1–24. doi:10.18637/jss.v020.i08

Kelley, K. (2007b). Methods for the behavioral, educational, and social sciences: An R package. Behavior Research Methods, 39(4), 979–984. doi:10.3758/BF03192993

Phillips, K. F. (1990). Power of the two one-sided tests procedure in bioequivalence. Journal of Pharmacokinetics and Biopharmaceutics, 18(2), 139–144. doi:10.1007/BF01063556

Schuirmann, D. J. (1987). A comparison of the two one-sided tests procedure and the power approach for assessing the equivalence of average bioavailability. Journal of Pharmacokinetics and Biopharmaceutics, 15(6), 657–680.

See also

power_equivalence_md, power_equivalence_md_plot

Other equivalence testing: equivalence_c(), equivalence_r(), equivalence_smd(), plot_equivalence(), power_equivalence_c(), power_equivalence_md(), power_equivalence_md_plot(), ss_power_equivalence_c()

Author

Ken Kelley kkelley@nd.edu

Examples

# Density at a single value of sigma:
power_density_equivalence_md(power_sigma = 0.10, alpha_level = .05,
                             theta1 = -.2, theta2 = .2, diff = .05,
                             sigma = .20, n = 24, nu = 22)
#>   power_sigma power_density
#> 1         0.1    0.02297896

# Vectorized over a grid:
grid <- power_density_equivalence_md(
  power_sigma = seq(0.01, 0.40, length.out = 50),
  alpha_level = .05, theta1 = -.2, theta2 = .2, diff = .05,
  sigma = .20, n = 24, nu = 22
)
head(grid)
#>   power_sigma power_density
#> 1  0.01000000  3.625481e-22
#> 2  0.01795918  7.451634e-17
#> 3  0.02591837  1.498398e-13
#> 4  0.03387755  3.634158e-11
#> 5  0.04183673  2.582758e-09
#> 6  0.04979592  8.170958e-08