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Power of the Two One-Sided Tests Procedure (TOST) for Equivalence

Source: R/power_equivalence_md.R
power_equivalence_md.Rd

Computes the power of the Schuirmann (1987) two one-sided tests procedure , the probability that a \((1 - 2\alpha)\) confidence interval for the mean difference (or ratio, on the log scale) lies entirely within the equivalence interval \([\theta_1, \theta_2]\), by numerical integration over the chi distribution of the sample standard deviation.

Usage

power_equivalence_md(
  alpha_level,
  logscale,
  ltheta1,
  ltheta2,
  ldiff,
  sigma,
  n,
  nu
)

Arguments

alpha_level

Type I error rate for each of the two one-sided tests (typically 0.05). The full equivalence test uses a \((1 - 2\alpha)\) confidence interval.

logscale

Logical. If TRUE, treatment means are compared on the logarithmic scale; ltheta1, ltheta2, and ldiff are expected as ratios (untransformed) and are log-transformed internally.

ltheta1

Lower limit of the equivalence interval (on the original scale; logged internally if logscale = TRUE).

ltheta2

Upper limit of the equivalence interval (on the original scale; logged internally if logscale = TRUE).

ldiff

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

sigma

\(\sqrt{\mathrm{error\ variance}}\); root-MSE from an ANOVA. On the log scale, this is the coefficient of variation.

n

Number of subjects per treatment (or total subjects in a crossover design).

nu

Degrees of freedom associated with sigma.

Value

A one-row data.frame with columns term ("power") and value (the computed power, in \([0, 1]\)).

Details

The computation conditions on the error standard deviation the study will actually observe. Given that value, whether the confidence interval fits inside the equivalence interval is an ordinary normal probability, and the power is that probability averaged over the chi distribution the error standard deviation follows on nu degrees of freedom. The averaging is carried out on a unit-free scale, with the equivalence limits expressed in standard errors and the error standard deviation as a multiple of sigma, so the power depends on the design rather than on the units of the response: multiplying ltheta1, ltheta2, ldiff, and sigma by a common factor leaves the answer unchanged.

For Phillips's (1990) original example (regular-scale two-period crossover with \(\theta_1 = -0.2\), \(\theta_2 = 0.2\), CV \(= 0.20\), \(\delta = 0.05\), \(n = 24\), \(\nu = 22\)), this function reproduces the published value of \(0.8029678\) (Phillips, 1990, Table 1, 5th row, 5th column).

Note

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

References

Diletti, E., Hauschke, D., & Steinijans, V. W. (1991). Sample size determination of bioequivalence assessment by means of confidence intervals. International Journal of Clinical Pharmacology, Therapy and Toxicology, 29(1), 1–8.

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_plot, power_density_equivalence_md

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

Author

Ken Kelley kkelley@nd.edu

Examples

# Phillips (1990) Table 1, 5th row, 5th column. Expected: 0.8029678.
power_equivalence_md(alpha_level = .05, logscale = FALSE,
                     ltheta1 = -.2, ltheta2 = .2, ldiff = .05,
                     sigma = .20, n = 24, nu = 22)
#>  term  value
#>  power 0.803

# Diletti (1991) Table 1, on the log scale (ratio of test to reference).
# Expected: 0.7922796.
power_equivalence_md(alpha_level = .05, logscale = TRUE,
                     ltheta1 = .8, ltheta2 = 1.25, ldiff = 1.05,
                     sigma = .20, n = 18, nu = 16)
#>  term  value
#>  power 0.792