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Finite Sample Adjustment for Average (Bio)Equivalence Assessment

Source: R/average_equivalence.R
ctost.Rd

Computes finite sample corrected versions of the standard (univariate or multivariate) TOST, as developed in Boulaguiem et al. (2024, https://doi.org/10.1002/sim.9993), Boulaguiem et al. (2024, https://doi.org/10.1002/sim.10258), and Insolia et al. (2025).

Usage

ctost(
  theta,
  sigma,
  nu,
  delta,
  alpha = 0.05,
  method = "optimal",
  B = 10^4,
  seed = 101010,
  correction = NULL,
  ...
)

Arguments

theta

A numeric value or vector representing the estimated difference(s) (e.g., between a generic and reference product).

sigma

A numeric value (univariate) or matrix (multivariate) corresponding to the estimated variance of theta.

nu

A numeric value specifying the degrees of freedom. In the multivariate case, it is assumed to be the same across all dimensions.

delta

A numeric value or vector defining the (bio)equivalence margin(s). The procedure assumes symmetry, i.e., the (bio)equivalence region is \((-\delta, \delta)\). In the multivariate case, it is assumed to be the same across all dimensions.

alpha

A numeric value specifying the significance level, which must be between 0 and 0.5 (default: alpha = 0.05).

method

A character string specifying the finite sample adjustment method. Available methods are: "unadjusted" (standard unadjusted TOST), "alpha" (alpha-TOST), "delta" (delta-TOST, not implemented for multivariate settings), and "optimal" (cTOST, default). See Details.

B

A numeric value specifying the number of Monte Carlo replications, required for some methods (default: B = 10^4).

seed

A numeric value specifying a seed for reproducibility (default: seed = 101010).

correction

A character string specifying the correction method. Available options are: "none" (no correction), "offline" (offline adjustment), and "bootstrap" (bootstrap adjustment). In univariate settings, the default is "offline"; in multivariate settings, the default is "bootstrap" if nu < 100, otherwise "none".

...

Additional parameters.

Value

An object of class tost with the following components:

  • decision: Logical; indicates whether (bio)equivalence is accepted.

  • ci: Confidence region at the \(1 - 2\alpha\) level.

  • theta: The estimated difference(s) used in the test.

  • sigma: The estimated variance of theta; a numeric value (univariate) or matrix (multivariate).

  • nu: The degrees of freedom used in the test.

  • alpha: The significance level used in the test.

  • corrected_alpha: The significance level after adjustment (if method = "alpha").

  • corrected_delta: The (bio)equivalence limits after adjustment (if method = "delta").

  • delta: The (bio)equivalence limits used in the test.

  • method: The adjustment method used (optimal, alpha-TOST, or delta-TOST).

  • setting: The setting used ("univariate" or "multivariate").

Details

In univariate settings, three adjustment methods are available: optimal (cTOST, method = "optimal") as proposed in Insolia et al. (2025), alpha-TOST (method = "alpha"), and delta-TOST (method = "delta"), both proposed in Boulaguiem et al. (2024, https://doi.org/10.1002/sim.9993, https://doi.org/10.1002/sim.10258). For multivariate settings, only cTOST and alpha-TOST are implemented.

The cTOST, alpha-TOST, and delta-TOST methods apply different finite sample adjustments. Alpha-TOST corrects the significance level, while delta-TOST adjusts the equivalence limits. The cTOST method is based on a more complex approach, and in small samples (typically less than 30), additional corrections may be beneficial. The correction argument further adjusts the test level to prevent liberal inference; see Insolia et al. (2025) for details.

Generally, cTOST outperforms other methods, with alpha-TOST performing better than delta-TOST. For this reason, delta-TOST is not implemented for multivariate settings and is not recommended.

Examples

data(skin)
theta_hat = diff(apply(skin, 2, mean))
nu = nrow(skin) - 1
sig_hat = var(apply(skin, 1, diff)) / nu

# alpha-TOST
atost = ctost(theta = theta_hat, sigma = sig_hat, nu = nu,
              alpha = 0.05, delta = log(1.25), method = "alpha")
atost
#> ✔ Accept (bio)equivalence
#> Equiv. Region:  |----------------0----------------|
#> Estim. Inter.:     (---------------x--------------)
#> CI =  (-0.17655 ; 0.22195)
#> 
#> Method: alpha-TOST
#> alpha = 0.05; Equiv. lim. = +/- 0.22314
#> Corrected alpha = 0.07865
#> Mean = 0.02270; Stand. dev. = 0.13428; df = 16
compare_to_tost(atost)
#> TOST:       
#> ✖ Can't accept (bio)equivalence
#> alpha-TOST: 
#> ✔ Accept (bio)equivalence
#> 
#> Equiv. Region:  |---------------0---------------|  
#> TOST:            (---------------x----------------)
#> alpha-TOST:        (-------------x--------------)  
#> 
#>                  CI - low      CI - high
#> TOST:            -0.21174       0.25715
#> alpha-TOST:      -0.17655       0.22195
#> 
#> Equiv. lim. = +/- 0.22314

# delta-TOST
dtost = ctost(theta = theta_hat, sigma = sig_hat, nu = nu,
              alpha = 0.05, delta = log(1.25), method = "delta")
dtost
#> ✖ Can't accept (bio)equivalence
#> Corr. Equiv. Region:  |----------------0----------------|
#>       Estim. Inter.:     (--------------x---------------)
#> CI =  (-0.21174 ; 0.25715)
#> 
#> Method: delta-TOST
#> alpha = 0.05; Equiv. lim. = +/- 0.22314
#> Corrected Equiv. lim. = +/- 0.25470
#> Mean = 0.02270; Stand. dev. = 0.13428; df = 16
compare_to_tost(dtost)
#> TOST:       
#> ✖ Can't accept (bio)equivalence
#> delta-TOST: 
#> ✖ Can't accept (bio)equivalence
#> 
#> Stand. Equiv. Region:    |--------------0--------------|  
#>  Corr. Equiv. Region:  |----------------0----------------|
#>                 TOST:     (--------------x---------------)
#>           delta-TOST:     (--------------x---------------)
#> 
#>  Standard Equiv. lim. = +/- 0.22314
#> Corrected Equiv. lim. = +/- 0.25470