cTOST: Finite Sample Corrected TOST for Equivalence Testing • cTOST

Overview

The cTOST package provides finite sample corrections for equivalence testing using the Two One-Sided Tests (TOST) procedure. Unlike traditional hypothesis testing that aims to detect differences, equivalence testing aims to demonstrate that two treatments are similar enough to be considered equivalent.

Key Features

Finite sample corrections: alpha-TOST, delta-TOST, and optimal cTOST for improved power
Univariate and multivariate equivalence testing
Quantile-based equivalence testing for distribution tails
Differential privacy equivalence testing for sensitive data
Comprehensive visualization and comparison tools

Methods Implemented

Method	Reference	Description
alpha-TOST	Boulaguiem et al. (2024)	Adjusts significance level
delta-TOST	Boulaguiem et al. (2024)	Adjusts equivalence bounds
Optimal cTOST	Insolia et al. (2025)	Best power (recommended)
alpha-qTOST	Wu et al. (2025)	Quantile-based testing
DP-TOST	Pareek et al. (2025)	Differential privacy

Installation

The cTOST package is available on cran and on GitHub. The version on GitHub is subject to ongoing updates that may lead to stability issues.

The package can from cran as follows:

install.packages("cTOST")

The package can also be install from GitHub using the devtools package:

install.packages("devtools")
devtools::install_github("stephaneguerrier/cTOST")

Note that Windows users are assumed that have Rtools installed (if this is not the case, please visit this link.

Quick Start

Here are a few examples demonstrating the usage of the cTOST package.

Univariate Equivalence Testing

To illustrate the use of the proposed method in the univariate settings, we consider the skin dataset analyzed in Boulaguiem et al. (2024a), which can be loaded as follows:

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

Standard TOST

The standard TOST can be used as follows:

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

Optimal cTOST

The cTOST can be used through the function ctost as follows:

opt_tost = ctost(theta = theta_hat, sigma = sig_hat, nu = nu,
              delta = log(1.25), method = "optimal")
opt_tost
#> ✔ Accept (bio)equivalence
#> Equiv. Region:  |----------------0----------------|
#> Estim. Inter.:     (---------------x--------------)
#> CI =  (-0.17492 ; 0.22032)
#> 
#> Method: cTOST
#> alpha = 0.05; Equiv. lim. = +/- 0.22314
#> Estimated c(0) = 0.02553
#> Finite sample correction: offline
#> Corrected alpha = 0.03853
#> Mean = 0.02270; Stand. dev. = 0.13428; df = 16

Alpha-TOST

The $\alpha$ -TOST can be used through the function ctost as follows:

atost = ctost(theta = theta_hat, sigma = sig_hat, nu = nu,
              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

It is possible to compare the results of the $\alpha$ -TOST (or $\delta$ -TOST, see below) with the standard TOST as follows:

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

The $\delta$ -TOST can be used through the function ctost as follows:

dtost = ctost(theta = theta_hat, sigma = sig_hat, nu = nu,
              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

Multivariate Equivalence Testing

data(ticlopidine)
n = nrow(ticlopidine)
p = ncol(ticlopidine)
nu = n-1
theta_hat = colMeans(ticlopidine)
Sigma_hat = cov(ticlopidine)/n

Standard Multivariate TOST

mtost = ctost(theta = theta_hat, sigma = Sigma_hat, nu = nu,
              delta = log(1.25), method = "unadjusted")
mtost
#> ✖ Can't accept (bio)equivalence
#> Equiv. Region:   |----------------0----------------|
#> t_half               (-----------x----------)       
#> AUC                 (-------x--------)              
#> AUC_inf             (--------x-------)              
#> C_max            (---------x---------)              
#> 
#> CIs:
#> t_half   (-0.15767 ; 0.12503)
#> ✔
#> AUC      (-0.18553 ; 0.00992)
#> ✔
#> AUC_inf  (-0.17914 ; 0.01620)
#> ✔
#> C_max    (-0.22379 ; 0.02154)
#> ✖
#> 
#> Method: TOST
#> alpha = 0.05; Equiv. lim. = +/- 0.22314

Multivariate Alpha-TOST

matost = ctost(theta = theta_hat, sigma = Sigma_hat, nu = nu, delta = log(1.25))
#> Warning in ctost(theta = theta_hat, sigma = Sigma_hat, nu = nu, delta =
#> log(1.25)): Available correction method for the multivariate cTOST ('optimal')
#> is only 'none' currently ('bootstrap' coming soon).
matost
#> ✔ Accept (bio)equivalence
#> Equiv. Region:   |----------------0----------------|
#> t_half                (----------x----------)       
#> AUC                 (-------x-------)               
#> AUC_inf              (-------x-------)              
#> C_max             (--------x---------)              
#> 
#> CIs:
#> t_half   (-0.14520 ; 0.11256)
#> ✔
#> AUC      (-0.17694 ; 0.00132)
#> ✔
#> AUC_inf  (-0.17055 ; 0.00761)
#> ✔
#> C_max    (-0.21300 ; 0.01075)
#> ✔
#> 
#> Method: cTOST
#> alpha = 0.05; Equiv. lim. = +/- 0.22314

4. Learn More

For detailed guides and examples, see the package vignettes:

Getting Started - Quick introduction and basic usage
Average Equivalence: Univariate - In-depth guide to univariate testing
Average Equivalence: Multivariate - Multivariate testing with examples
Quantile Equivalence Testing - Testing at specific quantiles
Differential Privacy Testing - Equivalence testing for sensitive data
Mathematical Background - Theory and mathematical details

Or browse the full reference documentation.

5. How to cite

@Manual{boulaguiem2024ctost,
  title = {cTOST: Finite Sample Correction of The TOST in The Univariate Framework},
  author = {Boulaguiem, Y. and Insolia, L. and Couturier, D.-L. and Guerrier, S.},
  year = {2024},
  note = {R package version 1.1.0},
  url = {https://github.com/stephaneguerrier},
}

6. License

The license this source code is released under is the GNU AFFERO GENERAL PUBLIC LICENSE (AGPL) v3.0. Please see the LICENSE file for full text. Otherwise, please consult GNU which will provide a synopsis of the restrictions placed upon the code.

7. References

Boulaguiem, Y., Quartier, J., Lapteva, M., Kalia, Y. N., Victoria-Feser, M. P., Guerrier, S. & Couturier, D. L., “Finite Sample Adjustments for Average Equivalence Testing”, Statistics in Medicine, 2024a, https://doi.org/10.1002/sim.9993.

Boulaguiem, Y., Insolia, L., Victoria-Feser, M. P., Couturier, D. L. & Guerrier, S., “Multivariate Adjustments for Average Equivalence Testing”, submitted manuscript, 2024b.