Title: Sample Size Estimation for Bio-Equivalence Trials Through Simulation
Version: 1.0.2
Description: Sample size estimation for bio-equivalence trials is supported through a simulation-based approach that extends the Two One-Sided Tests (TOST) procedure. The methodology provides flexibility in hypothesis testing, accommodates multiple treatment comparisons, and accounts for correlated endpoints. Users can model complex trial scenarios, including parallel and crossover designs, intra-subject variability, and different equivalence margins. Monte Carlo simulations enable accurate estimation of power and type I error rates, ensuring well-calibrated study designs. The statistical framework builds on established methods for equivalence testing and multiple hypothesis testing in bio-equivalence studies, as described in Schuirmann (1987) <doi:10.1007/BF01068419>, Mielke et al. (2018) <doi:10.1080/19466315.2017.1371071>, Shieh (2022) <doi:10.1371/journal.pone.0269128>, and Sozu et al. (2015) <doi:10.1007/978-3-319-22005-5>. Comprehensive documentation and vignettes guide users through implementation and interpretation of results.
License: Apache License (≥ 2)
Encoding: UTF-8
RoxygenNote: 7.3.2
Imports: MASS, Rcpp (≥ 1.0.13), data.table, matrixcalc, parallel
Suggests: PowerTOST, ggplot2, kableExtra, knitr, rmarkdown, testthat (≥ 3.0.0), tibble, tinytest
VignetteBuilder: knitr
Config/testthat/edition: 3
LinkingTo: Rcpp, RcppArmadillo
URL: https://smartdata-analysis-and-statistics.github.io/SimTOST/
NeedsCompilation: yes
Packaged: 2025-02-18 16:16:14 UTC; Thomas
Author: Thomas Debray ORCID iD [aut, cre], Johanna Munoz [aut], Dewi Amaliah [ctb], Wei Wei [ctb], Marian Mitroiu [ctb], Scott McDonald [ctb], Biogen Inc [cph, fnd]
Maintainer: Thomas Debray <tdebray@fromdatatowisdom.com>
Repository: CRAN
Date/Publication: 2025-02-18 23:40:08 UTC

Sample Size Estimation via Simulation

Description

SimTOST: A Package for Sample Size Simulations

Details

The SimTOST package provides tools for simulating sample sizes, calculating power, and assessing type-I error for various statistical scenarios.

Author(s)

Maintainer: Thomas Debray tdebray@fromdatatowisdom.com (ORCID)

Authors:

Other contributors:

References

Mielke, J., Jones, B., Jilma, B. & König, F. Sample Size for Multiple Hypothesis Testing in Biosimilar Development. Statistics in Biopharmaceutical Research 10, 39–49 (2018).

See Also

sampleSize

Check Equivalence for Multiple Endpoints

Description

This function evaluates whether equivalence criteria are met based on a predefined set of endpoints. It first checks whether all primary endpoints satisfy equivalence (if sequential testing is enabled). Then, it determines whether the required number of endpoints (k) meet the equivalence threshold. The function returns a binary decision indicating whether overall equivalence is established.

Usage

check_equivalence(typey, adseq, tbioq, k)

Arguments

typey

An integer vector specifying the hierarchy of each endpoint, where 1 denotes a primary endpoint and 2 denotes a secondary endpoint.

adseq

A boolean flag indicating whether sequential testing is enabled. If set to TRUE, all primary endpoints must pass equivalence before secondary endpoints are evaluated. If set to FALSE, primary and secondary endpoints are assessed independently.

tbioq

A matrix containing the equivalence test results for each endpoint, where 1 indicates that equivalence is met and 0 indicates that equivalence is not met.

k

An integer specifying the minimum number of endpoints required for overall equivalence.

Details

When sequential testing is enabled (adseq = TRUE), all primary endpoints must meet equivalence before secondary endpoints are considered. If sequential testing is disabled (adseq = FALSE), all endpoints are evaluated simultaneously without hierarchical constraints. The function then determines whether at least k endpoints meet the equivalence criteria. If the conditions are satisfied, the final equivalence decision (totaly) is 1; otherwise, it is 0.

Value

Returns a (1 × 1 matrix) containing a binary equivalence decision. A value of 1 indicates that equivalence is established, while 0 indicates that equivalence is not established.

Author(s)

Thomas Debray tdebray@fromdatatowisdom.com

Derive and Validate Treatment Allocation Rate (TAR)

Description

This function validates and adjusts the treatment allocation rate (TAR) to ensure it is correctly specified for the given number of treatment arms (n_arms). If TAR is missing or NULL, it is assigned a default vector of ones, ensuring equal allocation across all arms. The function also handles cases where TAR is shorter than n_arms, contains NA values, or has invalid values.

Usage

derive_allocation_rate(TAR = NULL, arm_names, verbose = FALSE)

Arguments

TAR

Optional numeric vector specifying the allocation rate for each treatment arm. If missing, a default equal allocation rate is assigned.

arm_names

Character vector specifying the names of the treatment arms. Used to name the elements of TAR.

verbose

Logical, if TRUE, displays messages about the status of TAR derivation or assignment.

Value

A named list representing the treatment allocation rate for each arm.

Author(s)

Thomas Debray tdebray@fromdatatowisdom.com

Derive or Assign Arm Names

Description

This function checks if arm_names is provided. If arm_names is missing, it attempts to derive names from mu_list. If mu_list does not contain names, it assigns default names ("A1", "A2", etc.) to each arm. Informational messages are displayed if verbose is set to TRUE.

Usage

derive_arm_names(arm_names, mu_list, verbose = FALSE)

Arguments

arm_names

Optional vector of arm names.

mu_list

Named list of means per treatment arm, from which arm names may be derived.

verbose

Logical, if TRUE, displays messages about the derivation process.

Value

A vector of arm names.

Author(s)

Thomas Debray tdebray@fromdatatowisdom.com

Derive Endpoint Names

Description

Derive Endpoint Names

Usage

derive_endpoint_names(ynames_list, mu_list, verbose = FALSE)

Arguments

ynames_list

Optional list of vectors with endpoint names for each arm.

mu_list

Named list of means per treatment arm, where names can be used as endpoint names.

verbose

Logical, if TRUE, displays messages about the derivation process.

Value

A list of endpoint names for each arm.

Author(s)

Thomas Debray tdebray@fromdatatowisdom.com

This function derives endpoint names (ynames_list) from mu_list if ynames_list is missing. If ynames_list is already provided, it confirms the names to the user when verbose is set to TRUE.

Derive Variance-Covariance Matrix List

Description

Constructs a list of variance-covariance matrices for multiple treatment arms based on provided standard deviations, means, and correlation structures.

Usage

derive_varcov_list(
  mu_list,
  sigma_list,
  ynames_list = NULL,
  varcov_list = NULL,
  cor_mat = NULL,
  rho = 0
)

Arguments

mu_list

A list of numeric vectors representing the means (\mu) for each treatment arm. Each element corresponds to one arm.

sigma_list

A list of numeric vectors representing the standard deviations (\sigma) for each treatment arm. Each element corresponds to one arm.

ynames_list

A list of character vectors specifying the names of the endpoints for each arm. Each element corresponds to one arm.

varcov_list

(Optional) A pre-specified list of variance-covariance matrices for each arm. If provided, it will override the construction of variance-covariance matrices.

cor_mat

(Optional) A correlation matrix to be used for constructing the variance-covariance matrices when there are multiple endpoints. If dimensions do not match the number of endpoints, a warning is issued.

rho

(Optional) A numeric value specifying the constant correlation coefficient to be used between all pairs of endpoints if no correlation matrix is provided. Default is 0 (uncorrelated endpoints).

Details

This function creates a list of variance-covariance matrices for multiple treatment arms. If the varcov_list is not provided, the function uses the sigma_list to compute the matrices. For single endpoints, the variance is simply the square of the standard deviation. For multiple endpoints, the function constructs the matrices using either a provided cor_mat or the constant correlation coefficient rho.

The function ensures that the lengths of mu_list, sigma_list, and ynames_list match for each arm. If dimensions mismatch, or if neither a variance-covariance matrix (varcov_list) nor a standard deviation list (sigma_list) is provided, an error is raised.

Value

A list of variance-covariance matrices, one for each treatment arm.

Author(s)

Thomas Debray tdebray@fromdatatowisdom.com

Parameter Configuration for Endpoints and Comparators

Description

Constructs and returns a list of key parameters (mean vectors, variance-covariance matrices, and allocation rates) required for input into the sampleSize function. This function ensures that the parameters for each endpoint and comparator are consistent, properly named, and formatted.

Usage

get_par(
  mu_list,
  varcov_list,
  TAR_list,
  type_y = NA,
  arm_names = NA,
  y_names = NA
)

Arguments

mu_list

A list of mean (\mu) vectors. Each element in the list represents a comparator, with the corresponding \mu vector having a length equal to the number of endpoints.

varcov_list

A list of variance-covariance matrices. Each element corresponds to a comparator, with a matrix of size (n \times n), where n is the number of endpoints.

TAR_list

A list of treatment allocation rates (TARs) for each comparator. Each element contains a numeric value (can be fractional or integer) representing the allocation rate for the respective comparator.

type_y

A numeric vector specifying the type of each endpoint. Use 1 for primary endpoints and 2 for secondary or other endpoints.

arm_names

(Optional) A character vector containing names of the arms. If not provided, default names (e.g., T1, T2, ...) will be generated.

y_names

(Optional) A character vector containing names of the endpoints. If not provided, default names (e.g., y1, y2, ...) will be generated.

Value

A named list with the following components:

mu

A list of mean vectors, named according to arm_names.

varcov

A list of variance-covariance matrices, named according to arm_names.

tar

A list of treatment allocation rates (TARs), named according to arm_names.

type_y

A vector specifying the type of each endpoint.

weight_seq

A weight sequence calculated from type_y, used for endpoint weighting.

y_names

A vector of names for the endpoints, named as per y_names.

#' @details This function ensures that all input parameters (mu_list, varcov_list, and TAR_list) are consistent across comparators and endpoints. It performs checks for positive semi-definiteness of variance-covariance matrices and automatically assigns default names for arms and endpoints if not provided.

Examples

mu_list <- list(c(0.1, 0.2), c(0.15, 0.25))
varcov_list <- list(matrix(c(1, 0.5, 0.5, 1), ncol = 2), matrix(c(1, 0.3, 0.3, 1), ncol = 2))
TAR_list <- list(0.5, 0.5)
get_par(mu_list, varcov_list, TAR_list, type_y = c(1, 2), arm_names = c("Arm1", "Arm2"))

Helper function for conditional messages

Description

This function displays a message if the verbose parameter is set to TRUE. It is useful for providing optional feedback to users during function execution.

Usage

info_msg(message, verbose)

Arguments

message

A character string containing the message to display.

verbose

Logical, if TRUE, the message is displayed; if FALSE, the message is suppressed.

Value

NULL (invisible). This function is used for side effects (displaying messages).

Author(s)

Thomas Debray tdebray@fromdatatowisdom.com

mcsapply

Description

An mc-version of the sapply function. https://stackoverflow.com/questions/31050556/parallel-version-of-sapply

Usage

mcsapply(X, FUN, ..., simplify = TRUE, USE.NAMES = TRUE)

Arguments

X

vector of iterations

FUN

function

...

additional parameters to pass

simplify

simplify array

USE.NAMES

use names in array

Value

vector output

Plot Power vs Sample Size for Simulation Results

Description

Generates a detailed plot showing the relationship between power and total sample size for each comparator and the overall combined comparators. The plot also includes confidence intervals for power estimates and highlights the target power with a dashed line for easy visual comparison.

Usage

## S3 method for class 'simss'
plot(x, ...)

Arguments

x

An object of class simss containing simulation results.

...

Additional arguments to be passed to the plot.simss function for customization.

Details

The plot dynamically adjusts to exclude unnecessary components, such as redundant endpoints or comparators with insufficient data, ensuring clarity and simplicity. The ggplot2 framework is used for visualizations, allowing further customization if needed.

Value

A ggplot object illustrating:

Author(s)

Johanna Muñoz johanna.munoz@fromdatatowisdom.com

Power Calculation for Hypothesis Testing in Equivalence Trials

Description

Estimates the power of hypothesis testing in equivalence trials using the method described by Mielke et al. This approach accounts for multiple endpoints, correlation structures, and multiplicity adjustments.

Usage

power_Mielke(
  N,
  m,
  k,
  R,
  sigma,
  true.diff,
  equi.tol = log(1.25),
  design,
  alpha = 0.05,
  adjust = "no",
  nsim = 10000
)

Arguments

N

Integer specifying the number of subjects per sequence.

m

Integer specifying the number of endpoints.

k

Integer specifying the number of endpoints that must meet equivalence to consider the test successful.

R

Matrix specifying the correlation structure between endpoints. This should be an m x m matrix, e.g., generated using variance.const.corr().

sigma

Numeric specifying the standard deviation of endpoints. Can be a vector of length m (one per endpoint) or a single value. In a 2x2 crossover design, this represents within-subject variance. In a parallel-group design, it represents the treatment group standard deviation.

true.diff

Numeric specifying the assumed true difference between test and reference. Can be a vector of length m or a single value.

equi.tol

Numeric specifying the equivalence margins, with the interval defined as (-equi.tol, +equi.tol). Default is log(1.25).

design

Character specifying the study design. Options are "22co" for a 2x2 crossover design or "parallel" for a parallel-group design.

alpha

Numeric specifying the significance level. Default is 0.05.

adjust

Character specifying the method for multiplicity adjustment. Options include "no" for no adjustment, "bon" for Bonferroni correction, and "k" for k-adjustment.

nsim

Integer specifying the number of simulations to perform. Default is 10,000.

Value

A numeric value representing the estimated power based on the simulations.

Calculate the power across all comparators

Description

Internal function to calculate the power across all comparators

Usage

power_cal(n, nsim, param, param.d, seed, ncores)

Arguments

n

sample size

nsim

number of simulated studies

param

list of parameters (mean,sd,tar)

param.d

design parameters

seed

main seed

ncores

number of cores

Value

power calculated from a global list of comparators

Power Calculation for Difference of Means (DOM) Hypothesis Test

Description

Computes the statistical power for testing the difference of means (DOM) between two groups using Monte Carlo simulations. The power is estimated based on specified sample sizes, means, standard deviations, and significance level.

Usage

power_dom(
  seed,
  mu_test,
  mu_control,
  sigma_test,
  sigma_control,
  N_test,
  N_control,
  lb,
  ub,
  alpha = 0.05,
  nsim = 10000
)

Arguments

seed

Integer. Seed for reproducibility.

mu_test

Numeric. Mean of the test group.

mu_control

Numeric. Mean of the control group.

sigma_test

Numeric. Standard deviation of the test group.

sigma_control

Numeric. Standard deviation of the control group.

N_test

Integer. Sample size of the test group.

N_control

Integer. Sample size of the control group.

lb

Numeric. Lower bound for the equivalence margin.

ub

Numeric. Upper bound for the equivalence margin.

alpha

Numeric. Significance level (default = 0.05).

nsim

Integer. Number of simulations (default = 10,000).

Value

Numeric. Estimated power (probability between 0 and 1).

Print Summary of Sample Size Estimation

Description

Prints the summary results of the sample size estimation for bioequivalence trials, including achieved power, total sample size, and power confidence intervals. The function also details the study design, primary endpoint comparisons, and applied multiplicity corrections.

Usage

## S3 method for class 'simss'
print(x, ...)

Arguments

x

An object of class "simss", typically generated by a sample size estimation function.

...

Optional arguments to be passed from or to other methods.

Details

This function displays key metrics from a sample size estimation analysis. It provides an overview of the study design, treatment comparisons, tested endpoints, significance level adjustments, and estimated sample size. For studies with multiple primary endpoints, it describes the multiplicity correction applied.

Value

No return value, called for side effects. The function prints the summary results of the sample size estimation to the console in a structured format.

Author(s)

Thomas Debray tdebray@fromdatatowisdom.com

Compute p-values for a t-distribution with Fixed Degrees of Freedom

Description

Computes p-values for a given set of random variables under a t-distribution with fixed degrees of freedom.

Usage

ptv(x, df, lower)

Arguments

x

A numeric matrix (or vector) representing the random variables.

df

A double specifying the degrees of freedom.

lower

A logical value indicating whether to compute the lower-tail probability (P(T <= x)). If FALSE, the function returns the upper-tail probability (P(T > x)).

Value

A numeric matrix containing the computed cumulative distribution function (CDF) values (p-values).

Calculate p-values using t-distribution with Variable Degrees of Freedom

Description

This function computes the cumulative distribution function (p-values) for a given random variable x and corresponding degrees of freedom df using the t-distribution. The function can compute the lower or upper tail probabilities depending on the value of the lower argument.

Usage

ptvdf(x, df, lower)

Arguments

x

arma::mat (vector) - A matrix or vector of random variable values for which the p-values will be calculated.

df

arma::mat (vector) - A matrix or vector of degrees of freedom for the t-distribution, matching the size of x.

lower

bool - If TRUE, calculates the lower-tail probability (P(T <= x)); if FALSE, calculates the upper-tail probability.

Value

arma::mat (vector) - A matrix containing the computed cumulative distribution function (p-values) for each element in x. The result is returned as a 1xN matrix, where N is the number of elements in x.

Run Simulations for a 2x2 Crossover Design with Difference of Means (DOM) test

Description

This function simulates a 2x2 crossover trial across multiple iterations. It evaluates equivalence across multiple endpoints using the Difference of Means (DOM) test.

Usage

run_simulations_2x2_dom(
  nsim,
  n,
  muT,
  muR,
  SigmaW,
  lequi_tol,
  uequi_tol,
  alpha,
  sigmaB,
  dropout,
  Eper,
  Eco,
  typey,
  adseq,
  k,
  arm_seed
)

Arguments

nsim

Integer. The number of simulations to run.

n

Integer. The sample size per period.

muT

Numeric vector. Mean outcomes for the active treatment.

muR

Numeric vector. Mean outcomes for the reference treatment.

SigmaW

Numeric matrix. Within-subject covariance matrix for endpoints.

lequi_tol

Numeric vector. Lower equivalence thresholds for each endpoint.

uequi_tol

Numeric vector. Upper equivalence thresholds for each endpoint.

alpha

Numeric vector. Significance levels for hypothesis testing across endpoints.

sigmaB

Numeric. Between-subject variance for the crossover model.

dropout

Numeric vector of size 2. Dropout rates for each sequence.

Eper

Numeric vector. Expected period effects for each sequence.

Eco

Numeric vector. Expected carryover effects for each sequence.

typey

Integer vector indicating the classification of each endpoint, where 1 corresponds to a primary endpoint and 2 corresponds to a secondary endpoint.

adseq

Logical. If TRUE, applies sequential (hierarchical) testing.

k

Integer. Minimum number of endpoints required for equivalence.

arm_seed

Integer vector. Random seed for each simulation.

Details

This function evaluates equivalence using the Difference of Means (DOM) test. Equivalence is determined based on predefined lower (lequi_tol) and upper (uequi_tol) equivalence thresholds, and hypothesis testing is conducted at the specified significance level (alpha). If adseq is TRUE, primary endpoints must establish equivalence before secondary endpoints are evaluated. The sample size per period is adjusted based on dropout rates, ensuring valid study conclusions. The simulation incorporates within-subject correlation using SigmaW and accounts for between-subject variance with sigmaB. Expected period effects (Eper) and carryover effects (Eco) are included in the model. A fixed random seed (arm_seed) is used to ensure reproducibility across simulations.

Value

A numeric matrix where each column stores simulation results: The first row (totaly) represents the overall equivalence decision (1 = success, 0 = failure). Subsequent rows contain equivalence decisions per endpoint, mean estimates for the treatment group, mean estimates for the reference group, standard deviations for treatment, and standard deviations for reference.

Author(s)

Thomas Debray tdebray@fromdatatowisdom.com

Run Simulations for a 2x2 Crossover Design with Ratio of Means (ROM) test

Description

This function simulates a 2x2 crossover trial across multiple iterations. It evaluates equivalence across multiple endpoints using the Ratio of Means (ROM) test.

Usage

run_simulations_2x2_rom(
  nsim,
  n,
  muT,
  muR,
  SigmaW,
  lequi_tol,
  uequi_tol,
  alpha,
  sigmaB,
  dropout,
  Eper,
  Eco,
  typey,
  adseq,
  k,
  arm_seed
)

Arguments

nsim

Integer. The number of simulations to run.

n

Integer. The sample size per period.

muT

Numeric vector. Mean outcomes for the active treatment.

muR

Numeric vector. Mean outcomes for the reference treatment.

SigmaW

Numeric matrix. Within-subject covariance matrix for endpoints.

lequi_tol

Numeric vector. Lower equivalence thresholds for each endpoint.

uequi_tol

Numeric vector. Upper equivalence thresholds for each endpoint.

alpha

Numeric vector. Significance levels for hypothesis testing across endpoints.

sigmaB

Numeric. Between-subject variance for the crossover model.

dropout

Numeric vector of size 2. Dropout rates for each sequence.

Eper

Numeric vector. Expected period effects for each sequence.

Eco

Numeric vector. Expected carryover effects for each sequence.

typey

Integer vector indicating the classification of each endpoint, where 1 corresponds to a primary endpoint and 2 corresponds to a secondary endpoint.

adseq

Logical. If TRUE, applies sequential (hierarchical) testing.

k

Integer. Minimum number of endpoints required for equivalence.

arm_seed

Integer vector. Random seed for each simulation.

Details

This function evaluates equivalence using the Ratio of Means (ROM) test. Equivalence is determined based on predefined lower lequi_tol and upper uequi_tol equivalence thresholds, and hypothesis testing is conducted at the specified significance level alpha. If adseq is TRUE, primary endpoints must establish equivalence before secondary endpoints are evaluated. The sample size per period is adjusted based on dropout rates, ensuring valid study conclusions. The simulation incorporates within-subject correlation using SigmaW and accounts for between-subject variance with sigmaB. Expected period effects Eper and carryover effects Eco are included in the model. A fixed random seed arm_seed is used to ensure reproducibility across simulations.//'

Value

A numeric matrix where each column stores simulation results: The first row (totaly) represents the overall equivalence decision (1 = success, 0 = failure). Subsequent rows contain equivalence decisions per endpoint, mean estimates for the treatment group, mean estimates for the reference group, standard deviations for treatment, and standard deviations for reference.

@author Thomas Debray tdebray@fromdatatowisdom.com

Run Simulations for a Parallel Design with Difference of Means (DOM) test

Description

This function simulates a parallel-group trial across multiple iterations. It evaluates equivalence across multiple endpoints using the Difference of Means (DOM) test.

Usage

run_simulations_par_dom(
  nsim,
  n,
  muT,
  muR,
  SigmaT,
  SigmaR,
  lequi_tol,
  uequi_tol,
  alpha,
  dropout,
  typey,
  adseq,
  k,
  arm_seed_T,
  arm_seed_R,
  TART,
  TARR,
  vareq
)

Arguments

nsim

Integer. The number of simulations to run.

n

Integer. The sample size per arm (before dropout).

muT

arma::vec. Mean vector for the treatment arm.

muR

arma::vec. Mean vector for the reference arm.

SigmaT

arma::mat. Covariance matrix for the treatment arm.

SigmaR

arma::mat. Covariance matrix for the reference arm.

lequi_tol

arma::rowvec. Lower equivalence thresholds for each endpoint.

uequi_tol

arma::rowvec. Upper equivalence thresholds for each endpoint.

alpha

arma::rowvec. Significance level for each endpoint.

dropout

arma::vec. Dropout rates for each arm (T, R).

typey

Integer vector indicating the classification of each endpoint, where 1 corresponds to a primary endpoint and 2 corresponds to a secondary endpoint.

adseq

Boolean. If TRUE, applies sequential (hierarchical) testing.

k

Integer. Minimum number of endpoints required for equivalence.

arm_seed_T

arma::ivec. Random seed vector for the treatment group (one per simulation).

arm_seed_R

arma::ivec. Random seed vector for the reference group (one per simulation).

TART

Double. Treatment allocation ratio (proportion of subjects in treatment arm).

TARR

Double. Reference allocation ratio (proportion of subjects in reference arm).

vareq

Boolean. If TRUE, assumes equal variances across treatment and reference groups.

Details

Equivalence testing uses either the Difference of Means (DOM) test, applying predefined equivalence thresholds and significance levels. When hierarchical testing (adseq) is enabled, all primary endpoints must demonstrate equivalence before secondary endpoints are evaluated. Dropout rates are incorporated into the sample size calculation to ensure proper adjustment. Randomization is controlled through separate random seeds for the treatment and reference groups, enhancing reproducibility.

Value

The function returns an arma::mat storing simulation results row-wise for consistency with R's output format. The first row (totaly) contains the overall equivalence decision (1 for success, 0 for failure). The subsequent rows include equivalence deicisons for each endpoint, mean estimates for both treatment and reference groups, and corresponding standard deviations.

Author(s)

Thomas Debray tdebray@fromdatatowisdom.com

Run Simulations for a Parallel Design with Ratio of Means (ROM) test

Description

This function simulates a parallel-group trial across multiple iterations. It evaluates equivalence across multiple endpoints using the Ratio of Means (ROM) test.

Usage

run_simulations_par_rom(
  nsim,
  n,
  muT,
  muR,
  SigmaT,
  SigmaR,
  lequi_tol,
  uequi_tol,
  alpha,
  dropout,
  typey,
  adseq,
  k,
  arm_seed_T,
  arm_seed_R,
  TART,
  TARR,
  vareq
)

Arguments

nsim

Integer. The number of simulations to run.

n

Integer. The sample size per arm (before dropout).

muT

arma::vec. Mean vector for the treatment arm.

muR

arma::vec. Mean vector for the reference arm.

SigmaT

arma::mat. Covariance matrix for the treatment arm.

SigmaR

arma::mat. Covariance matrix for the reference arm.

lequi_tol

arma::rowvec. Lower equivalence thresholds for each endpoint.

uequi_tol

arma::rowvec. Upper equivalence thresholds for each endpoint.

alpha

arma::rowvec. Significance level for each endpoint.

dropout

arma::vec. Dropout rates for each arm (T, R).

typey

Integer vector indicating the classification of each endpoint, where 1 corresponds to a primary endpoint and 2 corresponds to a secondary endpoint.

adseq

Boolean. If TRUE, applies sequential (hierarchical) testing.

k

Integer. Minimum number of endpoints required for equivalence.

arm_seed_T

arma::ivec. Random seed vector for the treatment group (one per simulation).

arm_seed_R

arma::ivec. Random seed vector for the reference group (one per simulation).

TART

Double. Treatment allocation ratio (proportion of subjects in treatment arm).

TARR

Double. Reference allocation ratio (proportion of subjects in reference arm).

vareq

Boolean. If TRUE, assumes equal variances across treatment and reference groups.

Details

Equivalence testing uses either the Ratio of Means (ROM) test, applying predefined equivalence thresholds and significance levels. When hierarchical testing (adseq) is enabled, all primary endpoints must demonstrate equivalence before secondary endpoints are evaluated. Dropout rates are incorporated into the sample size calculation to ensure proper adjustment. Randomization is controlled through separate random seeds for the treatment and reference groups, enhancing reproducibility.

Value

The function returns an arma::mat storing simulation results row-wise for consistency with R's output format. The first row (totaly) contains the overall equivalence decision (1 for success, 0 for failure). The subsequent rows include equivalence decisions for each endpoint, mean estimates for both treatment and reference groups, and corresponding standard deviations.

Author(s)

Thomas Debray tdebray@fromdatatowisdom.com

Sample Size Calculation for Bioequivalence and Multi-Endpoint Studies

Description

Computes the required sample size to achieve a target power in studies with multiple endpoints and treatment arms. The function employs modified root-finding algorithms to estimate sample size while accounting for correlation structures, variance assumptions, and equivalence bounds across endpoints. It is particularly useful for bioequivalence trials and multi-arm studies with complex endpoint structures.

Usage

sampleSize(
  mu_list,
  varcov_list = NA,
  sigma_list = NA,
  cor_mat = NA,
  sigmaB = NA,
  Eper,
  Eco,
  rho = 0,
  TAR = rep(1, length(mu_list)),
  arm_names = NA,
  ynames_list = NA,
  type_y = NA,
  list_comparator = NA,
  list_y_comparator = NA,
  power = 0.8,
  alpha = 0.05,
  lequi.tol = NA,
  uequi.tol = NA,
  list_lequi.tol = NA,
  list_uequi.tol = NA,
  dtype = "parallel",
  ctype = "ROM",
  vareq = TRUE,
  lognorm = TRUE,
  k = NA,
  adjust = "no",
  dropout = NA,
  nsim = 5000,
  seed = 1234,
  ncores = 1,
  optimization_method = "fast",
  lower = 2,
  upper = 500,
  step.power = 6,
  step.up = TRUE,
  pos.side = FALSE,
  maxiter = 1000,
  verbose = FALSE
)

Arguments

mu_list

Named list of arithmetic means per treatment arm. Each element is a vector representing expected outcomes for all endpoints in that arm.

varcov_list

List of variance-covariance matrices, where each element corresponds to a comparator. Each matrix has dimensions: number of endpoints × number of endpoints.

sigma_list

List of standard deviation vectors, where each element corresponds to a comparator and contains one standard deviation per endpoint.

cor_mat

Matrix specifying the correlation structure between endpoints, used along with sigma_list to calculate varcov_list if not provided.

sigmaB

Numeric. Between-subject variance for a 2×2 crossover design.

Eper

Optional numeric vector of length 2 specifying the period effect in a dtype = "2x2" design, applied as c(Period 0, Period 1). Defaults to c(0, 0). Ignored for dtype = "parallel".

Eco

Optional numeric vector of length 2 specifying the carry-over effect per arm in a dtype = "2x2" design, applied as c(Reference, Treatment). Defaults to c(0, 0). Ignored for dtype = "parallel".

rho

Numeric. Correlation parameter applied uniformly across all endpoint pairs. Used with sigma_list to compute varcov_list when cor_mat or varcov_list are not provided.

TAR

Numeric vector specifying treatment allocation rates per arm. The order must match arm_names. Defaults to equal allocation across arms if not provided.

arm_names

Optional character vector of treatment names. If not supplied, names are derived from mu_list.

ynames_list

Optional list of vectors specifying endpoint names per arm. If names are missing, arbitrary names are assigned based on order.

type_y

Integer vector indicating endpoint types: 1 for co-primary endpoints, 2 for secondary endpoints.

list_comparator

List of comparators. Each element is a vector of length 2 specifying the treatment names being compared.

list_y_comparator

List of endpoint sets per comparator. Each element is a vector containing endpoint names to compare. If not provided, all endpoints common to both comparator arms are used.

power

Numeric. Target power (default = 0.8).

alpha

Numeric. Significance level (default = 0.05).

lequi.tol

Numeric. Lower equivalence bounds (e.g., -0.5) applied uniformly across all endpoints and comparators.

uequi.tol

Numeric. Upper equivalence bounds (e.g., 0.5) applied uniformly across all endpoints and comparators.

list_lequi.tol

List of numeric vectors specifying lower equivalence bounds per comparator.

list_uequi.tol

List of numeric vectors specifying upper equivalence bounds per comparator.

dtype

Character. Trial design: "parallel" (default) for parallel-group or "2x2" for crossover (only for 2-arm studies).

ctype

Character. Hypothesis test type: "DOM" (Difference of Means) or "ROM" (Ratio of Means).

vareq

Logical. Assumes equal variances across arms if TRUE (default = FALSE).

lognorm

Logical. Whether data follows a log-normal distribution (TRUE or FALSE).

k

Integer vector. Minimum number of successful endpoints required for global bioequivalence per comparator. Defaults to all endpoints per comparator.

adjust

Character. Alpha adjustment method: "k" (K-fold), "bon" (Bonferroni), "sid" (Sidak), "no" (default, no adjustment), or "seq" (sequential).

dropout

Numeric vector specifying dropout proportion per arm.

nsim

Integer. Number of simulated studies (default = 5000).

seed

Integer. Seed for reproducibility.

ncores

Integer. Number of processing cores for parallel computation. Defaults to 1. Set to NA for automatic detection (ncores - 1).

optimization_method

Character. Sample size optimization method: "fast" (default, root-finding algorithm) or "step-by-step".

lower

Integer. Minimum sample size for search range (default = 2).

upper

Integer. Maximum sample size for search range (default = 500).

step.power

Numeric. Initial step size for sample size search, defined as 2^step.power. Used when optimization_method = "fast".

step.up

Logical. If TRUE (default), search increments upward from lower; if FALSE, decrements downward from upper. Used when optimization_method = "fast".

pos.side

Logical. If TRUE, finds the smallest integer i closest to the root such that f(i) > 0. Used when optimization_method = "fast".

maxiter

Integer. Maximum iterations allowed for sample size estimation (default = 1000). Used when optimization_method = "fast".

verbose

Logical. If TRUE, prints progress and messages during execution (default = FALSE).

Value

A list containing:

response

Array summarizing simulation results, including estimated sample sizes, achieved power, and confidence intervals.

table.iter

Data frame showing estimated sample sizes and calculated power at each iteration.

table.test

Data frame containing test results for all simulated trials.

param.u

Original input parameters.

param

Final adjusted parameters used in sample size calculation.

param.d

Trial design parameters used in the simulation.

Author(s)

Johanna Muñoz johanna.munoz@fromdatatowisdom.com

References

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. doi:10.1007/BF01068419

Mielke, J., Jones, B., Jilma, B., & König, F. (2018). Sample size for multiple hypothesis testing in biosimilar development. Statistics in Biopharmaceutical Research, 10(1), 39-49. doi:10.1080/19466315.2017.1371071

Berger, R. L., & Hsu, J. C. (1996). Bioequivalence trials, intersection-union tests, and equivalence confidence sets. Statistical Science, 283-302.

Sozu, T., Sugimoto, T., Hamasaki, T., & Evans, S. R. (2015). "Sample Size Determination in Clinical Trials with Multiple Endpoints." SpringerBriefs in Statistics. doi:10.1007/978-3-319-22005-5

Examples

mu_list <- list(SB2 = c(AUCinf = 38703, AUClast = 36862, Cmax = 127.0),
                EUREF = c(AUCinf = 39360, AUClast = 37022, Cmax = 126.2),
                USREF = c(AUCinf = 39270, AUClast = 37368, Cmax = 129.2))

sigma_list <- list(SB2 = c(AUCinf = 11114, AUClast = 9133, Cmax = 16.9),
                   EUREF = c(AUCinf = 12332, AUClast = 9398, Cmax = 17.9),
                   USREF = c(AUCinf = 10064, AUClast = 8332, Cmax = 18.8))

# Equivalent boundaries
lequi.tol <- c(AUCinf = 0.8, AUClast = 0.8, Cmax = 0.8)
uequi.tol <- c(AUCinf = 1.25, AUClast = 1.25, Cmax = 1.25)

# Arms to be compared
list_comparator <- list(EMA = c("SB2", "EUREF"),
                        FDA = c("SB2", "USREF"))

# Endpoints to be compared
list_y_comparator <- list(EMA = c("AUCinf", "Cmax"),
                          FDA = c("AUClast", "Cmax"))

# Equivalence boundaries for each comparison
lequi_lower <- c(AUCinf = 0.80, AUClast = 0.80, Cmax = 0.80)
lequi_upper <- c(AUCinf = 1.25, AUClast = 1.25, Cmax = 1.25)

# Run the simulation
sampleSize(power = 0.9, alpha = 0.05, mu_list = mu_list,
           sigma_list = sigma_list, list_comparator = list_comparator,
           list_y_comparator = list_y_comparator,
           list_lequi.tol = list("EMA" = lequi_lower, "FDA" = lequi_lower),
           list_uequi.tol = list("EMA" = lequi_upper, "FDA" = lequi_upper),
           adjust = "no", dtype = "parallel", ctype = "ROM", vareq = FALSE,
           lognorm = TRUE, ncores = 1, nsim = 50, seed = 1234)

Sample Size Estimation for Multiple Hypothesis Testing Using Mielke's Method

Description

Estimates the required sample size to achieve a specified power level for multiple hypothesis testing, using the approach described by Mielke et al. (2018). This function is particularly useful for bioequivalence or biosimilar studies with multiple correlated endpoints, where a minimum number of endpoints must meet equivalence criteria.

Usage

sampleSize_Mielke(
  power,
  Nmax,
  m,
  k,
  rho,
  sigma,
  true.diff,
  equi.tol,
  design,
  alpha,
  adjust = "no",
  seed = NULL,
  nsim = 10000
)

Arguments

power

Numeric. Desired statistical power.

Nmax

Integer. Maximum allowable sample size.

m

Integer. Total number of endpoints.

k

Integer. Number of endpoints that must meet the success criteria for overall study success.

rho

Numeric. Constant correlation coefficient among endpoints.

sigma

Numeric or vector. Standard deviation of each endpoint. If a single value is provided, it is assumed to be constant across all endpoints. In a 2x2 crossover design, this is the within-subject standard deviation; in a parallel design, it represents the treatment group’s standard deviation, assumed to be the same for both test and reference.

true.diff

Numeric or vector. Assumed true difference between test and reference for each endpoint. If a single value is provided, it is applied uniformly across all endpoints.

equi.tol

Numeric. Equivalence margin; the equivalence interval is defined as (-equi.tol, +equi.tol).

design

Character. Study design, either "22co" for a 2x2 crossover design or "parallel" for a parallel groups design.

alpha

Numeric. Significance level for the hypothesis test.

adjust

Character. Method for multiplicity adjustment; options are "no" (no adjustment), "bon" (Bonferroni adjustment), and "k" (k-adjustment).

seed

Integer. Random seed for reproducibility.

nsim

Integer. Number of simulations to run for power estimation (default: 10,000).

Details

This function uses the method proposed by Mielke et al. (2018) to estimate the sample size required to achieve the desired power level in studies with multiple correlated endpoints. The function iteratively increases sample size until the target power is reached or the maximum allowable sample size (Nmax) is exceeded. The approach accounts for endpoint correlation and supports adjustments for multiple testing using various correction methods.

Value

A named vector containing:

"power.a"

Achieved power with the estimated sample size.

"SS"

Required sample size per sequence to achieve the target power.

References

Mielke, J., Jones, B., Jilma, B. & König, F. Sample Size for Multiple Hypothesis Testing in Biosimilar Development. Statistics in Biopharmaceutical Research 10, 39–49 (2018).

Examples

# Example 1 from Mielke
sampleSize_Mielke(power = 0.8, Nmax = 1000, m = 5, k = 5, rho = 0,
                  sigma = 0.3, true.diff =  log(1.05), equi.tol = log(1.25),
                  design = "parallel", alpha = 0.05, adjust = "no",
                  seed = 1234, nsim = 100)

Simulated Test Statistic for Noninferiority/Equivalence Trials

Description

Simulates test statistics for multiple hypothesis testing in biosimilar development, following the approach described by Mielke et al. (2018). It calculates the necessary sample size for meeting equivalence criteria across multiple endpoints while considering correlation structures and applying multiplicity adjustments.

Usage

sign_Mielke(
  N,
  m,
  k,
  R,
  sigma,
  true.diff,
  equi.tol = log(1.25),
  design,
  alpha = 0.05,
  adjust = "no"
)

Arguments

N

Integer specifying the number of subjects per sequence.

m

Integer specifying the number of endpoints.

k

Integer specifying the number of endpoints that must meet equivalence to consider the test successful.

R

Matrix specifying the correlation structure between endpoints. This should be an m x m matrix, e.g., generated using variance.const.corr().

sigma

Numeric specifying the standard deviation of endpoints. Can be a vector of length m (one per endpoint) or a single value. In a 2x2 crossover design, this represents within-subject variance. In a parallel-group design, it represents the treatment group standard deviation.

true.diff

Numeric specifying the assumed true difference between test and reference. Can be a vector of length m or a single value.

equi.tol

Numeric specifying the equivalence margins. The interval is defined as (-equi.tol, +equi.tol).

design

Character specifying the study design. Options are "22co" for a 2x2 crossover design or "parallel" for a parallel-group design.

alpha

Numeric specifying the significance level.

adjust

Character specifying the method for multiplicity adjustment. Options include "no" for no adjustment, "bon" for Bonferroni correction, and "k" for k-adjustment.

Details

This function is designed for multiple-endpoint clinical trials, where success is defined as meeting equivalence criteria for at least a subset of tests. Simulated test statistics are based on multivariate normal distribution assumptions, and the function supports k-out-of-m success criteria for regulatory approval.

Type I error control is achieved through multiplicity adjustments as proposed by Lehmann and Romano (2005) to ensure rigorous error rate management. This approach is particularly relevant for biosimilar studies, where sample size estimation must account for multiple comparisons across endpoints, doses, or populations.

Value

A numeric vector representing a realization of the simulated test statistic for the given setting.

References

Kong, L., Kohberger, R. C., & Koch, G. G. (2004). Type I Error and Power in Noninferiority/Equivalence Trials with Correlated Multiple Endpoints: An Example from Vaccine Development Trials. Journal of Biopharmaceutical Statistics, 14(4), 893–907.

Lehmann, E. L., & Romano, J. P. (2005). Generalizations of the Familywise Error Rate. The Annals of Statistics, 33(2), 1138–1154.

Mielke, J., Jones, B., Jilma, B., & König, F. (2018). Sample Size for Multiple Hypothesis Testing in Biosimilar Development. Statistics in Biopharmaceutical Research, 10(1), 39–49.

Generate Simulated Endpoint Data for Parallel Group Design

Description

Generate simulated endpoint data for a parallel design, with options for normal and lognormal distributions.

Usage

simParallelEndpoints(
  n,
  mu.arithmetic,
  mu.geometric = NULL,
  Sigma,
  CV = NULL,
  seed,
  dist = "normal"
)

Arguments

n

Integer. The sample size for the generated data.

mu.arithmetic

Numeric vector. The arithmetic mean of the endpoints on the original scale.

mu.geometric

Numeric vector. The geometric mean of the endpoints on the original scale. Only used if dist = "lognormal".

Sigma

Matrix. Variance-covariance matrix of the raw data on the original scale. If dist = "lognormal", this matrix is transformed to the log scale.

CV

Numeric vector. Coefficient of variation (CV) of the raw data. Only used when dist = "lognormal", where it is transformed to the log scale.

seed

Integer. Seed for random number generation, ensuring reproducibility.

dist

Character. Assumed distribution of the endpoints: either "normal" or "lognormal".

Value

A matrix of simulated endpoint values for a parallel design, with dimensions n by the number of variables in mu.arithmetic or mu.geometric.

Author(s)

Thomas Debray tdebray@fromdatatowisdom.com

Simulate a 2x2 Crossover Design and Compute Difference of Means (DOM)

Description

Simulates a two-sequence, two-period (2x2) crossover design and evaluate equivalence for the difference of means (DOM).

Usage

test_2x2_dom(
  n,
  muT,
  muR,
  SigmaW,
  lequi_tol,
  uequi_tol,
  alpha,
  sigmaB,
  dropout,
  Eper,
  Eco,
  typey,
  adseq,
  k,
  arm_seed
)

Arguments

n

integer number of subjects per sequence

muT

vector mean of endpoints on treatment arm

muR

vector mean of endpoints on reference arm

SigmaW

matrix within subject covar-variance matrix across endpoints

lequi_tol

vector lower equivalence tolerance band across endpoints

uequi_tol

vector upper equivalence tolerance band across endpoints

alpha

vector alpha value across endpoints

sigmaB

double between subject variance (assumed same for all endpoints)

dropout

vector of size 2 with dropout proportion per sequence (0,1)

Eper

vector of size 2 with period effect on period (0,1)

Eco

vector of size 2 with carry over effect of arm c(Reference, Treatment).

typey

vector with positions of primary endpoints

adseq

boolean is used a sequential adjustment?

k

integer minimum number of equivalent endpoints

arm_seed

seed for the simulation

Value

A numeric matrix containing the simulated hypothesis test results. The first column represents the overall equivalence decision, where 1 indicates success and 0 indicates failure. The subsequent columns contain the hypothesis test results for each endpoint, followed by mean estimates for the reference and treatment groups, and standard deviations for the reference and treatment groups.

Simulate a 2x2 Crossover Design and Compute Ratio of Means (ROM)

Description

Simulates a two-sequence, two-period (2x2) crossover design and evaluate equivalence for the ratio of means (ROM).

Usage

test_2x2_rom(
  n,
  muT,
  muR,
  SigmaW,
  lequi_tol,
  uequi_tol,
  alpha,
  sigmaB,
  dropout,
  Eper,
  Eco,
  typey,
  adseq,
  k,
  arm_seed
)

Arguments

n

integer number of subjects per sequence

muT

vector mean of endpoints on treatment arm

muR

vector mean of endpoints on reference arm

SigmaW

matrix within subject covar-variance matrix across endpoints

lequi_tol

vector lower equivalence tolerance band across endpoints

uequi_tol

vector upper equivalence tolerance band across endpoints

alpha

vector alpha value across endpoints

sigmaB

double between subject variance (assumed same for all endpoints)

dropout

vector of size 2 with dropout proportion per sequence (0,1)

Eper

vector of size 2 with period effect on period (0,1)

Eco

vector of size 2 with carry over effect of arm c(Reference, Treatment).

typey

vector with positions of primary endpoints

adseq

boolean is used a sequential adjustment?

k

integer minimum number of equivalent endpoints

arm_seed

seed for the simulation

Value

A numeric matrix containing the simulated hypothesis test results. The first column represents the overall equivalence decision, where 1 indicates success and 0 indicates failure. The subsequent columns contain the hypothesis test results for each endpoint, followed by mean estimates for the reference and treatment groups, and standard deviations for the reference and treatment groups.

Simulate a Parallel Design and Test Difference of Means (DOM)

Description

Simulates a parallel-group design and performs equivalence testing using the difference of means (DOM) approach. This function evaluates whether the treatment and reference groups are equivalent based on predefined equivalence margins and hypothesis testing criteria.

Usage

test_par_dom(
  n,
  muT,
  muR,
  SigmaT,
  SigmaR,
  lequi_tol,
  uequi_tol,
  alpha,
  dropout,
  typey,
  adseq,
  k,
  arm_seedT,
  arm_seedR,
  TART,
  TARR,
  vareq
)

Arguments

n

integer number of subjects per arm

muT

vector mean of endpoints on treatment arm

muR

vector mean of endpoints on reference arm

SigmaT

matrix covar-variance matrix on treatment arm across endpoints

SigmaR

matrix covar-variance matrix on reference arm across endpoints

lequi_tol

vector lower equivalence tolerance band across endpoints

uequi_tol

vector upper equivalence tolerance band across endpoints

alpha

vector alpha value across endpoints

dropout

vector of size 2 with dropout proportion per arm (T,R)

typey

vector with positions of primary endpoints

adseq

boolean is used a sequential adjustment?

k

integer minimum number of equivalent endpoints

arm_seedT

integer seed for the simulation on treatment arm

arm_seedR

integer seed for the simulation on reference arm

TART

double treatment allocation rate for the treatment arm

TARR

double treatment allocation rate for the reference arm

vareq

boolean assumed equivalence variance between arms for the t-test

Details

The function simulates a parallel-group study design and evaluates equivalence using the difference of means (DOM) approach. It accounts for dropout rates and treatment allocation proportions while generating simulated data based on the specified covariance structure. The test statistics are computed, and a final equivalence decision is made based on the predefined number of required significant endpoints (k). If sequential testing (adseq) is enabled, primary endpoints must establish equivalence before secondary endpoints are evaluated. When vareq = TRUE, the test assumes equal variances between groups and applies Schuirmann's two one-sided tests (TOST).

Value

A numeric matrix containing the simulated hypothesis test results. The first column represents the overall equivalence decision, where 1 indicates success and 0 indicates failure. The subsequent columns contain the hypothesis test results for each endpoint, followed by mean estimates for the reference and treatment groups, and standard deviations for the reference and treatment groups.

Simulate a Parallel Design and Test Ratio of Means (ROM)

Description

Simulates a parallel-group design and performs equivalence testing using the ratio of means (ROM) approach. This function evaluates whether the treatment and reference groups are equivalent based on predefined equivalence margins and hypothesis testing criteria.

Usage

test_par_rom(
  n,
  muT,
  muR,
  SigmaT,
  SigmaR,
  lequi_tol,
  uequi_tol,
  alpha,
  dropout,
  typey,
  adseq,
  k,
  arm_seedT,
  arm_seedR,
  TART,
  TARR,
  vareq
)

Arguments

n

integer number of subjects per arm

muT

vector mean of endpoints on treatment arm

muR

vector mean of endpoints on reference arm

SigmaT

matrix covar-variance matrix on treatment arm across endpoints

SigmaR

matrix covar-variance matrix on reference arm across endpoints

lequi_tol

vector lower equivalence tolerance band across endpoints

uequi_tol

vector upper equivalence tolerance band across endpoints

alpha

vector alpha value across endpoints

dropout

vector of size 2 with dropout proportion per arm (T,R)

typey

vector with positions of primary endpoints

adseq

boolean is used a sequential adjustment?

k

integer minimum number of equivalent endpoints

arm_seedT

integer seed for the simulation on treatment arm

arm_seedR

integer seed for the simulation on reference arm

TART

double treatment allocation rate for the treatment arm

TARR

double treatment allocation rate for the reference arm

vareq

Boolean. If TRUE, assumes equal variance between arms and applies Schuirmann's two one-sided tests (TOST) for equivalence using a pooled variance.

Details

The function simulates a parallel-group study design and evaluates equivalence using the ratio of means (ROM) approach. It accounts for dropout rates and treatment allocation proportions while generating simulated data based on the specified covariance structure. The test statistics are computed, and a final equivalence decision is made based on the predefined number of required significant endpoints (k). If sequential testing (adseq) is enabled, primary endpoints must establish equivalence before secondary endpoints are evaluated. When vareq = TRUE, the test assumes equal variances between groups and applies Schuirmann's two one-sided tests (TOST).

Value

A numeric matrix containing the simulated hypothesis test results. The first column represents the overall equivalence decision, where 1 indicates success and 0 indicates failure. The subsequent columns contain the hypothesis test results for each endpoint, followed by mean estimates for the reference and treatment groups, and standard deviations for the reference and treatment groups.

test_studies

Description

Internal function to estimate the bioequivalence test for nsim simulated studies given a sample size n

Usage

test_studies(nsim, n, comp, param, param.d, arm_seed, ncores)

Arguments

nsim

number of simulated studies

n

sample size

comp

index comparator

param

list of parameters (mean,sd,tar)

param.d

design parameters

arm_seed

seed for each endpoint to get consistent in simulations across all comparators

ncores

number of cores used for the calculation

Value

a logical matrix of size (nsim) X (number of endpoints + 1) function only replicates test_bioq nsim times.

Optimizer for Uniroot Integer (Modified)

Description

A modified integer-based root-finding algorithm for determining the sample size required to achieve a target power. This function extends the uniroot integer search method to handle cases with stepwise power searches while considering constraints on search limits.

Usage

uniroot.integer.mod(
  f,
  power,
  lower = lower,
  upper = upper,
  step.power = step.power,
  step.up = step.up,
  pos.side = pos.side,
  maxiter = maxiter,
  ...
)

Arguments

f

Function for which a root is needed.

power

Numeric. Target power value.

lower

Integer. Minimum allowable root value.

upper

Integer. Maximum allowable root value.

step.power

Numeric. Initial step size defined as 2^step.power.

step.up

Logical. If TRUE, the search increments from lower; if FALSE, it decrements from upper.

pos.side

Logical. If TRUE, finds the closest integer i such that f(i) > 0.

maxiter

Integer. Maximum number of iterations allowed.

...

Additional arguments passed to f.

Value

A list containing:

root

The integer value closest to the root on the correct side.

f.root

Value of f at the estimated root.

iter

Number of function evaluations performed.

table.iter

A data frame showing estimated sample size (N) and corresponding power at each iteration.

table.test

A data frame containing endpoint-level test results for each simulation and corresponding N.

Validate Positive Semi-Definite Matrices

Description

Validates that all matrices in a list are symmetric and positive semi-definite.

Usage

validate_positive_definite(varcov_list)

Arguments

varcov_list

List of matrices. Each matrix is checked to ensure it is symmetric and positive semi-definite.

Value

NULL. If all matrices pass, the function returns nothing. If any matrix fails, it stops with an error message.

Author(s)

Thomas Debray tdebray@fromdatatowisdom.com

Check Sample Size Limits

Description

Validates that the upper and lower limits are numeric and that the upper limit is greater than the lower limit.

Usage

validate_sample_size_limits(lower, upper)

Arguments

lower

Numeric. The initial lower limit for the search range.

upper

Numeric. The initial upper limit for the search range.

Value

NULL. If the checks pass, the function returns nothing. If the checks fail, it stops execution with an error message.

Author(s)

Thomas Debray tdebray@fromdatatowisdom.com