SimTOST (version 1.0.2)

sampleSize: 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
)

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.

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).

Author

Johanna Muñoz

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)