R/SampleSize.R
sampleSize.RdThis function calculates the required sample size to achieve a target power in studies with multiple endpoints and treatment arms. The function leverages modified root-finding algorithms to estimate sample size while considering correlation structures, variance assumptions, and equivalence bounds across endpoints. It is especially useful for bioequivalence trials or multi-arm trials with complex endpoint structures.
sampleSize(
mu_list,
varcov_list = NA,
sigma_list = NA,
cor_mat = NA,
sigmaB = NA,
Eper,
Eco,
rho = 0,
TAR = NULL,
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 = T,
k = NA,
adjust = "no",
dropout = NA,
nsim = 5000,
seed = 1234,
ncores = NA,
optimization_method = "fast",
lower = 2,
upper = 500,
step.power = 6,
step.up = TRUE,
pos.side = FALSE,
maxiter = 1000,
verbose = FALSE
)Named list of arithmetic means per treatment arm. Each element contains a vector (i.e., one per treatment arm) with the expected outcomes for all endpoints of interest.
list of var-cov matrices, each element corresponds to a comparator with a varcov matrix of size number of endpoints X number of endpoints.
list of sigma vectors, each element corresponds to a comparator with a sigma vector of size number of endpoints.
matrix specifying the correlation matrix between endpoints, used along with sigma_list to calculate the varcov list in case it is not provided.
number between subject variance only for 2x2 design.
Optional. Vector of length 2 specifying the period effect in a dtype = "2x2" design, applied to c(Period 0, Period 1). Defaults to c(0, 0) if not provided. Ignored for dtype = "parallel".
Optional. Vector of length 2 specifying the carry-over effect for each arm in a dtype = "2x2" design, applied to c(Reference, Treatment). Defaults to c(0, 0) if not provided. Ignored for dtype = "parallel".
Correlation parameter applied uniformly across all endpoint pairs, used with sigma_list to calculate varcov if cor_mat or varcov_list are not provided.
Numeric vector. Treatment allocation rates for each arm, where the order of values corresponds to the order of arm_names. The length of TAR must match the number of arms. If not provided, a default equal allocation rate is assigned across all arms.
Optional vector with the treatment names. If not supplied, it will be derived from mu_list.
Optional list of vectors with Endpoint names on each arm. When not all endpoint names are provided for each arm, arbitrary names (assigned by vector order) are used.
vector with the type of endpoints: primary endpoint(1), otherwise (2).
list of comparators, i.e each comparator is a vector of size 1 X 2 where are specified the name of treatments
list of endpoints to be considered in each comparator. Each element of the list is a vector containing the names of the endpoints to compare. When it is not provided, all endpoints present in both compared arms are used.
target power (default = 0.8)
alpha level (default = 0.05)
lower equivalence bounds (e.g., -0.5) expressed in raw scale units (e.g., scalepoints) of endpoint repeated on all endpoints and comparators
upper equivalence bounds (e.g., -0.5) expressed in raw scale units (e.g., scalepoints) of endpoint repeated on all endpoints and comparators
list of lower equivalence bounds (e.g., -0.5) expressed in raw scale units (e.g., scalepoints) of endpoint in comparator
list of upper equivalence bounds (e.g., -0.5) expressed in raw scale units (e.g., scalepoints) of endpoint in comparator
Character. Design type for the trial: "parallel" (default) for parallel group design or "2x2" for crossover design (applicable only for trials with 2 arms).
Character. Specifies the type of hypothesis test for comparison: "DOM" for Difference of Means or "ROM" for Ratio of Means.
Logical indicating whether variances are assumed equal across arms (default = FALSE).
Is data log-normally distributed? (TRUE, FALSE)
Vector with the number of endpoints that must be successful (integer) for global bioequivalence for each comparator. If no k vector is provided, it will be set to the total number of endpoints on each comparator.
Character. Method for alpha adjustment: "k" (K-fold), "bon" (Bonferroni), "sid" (Sidak), "no" (no adjustment, default), or "seq" (sequential adjustment).
vector with proportion of total population with dropout per arm
number of simulated studies (default=5000)
main seed
Integer. Number of processing cores to use for parallel computation. Defaults to one less than the total number of detected cores.
Character. Method for determining the required sample size: "fast" (using modified root-finding algorithms) or "step-by-step". Defaults to "fast".
Integer. Initial value of N for the search range. Defaults to 2.
Integer. Maximum value of N for the search range. Defaults to 500.
Numeric. The initial step size for the sample size search, defined as 2^step.power. Relevant when optimization_method is "fast".
Logical. If TRUE (default), the sample size search increments upward from the lower limit; if FALSE, it decrements downward from the upper limit. Used only when optimization_method is "fast".
Logical. If TRUE, finds the smallest integer, i, closest to the root such that f(i) > 0. Used only when optimization_method is "fast".
Integer. Maximum number of iterations allowed for finding the sample size. Defaults to 1000. Used only when optimization_method is "fast".
Logical. If TRUE, the function displays progress and informational messages during execution. Defaults to FALSE.
An object simss that contains the following elements :
array with the sample sizes for each arm and aproximated achieved power with confidence intervals
data frame with the estimated sample size for each arm and power calculated at each searching iteration
data frame that collects the total information of the simulation at each iteration
parameters provided by the user
parameters used for the sample size calculation; as param.u are checked and modified in case of any inconsistent or missing information provided
parameters of design
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.
Berger, R. L., & Hsu, J. C. (1996). Bioequivalence trials, intersection-union tests and equivalence confidence sets. Statistical Science, 283-302.
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)
#> Given a 90% target power with 100(1-2*0.05)% confidence level.
#>
#> The total required sample size to achieve 90% power is 138 sample units.
#>
#> n_drop n_SB2 n_EUREF n_USREF n_total power power_LCI power_UCI
#> <num> <num> <num> <num> <num> <num> <num> <num>
#> 0 46 46 46 138 0.9 0.7740882 0.9625954