Bioequivalence Tests for Parallel Trial Designs with Log-Normal Data"
In SimTOST: Sample Size Estimation for Bio-Equivalence Trials Through Simulation

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In the SimTOST R package, which is specifically designed for sample size estimation for bioequivalence studies, hypothesis testing is based on the Two One-Sided Tests (TOST) procedure. [@sozu_sample_2015] In TOST, the equivalence test is framed as a comparison between the the null hypothesis of ‘new product is worse by a clinically relevant quantity’ and the alternative hypothesis of ‘difference between products is too small to be clinically relevant’. This vignette focuses on a parallel design, with 2 arms/treatments and 5 primary endpoints.

In the following two examples, we demonstrate the use of SimTOST for parallel trial designs with data assumed to follow a normal distribution on the log scale. We start by loading the package.

library(SimTOST)

Multiple Independent Co-Primary Endpoints

Here, we consider a bio-equivalence trial with 2 treatment arms and $m=5$ endpoints. The sample size is calculated to ensure that the test and reference products are equivalent with respect to all 5 endpoints. The true ratio between the test and reference products is assumed to be 1.05. It is assumed that the standard deviation of the log-transformed response variable is $\sigma = 0.3$, and that all tests are independent ($\rho = 0$). The equivalence limits are set at 0.80 and 1.25. The significance level is 0.05. The sample size is determined at a power of 0.8.

This example is adapted from @mielke_sample_2018, who employed a difference-of-means test on the log scale. The sample size calculation can be conducted using two approaches, both of which are illustrated below.

Approach 1: Using sampleSize_Mielke

In the first approach, we calculate the required sample size for 80% power using the sampleSize_Mielke() function. This method directly follows the approach described in @mielke_sample_2018, assuming a difference-of-means test on the log-transformed scale with specified parameters.

ssMielke <- 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 = 10000)
ssMielke

For 80\% power, r ssMielke["SS"] subjects per sequence (r ssMielke["SS"] * 2 in total) would be required.

Approach 2: Using sampleSize

Alternatively, the sample size calculation can be performed using the sampleSize() function. This method assumes that effect sizes are normally distributed on the log scale and uses a difference-of-means test (ctype = "DOM") with user-specified values for mu_list and sigma_list. This method allows for greater flexibility than Approach 1 in specifying parameter distributions.

mu_r <- setNames(rep(log(1.00), 5), paste0("y", 1:5))
mu_t <- setNames(rep(log(1.05), 5), paste0("y", 1:5))
sigma <- setNames(rep(0.3, 5), paste0("y", 1:5))
lequi_lower <- setNames(rep(log(0.8), 5), paste0("y", 1:5))
lequi_upper <- setNames(rep(log(1.25), 5), paste0("y", 1:5))

ss <- sampleSize(power = 0.8, alpha = 0.05,
                 mu_list = list("R" = mu_r, "T" = mu_t),
                 sigma_list = list("R" = sigma, "T" = sigma),
                 list_comparator = list("T_vs_R" = c("R", "T")),
                 list_lequi.tol = list("T_vs_R" = lequi_lower),
                 list_uequi.tol = list("T_vs_R" = lequi_upper),
                 dtype = "parallel", ctype = "DOM", lognorm = FALSE, 
                 adjust = "no", ncores = 1, nsim = 10000, seed = 1234)
ss

For 80\% power, a total of r ss$response$n_total subjects would be required.

Consider an alternative scenario in which the standard deviation of the log-transformed response variable is unknown, but the standard deviation on the original scale is known ($\sigma = 1$). In such cases, the sampleSize() function can still accommodate adjustments to handle these uncertainties by transforming parameters accordingly. We now provide all data on the raw scale, including the equivalence bounds, and set ctype = ROM and lognorm = TRUE.

Multiple Correlated Co-Primary Endpoints

In the second example, we set $k=m=5$, $\sigma = 0.3$ and $\rho = 0.8$. This example is also adapted from @mielke_sample_2018, who employed a difference-of-means test on the log scale. The sample size calculation can again be conducted using two approaches, both of which are illustrated below.

Approach 1: Using sampleSize_Mielke

ssMielke <- sampleSize_Mielke(power = 0.8, Nmax = 1000, m = 5, k = 5, rho = 0.8, 
                              sigma = 0.3, true.diff = log(1.05), 
                              equi.tol = log(1.25), design = "parallel", 
                              alpha = 0.05, adjust = "no", seed = 1234, 
                              nsim = 10000)
ssMielke

For 80\% power, r ssMielke["SS"] subjects per sequence (r ssMielke["SS"] * 2 in total) would be required.

Approach 2: Using sampleSize

Alternatively, the sample size calculation can be performed using the sampleSize() function. This method assumes that effect sizes are normally distributed on the log scale and uses a difference-of-means test (ctype = "DOM") with user-specified values for mu_list, sigma_list, and the correlation parameter rho.

mu_r <- setNames(rep(log(1.00), 5), paste0("y", 1:5))
mu_t <- setNames(rep(log(1.05), 5), paste0("y", 1:5))
sigma <- setNames(rep(0.3, 5), paste0("y", 1:5))
lequi_lower <- setNames(rep(log(0.8), 5), paste0("y", 1:5))
lequi_upper <- setNames(rep(log(1.25), 5), paste0("y", 1:5))

ss <- sampleSize(power = 0.8, alpha = 0.05,
                 mu_list = list("R" = mu_r, "T" = mu_t),
                 sigma_list = list("R" = sigma, "T" = sigma),
                 rho = 0.8, # high correlation between the endpoints
                 list_comparator = list("T_vs_R" = c("R", "T")),
                 list_lequi.tol = list("T_vs_R" = lequi_lower),
                 list_uequi.tol = list("T_vs_R" = lequi_upper),
                 dtype = "parallel", ctype = "DOM", lognorm = FALSE, 
                 adjust = "no", ncores = 1, k = 5, nsim = 10000, seed = 1234)
ss