Introduction
This vignette demonstrates advanced sample size calculationtechniques for parallel trial designs involving three arms and oneendpoint. Specifically, we calculate the required sample size to testbioequivalence between a new treatment (SB2) and a reference product(Remicade) administered in two different locations (“EU_Remicade” and“USA_Remicade”). The endpoint of interest is the Area Under the Curve(AUCinf), a commonly used pharmacokinetic measure.
In this example, we assume the endpoint follows a log-normaldistribution with equal variances across arms. The goal is to determinethe sample size needed to achieve 90% power while controlling the type Ierror rate at 5%. The equivalence margin is defined as\(E_L = 80\%\) and\(E_U = 125\%\) of the reference mean on theoriginal scale.
The methods presented in this vignette build on fundamentalbioequivalence testing concepts and extend them to multi-arm scenarios.These examples provide practical insights for designing robust paralleltrials with complex equivalence testing requirements.
We assume that the primary endpoint, AUCinf, is on the originalscale, with the mean and standard deviation for each arm available. Thisinformation is organized into a structured data table for furtheranalysis:
library(SimTOST)data<- data.table::data.table(arm =c("SB2","RemEU","RemUSA"),mean =c(37162.0,37705.0,37702.8),sd =c(11113.62172,12332.41615,12113.72))
Sample Size Calculation for AUCinf: Equivalence to EU Remicade
This example demonstrates how to calculate the required sample sizewhen testing the equivalence of SB2 to a reference drug, Remicade, asadministered in the EU. The goal is to determine the minimum number ofparticipants needed to ensure adequate power for the equivalencetest.
Hypotheses
Null Hypothesis (No Equivalence):\[H_0:\frac{\mu_{SB2}}{\mu_{RemEU}} \le E_L ~~ or~~\frac{\mu_{SB2}}{\mu_{RemEU}} \ge E_U\]
Alternative Hypothesis (Equivalence):
\[H_1:E_L<\frac{\mu_{SB2}}{\mu_{RemEU}} < E_U\]
Here,\(E_L\) and\(E_U\) represent the lower and upperequivalence margins, respectively.
Preparing the function arguments
To proceed with the sample size calculation, we first need toorganize the mean and standard deviation values of each arm (SB2, EURemicade, USA Remicade) as list objects.
Since this example focuses on a single endpoint, the mean list(mu_list) contains three scalar elements corresponding toeach arm, and the standard deviation list (sigma_list)contains three 1x1 matrix elements.
mu_list<-as.list(data$mean)# Organize mean values into a listsigma_list<-as.list(data$sd)# Organize standard deviation values into a list
Next, we define the comparison parameters, including the lower(lequi.tol) and upper (uequi.tol) equivalenceboundaries, as well as the list of comparators. Since we are onlycomparing two arms (SB2 and EU Remicade), the list of comparatorscontains a single element specifying these two arms:
list_comparator<-list("Comparison"=c("SB2","RemEU"))list_lequi.tol<-list("Comparison"=0.8)list_uequi.tol<-list("Comparison"=1/0.8)
Computing Sample Size
Finally, we use thesampleSize() function tocalculate the required sample size based on stochastic simulations ofthe trial. The function accepts several parameters, such as the desiredpower, confidence level, and design specifications. By default, itassumes:
- A parallel design,
- A test based on the ratio of means (ROM),
- Equal variances across arms,
- A lognormal distribution for the endpoint.
AUCinf_1comp<-sampleSize(power =0.9,# Target poweralpha =0.05,# Confidence levelarm_names = data$arm,# Names of trial armslist_comparator = list_comparator,# Comparator configurationmu_list = mu_list,# Mean valuessigma_list = sigma_list,# Standard deviation valueslist_lequi.tol = list_lequi.tol,# Lower equivalence boundarylist_uequi.tol = list_uequi.tol,# Upper equivalence boundarynsim =1000# Number of stochastic simulations)AUCinf_1comp#> Sample Size Calculation Results#> -------------------------------------------------------------#> Study Design: parallel trial targeting 90% power with a 5% type-I error.#>#> Comparisons:#> SB2 vs. RemEU#> - Endpoints Tested: y1#> -------------------------------------------------------------#> Parameter Value#> Total Sample Size 84#> Achieved Power 90.3#> Power Confidence Interval 88.3 - 92#> -------------------------------------------------------------
The required sample size for this scenario is 84, or 42 for eacharm.
Sample Size Calculation for AUCinf: Equivalence to US Remicade andEU Remicade
In this section, we calculate the sample size required to demonstrateequivalence with both the European reference product (RemEU) and the USreference product (RemUS). This scenario involves a more complexcomparison, as we must simultaneously establish equivalence with twodistinct reference arms.
Hypotheses
Null Hypothesis (No Equivalence):
\[H_0: \frac{\mu_{SB2}}{\mu_{RemEU}} \leE_L ~~ or~~ \frac{\mu_{SB2}}{\mu_{RemEU}} \ge E_U~~ or~~\frac{\mu_{SB2}}{\mu_{RemUSA}} \le E_L ~~ or~~\frac{\mu_{SB2}}{\mu_{RemUSA}} \ge E_U\]
Alternative Hypothesis (Equivalence):
\[H_1:E_L<\frac{\mu_{SB2}}{\mu_{RemEU}} < E_U~~and~~E_L<\frac{\mu_{SB2}}{\mu_{RemUSA}} < E_U\]KeyConsiderations
- The functionsampleSize()adjusts the alpha level for multiple comparisons within a singlecomparator (e.g., comparisons involving only RemEU or RemUSA).
- The function does not yet account for alpha adjustments betweencomparators when simultaneously comparing SB2 with both referenceproducts (RemEU and RemUSA). Thus, the null hypothesis is rejected ifany ROM falls outside the equivalence boundaries. Future package updateswill address this limitation and include proper alpha adjustments forcomparisons between comparators.
Implementation Details
Implementation of this scenario is similar to the initial example, inwhich thelist_comparator is modified to include multiplecomparators. The list contains two elements specifying the simultaneouscomparison between SB2 vs. RemEU and SB2 vs. RemUSA. This approachensures that the equivalence assessment accounts for both referenceproducts.
Additionally, we must specify equivalence boundaries for eachcomparison individually. These boundaries define the acceptable rangefor equivalence and are provided separately for each comparator.
list_comparator<-list("EMA"=c("SB2","RemEU"),"FDA"=c("SB2","RemUSA"))list_lequi.tol<-list("EMA"=0.8,"FDA"=0.8)# Lower equivalence boundarylist_uequi.tol<-list("EMA"=1/0.8,"FDA"=1/0.8)# Upper equivalence boundary
Computing Sample Size
We then pass these values into thesampleSize() function tocalculate the required sample size for multiple comparisons.
(AUCinf_2comp<-sampleSize(power =0.9,# Target poweralpha =0.05,# Confidence levelarm_names = data$arm,# Names of trial armslist_comparator = list_comparator,# Comparator configurationmu_list = mu_list,# Mean valuessigma_list = sigma_list,# Standard deviation valueslist_lequi.tol = list_lequi.tol,# Lower equivalence boundarylist_uequi.tol = list_uequi.tol,# Upper equivalence boundarynsim =1000# Number of stochastic simulations))#> Sample Size Calculation Results#> -------------------------------------------------------------#> Study Design: parallel trial targeting 90% power with a 5% type-I error.#>#> Comparisons:#> SB2 vs. RemEU#> - Endpoints Tested: y1#> SB2 vs. RemUSA#> - Endpoints Tested: y1#> -------------------------------------------------------------#> Parameter Value#> Total Sample Size 150#> Achieved Power 91#> Power Confidence Interval 89 - 92.7#> -------------------------------------------------------------
Results and Interpretation
The required total sample size for this scenario is 150. Notably, anadditional 8 patients per arm are required to achieve equivalence withboth reference products (RemEU and RemUSA), compared to the scenariowhere equivalence is required with only one reference arm (42).
This example demonstrates the added complexity and sample sizerequirements when multiple comparators are involved in an equivalencetrial.
The SimTOST package includes theplot() function, which isdesigned to permit visualization of the relationship between sample size(x-axis) and achieved power (y-axis) for all combinations of endpointsand comparators. In this example, we use theplot() function to generate aplot for theAUCinf_2comp object.
