# Bioequivalence Tests for Parallel Trial Designs: 3 Arms, 3 Endpoints

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 et al. 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 3 arms/treatments and 3 primary endpoints.

# Introduction

Similar to the example presented in Bioequivalence Tests for Parallel Trial Designs: 2 Arms, 3 Endpoints, where equivalence across multiple endpoints was assessed, this vignette extends the framework to trials involving two reference products. This scenario arises when regulators from different regions require comparisons with their locally sourced reference biosimilars.

In many studies, the aim is to evaluate equivalence across multiple primary variables. For example, the European Medicines Agency (EMA) recommends demonstrating equivalence for both the Area Under the Curve (AUC) and the maximum concentration (Cmax) when endpoints. In this scenario, we additionally evaluate equivalence across three treatment arms, including a test biosimilar and two reference products, each sourced from a different regulatory region.

This vignette demonstrates advanced techniques for calculating sample size in such trials, in which multiple endpoints and multiple reference products are considered. As an illustrative example, we use data from the phase 1 trial NCT01922336, which assessed pharmacokinetic (PK) endpoints following a single dose of SB2, its EU-sourced reference product (EU-INF), and its US-sourced reference product (US-INF) (Shin et al. 2015).

Primary PK measures between test and reference product. Data represent arithmetic mean +- standard deviation.
PK measure | SB2 | EU-INF | US-INF
--- | --- | --- | ---
AUCinf (\(\mu\)g*h/mL) | 38,703 \(\pm\) 11,114 | 39,360 \(\pm\) 12,332 | 39,270 \(\pm\) 10,064
AUClast (\(\mu\)g*h/mL) | 36,862 \(\pm\) 9133 | 37,022 \(\pm\) 9398 | 37,368 \(\pm\) 8332
Cmax (\(\mu\)g/mL) | 127.0 \(\pm\) 16.9 | 126.2 \(\pm\) 17.9 | 129.2 \(\pm\) 18.8

# Input Data

As in this vignette, the following inputs are required:

- Means for each endpoint and treatment arm (`mu_list`),
- Standard deviations for each endpoint and arm (`sigma_list`),
- Endpoint and arm names (`yname_list` and `arm_names`),
- Arms to be compared within each comparator (`list_comparator`),
- Endpoints to be compared within each comparator (`list_y_comparator`).

In this example, we simultaneously compare the trial drug (SB2) to each reference biosimilar:

- SB2 vs. EU-INF
- SB2 vs. US-INF

We define the required list objects:

    # Mean values for each endpoint and arm
    mu_list <- list(
      SB2 = c(AUCinf = 38703, AUClast = 36862, Cmax = 127.0),
      EUINF = c(AUCinf = 39360, AUClast = 37022, Cmax = 126.2),
      USINF = c(AUCinf = 39270, AUClast = 37368, Cmax = 129.2)
    )

    # Standard deviation values for each endpoint and arm
    sigma_list <- list(
      SB2 = c(AUCinf = 11114, AUClast = 9133, Cmax = 16.9),
      EUINF = c(AUCinf = 12332, AUClast = 9398, Cmax = 17.9),
      USINF = c(AUCinf = 10064, AUClast = 8332, Cmax = 18.8)
    )

# Simultaneous Testing of Independent Co-Primary Endpoints

In this analysis, we adopt the Ratio of Means (ROM) approach to evaluate bioequivalence across independent co-primary endpoints.

## Different Hypotheses across endpoints

In this example, we aim to demonstrate:

- Bioequivalence of SB2 vs. EU-INF for AUCinf and Cmax, and
- Bioequivalence of SB2 vs. US-INF for AUClast and Cmax.

The comparisons are specified as follows:

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

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

For all endpoints, bioequivalence is established if the 90% confidence intervals for the ratios of the geometric means fall within the equivalence range of 80.00% to 125.00%.

Below we define the equivalence boundaries for each comparison:

    list_lequi.tol <- list(
      "EMA" = c(AUCinf = 0.8, Cmax = 0.8),
      "FDA" = c(AUClast = 0.8, Cmax = 0.8)
    )

    list_uequi.tol <- list(
      "EMA" = c(AUCinf = 1.25, Cmax = 1.25),
      "FDA" = c(AUClast = 1.25, Cmax = 1.25)
    )

Here, the `list_comparator` parameter specifies the arms being compared, while `list_lequi.tol` and `list_uequi.tol` define the lower and upper equivalence boundaries for the two endpoints.

To calculate the required sample size for testing equivalence under the specified conditions, we use the sampleSize() function. This function computes the total sample size needed to achieve the target power while ensuring equivalence criteria are met.

    library(SimTOST)

    (N_ss <- sampleSize(power = 0.9,        # Target power
                        alpha = 0.05,      # Type I error rate
                        mu_list = mu_list, # Means for each endpoint and arm
                        sigma_list = sigma_list, # Standard deviations
                        list_comparator = list_comparator, # Comparator arms
                        list_y_comparator = list_y_comparator, # Endpoints to compare
                        list_lequi.tol = list_lequi.tol, # Lower equivalence boundaries
                        list_uequi.tol = list_uequi.tol, # Upper equivalence boundaries
                        dtype = "parallel", # Trial design type
                        ctype = "ROM",      # Test type: Ratio of Means
                        vareq = TRUE,       # Assume equal variances
                        lognorm = TRUE,     # Log-normal distribution assumption
                        nsim = 1000,        # Number of stochastic simulations
                        seed = 1234))       # Random seed for reproducibility
    #> Sample Size Calculation Results
    #> -------------------------------------------------------------
    #> Study Design: parallel trial targeting 90% power with a 5% type-I error.
    #>
    #> Comparisons:
    #>    SB2 vs. EUINF
    #>     - Endpoints Tested: AUCinf, Cmax
    #>       (multiple co-primary endpoints, m =  2 )
    #>    SB2 vs. USINF
    #>     - Endpoints Tested: AUClast, Cmax
    #>       (multiple co-primary endpoints, m =  2 )
    #> -------------------------------------------------------------
    #>                  Parameter       Value
    #>          Total Sample Size         117
    #>             Achieved Power          90
    #>  Power Confidence Interval 87.9 - 91.8
    #> -------------------------------------------------------------

A total sample size of 117 patients (or 39 per arm) is required to demonstrate equivalence.

# Simultaneous Testing of Independent Primary Endpoints

## Equivalence for At Least 2 of the 3 Endpoints with Bonferroni Adjustment

In this example, we aim to establish equivalence for at least two out of three primary endpoints while accounting for multiplicity using the Bonferroni adjustment. The following assumptions are made:

- Equality of Variances: variances are assumed to be equal across groups (`vareq = TRUE`).
- Testing Parameter: the Ratio of Means (ROM) is used as the testing parameter (`ctype = "ROM"`).
- Design: a parallel trial design is assumed (`dtype = "parallel"`).
- Distribution: endpoint data follows a log-normal distribution (`lognorm = TRUE`).
- Correlation: endpoints are assumed to be independent, with no correlation between them (default `rho = 0`).
- Multiplicity Adjustment: the Bonferroni correction is applied to control for Type I error (`adjust = "bon"`).
- Equivalence Criterion: equivalence is required for at least two of the three endpoints (`k = 2`).

The comparisons and equivalence boundaries are defined as follows:

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

    # Define lower equivalence boundaries for each comparator
    list_lequi.tol <- list(
      EMA = c(AUCinf = 0.8, AUClast = 0.8, Cmax = 0.8),
      FDA = c(AUCinf = 0.8, AUClast = 0.8, Cmax = 0.8)
    )

    # Define upper equivalence boundaries for each comparator
    list_uequi.tol <- list(
      EMA = c(AUCinf = 1.25, AUClast = 1.25, Cmax = 1.25),
      FDA = c(AUCinf = 1.25, AUClast = 1.25, Cmax = 1.25)
    )

Finally, we calculate the required sample size:

    (N_mp <- sampleSize(power = 0.9,       # Target power
                        alpha = 0.05,      # Type I error rate
                        mu_list = mu_list, # Means for each endpoint and arm
                        sigma_list = sigma_list, # Standard deviations
                        list_comparator = list_comparator, # Comparator arms
                        list_y_comparator = list_y_comparator, # Endpoints to compare
                        list_lequi.tol = list_lequi.tol, # Lower equivalence boundaries
                        list_uequi.tol = list_uequi.tol, # Upper equivalence boundaries
                        k = 2,              # Number of endpoints required to demonstrate equivalence
                        adjust = "bon",     # Bonferroni adjustment for multiple comparisons
                        dtype = "parallel", # Trial design type (parallel group)
                        ctype = "ROM",      # Test type: Ratio of Means
                        vareq = TRUE,       # Assume equal variances across arms
                        lognorm = TRUE,     # Log-normal distribution assumption
                        nsim = 1000,        # Number of stochastic simulations
                        seed = 1234))       # Random seed for reproducibility
    #> Sample Size Calculation Results
    #> -------------------------------------------------------------
    #> Study Design: parallel trial targeting 90% power with a 5% type-I error.
    #>
    #> Comparisons:
    #>    SB2 vs. EUINF
    #>     - Endpoints Tested: AUCinf, AUClast, Cmax
    #>       (multiple primary endpoints, k =  2 )
    #>     - Multiplicity Correction: Bonferroni
    #>       Adjusted Significance Levels: alpha = 0.0167
    #>
    #>    SB2 vs. USINF
    #>     - Endpoints Tested: AUCinf, AUClast, Cmax
    #>       (multiple primary endpoints, k =  2 )
    #>     - Multiplicity Correction: Bonferroni
    #>       Adjusted Significance Levels: alpha = 0.0167
    #>
    #> -------------------------------------------------------------
    #>                  Parameter       Value
    #>          Total Sample Size          93
    #>             Achieved Power          90
    #>  Power Confidence Interval 87.9 - 91.8
    #> -------------------------------------------------------------

## Unequal Allocation Rates Across Arms

In this example, we build upon the previous scenario but introduce unequal allocation rates across treatment arms. Specifically, we require that the number of patients in the new treatment arm is double the number in each of the reference arms.

This can be achieved by specifying the treatment allocation rate parameter (`TAR`). Rates are provided as a vector, for example: `TAR = c(2, 1, 1)`. This ensures that for every two patients assigned to the new treatment arm, one patient is assigned to each reference arm.

    (N_mp2 <- sampleSize(power = 0.9,       # Target power
                         alpha = 0.05,      # Type I error rate
                         mu_list = mu_list, # Means for each endpoint and arm
                         sigma_list = sigma_list, # Standard deviations
                         list_comparator = list_comparator, # Comparator arms
                         list_y_comparator = list_y_comparator, # Endpoints to compare
                         list_lequi.tol = list_lequi.tol, # Lower equivalence boundaries
                         list_uequi.tol = list_uequi.tol, # Upper equivalence boundaries
                         k = 2,              # Number of endpoints required to demonstrate equivalence
                         adjust = "bon",     # Bonferroni adjustment for multiple comparisons
                         TAR = c("SB2" = 2, "EUINF" = 1, "USINF" = 1), # Treatment allocation rates
                         dtype = "parallel", # Trial design type (parallel group)
                         ctype = "ROM",      # Test type: Ratio of Means
                         vareq = TRUE,       # Assume equal variances across arms
                         lognorm = TRUE,     # Log-normal distribution assumption
                         nsim = 1000,        # Number of stochastic simulations
                         seed = 1234))       # Random seed for reproducibility
    #> Sample Size Calculation Results
    #> -------------------------------------------------------------
    #> Study Design: parallel trial targeting 90% power with a 5% type-I error.
    #>
    #> Comparisons:
    #>    SB2 vs. EUINF
    #>     - Endpoints Tested: AUCinf, AUClast, Cmax
    #>       (multiple primary endpoints, k =  2 )
    #>     - Multiplicity Correction: Bonferroni
    #>       Adjusted Significance Levels: alpha = 0.0167
    #>
    #>    SB2 vs. USINF
    #>     - Endpoints Tested: AUCinf, AUClast, Cmax
    #>       (multiple primary endpoints, k =  2 )
    #>     - Multiplicity Correction: Bonferroni
    #>       Adjusted Significance Levels: alpha = 0.0167
    #>
    #> -------------------------------------------------------------
    #>                  Parameter       Value
    #>          Total Sample Size          96
    #>             Achieved Power          90
    #>  Power Confidence Interval 87.9 - 91.8
    #> -------------------------------------------------------------

Results indicate that 48 patients are required for the active treatment arm (SB2), and 24 patients are required for each reference arm. The total sample size required is 96, which is larger compared to the trial with an equal allocation ratio, for which the total sample size was 93.

## Accounting for Participant Dropout

In the examples above, the sample size calculations assumed that all enrolled patients complete the trial. However, in practice, a certain percentage of participants drop out, which can impact the required sample size. To account for this, we consider a 20% dropout rate across all treatment arms.

    (N_mp3 <- sampleSize(power = 0.9,       # Target power
                         alpha = 0.05,      # Type I error rate
                         mu_list = mu_list, # Means for each endpoint and arm
                         sigma_list = sigma_list, # Standard deviations
                         list_comparator = list_comparator, # Comparator arms
                         list_y_comparator = list_y_comparator, # Endpoints to compare
                         list_lequi.tol = list_lequi.tol, # Lower equivalence boundaries
                         list_uequi.tol = list_uequi.tol, # Upper equivalence boundaries
                         k = 2,              # Number of endpoints required to demonstrate equivalence
                         adjust = "bon",     # Bonferroni adjustment for multiple comparisons
                         TAR = c("SB2" = 2, "EUINF" = 1, "USINF" = 1), # Treatment allocation rates
                         dropout = c("SB2" = 0.20, "EUINF" = 0.20, "USINF" = 0.20), # Expected dropout rates
                         dtype = "parallel", # Trial design type (parallel group)
                         ctype = "ROM",      # Test type: Ratio of Means
                         vareq = TRUE,       # Assume equal variances across arms
                         lognorm = TRUE,     # Log-normal distribution assumption
                         nsim = 1000,        # Number of stochastic simulations
                         seed = 1234))       # Random seed for reproducibility
    #> Sample Size Calculation Results
    #> -------------------------------------------------------------
    #> Study Design: parallel trial targeting 90% power with a 5% type-I error.
    #>
    #> Comparisons:
    #>    SB2 vs. EUINF
    #>     - Endpoints Tested: AUCinf, AUClast, Cmax
    #>       (multiple primary endpoints, k =  2 )
    #>     - Multiplicity Correction: Bonferroni
    #>       Adjusted Significance Levels: alpha = 0.0167
    #>
    #>    SB2 vs. USINF
    #>     - Endpoints Tested: AUCinf, AUClast, Cmax
    #>       (multiple primary endpoints, k =  2 )
    #>     - Multiplicity Correction: Bonferroni
    #>       Adjusted Significance Levels: alpha = 0.0167
    #>
    #> -------------------------------------------------------------
    #>                  Parameter       Value
    #>          Total Sample Size         120
    #>             Achieved Power          90
    #>  Power Confidence Interval 87.9 - 91.8
    #> -------------------------------------------------------------

Considering a 20% dropout rate, the total sample size required is 120.

# References

Shin, Donghoon, Youngdoe Kim, Yoo Seok Kim, Thomas Körnicke, and Rainard Fuhr. 2015. “A Randomized, Phase I Pharmacokinetic Study Comparing SB2 and Infliximab Reference Product (Remicade) in Healthy Subjects.” BioDrugs 29 (6): 381–88. https://doi.org/10.1007/s40259-015-0150-5.
Sozu, Takashi, Tomoyuki Sugimoto, Toshimitsu Hamasaki, and Scott R. Evans. 2015. Sample Size Determination in Clinical Trials with Multiple Endpoints. SpringerBriefs in Statistics. Cham: Springer International Publishing. https://doi.org/10.1007/978-3-319-22005-5.