Power2Stage

• Supported Methods
  • Simulation-based
    • ‘Type 1’
    • ‘Type 2’
    • Blinded Sample Size Re-estimation in the Interim
    • Group Sequential Design
  • Inverse-Normal Combination
• Functions
  • Main
  • Helpers
• Examples
  • Method B
  • Method C
  • Inverse-Normal Combination
• Speed Comparisons
• Installation
• Session Information

The package contains functions to obtain the operational characteristics (power, type I error, percentage of studies proceeding to the second stage, average and quantiles of total sample sizes) of bioequivalence studies in adaptive sequential Two-Stage Designs (TSD) via simulations.

Version 0.5.4.9000 built 2022-07-02 with R 4.2.1 (development version not on CRAN).

Supported Methods

Simulation-based

Since the many letters denoting the methods given by various authors might be confusing, I classified the methods as two ‘types’:

• ‘Type 1’
  An adjusted α is used both in the interim as well as in the final analysis of pooled data.
• ‘Type 2’
  Whether an unadjusted or an adjusted α is used depends on interim power. An adjusted α is used in the final analysis of pooled data.

It should be noted that the adjusted alphas do not necessarily have to be the same in both stages. Below a summary of conditions used in the decision schemes of the published methods.

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‘Type 1’

• Potvin et al. (2008) ‘Method B’: α 0.0294 (θ₀ 0.95, target power 0.80).
• Fuglsang (2013) ‘Method B’: α 0.0284 (θ₀ 0.95, target power 0.90).
• Karalis (2013) ‘TSD-2’: α 0.0294 (θ₀ = PE, target power 0.80).
• Fuglsang (2014) ‘Method B’ (parallel design): α 0.0294 (θ₀ 0.95, target power 0.80).
• Zheng et al. (2015) ‘MSDBE’: α₁ 0.01, α₂ 0.04.
• Xu et al. (2016) ‘Method E’: (θ₀ 0.95, target power 0.80, n_max 42).
  • For CV 10–30%
    α₁ 0.0294, α₂ 0.0357, futility rule on CI {0.9374, 1/0.9374}.
  • For CV 30–55%
    α₁ 0.0254, α₂ 0.0363, futility rule on CI {0.9305, 1/0.9305}.
• Molins et al. (2017) ‘Type 1 modified Potvin B’: α 0.0301 (θ₀ 0.95, target power 0.80, min. n₂ = 1.5n₁, n_max 150).

‘Type 2’

• Potvin et al. (2008) ‘Method C’: α 0.0294 (θ₀ 0.95, target power 0.80).
• Montague et al. (2011) ‘Method D’: α 0.0280 (θ₀ 0.90, target power 0.80).
• Fuglsang (2013) ‘Method C/D’:
  α 0.0274 (θ₀ 0.95, target power 0.90).
  α 0.0269 (θ₀ 0.90, target power 0.90).
• Karalis and Macheras (2013) ‘TSD’: α 0.0294 (θ₀ = PE, target power 0.80).
• Karalis (2013) ‘TSD-1’: α 0.0280 (θ₀ = PE, target power 0.80).
• Xu et al. (2016) ‘Method F’: (θ₀ 0.95, target power 0.80, n_max 180).
  • For CV 10–30%
    α₁ 0.0248, α₂ 0.0364, futility rule on CI {0.9492, 1/0.9492}.
  • For CV 30–55%
    α₁ 0.0259, α₂ 0.0349, futility rule on CI {0.9350, 1/0.9350}.
• Molins et al. (2017) ‘Type 2 modified Potvin C’: α 0.0280 (θ₀ 0.95, target power 0.80, min. n₂ = 1.5n₁, n_max 150).

Blinded Sample Size Re-estimation in the Interim

Golkowski et al. (2014).

Group Sequential Design

Kieser and Rauch (2015).

Inverse-Normal Combination

König et al. (2014), Kieser and Rauch (2015), Wassmer and Brannath (2016), Maurer et al. (2018).

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Functions

Main

Defaults employed if not specified in the function call:

function	theta0	target power	usePE	Nmax	max.n	fCrit	fClower
power.tsd()	0.95	0.80	FALSE	Inf	–	–	–
power.tsd.fC()	0.95	0.80	FALSE	–	Inf	"PE"	0.80
power.tsd.KM()	0.95	0.80	–	150	–	–	–
power.tsd.ssr()	0.95	0.80	FALSE	–	Inf	–	–
power.tsd.GS()	0.95	–	–	–	–	"PE"	0.80
power.tsd.in()	0.95	0.80	FALSE	–	Inf	"CI"	0.95
power.tsd.p()	0.95	0.80	FALSE	Inf	–	–	–

All functions are for a 2×2×2 crossover design except power.tsd.p(), which is for a two-group parallel design.
If usePE = TRUE the point estimate in the interim is used in sample size estimation of the second stage.
If the estimated total sample size exceeds max.n the second stage will be forced to max.n - n1 (i.e., it is not a futility criterion).
The method used for interim power and sample size estimation is specified by the argument pmethod. It defaults to "nct" (approximation by the noncentral t-distribution) except in power.tsd.GS(), where the total sample size is already fixed.
The BE limits are specified by the arguments theta1 and theta2 (default to 0.80 and 1.25).
The number of simulations is specified by the argument nsims. It defaults to 1e5 if simulating power and to 1e6 if simulating the empiric type I error (i.e., theta0 set to the value of theta1 or theta2).

Futility Criteria in the Interim

• Nmax: The study will stop if the estimated total sample size exceeds Nmax.
• fCrit ("PE" or "CI"): The study will stop if outside fClower and 1/fClower.
  • "PE": fClower defaults to 0.80.
  • "CI": fClower defaults to 0.925 (except in function power.tsd.in(), where it defaults to 0.95).

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Helpers

• sampleN2.TOST()
  Estimates the sample size of stage 2 to achieve at least the target power.
• interim.tsd.in()
  Interim analysis based on the Inverse-Normal Combination method.
• final.tsd.in()
  Final analysis based on the Inverse-Normal Combination method.

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Examples

Before running the examples attach the library.

library(Power2Stage)

If not noted otherwise, defaults are employed.

Method B

Power estimation by the shifted central t-distribution.

power.tsd(CV = 0.20, n1 = 12, pmethod = "shifted")
# TSD with 2x2 crossover 
# Method B: alpha (s1/s2) = 0.0294 0.0294 
# Target power in power monitoring and sample size est. = 0.8
# Power calculation via shifted central t approx. 
# CV1 and GMR = 0.95 in sample size est. used
# No futility criterion
# BE acceptance range = 0.8 ... 1.25
# 
# CV = 0.2; n(stage 1) = 12; GMR = 0.95
# 
# 1e+05 sims at theta0 = 0.95 (p(BE) = 'power').
# p(BE)    = 0.84454
# p(BE) s1 = 0.41333
# Studies in stage 2 = 56.45%
# 
# Distribution of n(total)
# - mean (range) = 20.7 (12 ... 82)
# - percentiles
#  5% 50% 95% 
#  12  18  40

Explore the empiric type I error at the upper BE-limit.

power.tsd(CV = 0.20, n1 = 12, pmethod = "shifted",
          theta0 = 1.25)[["pBE"]]
# [1] 0.046352

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Method C

Power estimation by the shifted central t-distribution.

power.tsd(method = "C", CV = 0.20, n1 = 12, pmethod = "shifted")
# TSD with 2x2 crossover 
# Method C: alpha0 = 0.05, alpha (s1/s2) = 0.0294 0.0294 
# Target power in power monitoring and sample size est. = 0.8
# Power calculation via shifted central t approx. 
# CV1 and GMR = 0.95 in sample size est. used
# No futility criterion
# BE acceptance range = 0.8 ... 1.25
# 
# CV = 0.2; n(stage 1) = 12; GMR = 0.95
# 
# 1e+05 sims at theta0 = 0.95 (p(BE) = 'power').
# p(BE)    = 0.8496
# p(BE) s1 = 0.42656
# Studies in stage 2 = 53.7%
# 
# Distribution of n(total)
# - mean (range) = 20.6 (12 ... 82)
# - percentiles
#  5% 50% 95% 
#  12  18  40

Slightly better than ‘Method B’ in terms of power in both stages and fewer studies are expected to proceed to the second stage.

Explore the empiric type I error at the upper BE-limit (1 milion simulations).

power.tsd(method = "C", CV = 0.20, n1 = 12, pmethod = "shifted",
          theta0 = 1.25)[["pBE"]]
# [1] 0.051238

Slight inflation of the type I error (although considered negligible by the authors). However, more adjustment (adjusted α 0.0280) controls the type I error.

power.tsd(method = "C", alpha = rep(0.0280, 2), CV = 0.20,
          n1 = 12, pmethod = "shifted", theta0 = 1.25)[["pBE"]]
# [1] 0.049903

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Inverse-Normal Combination

Data given by Potvin et al. in Example 2: 12 subjects in stage 1, PE 1.0876, CV 0.18213, all defaults of the function used.

interim.tsd.in(GMR = 0.95, GMR1 = 1.0876, CV1 = 0.18213, n1 = 12)
# TSD with 2x2 crossover
# Inverse Normal approach
#  - Maximum combination test with weights for stage 1 = 0.5 0.25 
#  - Significance levels (s1/s2) = 0.02635 0.02635 
#  - Critical values (s1/s2) = 1.93741 1.93741 
#  - BE acceptance range = 0.8 ... 1.25
#  - Observed point estimate from stage 1 is not used for SSR
#  - With conditional error rates and conditional estimated target power
# 
# Interim analysis after first stage
# - Derived key statistics:
#   z1 = 3.10000, z2 = 1.70344
#   Repeated CI = (0.92491, 1.27891)
#   Median unbiased estimate = NA
# - No futility criterion met
# - Test for BE not positive (not considering any futility rule)
# - Calculated n2 = 6
# - Decision: Continue to stage 2 with 6 subjects

The second stage should be initiated with 6 subjects. Note that with interim.tsd.in(..., fCrit = "No", ssr.conditional = "no") 8 subjects would be required like in the Methods of Potvin et al.

The second stage is performed in 8 subjects, PE 0.9141, CV 0.25618.

final.tsd.in(GMR1 = 1.0876, CV1 = 0.18213, n1 = 12,
             GMR2 = 0.9141, CV2 = 0.25618, n2 = 8)
# TSD with 2x2 crossover
# Inverse Normal approach
#  - Maximum combination test with weights for stage 1 = 0.5 0.25 
#  - Significance levels (s1/s2) = 0.02635 0.02635 
#  - Critical values (s1/s2) = 1.93741 1.93741 
#  - BE acceptance range = 0.8 ... 1.25
# 
# Final analysis after second stage
# - Derived key statistics:
#   z1 = 2.87952, z2 = 2.60501
#   Repeated CI = (0.87690, 1.17356)
#   Median unbiased estimate = 1.0135
# - Decision: BE achieved

The study passed with a (repeated) CI of 87.69–117.36%. Although slightly more conservative, same conclusion like based on the 94.12% CI of 88.45–116.38% reported by Potvin et al.

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Speed Comparisons

Performed on a Xeon E3-1245v3 3.4 GHz, 8 MB cache, 16 GB RAM, R 4.2.1 on Windows 7.

‘Method B’ (CV 0.20, n₁ 12).

#   method   power seconds
#  shifted 0.84454    1.09
#      nct 0.84266    1.61
#    exact 0.84260   31.98

Despite being the fastest, the shifted central t-distribution should only be used in order to compare with published methods. The noncentral t-distribution is a good compromise between speed and accuracy and hence, the default in all functions. The exact method based on Owen’s Q-function is time-consuming and therefore, not recommended in validating a custom method in a narrow grid of n₁/CV-combinations. However, in designing a new study it is the method of choice.

Blinded sample size re-estimation (α 0.03505, CV 0.239, n₁ 10, target power 0.90), 1 million simulations for the empiric type I error.

#   method      TIE seconds
#       ls 0.049054    3.67
#  shifted 0.046106   12.85
#      nct 0.046319   18.24
#    exact 0.046319  429.10

The crude large sample approximation (pmethod = "ls") should only be used to compare with the published method.

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Installation

You can install the released version of Power2Stage from CRAN with …

package <- "Power2Stage"
inst    <- package %in% installed.packages()
if (length(package[!inst]) > 0) install.packages(package[!inst])

… and the development version from GitHub with

# install.packages("devtools")
devtools::install_github("Detlew/Power2Stage")

Skips installation from a github remote if the SHA-1 has not changed since last install. Use force = TRUE to force installation.

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Session Information

Inspect this information for reproducibility. Of particular importance are the versions of R and the packages used to create this workflow. It is considered good practice to record this information with every analysis.
Version 0.5.4.9000 built 2022-07-02 with R 4.2.1.

options(width = 66)
sessionInfo()
# R version 4.2.1 (2022-06-23 ucrt)
# Platform: x86_64-w64-mingw32/x64 (64-bit)
# Running under: Windows 10 x64 (build 22000)
# 
# Matrix products: default
# 
# locale:
# [1] LC_COLLATE=German_Austria.utf8 
# [2] LC_CTYPE=German_Austria.utf8   
# [3] LC_MONETARY=German_Austria.utf8
# [4] LC_NUMERIC=C                   
# [5] LC_TIME=German_Austria.utf8    
# 
# attached base packages:
# [1] stats     graphics  grDevices utils     datasets  methods  
# [7] base     
# 
# other attached packages:
# [1] Power2Stage_0.5-4.9000
# 
# loaded via a namespace (and not attached):
#  [1] Rcpp_1.0.8.3       mvtnorm_1.1-3      digest_0.6.29     
#  [4] PowerTOST_1.5-4    magrittr_2.0.3     evaluate_0.15     
#  [7] TeachingDemos_2.12 rlang_1.0.3        stringi_1.7.6     
# [10] cli_3.3.0          cubature_2.0.4.4   rstudioapi_0.13   
# [13] rmarkdown_2.14     tools_4.2.1        stringr_1.4.0     
# [16] xfun_0.31          yaml_2.3.5         fastmap_1.1.0     
# [19] compiler_4.2.1     htmltools_0.5.2    knitr_1.39

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