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README.Rmd
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---
title: "Power2Stage"
output:
github_document:
toc: true
toc_depth: 4
---
<!-- README.md is generated from README.Rmd. Please edit that file -->
[](https://www.gnu.org/licenses/gpl-2.0) [](https://www.gnu.org/licenses/gpl-3.0)      [](https://cran.r-project.org/web/checks/check_results_Power2Stage.html)  [](https://r-pkg.org/pkg/Power2Stage) [](https://r-pkg.org/pkg/Power2Stage)
```{r, echo = FALSE}
knitr::opts_chunk$set(
collapse = TRUE,
comment = "#"
)
```
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.
```{r, echo = FALSE, results = "asis"}
txt <- paste0("Version ", packageVersion("Power2Stage"), " built ",
packageDate("Power2Stage", date.fields = "Built"),
" with R ", substr(packageDescription("Power2Stage", fields = "Built"), 3, 7))
if (grepl("900", as.character(packageVersion("Power2Stage")))) {
txt <- paste(txt, "\n(development version not on CRAN).")
} else {
txt <- paste0(txt, "\n(stable release on CRAN ",
packageDate("Power2Stage", date.fields = "Date/Publication"), ").")
}
cat(txt)
```
## Supported Methods
### Simulation-based
Since the many letters denoting the methods given by various authors might be confusing, [I classified](https://doi.org/10.1007/s00228-015-1806-2) 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.
<small>[TOC ↩](#power2stage)</small>
#### ‘Type 1’
- [Potvin *et al.*](https://doi.org/10.1002/pst.294) (2008) ‘Method B’: *α* 0.0294 (*θ*~0~ 0.95, target power 0.80).
- [Fuglsang](https://doi.org/10.1208/s12248-013-9475-5) (2013) ‘Method B’: *α* 0.0284 (*θ*~0~ 0.95, target power 0.90).
- [Karalis](https://doi.org/10.1016/j.ijpharm.2013.08.013) (2013) ‘TSD-2’: *α* 0.0294 (*θ*~0~ = PE, target power 0.80).
- [Fuglsang](https://doi.org/10.1208/s12248-014-9571-1) (2014) ‘Method B’ (parallel design): *α* 0.0294 (*θ*~0~ 0.95, target power 0.80).
- [Zheng *et al.*](https://doi.org/10.1002/pst.1672) (2015) ‘MSDBE’: *α*~1~ 0.01, *α*~2~ 0.04.
- [Xu *et al.*](https://doi.org/10.1002/pst.1721) (2016) ‘Method E’: (*θ*~0~ 0.95, target power 0.80, *n*~max~ 42).\
- For *CV* 10--30%\
*α*~1~ 0.0294, *α*~2~ 0.0357, futility rule on CI {0.9374, 1/0.9374}.
- For *CV* 30--55%\
*α*~1~ 0.0254, *α*~2~ 0.0363, futility rule on CI {0.9305, 1/0.9305}.
- [Molins *et al.*](https://doi.org/10.1002/sim.7452) (2017) ‘Type 1 modified Potvin B’: *α* 0.0301 (*θ*~0~ 0.95, target power 0.80, min. *n~2~* = 1.5*n~1~*, *n~max~* 150).
#### ‘Type 2’
- [Potvin *et al.*](https://doi.org/10.1002/pst.294) (2008) ‘Method C’: *α* 0.0294 (*θ*~0~ 0.95, target power 0.80).
- [Montague *et al.*](https://doi.org/10.1002/pst.483) (2011) ‘Method D’: *α* 0.0280 (*θ*~0~ 0.90, target power 0.80).
- [Fuglsang](https://doi.org/10.1208/s12248-013-9475-5) (2013) ‘Method C/D’:\
*α* 0.0274 (*θ*~0~ 0.95, target power 0.90).\
*α* 0.0269 (*θ*~0~ 0.90, target power 0.90).
- [Karalis and Macheras](https://doi.org/10.1007/s11095-013-1026-3) (2013) ‘TSD’: *α* 0.0294 (*θ*~0~ = PE, target power 0.80).
- [Karalis](https://doi.org/10.1016/j.ijpharm.2013.08.013) (2013) ‘TSD-1’: *α* 0.0280 (*θ*~0~ = PE, target power 0.80).
- [Xu *et al.*](https://doi.org/10.1002/pst.1721) (2016) ‘Method F’: (*θ*~0~ 0.95, target power 0.80, *n~max~* 180).\
- For *CV* 10--30%\
*α*~1~ 0.0248, *α*~2~ 0.0364, futility rule on CI {0.9492, 1/0.9492}.
- For *CV* 30--55%\
*α*~1~ 0.0259, *α*~2~ 0.0349, futility rule on CI {0.9350, 1/0.9350}.
- [Molins *et al.*](https://doi.org/10.1002/sim.7452) (2017) ‘Type 2 modified Potvin C’: *α* 0.0280 (*θ*~0~ 0.95, target power 0.80, min. *n~2~* = 1.5*n~1~*, *n~max~* 150).
#### Blinded Sample Size Re-estimation in the Interim
[Golkowski *et al.*](https://doi.org/10.1002/pst.1617) (2014).
#### Group Sequential Design
[Kieser and Rauch](https://doi.org/10.1002/sim.6487) (2015).
### Inverse-Normal Combination
[König *et al.*](https://doi.org/10.13140/RG.2.1.5190.0967) (2014), [Kieser and Rauch](https://doi.org/10.1002/sim.6487) (2015),
[Wassmer and Brannath](https://doi.org/10.1007/978-3-319-32562-0) (2016), [Maurer *et al.*](https://doi.org/10.1002/sim.7614) (2018).
<small>[TOC ↩](#power2stage)</small>
## 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).
<small>[TOC ↩](#power2stage)</small>
### 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.
<small>[TOC ↩](#power2stage)</small>
## Examples
Before running the examples attach the library.
```{r attach_library}
library(Power2Stage)
```
If not noted otherwise, defaults are employed.
### Method B
Power estimation by the shifted central *t*-distribution.
```{r methodB_exp1}
power.tsd(CV = 0.20, n1 = 12, pmethod = "shifted")
```
Explore the empiric type I error at the upper BE-limit.
```{r methodB_exp2}
power.tsd(CV = 0.20, n1 = 12, pmethod = "shifted",
theta0 = 1.25)[["pBE"]]
```
<small>[TOC ↩](#power2stage)</small>
### Method C
Power estimation by the shifted central *t*-distribution.
```{r methodC_exp1}
power.tsd(method = "C", CV = 0.20, n1 = 12, pmethod = "shifted")
```
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).
```{r methodC_exp2}
power.tsd(method = "C", CV = 0.20, n1 = 12, pmethod = "shifted",
theta0 = 1.25)[["pBE"]]
```
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.
```{r methodC_exp3}
power.tsd(method = "C", alpha = rep(0.0280, 2), CV = 0.20,
n1 = 12, pmethod = "shifted", theta0 = 1.25)[["pBE"]]
```
<small>[TOC ↩](#power2stage)</small>
### 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.
```{r inc_interim}
interim.tsd.in(GMR = 0.95, GMR1 = 1.0876, CV1 = 0.18213, n1 = 12)
```
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.
```{r inc_final}
final.tsd.in(GMR1 = 1.0876, CV1 = 0.18213, n1 = 12,
GMR2 = 0.9141, CV2 = 0.25618, n2 = 8)
```
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.*
<small>[TOC ↩](#power2stage)</small>
## Speed Comparisons
Performed on a Xeon E3-1245v3 3.4 GHz, 8 MB cache, 16 GB RAM, R `r getRversion()` on Windows 7.
‘Method B’ (*CV* 0.20, *n*~1~ 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 *n*~1~/*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, *n*~1~ 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.
<small>[TOC ↩](#power2stage)</small>
## Installation
You can install the released version of Power2Stage from [CRAN](https://CRAN.R-project.org) with …
```{r inst}
package <- "Power2Stage"
inst <- package %in% installed.packages()
if (length(package[!inst]) > 0) install.packages(package[!inst])
```
… and the development version from [GitHub](https://github.com/) with
```
# install.packages("devtools")
devtools::install_github("Detlew/Power2Stage")
```
Skips installation from a github remote if the [SHA-1](https://en.wikipedia.org/wiki/SHA-1) has not changed since last install. Use `force = TRUE` to force installation.
<small>[TOC ↩](#power2stage)</small>
## 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 `r packageVersion("Power2Stage")` built `r packageDate("Power2Stage", date.fields = "Built")` with R `r substr(packageDescription("Power2Stage", fields = "Built"), 3, 7)`.
```{r, sessioninfo}
options(width = 66)
sessionInfo()
```
<small>[TOC ↩](#power2stage)</small>