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---
title: "PowerTOST"
output:
github_document:
toc: true
toc_depth: 3
---
<!-- 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_PowerTOST.html)   [](https://r-pkg.org/pkg/PowerTOST) [](https://r-pkg.org/pkg/PowerTOST)
```{r, echo = FALSE}
knitr::opts_chunk$set(
collapse = TRUE,
comment = "#"
)
```
```{r, echo = FALSE, results = "asis"}
txt <- paste0("Version ", packageVersion("PowerTOST"), " built ",
packageDate("PowerTOST", date.fields = "Built"),
" with R ", substr(packageDescription("PowerTOST", fields = "Built"), 3, 7))
# date.fields = "Date/Publication") is available if package is already on CRAN!
# But we want submit it to CRAN as stable. Thus the next doesn't do the job.
# if (grepl("900", as.character(packageVersion("PowerTOST"))) |
# is.na(packageDate("PowerTOST", date.fields = "Date/Publication"))) {
# txt <- paste(txt, "\n(development version not on CRAN).")
# } else {
# txt <- paste0(txt, "\n(stable release on CRAN ",
# packageDate("PowerTOST", date.fields = "Date/Publication"), ").")
# }
cat(txt)
```
## Introduction{#introduction}
The package contains functions to calculate power and estimate sample size for various study designs used in (not only bio-) equivalence studies.
## Supported Designs{#designs}
```{r, echo = FALSE}
library(PowerTOST)
designs <- known.designs()
print(designs[, c(2, 9, 3)], row.names = FALSE)
```
Codes of designs follow this pattern: `treatments x sequences x periods`.
Although some replicate designs are more ‘popular’ than others, sample size estimations are valid for *all* of the following designs:
|<small>design</small>|<small>type</small>|<small>sequences</small>||<small>periods</small>|
|:----:|:----:|:-:|---------|:-:|
|`2x2x4`|<small>full|<small>2</small>|<small>TRTR\|RTRT</small>|<small>4</small>|
|`2x2x4`|<small>full|<small>2</small>|<small>TRRT\|RTTR</small>|<small>4</small>|
|`2x2x4`|<small>full|<small>2</small>|<small>TTRR\|RRTT</small>|<small>4</small>|
|`2x4x4`|<small>full|<small>4</small>|<small>TRTR\|RTRT\|TRRT\|RTTR</small>|<small>4</small>|
|`2x4x4`|<small>full|<small>4</small>|<small>TRRT\|RTTR\|TTRR\|RRTT</small>|<small>4</small>|
|`2x2x3`|<small>full|<small>2</small>|<small>TRT\|RTR</small>|<small>3</small>|
|`2x2x3`|<small>full|<small>2</small>|<small>TRR\|RTT</small>|<small>3</small>|
|`2x4x2`|<small>full|<small>4</small>|<small>TR\|RT\|TT\|RR</small>|<small>2</small>|
|`2x3x3`|<small>partial|<small>3</small>|<small>TRR\|RTR\|RRT</small>|<small>3</small>|
|`2x2x3`|<small>partial|<small>2</small>|<small>TRR\|RTR</small>|<small>3</small>|
Balaam’s design TR|RT|TT|RR should be avoided due to its poor power characteristics. The three period partial replicate design with two sequences TRR|RTR (<span title="also known as">a.k.a.</span> extra-reference design) should be avoided because it is biased in the presence of period effects.
<small>[TOC ↩](#powertost)</small>
## Purpose{#purpose}
For various methods power can be *calculated* based on
- nominal *α*, coefficient of variation (*CV*), deviation of test from reference (*θ*~0~), acceptance limits {*θ*~1~, *θ*~2~}, sample size (*n*), and design.
For all methods the sample size can be *estimated* based on
- nominal *α*, coefficient of variation (*CV*), deviation of test from reference (*θ*~0~), acceptance limits {*θ*~1~, *θ*~2~}, target (*i.e.*, desired) power, and design.
<small>[TOC ↩](#powertost)</small>
## Supported{#supported}
### Power and Sample Size{#power_sample_size}
Power covers balanced as well as unbalanced sequences in crossover or replicate designs and equal/unequal group sizes in two-group parallel designs. Sample sizes are always rounded up to achieve balanced sequences or equal group sizes.
- Average Bioequivalence (with arbitrary *fixed* limits).
- <span title="Average Bioequivalence">ABE</span> for Highly Variable Narrow Therapeutic Index Drugs by simulations: U.S. <span title="Food and Drug Administration">FDA</span>, China <span title="Center for Drug Evaluation">CDE</span>.
- Scaled Average Bioequivalence based on simulations.
- Average Bioequivalence with Expanding Limits (ABEL) for Highly Variable Drugs / Drug Products: <span title="European Medicines Agency">EMA</span>, <span title="World Health Organization">WHO</span> and many others.
- Average Bioequivalence with *fixed* widened limits of 75.00--133.33\% if *CV*~wR~ >30\%: Gulf Cooperation Council.
- Reference-scaled Average Bioequivalence (RSABE) for <span title="Highly Variable Drugs / Drug Products">HVDP(s)</span>: U.S. <span title="Food and Drug Administration">FDA</span>, China <span title="Center for Drug Evaluation">CDE</span>.
- Iteratively adjust *α* to control the type I error in <span title="Average Bioequivalence with Expanding Limits">ABEL</span> and <span title="Reference-scaled Average Bioequivalence">RSABE</span> for <span title="Highly Variable Drugs / Drug Products">HVDP(s)</span>.
- <span title="Reference-scaled Average Bioequivalence">RSABE</span> for <span title="Narrow Therapeutic Index Drugs">NTIDs</span>: U.S. <span title="Food and Drug Administration">FDA</span>, China <span title="Center for Drug Evaluation">CDE</span>.
- Two simultaneous <span title="Two One-Sided Tests">TOST</span> procedures.
- Non-inferiority *t*-test.
- Ratio of two means with normally distributed data on the original scale based on Fieller’s (‘fiducial’) confidence interval.
- ‘Expected’ power in case of uncertain (estimated) variability and/or uncertain *θ*~0~.
- Dose-Proportionality using the power model.
<small>[TOC ↩](#powertost)</small>
### Methods{#methods}
- Exact
- Owen’s Q.
- Direct integration of the bivariate non-central *t*-distribution.
- Approximations
- Non-central *t*-distribution.
- ‘Shifted’ central *t*-distribution.
<small>[TOC ↩](#powertost)</small>
### Helpers{#helpers}
- Calculate *CV* from *MSE* or *SE* (and vice versa).
- Calculate *CV* from given confidence interval.
- Calculate *CV*~wR~ from the upper expanded limit of an ABEL study.
- Confidence interval of *CV*.
- Pool *CV* from several studies.
- Confidence interval for given *α*, *CV*, point estimate, sample size, and design.
- Calculate *CV*~wT~ and *CV*~wR~ from a (pooled) *CV*~w~ assuming a ratio of intra-subject variances.
- *p*-values of the <span title="Two One-Sided Tests">TOST</span> procedure.
- Analysis tool for exploration/visualization of the impact of expected values (*CV*, *θ*~0~, reduced sample size due to dropouts) on power of BE decision.
- Construct design matrices of incomplete block designs.
<small>[TOC ↩](#powertost)</small>
## Defaults {#defaults}
* *α* 0.05, {*θ*~1~, *θ*~2~} (0.80, 1.25), target power 0.80. Details of the sample size search (and the regulatory settings in reference-scaled average bioequivalence) are shown in the console.
* Note: In all functions values have to be given as ratios, not in percent.
### Average Bioequivalence{#ABE.defaults}
#### Conventional (unscaled){#conventional}
Design `"2x2"` (TR|RT), exact method (Owen’s Q).
#### Highly Variable NTIDs (FDA, CDE){#NTID.defaults}
Design `"2x2x4"` (TRTR|RTRT), upper limit of the confidence interval of *σ*~wT~/*σ*~wR~ ≤2.5, approximation by the non-central *t*-distribution, 100,000 simulations.
### Reference-Scaled Average Bioequivalence{#RSABE.defaults}
Point estimate constraints (0.80, 1.25), homoscedasticity (*CV*~wT~ = *CV*~wR~), scaling is based on *CV*~wR~, design `"2x3x3"` (TRR|RTR|RRT), approximation by the non-central *t*-distribution, 100,000 simulations.
- <span title="European Medicines Agency">EMA</span>, <span title="World Health Organization">WHO</span>, Health Canada, and many other jurisdictions: Average Bioequivalence with Expanding Limits (ABEL).
- U.S. <span title="Food and Drug Administration">FDA</span>, China <span title="Center for Drug Evaluation">CDE</span>: RSABE.
#### Highly Variable Drugs / Drug Products{#HVDP.defaults}
*θ*~0~ 0.90.<sup id="a1">[1](#f1)</sup>
###### EMA and many others{#ABEL.EMA.defaults}
Regulatory constant `0.760`, upper cap of scaling at *CV*~wR~ 50\%, evaluation by <span title="Analysis of Variance">ANOVA</span>.
###### Health Canada{#ABEL.HC.defaults}
Regulatory constant `0.760`, upper cap of scaling at *CV*~wR~ ~57.4\%, evaluation by intra-subject contrasts.
###### Gulf Cooperation Council{#ABEL.GCC.defaults}
Regulatory constant `log(1/0.75)/sqrt(log(0.3^2+1))`, widened limits 75.00--133.33\% if *CV*~wR~ >30\%, no upper cap of scaling, evaluation by <span title="Analysis of Variance">ANOVA</span>.
###### FDA, CDE{#RSABE.FDA.defaults}
Regulatory constant `log(1.25)/0.25`, no upper cap of scaling, evaluation by linearized scaled <span title="Average Bioequivalence">ABE</span> (Howe’s approximation).
#### Narrow Therapeutic Index Drugs (FDA, CDE){#NTID.FDA.defaults}
*θ*~0~ 0.975, regulatory constant `log(1.11111)/0.1`, implicit upper cap of scaling at *CV*~wR~ ~21.4\%, design `"2x2x4"` (TRTR|RTRT), evaluation by linearized scaled <span title="Average Bioequivalence">ABE</span> (Howe’s approximation), upper limit of the confidence interval of *σ*~wT~/*σ*~wR~ ≤2.5.
### Dose-Proportionality{#DP}
*β*~0~ (slope) `1+log(0.95)/log(rd)` where `rd` is the ratio of the highest and lowest dose, target power 0.80, crossover design, details of the sample size search suppressed.
### Power Analysis{#PA}
Minimum acceptable power 0.70. *θ*~0~; design, conditions, and sample size method depend on defaults of the respective approaches (ABE, ABEL, RSABE, NTID, HVNTID).
<small>[TOC ↩](#powertost)</small>
## Examples{#examples}
Before running the examples attach the library.
```{r attach_library}
library(PowerTOST)
```
If not noted otherwise, the functions’ [defaults](#defaults) are employed.
### Parallel Design{#parallel}
Power for total *CV* 0.35 (35\%), group sizes 52 and 49.
```{r parallel}
power.TOST(CV = 0.35, n = c(52, 49), design = "parallel")
```
### Crossover Design{#crossover}
Sample size for assumed within- (intra-) subject *CV* 0.20 (20\%).
```{r crossover_ABE1}
sampleN.TOST(CV = 0.20)
```
<small>[TOC ↩](#powertost)</small>
Sample size for assumed within- (intra-) subject *CV* 0.40 (40\%), *θ*~0~ 0.90, four period full replicate study (any of TRTR|RTRT, TRRT|RTTR, TTRR|RRTT). Wider acceptance range for *C*~max~ (South Africa).
```{r crossover_ABE2}
sampleN.TOST(CV = 0.40, theta0 = 0.90, theta1 = 0.75, design = "2x2x4")
```
<small>[TOC ↩](#powertost)</small>
Sample size for assumed within- (intra-) subject *CV* 0.125 (12.5\%), *θ*~0~ 0.975. Narrower acceptance range for <span title="Narrow Therapeutic Index Drugs">NTIDs</span> (most jurisdictions).
```{r crossover_ABE3}
sampleN.TOST(CV = 0.125, theta0 = 0.975, theta1 = 0.90)
```
<small>[TOC ↩](#powertost)</small>
Sample size for equivalence of the ratio of two means with normality on the original scale based on Fieller’s (‘fiducial’) confidence interval.<sup id="a2">[2](#f2)</sup> Within- (intra-) subject *CV*~w~ 0.20 (20\%), between- (inter-) subject *CV*~b~ 0.40 (40\%).
Note the default *α* 0.025 (95\% CI) of this function because it is intended for studies with clinical endpoints.
```{r crossover_fieller}
sampleN.RatioF(CV = 0.20, CVb = 0.40)
```
<small>[TOC ↩](#powertost)</small>
### Replicate Designs{#replicate}
#### ABE{#repl.ABE}
##### Conventional (unscaled){#repl.ABE.conv}
Sample size for assumed within- (intra-) subject *CV* 0.45 (45\%), *θ*~0~ 0.90, three period full replicate study (TRT|RTR *or* TRR|RTT).
```{r replicate_ABE1}
sampleN.TOST(CV = 0.45, theta0 = 0.90, design = "2x2x3")
```
Note that the conventional model assumes homoscedasticity (equal variances of treatments). For heteroscedasticity we can ‘switch off’ all conditions of one of the methods for reference-scaled <span title="Average Bioequivalence">ABE</span>. We assume a *σ*^2^-ratio of ⅔ (*i.e.*, the test has a lower variability than the reference). Only relevant columns of the data frame shown.
```{r replicate_ABE2}
reg <- reg_const("USER", r_const = NA, CVswitch = Inf,
CVcap = Inf, pe_constr = FALSE)
CV <- CVp2CV(CV = 0.45, ratio = 2/3)
res <- sampleN.scABEL(CV=CV, design = "2x2x3", regulator = reg,
details = FALSE, print = FALSE)
print(res[c(3:4, 8:9)], digits = 5, row.names = FALSE)
```
Similar sample size because the pooled *CV*~w~ is still 0.45.
<small>[TOC ↩](#powertost)</small>
##### Highly Variable Narrow Therapeutic Index Drug{#HVNTID}
Sample size assuming heteroscedasticity (*CV*~w~ 0.45, variance-ratio 2.5, *i.e.*, the test treatment has a substantially higher variability than the reference). TRTR|RTRT according to the FDA’s guidances.<sup id="a3">[3,](#f3)</sup><sup id="a4">[4,](#f4)</sup><sup id="a5">[5](#f5)</sup> Assess additionally which one of the components (<span title="Average Bioequivalence">ABE</span>, *s*~wT~/*s*~wR~-ratio) drives the sample size.
```{r HVNTID}
CV <- signif(CVp2CV(CV = 0.45, ratio = 2.5), 4)
n <- sampleN.HVNTID(CV = CV, details = FALSE)[["Sample size"]]
suppressMessages(power.HVNTID(CV = CV, n = n, details = TRUE))
```
The ABE component shows a lower probability to demonstrate BE than the *s*~wT~/*s*~wR~ component and hence, drives the sample size.
<small>[TOC ↩](#powertost)</small>
#### ABEL{#ABEL}
Sample size assuming homoscedasticity (*CV*~wT~ = *CV*~wR~ = 0.45).
```{r ABEL_1}
sampleN.scABEL(CV = 0.45)
```
<small>[TOC ↩](#powertost)</small>
Iteratively adjust *α* to control the Type I Error.<sup id="a6">[6](#f6)</sup> Heteroscedasticity (*CV*~wT~ 0.30, *CV*~wR~ 0.40, *i.e.*, variance-ratio ~0.58), four period full replicate study (any of TRTR|RTRT, TRRT|RTTR, TTRR|RRTT), 24 subjects, balanced sequences.
```{r ABEL_2}
scABEL.ad(CV = c(0.30, 0.40), design = "2x2x4", n = 24)
```
With the nominal *α* 0.05 the Type I Error will be inflated (0.05953). With the adjusted *α* 0.03997 (*i.e.*, a ~92\% <span title="Confidence Interval">CI</span>) the <span title="Type I Error">TIE</span> will be controlled, although with a slight loss in power (decreases from 0.805 to 0.778).
Consider `sampleN.scABEL.ad(CV = c(0.30, 0.35), design = "2x2x4")` to estimate the sample size preserving both the <span title="Type I Error">TIE</span> and target power. In this example 26 subjects would be required.
<small>[TOC ↩](#powertost)</small>
<span title="Average Bioequivalence with Expanded Limits">ABEL</span> cannot be applied for *AUC* (except for the <span title="World Health Organization">WHO<span>). Hence, in many cases <span title="Average Bioequivalence">ABE</span> drives the sample size. Four period full replicate study (any of TRTR|RTRT, TRRT|RTTR, TTRR|RRTT).
```{r ABEL_3}
PK <- c("Cmax", "AUC")
CV <- c(0.45, 0.30)
# extract sample sizes and power
r1 <- sampleN.scABEL(CV = CV[1], design = "2x2x4",
print = FALSE, details = FALSE)[8:9]
r2 <- sampleN.TOST(CV = CV[2], theta0 = 0.90, design = "2x2x4",
print = FALSE, details = FALSE)[7:8]
n <- as.numeric(c(r1[1], r2[1]))
pwr <- signif(as.numeric(c(r1[2], r2[2])), 5)
# compile results
res <- data.frame(PK = PK, method = c("ABEL", "ABE"),
n = n, power = pwr)
print(res, row.names = FALSE)
```
*AUC* drives the sample size.
For Health Canada it is the opposite (ABE for *C*~max~ and ABEL for *AUC*).
```{r ABEL_4}
PK <- c("Cmax", "AUC")
CV <- c(0.45, 0.30)
# extract sample sizes and power
r1 <- sampleN.TOST(CV = CV[1], theta0 = 0.90, design = "2x2x4",
print = FALSE, details = FALSE)[7:8]
r2 <- sampleN.scABEL(CV = CV[2], design = "2x2x4",
print = FALSE, details = FALSE)[8:9]
n <- as.numeric(c(r1[1], r2[1]))
pwr <- signif(as.numeric(c(r1[2], r2[2])), 5)
# compile results
res <- data.frame(PK = PK, method = c("ABE", "ABEL"),
n = n, power = pwr)
print(res, row.names = FALSE)
```
Here *C*~max~ drives the sample size.
<small>[TOC ↩](#powertost)</small>
Sample size assuming homoscedasticity (*CV*~wT~ = *CV*~wR~ = 0.45) for the widened limits of the Gulf Cooperation Council.
```{r ABEL_5}
sampleN.scABEL(CV = 0.45, regulator = "GCC", details = FALSE)
```
<small>[TOC ↩](#powertost)</small>
#### RSABE{#RSABE}
#### HVD(P)s{#RSABE.HVDP}
Sample size for a four period full replicate study (any of TRTR|RTRT, TRRT|RTTR, TTRR|RRTT) assuming heteroscedasticity (*CV*~wT~ 0.40, *CV*~wR~ 0.50, *i.e.*, variance-ratio ~0.67). Details of the sample size search suppressed.
```{r RSABE}
sampleN.RSABE(CV = c(0.40, 0.50), design = "2x2x4", details = FALSE)
```
<small>[TOC ↩](#powertost)</small>
#### NTIDs (FDA, CDE){#RSABE.HVNTID}
Sample size assuming heteroscedasticity (*CV*~w~ 0.10, variance-ratio 2.5, *i.e.*, the test treatment has a substantially higher variability than the reference). TRTR|RTRT according to the FDA’s guidance.<sup id="a7">[7](#f7)</sup> Assess additionally which one of the three components (scaled <span title="Average Bioequivalence">ABE</span>, conventional <span title="Average Bioequivalence">ABE</span>, *s*~wT~/*s*~wR~-ratio) drives the sample size.
```{r NTID2}
CV <- signif(CVp2CV(CV = 0.10, ratio = 2.5), 4)
n <- sampleN.NTID(CV = CV)[["Sample size"]]
suppressMessages(power.NTID(CV = CV, n = n, details = TRUE))
```
The *s*~wT~/*s*~wR~ component shows the lowest probability to demonstrate <span title="Bioequivalence">BE</span> and hence, drives the sample size.
<small>[TOC ↩](#powertost)</small>
Compare that with homoscedasticity (*CV*~wT~ = *CV*~wR~ = 0.10):
```{r NTID3}
CV <- 0.10
n <- sampleN.NTID(CV = CV, details = FALSE)[["Sample size"]]
suppressMessages(power.NTID(CV = CV, n = n, details = TRUE))
```
Here the scaled <span title="Average Bioequivalence">ABE</span> component shows the lowest probability to demonstrate <span title="Bioequivalence">BE</span> and drives the sample size – which is much lower than in the previous example.
<small>[TOC ↩](#powertost)</small>
Comparison with *fixed* narrower limits applicable in other jurisdictions. Note that a replicate design is not mandatory -- reducing the chance of dropouts and requiring less administrations
```{r NTID4}
CV <- 0.10
# extract sample sizes and power
r1 <- sampleN.NTID(CV = CV, print = FALSE, details = FALSE)[8:9]
r2 <- sampleN.TOST(CV = CV, theta0 = 0.975, theta1 = 0.90,
design = "2x2x4", print = FALSE, details = FALSE)[7:8]
r3 <- sampleN.TOST(CV = CV, theta0 = 0.975, theta1 = 0.90,
design = "2x2x3", print = FALSE, details = FALSE)[7:8]
r4 <- sampleN.TOST(CV = CV, theta0 = 0.975, theta1 = 0.90,
print = FALSE, details = FALSE)[7:8]
n <- as.numeric(c(r1[1], r2[1], r3[1], r4[1]))
pwr <- signif(as.numeric(c(r1[2], r2[2], r3[2], r4[2])), 5)
# compile results
res <- data.frame(method = c("FDA/CDE", rep ("fixed narrow", 3)),
design = c(rep("2x2x4", 2), "2x2x3", "2x2x2"),
n = n, power = pwr, a = n * c(4, 4, 3, 2))
names(res)[5] <- "adm. #" # number of administrations
print(res, row.names = FALSE)
```
<small>[TOC ↩](#powertost)</small>
### Dose-Proportionality
*CV* 0.20 (20\%), doses 1, 2, and 8 units, assumed slope *β*~0~ 1, target power 0.90.
```{r DP}
sampleN.dp(CV = 0.20, doses = c(1, 2, 8), beta0 = 1, targetpower = 0.90)
```
Note that the acceptance range of the slope depends on the ratio of the highest and lowest doses (*i.e.*, it gets tighter for wider dose ranges and therefore, higher sample sizes will be required).
In an exploratory setting wider equivalence margins {*θ*~1~, *θ*~2~} (0.50, 2.00) were proposed,<sup id="a8">[8](#f8)</sup> translating in this example to an acceptance range of `0.66667 ... 1.3333` and a sample size of only six subjects.
<small>[TOC ↩](#powertost)</small>
### Power Analysis
Explore impact of deviations from assumptions (higher *CV*, higher deviation of *θ*~0~ from 1, dropouts) on power. Assumed within-subject *CV* 0.20 (20\%), target power 0.90. Plot suppressed.
```{r PA}
res <- pa.ABE(CV = 0.20, targetpower = 0.90)
print(res, plotit = FALSE)
```
If the study starts with 26 subjects (power \~0.92), the *CV* can increase to \~0.27 **or** *θ*~0~ decrease to \~0.90 **or** the sample size decrease to 10 whilst power will still be ≥0.70.
However, this is **not** a substitute for the ‘Sensitivity Analysis’ recommended in ICH-E9,<sup id="a9">[9](#f9)</sup> since in a real study a combination of all effects occurs simultaneously. It is up to *you* to decide on reasonable combinations and analyze their respective power.
<small>[TOC ↩](#powertost)</small>
### Speed Comparisons
Performed on a Xeon E3-1245v3 3.4 GHz, 8 MB cache, 16 GB RAM, R `r getRversion()` 64 bit on Windows 7.
#### ABE
2×2 crossover design, *CV* 0.17. Sample sizes and achieved power for the supported methods (the 1<sup>st</sup> one is the default).
```
method n power time (s)
owenq 14 0.80568 0.00128
mvt 14 0.80569 0.11778
noncentral 14 0.80568 0.00100
shifted 16 0.85230 0.00096
```
The 2<sup>nd</sup> exact method is substantially slower than the 1<sup>st</sup>. The approximation based on the noncentral *t*-distribution is slightly faster but matches the 1<sup>st</sup> exact method closely. Though the approximation based on the shifted central *t*-distribution is the fastest, it *might* estimate a larger than necessary sample size. Hence, it should be used only for comparative purposes.
#### ABEL
Four period full replicate study (any of TRTR|RTRT, TRRT|RTTR, TTRR|RRTT), homogenicity (*CV*~wT~ = *CV*~wR~ 0.45). Sample sizes and achieved power for the supported methods.
```
function method n power time (s)
sampleN.scABEL ‘key’ statistics 28 0.81116 0.1348
sampleN.scABEL.sdsims subject simulations 28 0.81196 2.5377
```
Simulating via the ‘key’ statistics is the method of choice for speed reasons.
However, subject simulations are recommended **if**
- the partial replicate design (TRR|RTR|RRT) is planned **and**
- the special case of heterogenicity *CV*~wT~ > *CV*~wR~ is expected.
<small>[TOC ↩](#powertost)</small>
## Installation
You can install the released version of PowerTOST from [CRAN](https://CRAN.R-project.org) with
```{r inst}
package <- "PowerTOST"
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("remotes")
remotes::install_github("Detlew/PowerTOST")
```
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 ↩](#powertost)</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("PowerTOST")` built `r packageDate("PowerTOST", date.fields = "Built")` with R `r substr(packageDescription("PowerTOST", fields = "Built"), 3, 7)`.
```{r, sessioninfo}
options(width = 66)
sessionInfo()
```
<small>[TOC ↩](#powertost)</small>
***
<span id="f1" style="font-size:smaller">1. Tóthfalusi L, Endrényi L. *Sample Sizes for Designing
Bioequivalence Studies for Highly Variable Drugs.* J Pharm Pharmacol Sci. 2012;
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<span id="f5" style="font-size:smaller">5. U.S. Food and Drug Administration, Office of Generic Drugs. *Draft Guidance on Edoxaban Tosylate.* Recommended May 2017; Revised Mar 2020. [Online](https://www.accessdata.fda.gov/drugsatfda_docs/psg/PSG_206316.pdf).</span> [↩](#a5)\
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<span id="f8" style="font-size:smaller">8. Hummel J, McKendrick S, Brindley C, French R. *Exploratory assessment of dose proportionality: review of current approaches and proposal for a practical criterion.* Pharm. Stat. 2009; 8(1): 38--49. [doi:10.1002/pst.326](https://doi.org/10.1002/pst.326).</span> [↩](#a8)\
<span id="f9" style="font-size:smaller">9. International Conference on
Harmonisation of Technical Requirements for Registration of
Pharmaceuticals for Human Use. *ICH Harmonised Tripartite
Guideline. E9. Statistical Principles for Clinical Trials.* 5 February 1998.
[Online](https://database.ich.org/sites/default/files/E9_Guideline.pdf).</span> [↩](#a9)