PowerTOST

• Supported Designs
• Purpose
• Supported
  • Power and Sample Size
  • Methods
  • Helpers
• Defaults
  • Average Bioequivalence
  • Reference-Scaled Average Bioequivalence
  • Dose-Proportionality
  • Power Analysis
• Examples
  • Parallel Design
  • Crossover Design
  • Replicate Designs
  • Dose-Proportionality
  • Power Analysis
  • Speed Comparisons
• Installation
• Session Information

The package contains functions to calculate power and estimate sample size for various study designs used in (not only bio-) equivalence studies.

# Version 1.5.1.9999 built 2020-10-26 with R 4.0.3 
# (development version not on CRAN).

Supported Designs

#    design                        name    df
#  parallel           2 parallel groups   n-2
#       2x2               2x2 crossover   n-2
#     2x2x2             2x2x2 crossover   n-2
#       3x3               3x3 crossover 2*n-4
#     3x6x3             3x6x3 crossover 2*n-4
#       4x4               4x4 crossover 3*n-6
#     2x2x3   2x2x3 replicate crossover 2*n-3
#     2x2x4   2x2x4 replicate crossover 3*n-4
#     2x4x4   2x4x4 replicate crossover 3*n-4
#     2x3x3   partial replicate (2x3x3) 2*n-3
#     2x4x2            Balaam's (2x4x2)   n-2
#    2x2x2r Liu's 2x2x2 repeated x-over 3*n-2
#    paired                paired means   n-1

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:

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 (a.k.a. extra-reference design) should be avoided because it is biased in the presence of period effects.

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Purpose

For various methods power can be calculated based on

• nominal α, coefficient of variation (CV), deviation of test from reference (θ0), acceptance limits {θ1, θ2}, 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 (θ0), acceptance limits {θ1, θ2}, target (i.e., desired) power, and design.

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Supported

Power and 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).
• Two simultaneous TOST 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 θ0.
• Reference-scaled bioequivalence based on simulations.
  • EMA: Average Bioequivalence with Expanding Limits (ABEL).
  • U.S. FDA, China CDE: Reference-scaled Average Bioequivalence (RSABE) for Highly Variable Drugs / Drug Products and Narrow Therapeutic Index Drugs (NTIDs). In China the former is required and the latter currently under discussion.
• Iteratively adjust α to control the type I error in ABEL and RSABE.
• U.S. FDA: ABE for highly variable NTIDs by simulations.
• Dose-Proportionality using the power model.

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Methods

• Exact
  • Owen’s Q.
  • Direct integration of the bivariate non-central t-distribution.
• Approximations
  • Non-central t-distribution.
  • ‘Shifted’ central t-distribution.

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Helpers

• Calculate CV from MSE or SE (and vice versa).
• Calculate CV from given confidence interval.
• Calculate CVwR 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 CVwT and CVwR from a (pooled) CVw assuming a ratio of intra-subject variances.
• p-values of the TOST procedure.
• Analysis tool for exploration/visualization of the impact of expected values (CV, θ0, reduced sample size due to dropouts) on power of BE decision.
• Construct design matrices of incomplete block designs.

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Defaults

• α 0.05, {θ1, θ2} (0.80, 1.25). 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

θ0 0.95, target power 0.80, design "2x2" (TR|RT), exact method (Owen’s Q).

Reference-Scaled Average Bioequivalence

α 0.05, point estimate constraint (0.80, 1.25), homoscedasticity (CVwT = CVwR), scaling is based on CVwR, target power 0.80, design "2x3x3" (TRR|RTR|RRT), approximation by the non-central t-distribution, 100,000 simulations.

• EMA, WHO, Health Canada, and many other jurisdictions: Average Bioequivalence with Expanding Limits (ABEL).
• U.S. FDA, China CDE: RSABE.

Highly Variable Drugs / Drug Products

θ0 0.90 as recommended by Tóthfalusi and Endrényi (2011).

EMA

Regulatory constant 0.76, upper cap of scaling at CVwR 50%, evaluation by ANOVA.

Health Canada

Regulatory constant 0.76, upper cap of scaling at CVwR ~57.4%, evaluation by intra-subject contrasts.

FDA

Regulatory constant log(1.25)/0.25, linearized scaled ABE (Howe’s approximation).

Narrow Therapeutic Index Drugs (FDA)

θ0 0.975, regulatory constant log(1.11111)/0.1, upper cap of scaling at CVwR ~21.4%, design "2x2x4" (TRTR|RTRT), linearized scaled ABE (Howe’s approximation), upper limit of the confidence interval of swT/swR ≤2.5.

Dose-Proportionality

β0 (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

Minimum acceptable power 0.70. θ0, design, conditions, and sample size method depend on defaults of the respective approaches (ABE, ABEL, RSABE, NTID, HVNTID).

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Examples

Before running the examples attach the library.

library(PowerTOST)

If not noted otherwise, the functions’ defaults are employed.

Parallel Design

Power for total CV 0.35 (35%), group sizes 52 and 49, design "parallel".

power.TOST(CV = 0.35, n = c(52, 49), design = "parallel")
# [1] 0.8011186

Crossover Design

Sample size for assumed within- (intra-) subject CV 0.20 (20%).

sampleN.TOST(CV = 0.20)
# 
# +++++++++++ Equivalence test - TOST +++++++++++
#             Sample size estimation
# -----------------------------------------------
# Study design: 2x2 crossover 
# log-transformed data (multiplicative model)
# 
# alpha = 0.05, target power = 0.8
# BE margins = 0.8 ... 1.25 
# True ratio = 0.95,  CV = 0.2
# 
# Sample size (total)
#  n     power
# 20   0.834680

Sample size for assumed within- (intra-) subject CV 0.40 (40%), θ0 0.90, four period full replicate “2x2x4” study. Wider acceptance range for Cmax (Gulf Cooperation Council, South Africa).

sampleN.TOST(CV = 0.40, theta0 = 0.90, theta1 = 0.75, design = "2x2x4")
# 
# +++++++++++ Equivalence test - TOST +++++++++++
#             Sample size estimation
# -----------------------------------------------
# Study design: 2x2x4 (4 period full replicate) 
# log-transformed data (multiplicative model)
# 
# alpha = 0.05, target power = 0.8
# BE margins = 0.75 ... 1.333333 
# True ratio = 0.9,  CV = 0.4
# 
# Sample size (total)
#  n     power
# 30   0.822929

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Sample size for assumed within- (intra-) subject CV 0.125 (12.5%), θ0 0.975. Acceptance range for NTIDs (most jurisdictions).

sampleN.TOST(CV = 0.125, theta0 = 0.975, theta1 = 0.90)
# 
# +++++++++++ Equivalence test - TOST +++++++++++
#             Sample size estimation
# -----------------------------------------------
# Study design: 2x2 crossover 
# log-transformed data (multiplicative model)
# 
# alpha = 0.05, target power = 0.8
# BE margins = 0.9 ... 1.111111 
# True ratio = 0.975,  CV = 0.125
# 
# Sample size (total)
#  n     power
# 32   0.800218

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Sample size for equivalence of the ratio of two means with normality on the original scale based on Fieller’s (‘fiducial’) confidence interval. Within- (intra-) subject CVw 0.20 (20%), between- (inter-) subject CVb 0.40 (40%).
Note the default α 0.025 (95% CI) of this function because it is intended for studies with clinical endpoints.

sampleN.RatioF(CV = 0.20, CVb = 0.40)
# 
# +++++++++++ Equivalence test - TOST +++++++++++
#     based on Fieller's confidence interval
#             Sample size estimation
# -----------------------------------------------
# Study design: 2x2 crossover
# Ratio of means with normality on original scale
# alpha = 0.025, target power = 0.8
# BE margins = 0.8 ... 1.25 
# True ratio = 0.95,  CVw = 0.2,  CVb = 0.4
# 
# Sample size
#  n     power
# 28   0.807774

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Replicate Designs

ABE

Sample size for assumed within- (intra-) subject CV 0.45 (45%), θ0 0.90, three period full replicate study "2x2x3" (TRT|RTR or TRR|RTT).

sampleN.TOST(CV = 0.45, theta0 = 0.90, design = "2x2x3")
# 
# +++++++++++ Equivalence test - TOST +++++++++++
#             Sample size estimation
# -----------------------------------------------
# Study design: 2x2x3 (3 period full replicate) 
# log-transformed data (multiplicative model)
# 
# alpha = 0.05, target power = 0.8
# BE margins = 0.8 ... 1.25 
# True ratio = 0.9,  CV = 0.45
# 
# Sample size (total)
#  n     power
# 124   0.800125

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 ABE. We assume a σ2 ratio of ⅔ (i.e., the test has a lower variability than the reference). Only relevant columns of the data.frame shown.

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)
#    CVwT    CVwR Sample size Achieved power
#  0.3987 0.49767         126         0.8052

Similar sample size because the pooled CV is still 0.45.

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ABEL

Sample size assuming homoscedasticity (CVwT = CVwR = 0.45).

sampleN.scABEL(CV = 0.45)
# 
# +++++++++++ scaled (widened) ABEL +++++++++++
#             Sample size estimation
#    (simulation based on ANOVA evaluation)
# ---------------------------------------------
# Study design: 2x3x3 (partial replicate)
# log-transformed data (multiplicative model)
# 1e+05 studies for each step simulated.
# 
# alpha  = 0.05, target power = 0.8
# CVw(T) = 0.45; CVw(R) = 0.45
# True ratio = 0.9
# ABE limits / PE constraint = 0.8 ... 1.25 
# EMA regulatory settings
# - CVswitch            = 0.3 
# - cap on scABEL if CVw(R) > 0.5
# - regulatory constant = 0.76 
# - pe constraint applied
# 
# 
# Sample size search
#  n     power
# 36   0.7755 
# 39   0.8059

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Iteratively adjust α to control the Type I Error (Labes, Schütz). Slight heteroscedasticity (CVwT 0.30, CVwR 0.35), four period full replicate "2x2x4" study, 30 subjects, balanced sequences.

scABEL.ad(CV = c(0.30, 0.35), design = "2x2x4", n = 30)
# +++++++++++ scaled (widened) ABEL ++++++++++++
#          iteratively adjusted alpha
#    (simulations based on ANOVA evaluation)
# ----------------------------------------------
# Study design: 2x2x4 (4 period full replicate)
# log-transformed data (multiplicative model)
# 1,000,000 studies in each iteration simulated.
# 
# CVwR 0.35, CVwT 0.3, n(i) 15|15 (N 30)
# Nominal alpha                 : 0.05 
# True ratio                    : 0.9000 
# Regulatory settings           : EMA (ABEL)
# Switching CVwR                : 0.3 
# Regulatory constant           : 0.76 
# Expanded limits               : 0.7723 ... 1.2948
# Upper scaling cap             : CVwR > 0.5 
# PE constraints                : 0.8000 ... 1.2500
# Empiric TIE for alpha 0.0500  : 0.06651
# Power for theta0 0.9000       : 0.814
# Iteratively adjusted alpha    : 0.03540
# Empiric TIE for adjusted alpha: 0.05000
# Power for theta0 0.9000       : 0.771

With the nominal α 0.05 the Type I Error will be inflated (0.0665). With the adjusted α 0.0354 (i.e., a 92.92% CI) the TIE will be controlled, although with a slight loss in power (decreases from 0.814 to 0.771).
Consider sampleN.scABEL.ad(CV = c(0.30, 0.35), design = "2x2x4") to estimate the sample size which both controls the TIE and maintains the target power. In this example 34 subjects would be required.

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ABEL cannot be applied for AUC (except for the WHO). Hence, in many cases ABE drives the sample size. Three period full replicate "2x2x3" study (TRT|RTR or TRR|RTT).

PK  <- c("Cmax", "AUC")
CV  <- c(0.45, 0.30)
# extract sample sizes and power
r1  <- sampleN.scABEL(CV = CV[1], theta0 = 0.90, design = "2x2x3",
                      print = FALSE, details = FALSE)[8:9]
r2  <- sampleN.TOST(CV = CV[2], theta0 = 0.90, design = "2x2x3",
                    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)
#    PK method  n   power
#  Cmax   ABEL 42 0.80017
#   AUC    ABE 60 0.81002

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RSABE

HVD(P)s

Sample size for a four period full replicate "2x2x4" study (any of TRTR|RTRT, TRRT|RTTR, TTRR|RRTT) assuming heteroscedasticity (CVwT 0.40, CVwR 0.50). Details of the sample size search suppressed.

sampleN.RSABE(CV = c(0.40, 0.50), design = "2x2x4", details = FALSE)
# 
# ++++++++ Reference scaled ABE crit. +++++++++
#            Sample size estimation
# ---------------------------------------------
# Study design: 2x2x4 (4 period full replicate)
# log-transformed data (multiplicative model)
# 1e+05 studies for each step simulated.
# 
# alpha  = 0.05, target power = 0.8
# CVw(T) = 0.4; CVw(R) = 0.5
# True ratio = 0.9
# ABE limits / PE constraints = 0.8 ... 1.25 
# Regulatory settings: FDA 
# 
# Sample size
#  n    power
# 20   0.81509

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NTIDs

Sample size assuming heteroscedasticity (CVw 0.10, σ2 ratio 2.5, i.e., the test treatment has a substantially higher variability than the reference). TRTR|RTRT according to the FDA’s guidance. Assess additionally which one of the three components (scaled ABE, conventional ABE, swT/swR ratio) drives the sample size.

CV <- signif(CVp2CV(CV = 0.10, ratio = 2.5), 4)
n  <- sampleN.NTIDFDA(CV = CV)[["Sample size"]]
# 
# +++++++++++ FDA method for NTIDs ++++++++++++
#            Sample size estimation
# ---------------------------------------------
# Study design:  2x2x4 (TRTR|RTRT) 
# log-transformed data (multiplicative model)
# 1e+05 studies for each step simulated.
# 
# alpha  = 0.05, target power = 0.8
# CVw(T) = 0.1197, CVw(R) = 0.07551
# True ratio     = 0.975 
# ABE limits     = 0.8 ... 1.25 
# Implied scABEL = 0.9236 ... 1.0827 
# Regulatory settings: FDA 
# - Regulatory const. = 1.053605 
# - 'CVcap'           = 0.2142 
# 
# Sample size search
#  n     power
# 32   0.699120 
# 34   0.730910 
# 36   0.761440 
# 38   0.785910 
# 40   0.809580
suppressMessages(power.NTIDFDA(CV = CV, n = n, details = TRUE))
#        p(BE)  p(BE-sABEc)    p(BE-ABE) p(BE-sratio) 
#      0.80958      0.90966      1.00000      0.87447

The swT/swR component shows the lowest probability to demonstrate BE and hence, drives the sample size.

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Compare that with homoscedasticity (CVwT = CVwR = 0.10):

CV <- 0.10
n  <- sampleN.NTIDFDA(CV = CV, details = FALSE)[["Sample size"]]
# 
# +++++++++++ FDA method for NTIDs ++++++++++++
#            Sample size estimation
# ---------------------------------------------
# Study design:  2x2x4 (TRTR|RTRT) 
# log-transformed data (multiplicative model)
# 1e+05 studies for each step simulated.
# 
# alpha  = 0.05, target power = 0.8
# CVw(T) = 0.1, CVw(R) = 0.1
# True ratio     = 0.975 
# ABE limits     = 0.8 ... 1.25 
# Regulatory settings: FDA 
# 
# Sample size
#  n     power
# 18   0.841790
suppressMessages(power.NTIDFDA(CV = CV, n = n, details = TRUE))
#        p(BE)  p(BE-sABEc)    p(BE-ABE) p(BE-sratio) 
#      0.84179      0.85628      1.00000      0.97210

Here the scaled ABE component shows the lowest probability to demonstrate BE and drives the sample size – which is much lower than in the previous example.

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Comparison with fixed narrower limits applicable in other jurisdictions. Note that a replicate design is not required, reducing the chance of dropouts.

CV  <- 0.10
# extract sample sizes and power
r1  <- sampleN.NTIDFDA(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 scaled", 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. #"
print(res, row.names = FALSE)
#        method design  n   power adm. #
#    FDA scaled  2x2x4 18 0.84179     72
#  fixed narrow  2x2x4 12 0.85628     48
#  fixed narrow  2x2x3 16 0.81393     48
#  fixed narrow  2x2x2 22 0.81702     44

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Dose-Proportionality

CV 0.20 (20%), doses 1, 2, and 8 units, assumed slope β0 1, target power 0.90.

sampleN.dp(CV = 0.20, doses = c(1, 2, 8), beta0 = 1, targetpower = 0.90)
# 
# ++++ Dose proportionality study, power model ++++
#             Sample size estimation
# -------------------------------------------------
# Study design: crossover (3x3 Latin square) 
# alpha = 0.05, target power = 0.9
# Equivalence margins of R(dnm) = 0.8 ... 1.25 
# Doses = 1 2 8 
# True slope = 1, CV = 0.2
# Slope acceptance range = 0.89269 ... 1.1073 
# 
# Sample size (total)
#  n     power
# 18   0.915574

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 {θ1, θ2} (0.50, 2.00) were proposed, translating in this example to an acceptance range of 0.66667 ... 1.3333 and a sample size of only six subjects.

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Power Analysis

Explore impact of deviations from assumptions (higher CV, higher deviation of θ0 from 1, dropouts) on power. Assumed within-subject CV 0.20 (20%), target power 0.90. Plot suppressed.

res <- pa.ABE(CV = 0.20, targetpower = 0.90)
print(res, plotit = FALSE)
# Sample size plan ABE
#  Design alpha  CV theta0 theta1 theta2 Sample size Achieved power
#     2x2  0.05 0.2   0.95    0.8   1.25          26      0.9176333
# 
# Power analysis
# CV, theta0 and number of subjects which lead to min. acceptable power of at least 0.7:
#  CV= 0.2729, theta0= 0.9044
#  n = 16 (power= 0.7354)

If the study starts with 26 subjects (power ~0.92), the CV can increase to ~0.27 or θ0 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, 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.

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Speed Comparisons

Performed on a Xeon E3-1245v3 3.4 GHz, 8 MB cache, 16 GB RAM, R 4.0.3 64 bit on Windows 7.

ABE

"2x2" crossover design, CV 0.17. Sample sizes and achieved power for the supported methods (the 1st 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 2nd exact method is substantially slower than the 1st. The approximation based on the noncentral t-distribution is slightly faster but matches the 1st 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, homogenicity (CVwT = CVwR 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 CVwT > CVwR is expected.

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Installation

You can install the released version of PowerTOST from CRAN with

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

… and the development version from GitHub with

# install.packages("remotes")
remotes::install_github("Detlew/PowerTOST")

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 1.5.1.9999 built 2020-10-26 with R 4.0.3.

options(width = 80)
devtools::session_info()
# - Session info ---------------------------------------------------------------
#  setting  value                       
#  version  R version 4.0.3 (2020-10-10)
#  os       Windows 7 x64 SP 1          
#  system   x86_64, mingw32             
#  ui       RTerm                       
#  language EN                          
#  collate  German_Germany.1252         
#  ctype    German_Germany.1252         
#  tz       Europe/Vienna               
#  date     2020-10-26                  
# 
# - Packages -------------------------------------------------------------------
#  package       * version    date       lib source        
#  assertthat      0.2.1      2019-03-21 [1] CRAN (R 4.0.0)
#  backports       1.1.10     2020-09-15 [1] CRAN (R 4.0.2)
#  callr           3.5.1      2020-10-13 [1] CRAN (R 4.0.3)
#  cli             2.1.0      2020-10-12 [1] CRAN (R 4.0.3)
#  crayon          1.3.4      2017-09-16 [1] CRAN (R 4.0.0)
#  cubature        2.0.4.1    2020-07-06 [1] CRAN (R 4.0.2)
#  desc            1.2.0      2018-05-01 [1] CRAN (R 4.0.0)
#  devtools        2.3.2      2020-09-18 [1] CRAN (R 4.0.2)
#  digest          0.6.25     2020-02-23 [1] CRAN (R 4.0.0)
#  ellipsis        0.3.1      2020-05-15 [1] CRAN (R 4.0.0)
#  evaluate        0.14       2019-05-28 [1] CRAN (R 4.0.0)
#  fansi           0.4.1      2020-01-08 [1] CRAN (R 4.0.0)
#  fs              1.5.0      2020-07-31 [1] CRAN (R 4.0.2)
#  glue            1.4.2      2020-08-27 [1] CRAN (R 4.0.2)
#  htmltools       0.5.0      2020-06-16 [1] CRAN (R 4.0.0)
#  knitr           1.30       2020-09-22 [1] CRAN (R 4.0.2)
#  magrittr        1.5        2014-11-22 [1] CRAN (R 4.0.0)
#  memoise         1.1.0      2017-04-21 [1] CRAN (R 4.0.0)
#  mvtnorm         1.1-1      2020-06-09 [1] CRAN (R 4.0.0)
#  pkgbuild        1.1.0      2020-07-13 [1] CRAN (R 4.0.2)
#  pkgload         1.1.0      2020-05-29 [1] CRAN (R 4.0.0)
#  PowerTOST     * 1.5-1.9999 2020-10-26 [1] local         
#  prettyunits     1.1.1      2020-01-24 [1] CRAN (R 4.0.0)
#  processx        3.4.4      2020-09-03 [1] CRAN (R 4.0.2)
#  ps              1.4.0      2020-10-07 [1] CRAN (R 4.0.2)
#  R6              2.4.1      2019-11-12 [1] CRAN (R 4.0.0)
#  Rcpp            1.0.5      2020-07-06 [1] CRAN (R 4.0.2)
#  remotes         2.2.0      2020-07-21 [1] CRAN (R 4.0.2)
#  rlang           0.4.8      2020-10-08 [1] CRAN (R 4.0.3)
#  rmarkdown       2.5        2020-10-21 [1] CRAN (R 4.0.3)
#  rprojroot       1.3-2      2018-01-03 [1] CRAN (R 4.0.0)
#  sessioninfo     1.1.1      2018-11-05 [1] CRAN (R 4.0.0)
#  stringi         1.5.3      2020-09-09 [1] CRAN (R 4.0.2)
#  stringr         1.4.0      2019-02-10 [1] CRAN (R 4.0.0)
#  TeachingDemos   2.12       2020-04-07 [1] CRAN (R 4.0.0)
#  testthat        2.3.2      2020-03-02 [1] CRAN (R 4.0.0)
#  usethis         1.6.3      2020-09-17 [1] CRAN (R 4.0.2)
#  withr           2.3.0      2020-09-22 [1] CRAN (R 4.0.2)
#  xfun            0.18       2020-09-29 [1] CRAN (R 4.0.2)
#  yaml            2.2.1      2020-02-01 [1] CRAN (R 4.0.0)
# 
# [1] D:/Program Files/R/R-4.0.3/library

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