Version 1.5.6 built 2024-03-18 with R 4.3.3
The package contains functions to calculate power and estimate sample size for various study designs used in (not only bio-) equivalence studies.
# 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:
design | type | sequences | periods | |
---|---|---|---|---|
2x2x4 |
full | 2 | TRTR|RTRT | 4 |
2x2x4 |
full | 2 | TRRT|RTTR | 4 |
2x2x4 |
full | 2 | TTRR|RRTT | 4 |
2x4x4 |
full | 4 | TRTR|RTRT|TRRT|RTTR | 4 |
2x4x4 |
full | 4 | TRRT|RTTR|TTRR|RRTT | 4 |
2x2x3 |
full | 2 | TRT|RTR | 3 |
2x2x3 |
full | 2 | TRR|RTT | 3 |
2x4x2 |
full | 4 | TR|RT|TT|RR | 2 |
2x3x3 |
partial | 3 | TRR|RTR|RRT | 3 |
2x2x3 |
partial | 2 | TRR|RTR | 3 |
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.
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.
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).
- ABE for Highly Variable Narrow Therapeutic Index Drugs by simulations: U.S. FDA, China CDE.
- Scaled Average Bioequivalence based on simulations.
- Average Bioequivalence with Expanding Limits (ABEL) for Highly Variable Drugs / Drug Products: EMA, WHO and many others.
- Average Bioequivalence with fixed widened limits of 75.00–133.33% if CVwR >30%: Gulf Cooperation Council.
- Reference-scaled Average Bioequivalence (RSABE) for HVDP(s): U.S. FDA, China CDE.
- Iteratively adjust α to control the type I error in ABEL and RSABE for HVDP(s).
- RSABE for NTIDs: U.S. FDA, China CDE.
- 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.
- Dose-Proportionality using the power model.
- Exact
- Owen’s Q.
- Direct integration of the bivariate non-central t-distribution.
- Approximations
- Non-central t-distribution.
- ‘Shifted’ central t-distribution.
- 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.
- α 0.05, {θ1, θ2} (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.
Design "2x2"
(TR|RT), exact method (Owen’s Q).
Design "2x2x4"
(TRTR|RTRT), upper limit of the confidence interval of
σwT/σwR ≤2.5, approximation by the non-central
t-distribution, 100,000 simulations.
Point estimate constraints (0.80, 1.25), homoscedasticity
(CVwT = CVwR), scaling is based on
CVwR, 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.
θ0 0.90.1
Regulatory constant 0.760
, upper cap of scaling at CVwR
50%, evaluation by ANOVA.
Regulatory constant 0.760
, upper cap of scaling at CVwR
~57.4%, evaluation by intra-subject contrasts.
Regulatory constant log(1/0.75)/sqrt(log(0.3^2+1))
, widened limits
75.00–133.33% if CVwR >30%, no upper cap of scaling,
evaluation by ANOVA.
Regulatory constant log(1.25)/0.25
, no upper cap of scaling,
evaluation by linearized scaled
ABE (Howe’s approximation).
θ0 0.975, regulatory constant log(1.11111)/0.1
, implicit
upper cap of scaling at CVwR ~21.4%, design "2x2x4"
(TRTR|RTRT), evaluation by linearized scaled
ABE (Howe’s approximation),
upper limit of the confidence interval of
σwT/σwR ≤2.5.
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.
Minimum acceptable power 0.70. θ0; design, conditions, and sample size method depend on defaults of the respective approaches (ABE, ABEL, RSABE, NTID, HVNTID).
Before running the examples attach the library.
library(PowerTOST)
If not noted otherwise, the functions’ defaults are employed.
Power for total CV 0.35 (35%), group sizes 52 and 49.
power.TOST(CV = 0.35, n = c(52, 49), design = "parallel")
# [1] 0.8011186
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 study (any of TRTR|RTRT, TRRT|RTTR, TTRR|RRTT). Wider acceptance range for Cmax (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
Sample size for assumed within- (intra-) subject CV 0.125 (12.5%), θ0 0.975. Narrower 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
Sample size for equivalence of the ratio of two means with normality on
the original scale based on Fieller’s (‘fiducial’) confidence
interval.2 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
Sample size for assumed within- (intra-) subject CV 0.45 (45%), θ0 0.90, three period full replicate study (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 CVw is still 0.45.
Sample size assuming heteroscedasticity (CVw 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.3,4,5 Assess additionally which one of the components (ABE, swT/swR-ratio) drives the sample size.
CV <- signif(CVp2CV(CV = 0.45, ratio = 2.5), 4)
n <- sampleN.HVNTID(CV = CV, details = FALSE)[["Sample size"]]
#
# +++++++++ FDA method for HV 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) = 0.549, CVw(R) = 0.3334
# True ratio = 0.95
# ABE limits = 0.8 ... 1.25
#
# Sample size
# n power
# 50 0.812820
suppressMessages(power.HVNTID(CV = CV, n = n, details = TRUE))
# p(BE) p(BE-ABE) p(BE-sratio)
# 0.81282 0.87052 0.93379
The ABE component shows a lower probability to demonstrate BE than the swT/swR component and hence, drives the sample size.
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
Iteratively adjust α to control the Type I Error.6 Heteroscedasticity (CVwT 0.30, CVwR 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.
scABEL.ad(CV = c(0.30, 0.40), design = "2x2x4", n = 24)
# +++++++++++ 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 0.4, CVwT 0.3, n(i) 12|12 (N 24)
# Nominal alpha : 0.05
# True ratio : 0.9000
# Regulatory settings : EMA (ABEL)
# Switching CVwR : 0.3
# Regulatory constant : 0.76
# Expanded limits : 0.7462 ... 1.3402
# Upper scaling cap : CVwR > 0.5
# PE constraints : 0.8000 ... 1.2500
# Empiric TIE for alpha 0.0500 : 0.05953
# Power for theta0 0.9000 : 0.805
# Iteratively adjusted alpha : 0.03997
# Empiric TIE for adjusted alpha: 0.05000
# Power for theta0 0.9000 : 0.778
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%
CI) the
TIE 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
TIE and target power. In this example
26 subjects would be required.
ABEL cannot be applied for AUC (except for the WHO). Hence, in many cases ABE drives the sample size. Four period full replicate study (any of TRTR|RTRT, TRRT|RTTR, TTRR|RRTT).
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)
# PK method n power
# Cmax ABEL 28 0.81116
# AUC ABE 40 0.80999
AUC drives the sample size.
For Health Canada it is the opposite (ABE for Cmax and ABEL for AUC).
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)
# PK method n power
# Cmax ABE 84 0.80569
# AUC ABEL 34 0.80281
Here Cmax drives the sample size.
Sample size assuming homoscedasticity (CVwT = CVwR = 0.45) for the widened limits of the Gulf Cooperation Council.
sampleN.scABEL(CV = 0.45, regulator = "GCC", details = FALSE)
#
# +++++++++++ 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
# Widened limits = 0.75 ... 1.333333
# Regulatory settings: GCC
#
# Sample size
# n power
# 54 0.8123
Sample size for a four period full replicate study (any of TRTR|RTRT, TRRT|RTTR, TTRR|RRTT) assuming heteroscedasticity (CVwT 0.40, CVwR 0.50, i.e., variance-ratio ~0.67). 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
Sample size assuming heteroscedasticity (CVw 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.7 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.NTID(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.NTID(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.
Compare that with homoscedasticity (CVwT = CVwR = 0.10):
CV <- 0.10
n <- sampleN.NTID(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.NTID(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.
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
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)
# method design n power adm. #
# FDA/CDE 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
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,8
translating in this example to an acceptance range of
0.66667 ... 1.3333
and a sample size of only six subjects.
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 leading to min. acceptable power of ~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-E9,9 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.
Performed on a Xeon E3-1245v3 3.4 GHz, 8 MB cache, 16 GB RAM, R 4.3.3 64 bit on Windows 7.
2×2 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.
Four period full replicate study (any of TRTR|RTRT, TRRT|RTTR, TTRR|RRTT), 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.
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.
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 1.5.6 built 2024-03-18 with R 4.3.3.
options(width = 66)
sessionInfo()
# R version 4.3.3 (2024-02-29 ucrt)
# Platform: x86_64-w64-mingw32/x64 (64-bit)
# Running under: Windows 10 x64 (build 19045)
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# Matrix products: default
#
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# locale:
# [1] LC_COLLATE=German_Germany.utf8
# [2] LC_CTYPE=German_Germany.utf8
# [3] LC_MONETARY=German_Germany.utf8
# [4] LC_NUMERIC=C
# [5] LC_TIME=German_Germany.utf8
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# time zone: Europe/Berlin
# tzcode source: internal
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# attached base packages:
# [1] stats graphics grDevices utils datasets methods
# [7] base
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# other attached packages:
# [1] PowerTOST_1.5-6
#
# loaded via a namespace (and not attached):
# [1] cubature_2.1.0 compiler_4.3.3 fastmap_1.1.1
# [4] cli_3.6.2 tools_4.3.3 htmltools_0.5.7
# [7] rstudioapi_0.15.0 yaml_2.3.8 Rcpp_1.0.12
# [10] mvtnorm_1.2-4 rmarkdown_2.26 knitr_1.45
# [13] xfun_0.42 digest_0.6.35 rlang_1.1.3
# [16] evaluate_0.23
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