Version 1.5.4 built 2022-02-20 with R 4.1.2 (development version not on CRAN).

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 (*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.

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*CV*_{wR}>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
*θ*_{0}. - 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.

- Non-central

- 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 TOST 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.

*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.

Design `"2x2"`

(TR|RT), exact method (Owen’s Q).

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.

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.

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

*θ*_{0} 0.90.^{1}

Regulatory constant `0.760`

, upper cap of scaling at *CV*_{wR} 50%, evaluation by ANOVA.

Regulatory constant `0.760`

, upper cap of scaling at *CV*_{wR} ~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 *CV*_{wR} >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).

*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 ABE (Howe’s approximation), upper limit of the confidence interval of *σ*_{wT}/*σ*_{wR} ≤2.5.

*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.

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).

Before running the examples attach the library.

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

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

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%), *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).

```
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%), *θ*_{0} 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 *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.

```
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%), *θ*_{0} 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 *σ*^{2}-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*_{w} is still 0.45.

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.^{3,}^{4,}^{5} Assess additionally which one of the components (ABE, *s*_{wT}/*s*_{wR}-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 *s*_{wT}/*s*_{wR} component and hence, drives the sample size.

Sample size assuming homoscedasticity (*CV*_{wT} = *CV*_{wR} = 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 (*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.

```
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 *C*_{max} 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 *C*_{max} drives the sample size.

Sample size assuming homoscedasticity (*CV*_{wT} = *CV*_{wR} = 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 (*CV*_{wT} 0.40, *CV*_{wR} 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 (*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.^{7} Assess additionally which one of the three components (scaled ABE, conventional ABE, *s*_{wT}/*s*_{wR}-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 *s*_{wT}/*s*_{wR} component shows the lowest probability to demonstrate BE and hence, drives the sample size.

Compare that with homoscedasticity (*CV*_{wT} = *CV*_{wR} = 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 *β*_{0} 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 {*θ*_{1}, *θ*_{2}} (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 *θ*_{0} 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** *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,^{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 4.1.2 64 bit on Windows 7.

2×2 crossover design, *CV* 0.17. Sample sizes and achieved power for the supported methods (the 1^{st} 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^{nd} exact method is substantially slower than the 1^{st}. The approximation based on the noncentral *t*-distribution is slightly faster but matches the 1^{st} 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 (*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.

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 1.5.4 built 2022-02-20 with R 4.1.2.

```
options(width = 66)
sessionInfo()
# R version 4.1.2 (2021-11-01)
# Platform: x86_64-w64-mingw32/x64 (64-bit)
# Running under: Windows 7 x64 (build 7601) Service Pack 1
#
# Matrix products: default
#
# locale:
# [1] LC_COLLATE=German_Germany.1252
# [2] LC_CTYPE=German_Germany.1252
# [3] LC_MONETARY=German_Germany.1252
# [4] LC_NUMERIC=C
# [5] LC_TIME=German_Germany.1252
#
# attached base packages:
# [1] stats graphics grDevices utils datasets methods
# [7] base
#
# other attached packages:
# [1] PowerTOST_1.5-4
#
# loaded via a namespace (and not attached):
# [1] Rcpp_1.0.8 mvtnorm_1.1-3 digest_0.6.29
# [4] magrittr_2.0.2 evaluate_0.15 TeachingDemos_2.12
# [7] rlang_1.0.1 stringi_1.7.6 cli_3.1.1
# [10] cubature_2.0.4.2 rstudioapi_0.13 rmarkdown_2.11
# [13] tools_4.1.2 stringr_1.4.0 xfun_0.29
# [16] yaml_2.2.2 fastmap_1.1.0 compiler_4.1.2
# [19] htmltools_0.5.2 knitr_1.37
```

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2. Fieller EC. *Some Problems In Interval Estimation.* J Royal Stat Soc B. 1954; 16(2): 175–85. JSTOR:2984043. ↩︎

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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. ↩︎

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. ↩︎