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Examples in this article were generated with  4.4.1 by the package PowerTOST.1

More examples are given in the respective vignette.2 See also the README on GitHub for an overview and the online manual3 for details.

The right-hand badges give the respective section’s ‘level’.

1. Basics about sample size methodology – requiring no or only limited statistical expertise.

1. These sections are the most important ones. They are – hopefully – easily comprehensible even for novices.

1. A somewhat higher knowledge of statistics and/or R is required. May be skipped or reserved for a later reading.

1. An advanced knowledge of statistics and/or R is required. Not recommended for beginners in particular.

1. If you are not a neRd or statistics afficionado, skipping is recommended. Suggested for experts but might be confusing for others.
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Abbreviations are given at the end.

# Introduction

What is Average Bioequivalence with Expanding Limits?

For details about inferential statistics and hypotheses in equivalence see another article.

Definitions:

1. A Highy Variable Drug (HVD) shows a within-subject Coefficient of Variation $$\small{(CV_\text{wR})}$$ $$\small{>30\%}$$ if administered as a solution in a replicate design. The high variability is an intrinsic property of the drug (absorption/permeation, clearance).
Agencies are only interested in $$\small{CV_\text{wR}}$$. For an applicant $$\small{CV_\text{wT}}$$ is ‘nice to know’ as well.
1. A Highy Variable Drug Product (HVDP) shows a $$\small{CV_\text{wR}>30\%}$$ in a replicate design.4

The concept of Scaled Average Bioequivalence (SABE) for HVDPs is based on the following considerations:

1. HVDPs are safe and efficacious despite their high variability because:
• They have a wide therapeutic index (i.e., a flat dose-response curve). Consequently, even substantial changes in concentrations have only a limited impact on the effect.
If they would have a narrow therapeutic index, adverse effects (due to high concentrations) and lacking effects (due to low concentrations) would have been observed in Phase III and consequently, the originator’s product should not have been approved in the first place.5
• Once approved, the product has a documented safety / efficacy record in phase IV and in clinical practice – despite its high variability.
If problems were evident, the product would have been taken off the market.
2. Given that, the conventional ‘clinically relevant difference’ $$\small{\Delta}$$ of 20% (leading to the limits of 80.00 – 125.00% in Average Bioequivalence) is overly conservative and thus, leading to large sample sizes.
3. Hence, a more relaxed $$\small{\Delta>20\%}$$ was proposed. A natural approach is to scale (expand / widen) the limits based on the within-subject variability of the reference product $$\small{s_\text{wR}}$$. In this approach, power for a given sample size would be ideally independent from the variability.

The conventional confidence interval inclusion approach in ABE $\begin{matrix}\tag{1} \theta_1=1-\Delta,\theta_2=\left(1-\Delta\right)^{-1}\\ H_0:\;\frac{\mu_\text{T}}{\mu_\text{R}}\ni\left\{\theta_1,\theta_2\right\}\;vs\;H_1:\;\theta_1<\frac{\mu_\text{T}}{\mu_\text{R}}<\theta_2\textsf{,}, \end{matrix}$ where $$\small{H_0}$$ is the null hypothesis of inequivalence and $$\small{H_1}$$ the alternative hypothesis, $$\small{\theta_1}$$ and $$\small{\theta_2}$$ are the fixed lower and upper limits of the acceptance range, and $$\small{\mu_\text{T}}$$ are the geometric least squares means of $$\small{\text{T}}$$ and $$\small{\text{R}}$$, respectively is in Scaled Average Bioequivalence (SABE)6 modified to $H_0:\;\frac{\mu_\text{T}}{\mu_\text{R}}\Big{/}\sigma_\text{wR}\ni\left\{\theta_{\text{s}_1},\theta_{\text{s}_2}\right\}\;vs\;H_1:\;\theta_{\text{s}_1}<\frac{\mu_\text{T}}{\mu_\text{R}}\Big{/}\sigma_\text{wR}<\theta_{\text{s}_2}\textsf{,}\tag{2}$ where $$\small{\sigma_\text{wR}}$$ is the standard deviation of the reference and the scaled limits $$\small{\left\{\theta_{\text{s}_1},\theta_{\text{s}_2}\right\}}$$ of the acceptance range depend on conditions given by the agency.

Average Bioequivalence with Expanding Limits (ABEL)7 for HVDPs is acceptable in numerous jurisdictions (Fig. 2), whereas directly widening the limits was recommended 2016 – 2022 in the member states of the Gulf Co­ope­ra­tion Council8 (Fig. 3). Alas, we are far away from global harmonization. Work on the ICH’s M13C guideline9 has not started with July 2024 (expected after finalization of M13B).

Expanding the limits is acceptable for different pharmacokinetic metrics.

PK metric Jurisdiction
Cmax The EEA,10 the WHO,11 12 Australia,13 Belarus,14 the EAC,15 the ASEAN states,16 the Russian Fe­de­ration,17 the EEU,18 Egypt,19 New Zealand,20 Chile,21 Brazil,22 members of the GCC,23 Mexico.24
AUC WHO (if in a 4-period full replicate design),11 Canada.25
Cmin, Cτ The EEA (controlled release products in steady state).26
partial AUC The EEA (controlled release products).26

Based on the switching $$\small{CV_0=30\%}$$ we get the switching standard deviation $$\small{s_0=\sqrt{\log_{e}(CV_{0}^{2}+1)}\approx0.2935604\ldots}$$, the (rounded) regulatory constant $$\small{k=\frac{\log_{e}1.25}{s_0}\sim0.76}$$, and finally the expanded limits $$\small{\left\{\theta_{\text{s}_1},\theta_{\text{s}_2}\right\}=}$$ $$\small{100\left(\exp(\mp0.76\cdot s_{\text{wR}})\right)}$$. Note that in the guidelines the terminology $$\small{\left\{L,U\right\}}$$ is used instead.

In order to apply the methods following conditions have to be fulfilled:

1. The study has to be performed in a replicate design (i.e., at least the reference product has to be administered twice).
2. A clinical justification has to be provided that the expanded limits will not impact safety / efficacy.
3. The observed within-subject variability of the reference $$\small{(CV_\text{wR})}$$ has to be > 30%.
4. Except in Chile and Brazil, it has to be demonstrated that the high variability is not caused by ‘outliers’. This topic will be covered in another article.

The wording in the guidelines gives the false impression that the methods are straightforward. In fact they are decision schemes (frameworks), which hinge on the estimated standard deviation of the reference treatment $$\small{s_{\text{wR}}}$$.

If $$\small{CV_\text{wR}\leq30\%}$$ the study has to be assessed for ABE (left branch) or for ABEL (right branch) otherwise.

In the ABEL-branch there is an ‘upper cap’ of scaling ($$\small{uc=50\%}$$ in all jurisdictions except for Health Canada, where $$\small{uc\approx 57.382\%}$$). Furthermore, the point estimate ($$\small{PE}$$) has to lie within 80.00 – 125.00%.

From March 2016 to August 2022 the GCC recommended a simplified variant with fixed widened limits of 75.00 – 133.33% if $$\small{CV_\text{wR}>30\%}$$.8 Like in ABEL the $$\small{PE}$$ had to lie within 80.00 – 125.00%. As in the FDA’s RSABE there was no ‘upper cap’.

Realization: Ob­servations (in a sample) of a random variable (of the population).

Since the applicability of these approaches hinges on the realized values ($$\small{CV_\text{wR}}$$, $$\small{PE}$$) in the particular study – which are random variables and as such unknown before­hand – analytical solutions for power (and hence, the sample size) do not exist. Therefore, extensive simulations of potential combinations have to be employed.

Whereas ABE is bijective (if $$\small{T}$$ is equivalent to $$\small{R}$$, $$\small{R}$$ is also equivalent to $$\small{T}$$), this holds in ABEL only if $$\small{CV_\text{wT}=}$$ $$\small{CV_\text{wR}}$$. Therefore, switching from $$\small{R}$$ to $$\small{T}$$ is tolerable if $$\small{CV_\text{wT}<CV_\text{wR}}$$ but switching from $$\small{T}$$ to $$\small{R}$$ can be pro­ble­ma­tic. Individual Bioequivalence (IBE) would not only compare the averages of products but also their variances and thus, support ‘Switch­abi­lity’.27 However, IBE was never implemented in regulatory practice. For details see this article.

Cave: Under certain conditions the methods may lead to an inflated Type I Error (increased patient’s risk).28 29 It is elaborated in another article.

For the evaluation I recommend the package replicateBE.30

Interlude 1

Where do these numbers come from?

With the regulatory constant $$\small{k=0.76}$$ one gets in the $$\small{\log_{e}}$$-scale a linear expansion from $$\small{CV_\text{wR}=30\%}$$ to $$\small{uc}$$ based on $$\small{s_\text{wR}=\sqrt{\log_{e}(CV_\text{wR}^{2})+1}}$$ and $$\small{100\left(\exp(\mp k\cdot s_\text{wR})\right)}$$.
Hence, with $$\small{CV_\text{wR}=30\%}$$ one gets
$$\small{\;\;\;\left\{\theta_1,\theta_2\right\}=100\left(\exp(\mp0.76\cdot 0.2935604)\right)=\left\{80.00,\,125.00\right\}}$$.
With $$\small{uc=50\%}$$ one gets
$$\small{\;\;\;\left\{\theta_{\text{s}_1},\,\theta_{\text{s}_2}\right\}=100\left(\exp(\mp0.76\cdot 0.4723807)\right)=\left\{69.83678,\,143.19102\right\}}$$.
For Health Canada with $$\small{uc=57.382\%}$$ one gets $$\small{\;\;\;\left\{\theta_{\text{s}_1},\,\theta_{\text{s}_2}\right\}=100\left(\exp(\mp0.76\cdot 0.5335068)\right)=\left\{66.66667,\,150.00000\right\}}$$.

As stated above, the regulatory constant $$\small{k}$$ is given in all guidelines rounded to two significant digits. Based on the switching $$\small{CV_\text{w0}=30\%}$$ one gets $$\small{s_\text{w0}=\sqrt{\log_{e}(CV_\text{w0}^2+1)}\approx 0.2935604}$$ and, therefore, the exact value would be $$\small{k=\frac{\log_{e}1.25}{s_\text{w0}}\approx 0.7601283}$$. For a rant see Karalis et al.31 The wisdom of regulators is unfathomable.

A clinically not relevant $$\small{\Delta}$$ of 30% (leading a fixed range of 70.00 – 142.86%) was acceptable in Europe (in exceptional cases even for AUC) if pre-specified in the protocol. A replicate design was not required.32 33 34
A clinically not relevant $$\small{\Delta}$$ of 25% for Cmax (75.00 – 133.33%) was acceptable for the EMA if the study was performed in a replicate design and $$\small{CV_\text{wR}>30\%}$$.35 A $$\small{\Delta}$$ of 25% for Cmax (75.00 – 133.33%) is currently acceptable in South Africa,36 by the League of Arab States,37 and was acceptable in Kazakhstan.38
Hence, this excursion into history may explain the upper cap of scaling in ABEL and the (former) widened limits for the GCC. I assume that Health Canada’s 66.7 – 150.0% are no more than ‘nice numbers’ – as usual.39

top of section ↩︎

## Preliminaries

A basic knowledge of R is required. To run the scripts at least version 1.4.8 (2019-08-29) of PowerTOST is required and at least version 1.5.3 (2021-01-18) suggested. Any version of R would likely do, though the cur­rent release of PowerTOST was only tested with R 4.3.3 (2024-02-29) and later. All scripts were run on a Xeon E3-1245v3 @ 3.40GHz (1/4 cores) 16GB RAM with R 4.4.1 on Windows 7 build 7601, Service Pack 1.

Note that in all functions of PowerTOST the arguments (say, the assumed T/R-ratio theta0, the assumed coefficient of variation CV, etc.) have to be given as ratios and not in percent.

sampleN.scABEL() gives balanced sequences (i.e., an equal number of subjects is allocated to all sequences). Furthermore, the estimated sample size is the total number of subjects – and not the number of subjects per sequence like in some other software packages.

All examples deal with studies where the response variables likely follow a log-normal distribution, i.e., we assume a multiplicative model (ratios instead of differences). We work with $$\small{\log_{e}}$$-transformed data in order to allow analysis by the t-test (requiring differences).

## Terminology

It may sound picky but ‘sample size calculation’ (as used in most guidelines and alas, in some publications and textbooks) is sloppy terminology. In order to get prospective power (and hence, a sample size), we need five values:

1. The level of the test $$\small{\alpha}$$ (in BE commonly 0.05),
2. the (potentially expanded) BE-margin,
3. the desired (or target) power $$\small{\pi}$$,
4. the variance (commonly expressed as a coefficient of variation), and
5. the deviation of the test from the reference treatment,

where

1. is fixed by the agency,
2. depends on the observed40 $$\small{CV_\text{wR}}$$ (which is unknown beforehand),
3. is set by the sponsor (commonly to 0.80 – 0.90),
4. is an uncertain assumption,
5. is an uncertain assumption.

In other words, obtaining a sample size is not an exact calculation like $$\small{2\times2=4}$$ but always just an estimation.

Power Calculation – A guess masquerading as mathematics.
Stephen Senn (2020)41

Of note, it is extremely unlikely that all assumptions will be exactly realized in a particular study. Hence, calculating retrospective (a.k.a. estimated, post hoc, a posteriori) power is not only futile but plain nonsense.42 For details see another article.

Carryover: A resi­dual effect of a previous period.

It is a prerequisite that no carryover from one period to the next exists. Only then the comparison of treatments will be unbiased.43 Carryover is elaborated in another article.

Except for Health Canada (where a mixed-effects model is required) the recommended evaluation by an ANOVA assumes homoscedasticity ($$\small{s_\text{wT}^2=s_\text{wR}^2}$$), which is – more often than not – wrong.

# Power → Sample size

The sample size cannot be directly estimated, in ABEL
only power simulated for an already given sample size.

In contrast to ABE, where power can be exactly calculated and the sample size is determined in an iterative procedure, ABEL is a decision scheme and therefore, simulations must to performed for different combinations of assumed T/R-ratios and CVs.

Power. That which statisticians are always calculating but never have.
Stephen Senn (2007)45

Let’s start with PowerTOST.

library(PowerTOST) # attach it to run the examples

The sample size functions’ defaults are:

Argument Default Meaning
alpha 0.05 Nominal level of the test
targetpower 0.80 Target (desired) power
theta0 0.90 Assumed T/R-ratio
theta1 0.80 Lower BE-limit in ABE and lower PE-constraint in ABEL
theta2 1.25 Upper BE-limit in ABE and upper PE-constraint in ABEL
design "2x3x3" Treatments × Sequences × Periods
regulator "EMA" Guess…
nsims 1e05 Number of simulations
print TRUE Output to the console
details TRUE Show regulatory settings and sample size search
setseed TRUE Set a fixed seed (recommended for reproducibility)

Implemented designs are "2x3x3" (the three sequence, three period  – partial – replicate design TRR|RTR|RRT), "2x2x4" (two sequence, four period full replicate designs TRTR|RTRT, TRRT|RTTR, and TTRR|RRTT), and "2x2x3" (two sequence, three period full replicate designs TRT|RTR and TRR|RTT).

For a quick overview of the regulatory limits use the function scABEL() – for once in percent according to the guidelines. Note that Health Canada uses rounding to the first decimal place.

df1 <- data.frame(regulator = "EMA", CV = c(30, 50),
L = NA, U = NA,
cap = c("lower", "upper"))
df2 <- data.frame(regulator = "HC",
CV = c(30, 57.382),
L = NA, U = NA,
cap = c("lower", "upper"))
for (i in 1:2) {
df1[i, 3:4] <- sprintf("%.2f%%", 100*scABEL(df1$CV[i]/100)) df2[i, 3:4] <- sprintf("%.1f %%", 100*scABEL(df2$CV[i]/100,
regulator = "HC"))
}
df1$CV <- sprintf("%.3f%%", df1$CV)
df2$CV <- sprintf("%.3f%%", df2$CV)
if (packageVersion("PowerTOST") >= "1.5.3") {
df3 <- data.frame(regulator = "GCC <2022",
CV = c(30, 50), L = NA, U = NA,
cap = c("lower", "none"))
for (i in 1:2) {
df3[i, 3:4] <- sprintf("%.2f%%", 100*scABEL(df3$CV[i]/100, regulator = "GCC")) } df3$CV <- sprintf("%.3f%%", df3$CV) } if (packageVersion("PowerTOST") >= "1.5-3") { print(df1, row.names = F); print(df2, row.names = F); print(df3, row.names = F) } else { print(df1, row.names = F); print(df2, row.names = F) } # regulator CV L U cap # EMA 30.000% 80.00% 125.00% lower # EMA 50.000% 69.84% 143.19% upper # regulator CV L U cap # HC 30.000% 80.0 % 125.0 % lower # HC 57.382% 66.7 % 150.0 % upper # regulator CV L U cap # GCC <2022 30.000% 80.00% 125.00% lower # GCC <2022 50.000% 75.00% 133.33% none The sample size functions of PowerTOST use a modification of Zhang’s method46 for the first guess. # Note that theta0 = 0.90 is the default sampleN.scABEL(CV = 0.45, targetpower = 0.80, design = "2x2x4", details = TRUE) # # +++++++++++ scaled (widened) ABEL +++++++++++ # Sample size estimation # (simulation based on ANOVA evaluation) # --------------------------------------------- # 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) = 0.45; 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 # 24 0.7539 # 26 0.7846 # 28 0.8112 An alternative to simulating the ‘key statistics’ is by subject data simulations via the function sampleN.scABEL.sdsims(). However, it comes with a price, speed. # method n power rel.speed # ‘key statistics’ 28 0.81116 1.0 # subject simulations 28 0.81196 21.5 # Examples Throughout the examples I’m referring to studies in a single center – not multiple groups within them or multicenter studies. That’s another pot of tea. ## A Simple Case We assume a CV of 0.45, a T/R-ratio of 0.90, a target a power of 0.80, and want to perform the study in a two sequence four period full replicate study (TRTR|RTRT or TRRT|RTTR or TTRR|RRTT) for the EMA’s ABEL. Since theta0 = 0.90,47 targetpower = 0.80, and regulator = "EMA" are defaults of the function, we don’t have to give them explicitely. As usual in bioequivalence, alpha = 0.05 is employed (we will assess the study by a $$\small{100\,(1-2\,\alpha)=90\%}$$ confidence interval). Hence, you need to specify only the CV (assuming $$\small{CV_\text{wT}=CV_\text{wR}}$$) and design = "2x2x4". To shorten the output, use the argument details = FALSE. sampleN.scABEL(CV = 0.45, design = "2x2x4", details = FALSE) # # +++++++++++ scaled (widened) ABEL +++++++++++ # Sample size estimation # (simulation based on ANOVA evaluation) # --------------------------------------------- # 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) = 0.45; CVw(R) = 0.45 # True ratio = 0.9 # ABE limits / PE constraint = 0.8 ... 1.25 # Regulatory settings: EMA # # Sample size # n power # 28 0.8112 Sometimes we are not interested in the entire output and want to use only a part of the results in subsequent calculations. We can suppress the output by stating the additional argument print = FALSE and assign the result to a data frame (here x). x <- sampleN.scABEL(CV = 0.45, design = "2x2x4", details = FALSE, print = FALSE) Let’s retrieve the column names of x: names(x) # [1] "Design" "alpha" "CVwT" # [4] "CVwR" "theta0" "theta1" # [7] "theta2" "Sample size" "Achieved power" # [10] "Target power" "nlast" Now we can access the elements of x by their names. Note that double square brackets and double or single quotes ([["…"]], [['…']]) have to be used. x[["Sample size"]] # [1] 28 x[['Achieved power']] # [1] 0.81116 Although you could access the elements by the number of the column(s), I don’t recommend that, since in various other functions of PowerTOST these numbers are different and hence, difficult to remember. If you insist in accessing elements by column-numbers, use single square brackets […]. x[8:9] # Sample size Achieved power # 1 28 0.81116 With 28 subjects (14 per sequence) we achieve the power we desire. What happens if we have one dropout? power.scABEL(CV = 0.45, design = "2x2x4", n = x[["Sample size"]] - 1) # Unbalanced design. n(i)=14/13 assumed. # [1] 0.79848 Below the 0.80 we desire. Since dropouts are common, it makes sense to include / dose more subjects in order to end up with a number of eligible subjects which is not lower than our initial estimate. Let us explore that in the next section. ## Dropouts We define two supportive functions: 1. Provide equally sized sequences, i.e., any total sample size n will be rounded up to achieve balance. balance <- function(n, n.seq) { return(as.integer(n.seq * (n %/% n.seq + as.logical(n %% n.seq)))) } 1. Provide the adjusted sample size based on the original sample size n and the anticipated droput-rate do.rate. nadj <- function(n, do.rate, n.seq) { return(as.integer(balance(n / (1 - do.rate), n.seq))) } In order to come up with a suggestion we have to anticipate a (realistic!) dropout rate. Note that this not the job of the statistician; ask the Principal Investigator. It is a capital mistake to theorise before one has data. Arthur Conan Doyle (1892)48 ### Dropout-rate The dropout-rate is calculated from the eligible and dosed subjects or simply $$$\tag{3} do.rate=1-n_\text{eligible}/n_\text{dosed}$$$ Of course, we know it only after the study was performed. By substituting $$n_\text{eligible}$$ with the estimated sample size $$n$$, providing an anticipated dropout-rate and rearrangement to find the adjusted number of dosed subjects $$n_\text{adj}$$ we should use $$$\tag{4} n_\text{adj}=\;\upharpoonleft n\,/\,(1-do.rate)$$$ where $$\upharpoonleft$$ denotes rounding up to the complete sequence as implemented in the functions above. An all too common mistake is to increase the estimated sample size $$n$$ by the drop­out-rate according to $$$\tag{5} n_\text{adj}=\;\upharpoonleft n\times(1+do.rate)$$$ If you used $$\small{(5)}$$ in the past – you are not alone. In a small survey a whopping 29% of respondents reported to use it.49 Consider changing your routine. There are no routine statistical questions, only questionable statistical routines. ### Adjusted Sample Size In the following I specified more arguments to make the function more flexible. Note that I wrapped the function power.scABEL() in suppressMessages(). Otherwise, the function will throw for any odd sample size a message telling us that the design is unbalanced. Well, we know that. CV <- 0.45 # within-subject CV target <- 0.80 # target (desired) power theta0 <- 0.90 # assumed T/R-ratio design <- "2x2x4" do.rate <- 0.15 # anticipated dropout-rate 15% # might be realively high due # to the 4 periods n.seq <- as.integer(substr(design, 3, 3)) lims <- scABEL(CV) # expanded limits df <- sampleN.scABEL(CV = CV, theta0 = theta0, targetpower = target, design = design, details = FALSE, print = FALSE) # calculate the adjusted sample size n.adj <- nadj(df[["Sample size"]], do.rate, n.seq) # (decreasing) vector of eligible subjects n.elig <- n.adj:df[["Sample size"]] info <- paste0("Assumed CV : ", CV, "\nAssumed T/R ratio : ", theta0, "\nExpanded limits : ", sprintf("%.4f\u2026%.4f", lims[1], lims[2]), "\nPE constraints : ", sprintf("%.4f\u2026%.4f", 0.80, 1.25), # fixed in ABEL "\nTarget (desired) power : ", target, "\nAnticipated dropout-rate: ", do.rate, "\nEstimated sample size : ", df[["Sample size"]], " (", df[["Sample size"]]/n.seq, "/sequence)", "\nAchieved power : ", signif(df[["Achieved power"]], 4), "\nAdjusted sample size : ", n.adj, " (", n.adj/n.seq, "/sequence)", "\n\n") # explore the potential outcome for # an increasing number of dropouts do.act <- signif((n.adj - n.elig) / n.adj, 4) df <- data.frame(dosed = n.adj, eligible = n.elig, dropouts = n.adj - n.elig, do.act = do.act, power = NA) for (i in 1:nrow(df)) { df$power[i] <- suppressMessages(
power.scABEL(CV = CV,
theta0 = theta0,
design = design,
n = dfeligible[i])) } cat(info) print(round(df, 4), row.names = FALSE) # Assumed CV : 0.45 # Assumed T/R ratio : 0.9 # Expanded limits : 0.7215…1.3859 # PE constraints : 0.8000…1.2500 # Target (desired) power : 0.8 # Anticipated dropout-rate: 0.15 # Estimated sample size : 28 (14/sequence) # Achieved power : 0.8112 # Adjusted sample size : 34 (17/sequence) # # dosed eligible dropouts do.act power # 34 34 0 0.0000 0.8720 # 34 33 1 0.0294 0.8630 # 34 32 2 0.0588 0.8553 # 34 31 3 0.0882 0.8456 # 34 30 4 0.1176 0.8340 # 34 29 5 0.1471 0.8237 # 34 28 6 0.1765 0.8112 In the worst case (six dropouts) we end up with the originally estimated sample size of 28. Power preserved, mission accomplished. If we have less dropouts, splendid – we gain power. If we would have adjusted the sample size according to $$\small{(5)}$$ we would have dosed also 34 subjects. Cave: This might not always be the case… If the anticipated dropout rate of 15% is realized in the study, we would have also 28 eligible subjects (power 0.8112). In this example we achieve still more than our target power but the loss might be relevant in other cases. ### Post hoc Power As said in the preliminaries, calculating post hoc power is futile. There is simple intuition behind results like these: If my car made it to the top of the hill, then it is powerful enough to climb that hill; if it didn’t, then it obviously isn’t powerful enough. Retrospective power is an obvious answer to a rather uninteresting question. A more meaningful question is to ask whether the car is powerful enough to climb a particular hill never climbed before; or whether a different car can climb that new hill. Such questions are prospective, not retrospective. Russell V. Lenth (2000)50 However, sometimes we are interested in it for planning the next study. If you give and odd total sample size n, power.scABEL() will try to keep sequences as balanced as possible and show in a message how that was done. n.act <- 27 signif(power.scABEL(CV = 0.45, n = n.act, design = "2x2x4"), 6) # Unbalanced design. n(i)=14/13 assumed. # [1] 0.79848 Say, our study was more unbalanced. Let us assume that we dosed 34 subjects, the total number of subjects was also 27 but all drop­outs occured in one sequence (unlikely but possible). Instead of the total sample size n we can give the number of subjects of each sequence as a vector (the order is generally51 not relevant, i.e., it does not matter which element refers to which sequence). By setting details = TRUE we can retrieve the components of the simulations (probability to pass each test). design <- "2x2x4" CV <- 0.45 n.adj <- 34 n.act <- 27 n.s1 <- n.adj / 2 n.s2 <- n.act - n.s1 theta0 <- 0.90 post.hoc <- suppressMessages( power.scABEL(CV = CV, n = c(n.s1, n.s2), theta0 = theta0, design = design, details = TRUE)) ABE.xact <- power.TOST(CV = CV, n = c(n.s1, n.s2), theta0 = theta0, design = design) sig.dig <- nchar(as.character(n.adj)) fmt <- paste0("%", sig.dig, ".0f (%", sig.dig, ".0f dropouts)") cat(paste0("Dosed subjects: ", sprintf("%2.0f", n.adj), "\nEligible : ", sprintf(fmt, n.act, n.adj - n.act), "\n Sequence 1 : ", sprintf(fmt, n.s1, n.adj / 2 - n.s1), "\n Sequence 1 : ", sprintf(fmt, n.s2, n.adj / 2 - n.s2), "\nPower overall : ", sprintf("%.5f", post.hoc[1]), "\n p(ABEL) : ", sprintf("%.5f", post.hoc[2]), "\n p(PE) : ", sprintf("%.5f", post.hoc[3]), "\n p(ABE) : ", sprintf("%.5f", post.hoc[4]), "\n p(ABE) exact: ", sprintf("%.5f", ABE.xact), "\n")) # Dosed subjects: 34 # Eligible : 27 ( 7 dropouts) # Sequence 1 : 17 ( 0 dropouts) # Sequence 1 : 10 ( 7 dropouts) # Power overall : 0.77670 # p(ABEL) : 0.77671 # p(PE) : 0.91595 # p(ABE) : 0.37628 # p(ABE) exact: 0.37418 The components of overall power are: • p(ABEL) is the probability that the confidence interval is within the expanded / widened limits. • p(PE) is the probability that the point estimate is within 80.00–125.00%. • p(ABE) is the probability of passing conventional Average Bioequivalence. The line below gives the exact result obtained by power.TOST() – confirming the simulation’s result. Of course, in a particular study you will provide the numbers in the n vector directly. ## Lost in Assumptions The CV and the T/R-ratio are only assumptions. Whatever their origin might be (literature, previous studies), they carry some degree of uncertainty. Hence, believing52 that they are the true ones may be risky. Some statisticians call that the ‘Carved-in-Stone’ approach. Say, we performed a pilot study in 16 subjects and estimated the CV as 0.45. The $$\small{\alpha}$$ confidence interval of the CV is obtained via the $$\small{\chi^2}$$-distribution of its error variance $$\small{\sigma^2}$$ with $$\small{n-2}$$ degrees of freedom. \eqalign{\tag{6} s_\text{w}^2&=\log_{e}(CV_\text{w}^2+1)\\ L=\frac{(n-1)\,s_\text{w}^2}{\chi_{\alpha/2,\,n-2}^{2}}&\leq\sigma_\text{w}^2\leq\frac{(n-1)\,s_\text{w}^2}{\chi_{1-\alpha/2,\,n-2}^{2}}=U\\ \left\{lower\;CL,\;upper\;CL\right\}&=\left\{\sqrt{\exp(L)-1},\sqrt{\exp(U)-1}\right\} } Let’s calculate the 95% confidence interval of the CV to get an idea. m <- 16 # pilot study ci <- CVCL(CV = 0.45, df = m - 2, side = "2-sided", alpha = 0.05) ci <- c(ci, mean(ci)) names(ci)[3] <- "midpoint" signif(ci, 4) # lower CL upper CL midpoint # 0.3223 0.7629 0.5426 Surprised? Although 0.45 is the best estimate for planning the next study, there is no guarantee that we will get exactly the same outcome. Since the $$\small{\chi^2}$$-distribution is skewed to the right, it is more likely that we will face a higher CV than a lower one in the planned study. If we plan the study based on 0.45, we would opt for 28 subjects like in the examples before (not adjusted for the dropout-rate). If the CV will be lower, we loose power (less expansion). But what if it will be higher? Depends. Since we may expand more, we gain power. However, if we cross the upper cap of scaling (50% for the EMA), we will loose power. But how much? Let’s explore what might happen at the confidence limits of the CV. m <- 16 ci <- CVCL(CV = 0.45, df = m - 2, side = "2-sided", alpha = 0.05) n <- 28 comp <- data.frame(CV = c(ci[["lower CL"]], 0.45, mean(ci), ci[["upper CL"]]), power = NA) for (i in 1:nrow(comp)) { comppower[i] <- power.scABEL(CV = comp$CV[i], design = "2x2x4", n = n) } comp[, 1] <- signif(comp[, 1], 4) comp[, 2] <- signif(comp[, 2], 6) print(comp, row.names = FALSE) # CV power # 0.3223 0.73551 # 0.4500 0.81116 # 0.5426 0.79868 # 0.7629 0.60158 Might hurt. What can we do? The larger the previous study was, the larger the degrees of freedom and hence, the narrower the confidence interval of the CV. In simple terms: The estimate is more certain. On the other hand, it also means that very small pilot studies are practically useless. What happens when we plan the study based on the confidence interval of the CV? m <- seq(12, 30, 6) df <- data.frame(n.pilot = m, CV = 0.45, l = NA, u = NA, n.low = NA, n.CV = NA, n.hi = NA) for (i in 1:nrow(df)) { df[i, 3:4] <- CVCL(CV = 0.45, df = m[i] - 2, side = "2-sided", alpha = 0.05) df[i, 5] <- sampleN.scABEL(CV = df$l[i], design = "2x2x4",
details = FALSE,
print = FALSE)[["Sample size"]]
df[i, 6] <- sampleN.scABEL(CV = 0.45, design = "2x2x4",
details = FALSE,
print = FALSE)[["Sample size"]]
df[i, 7] <- sampleN.scABEL(CV = df$u[i], design = "2x2x4", details = FALSE, print = FALSE)[["Sample size"]] } df[, 3:4] <- signif(df[, 3:4], 4) names(df)[3:4] <- c("lower CL", "upper CL") print(df, row.names = FALSE) # n.pilot CV lower CL upper CL n.low n.CV n.hi # 12 0.45 0.3069 0.8744 36 28 56 # 18 0.45 0.3282 0.7300 34 28 42 # 24 0.45 0.3415 0.6685 34 28 38 # 30 0.45 0.3509 0.6334 34 28 34 One leading generic company has an internal rule to perform pilot studies of HVDPs in a four period full replicate design and at least 24 subjects. Makes sense. Furthermore, we don’t know where the true T/R-ratio lies but we can calculate the lower 95% confidence limit of the pilot study’s point estimate to get an idea about a worst case. Say, it was 0.90. m <- 16 CV <- 0.45 pe <- 0.90 ci <- round(CI.BE(CV = CV, pe = 0.90, n = m, design = "2x2x4"), 4) if (pe <= 1) { cl <- ci[["lower"]] } else { cl <- ci[["upper"]] } print(cl) # [1] 0.7515 Explore the impact of a relatively 5% lower CV (less expansion) and a relatively 5% lower T/R-ratio on power for the given sample size. n <- 28 CV <- 0.45 theta0 <- 0.90 comp1 <- data.frame(CV = c(CV, CV*0.95), power = NA) comp2 <- data.frame(theta0 = c(theta0, theta0*0.95), power = NA) for (i in 1:2) { comp1$power[i] <- power.scABEL(CV = comp1$CV[i], theta0 = theta0, design = "2x2x4", n = n) } comp1$power <- signif(comp1$power, 5) for (i in 1:2) { comp2$power[i] <- power.scABEL(CV = CV,
theta0 = comp2$theta0[i], design = "2x2x4", n = n) } comp2$power <- signif(comp2$power, 5) print(comp1, row.names = FALSE) print(comp2, row.names = FALSE) # CV power # 0.4500 0.81116 # 0.4275 0.80095 # theta0 power # 0.900 0.81116 # 0.855 0.61952 Interlude 2 Note the log-scale of the x-axis. It demonstrates that power curves are symmetrical around 1 ($$\small{\log_{e}(1)=0}$$, where $$\small{\log_{e}(\theta_2)=\left|\log_{e}(\theta_1)\right|}$$) and we will achieve the same power for $$\small{\theta_0}$$ and $$\small{1/\theta_0}$$ (e.g., for 0.90 and 1.1111). Contrary to ABE, power is maintained, unless we cross the upper scaling limit, where additionally the PE-constraint becomes increasingly important. <nitpick> • A common flaw in protocols is the phrase »The sample size was calculated [sic] based on a T/R-ratio of 0.90 – 1.10« If you assume a deviation of 10% of the test from the reference and are not sure about its direction (lower or higher than 1), always use the lower T/R-ratio. If you would use the upper T/R-ratio, power would be only preserved down to 1/1.10 = 0.9091. Given, sometimes you will need a higher sample size with the lower T/R-ratio. There’s no free lunch. CV <- 0.45 delta <- 0.10 # direction unknown design <- "2x2x4" theta0s <- c(1 - delta, 1 / (1 + delta), 1 + delta, 1 / (1 - delta)) n <- sampleN.scABEL(CV = CV, theta0 = 1 - delta, design = design, details = FALSE, print = FALSE)[["Sample size"]] comp1 <- data.frame(CV = CV, theta0 = theta0s, base = c(TRUE, rep(FALSE, 3)), n = n, power = NA) for (i in 1:nrow(comp1)) { comp1$power[i] <- power.scABEL(CV = CV,
theta0 = comp1$theta0[i], design = design, n = n) } n <- sampleN.scABEL(CV = CV, theta0 = 1 + delta, design = design, details = FALSE, print = FALSE)[["Sample size"]] comp2 <- data.frame(CV = CV, theta0 = theta0s, base = c(FALSE, FALSE, TRUE, FALSE), n = n, power = NA) for (i in 1:nrow(comp2)) { comp2$power[i] <- power.scABEL(CV = CV,
theta0 = comp2$theta0[i], design = design, n = n) } comp1[, c(2, 5)] <- signif(comp1[, c(2, 5)] , 4) comp2[, c(2, 5)] <- signif(comp2[, c(2, 5)] , 4) print(comp1, row.names = FALSE) print(comp2, row.names = FALSE) # CV theta0 base n power # 0.45 0.9000 TRUE 28 0.8112 # 0.45 0.9091 FALSE 28 0.8388 # 0.45 1.1000 FALSE 28 0.8397 # 0.45 1.1110 FALSE 28 0.8101 # CV theta0 base n power # 0.45 0.9000 FALSE 26 0.7846 # 0.45 0.9091 FALSE 26 0.8149 # 0.45 1.1000 TRUE 26 0.8140 # 0.45 1.1110 FALSE 26 0.7837 </nitpick> Essentially this leads to the murky waters of prospective sensitivity analyses, which is covered in another article. An appetizer to show the maximum deviations (CV, T/R-ratio and decreased sample size due to dropouts) which give still a minimum acceptable power of ≥ 0.70: CV <- 0.45 theta0 <- 0.90 target <- 0.80 minpower <- 0.70 pa <- pa.scABE(CV = CV, theta0 = theta0, targetpower = target, minpower = minpower, design = "2x2x4") change.CV <- 100*(tail(pa$paCV[["CV"]], 1) -
pa$plan[["CVwR"]]) / pa$plan[["CVwR"]]
change.theta0 <- 100*(head(pa$paGMR$theta0, 1) -
pa$plan$theta0) /
pa$plan[["theta0"]] change.n <- 100*(tail(pa$paN[["N"]], 1) -
pa$plan[["Sample size"]]) / pa$plan[["Sample size"]]
comp     <- data.frame(parameter = c("CV", "theta0", "n"),
change = c(change.CV,
change.theta0,
change.n))
comp$change <- sprintf("%+.2f%%", comp$change)
names(comp)[2] <- "rel. change"
print(pa, plotit = FALSE)
print(comp, row.names = FALSE)
# Sample size plan scABE (EMA/ABEL)
#  Design alpha CVwT CVwR theta0 theta1 theta2 Sample size
#   2x2x4  0.05 0.45 0.45    0.9    0.8   1.25          28
#  Achieved power Target power
#         0.81116          0.8
#
# Power analysis
# CV, theta0 and number of subjects leading to min. acceptable power of ~0.7:
#  CV= 0.6629, theta0= 0.8719
#  n = 22 (power= 0.7185)
#
#  parameter rel. change
#         CV     +47.32%
#     theta0      -3.12%
#          n     -21.43%

Confirms what we have seen above. As expect the method is robust to changes of the CV. The sample size is also not very sensitive; many overrate the impact of dropouts on power.

## Heteroscedasticity

As we have seen already above, for an ANOVA we have to assume homoscedasticity.

I recommend to perform pilot studies in one of the fully replicated designs. When you are concerned about drop­outs in a four period design or the bioanalytical method requires large sample volumes, opt for one the two sequence three period designs (TRT|RTR or TRR|RTT).
Contrary to the partial replicate design (TRR|RTR|RRT) we get estimates of both $$\small{CV_\text{wT}}$$ and $$\small{CV_\text{wR}}$$. Since pharmaceutical technology improves, it is not uncommon that $$\small{CV_\text{wT}<CV_\text{wR}}$$. If this is the case, we get an incentive in the sample size $$\small{n}$$ of the planned pivotal study (expanding the limits is based on $$\small{CV_\text{wR}}$$ but the 90% CI on the – pooled – $$\small{s_\text{w}^{2}}$$).

\eqalign{\tag{7} s_\text{wT}^{2}&=\log_{e}(CV_\text{wT}^{2}+1)\\ s_\text{wR}^{2}&=\log_{e}(CV_\text{wR}^{2}+1)\\ s_\text{w}^{2}&=\left(s_\text{wT}^{2}+s_\text{wR}^{2}\right)/2\\ CV_\text{w}&=\sqrt{\exp(s_\text{w}^{2})-1}}

Say, we performed two pilot studies. In the partial replicate we estimated the $$\small{CV_\text{wR}}$$ with 0.45 (estimation of $$\small{CV_\text{wT}}$$ is not possible). In the full replicate we estimated $$\small{CV_\text{wR}}$$ with 0.484 and $$\small{CV_\text{wT}}$$ with 0.414. Note that the $$\small{CV_\text{w}}$$ according to $$\small{(7)}$$ is 0.45 as well. How will that impact the sample size $$\small{n}$$ of the pivotal 4-period full replicate design?

comp <- data.frame(pilot = c("TRR|RTR|RRT", "TRT|RTR"),
CVwT = c(0.45, 0.414),
CVwR = c(0.45, 0.484),
CVw = NA,
n = NA, power = NA)
for (i in 1:nrow(comp)) {
comp[i, 4]   <- signif(
mse2CV((CV2mse(comp$CVwT[i]) + CV2mse(comp$CVwR[i])) / 2), 3)
comp[i, 5:6] <- sampleN.scABEL(CV = c(comp$CVwT[i], comp$CVwR[i]),
design = "2x2x4", details = FALSE,
print = FALSE)[8:9]
}
print(comp, row.names = FALSE)
#        pilot  CVwT  CVwR  CVw  n   power
#  TRR|RTR|RRT 0.450 0.450 0.45 28 0.81116
#      TRT|RTR 0.414 0.484 0.45 24 0.80193

Since bioanalytics drives study costs to a great extent, we may save ~14%.

Note that when you give CV as two-element vector, the first element has to be $$\small{CV_\text{wT}}$$ and the second $$\small{CV_\text{wR}}$$.

Although according to the guidelines it is not required to estimate $$\small{CV_\text{wT}}$$, its value is ‘nice to know’. Sometimes studies fail only due to the large $$\small{CV_\text{wR}}$$ thus inflating the confidence interval. In such a case you have at least ammunation to start an argument.

Even if you plan the pivotal study in a partial replicate design (why on earth?) knowing both $$\small{CV_\text{wT}}$$ and $$\small{CV_\text{wR}}$$ is useful.

comp <- data.frame(pilot = c("TRR|RTR|RRT", "TRT|RTR"),
CVwT = c(0.45, 0.414),
CVwR = c(0.45, 0.484),
CVw = NA,
n = NA, power = NA)
for (i in 1:nrow(comp)) {
comp[i, 4]   <- signif(
mse2CV((CV2mse(comp$CVwT[i]) + CV2mse(comp$CVwR[i])) / 2), 3)
comp[i, 5:6] <- sampleN.scABEL(CV = c(comp$CVwT[i], comp$CVwR[i]),
design = "2x3x3", details = FALSE,
print = FALSE)[8:9]
}
print(comp, row.names = FALSE)
#        pilot  CVwT  CVwR  CVw  n   power
#  TRR|RTR|RRT 0.450 0.450 0.45 39 0.80588
#      TRT|RTR 0.414 0.484 0.45 36 0.80973

Again, a smaller sample size is possible and we may still save ~7.7%.

Note that sampleN.scABEL() is inaccurate for the partial replicate design if and only if $$\small{CV_\text{wT}>CV_\text{wR}}$$. Let’s reverse the values and compare the results with subject simulations by the function sampleN.scABEL.sdsims().

comp <- data.frame(method = c("key", "subj"), n = NA,
power = NA, rel.speed = NA)
st   <- proc.time()[[3]]
comp[1, 2:3] <- sampleN.scABEL(CV = c(0.484, 0.414),
design = "2x3x3",
details = FALSE,
print = FALSE)[8:9]
et   <- proc.time()[[3]]
comp$rel.speed[1] <- et - st st <- proc.time()[[3]] comp[2, 2:3] <- sampleN.scABEL.sdsims(CV = c(0.484, 0.414), design = "2x3x3", details = FALSE, print = FALSE)[8:9] et <- proc.time()[[3]] comp$rel.speed[2] <- et - st
comp$rel.speed <- signif(comp$rel.speed /
comp$rel.speed[1], 3) comp[1] <- c("\u2018key statistics\u2019", "subject simulations") print(comp, row.names = FALSE) # method n power rel.speed # ‘key statistics’ 45 0.80212 1.0 # subject simulations 48 0.80938 84.6 In such a case, always use the function sampleN.scABEL.sdsims(). ## Multiple Endpoints For demonstrating bioequivalence the pharmacokinetic metrics Cmax and AUC0–t are mandatory (in some jurisdictions like the FDA additionally AUC0–∞). We don’t have to worry about multiplicity issues (inflated Type I Error), since if all tests must pass at level $$\alpha$$, we are protected by the intersection-union principle.53 54 We design the study always for the worst case combination, i.e., based on the PK metric requiring the largest sample size. In most jurisdictions wider BE limits are acceptable only for Cmax. Let’s explore that with different CVs and T/R-ratios. metrics <- c("Cmax", "AUC") methods <- c("ABEL", "ABE") CV <- c(0.45, 0.35) design <- "2x2x4" theta0 <- c(0.90, 0.925) df <- data.frame(metric = metrics, method = methods, CV = CV, theta0 = theta0, n = NA, power = NA) df[1, 5:6] <- sampleN.scABEL(CV = CV[1], theta0 = theta0[1], design = design, details = FALSE, print = FALSE)[8:9] df[2, 5:6] <- sampleN.TOST(CV = CV[2], theta0 = theta0[2], design = design, print = FALSE)[7:8] df$power <- signif(df$power, 5) txt <- paste0("Sample size is driven by ", df$metric[df$n == max(df$n)], ".\n")
print(df, row.names = FALSE); cat(txt)
#  metric method   CV theta0  n   power
#    Cmax   ABEL 0.45  0.900 28 0.81116
#     AUC    ABE 0.35  0.925 36 0.81604
# Sample size is driven by AUC.

Commonly the PK metric evaluated by ABE drives the sample size. Consequently, with 36 subjects the study will be ‘overpowered’ (~0.886) for Cmax.

Let us assume the same T/R-ratios for both metrics. Which are the extreme T/R-ratios (largest deviations of T from R) for Cmax giving still the target power?

opt <- function(x) {
power.scABEL(theta0 = x, CV = df$CV[1], design = design, n = df$n[2]) - target
}
metrics <- c("Cmax", "AUC")
methods <- c("ABEL", "ABE")
CV      <- c(0.45, 0.30)
theta0  <- c(0.90, 0.90)
design  <- "2x2x4" # must be a replicate
target  <- 0.80
if (!design %in% c("2x2x4", "2x3x3", "2x2x3"))
stop("design = \"", design, "\" is not supported.")
df      <- data.frame(metric = metrics, method = methods,
CV = CV, theta0 = theta0, n = NA,
power = NA)
df[1, 5:6] <- sampleN.scABEL(CV = CV[1], theta0 = theta0[1],
design = design, details = FALSE,
print = FALSE)[8:9]
df[2, 5:6] <- sampleN.TOST(CV = CV[1], theta0 = theta0[2],
design = design, print = FALSE)[7:8]
df$power <- signif(df$power, 5)
if (theta0[1] < 1) {
res <- uniroot(opt, tol = 1e-8,
interval = c(0.80 + 1e-4, theta0[1]))
} else {
res <- uniroot(opt, tol = 1e-8,
interval = c(theta0[1], 1.25 - 1e-4))
}
res     <- unlist(res)
theta0s <- c(res[["root"]], 1/res[["root"]])
txt     <- paste0("Target power for ", metrics[1],
" and sample size ",
df$n[2], "\nachieved for theta0 ", sprintf("%.4f", theta0s[1]), " or ", sprintf("%.4f", theta0s[2]), ".\n") print(df, row.names = FALSE); cat(txt) # metric method CV theta0 n power # Cmax ABEL 0.45 0.9 28 0.81116 # AUC ABE 0.30 0.9 84 0.80569 # Target power for Cmax and sample size 84 # achieved for theta0 0.8340 or 1.1990. That means, although we assumed for Cmax the same T/R-ratio as for AUC– with the sample size of 84 required AUC, for Cmax it can be as low as 0.834 or as high as 1.199, which is a soothing side-effect. For HVD(P)s Health Canada accepts ABEL only for AUC, whereas for Cmax the T/R-ratio has to lie within 80.0 – 125.0% (i.e., without assessing its CI). metrics <- c("Cmax", "AUC") methods <- c("PE", "ABE(L)") alpha <- c(0.50, 0.05) CV <- c(0.65, 0.35) theta0 <- c(0.90, 0.90) design <- "2x2x4" # must be a replicate target <- 0.90 if (!design %in% c("2x2x4", "2x3x3", "2x2x3")) stop("design = \"", design, "\" is not supported.") df <- data.frame(design = design, metric = metrics, method = methods, CV = CV, theta0 = theta0, n = NA, power = NA) # alpha = 0.5 leads to a zero-width confidence # interval, i.e., only the T/R-ratio is relevant df[1, 6] <- sampleN.TOST(alpha = alpha[1], CV = CV[1], theta0 = theta0[1], design = design, targetpower = target, print = FALSE)[["Sample size"]] if (df[1, 6] < 12) df[1, 6] <- 12 # acc. to the guidance df[1, 7] <- power.TOST(alpha = alpha[1], CV = CV[1], theta0 = theta0[1], design = design, n = df[1, 6]) df[2, 6] <- sampleN.scABEL(alpha = alpha[2], CV = CV[2], theta0 = theta0[2], design = design, regulator = "HC", targetpower = target, details = FALSE, print = FALSE)[["Sample size"]] if (df[2, 6] < 12) df[2, 6] <- 12 # acc. to the guidance df[2, 7] <- power.scABEL(alpha = alpha[2], CV = CV[2], theta0 = theta0[2], regulator = "HC", design = design, n = df[2, 6]) driving <- which(df$n == max(df$n)) easy <- which(df$n != max(df$n)) if (driving == 1) { easy.pwr <- power.TOST(alpha = alpha[easy], CV = CV[easy], theta0 = theta0[easy], design = design, n = df$n[driving])
} else {
easy.pwr <- power.scABEL(alpha = alpha[easy],
CV = CV[easy],
theta0 = theta0[easy],
regulator = "HC",
design = design,
n = df$n[driving]) } df$power <- signif(df$power, 5) names(df)[5] <- "T/R" txt <- paste0(sprintf("Sample size is driven by %s.\n", df$metric[driving]),
sprintf("With a sample size of %.0f power for %s = %.5f.\n",
df\$n[driving], metrics[easy], easy.pwr))
print(df, row.names = FALSE); cat(txt)
#  design metric method   CV T/R  n   power
#   2x2x4   Cmax     PE 0.65 0.9 42 0.90058
#   2x2x4    AUC ABE(L) 0.35 0.9 50 0.90897
# Sample size is driven by AUC.
# With a sample size of 50 power for Cmax = 0.91980.

Even for a CV of Cmax which is substantially higher than the one of AUC, the latter still drives the sample size.

# Q & A

• Q: Can we use R in a regulated environment and is PowerTOST validated?
A: See this document55 about the acceptability of Base R and its Software Development Life Cycle.56
R is updated every couple of months with documented changes57 and maintaining a bug-tracking system.58 I recommed to use always the latest release.
The authors of PowerTOST tried to do their best to provide reliable and valid results. Its ‘NEWS’ do­cum­ents the development of the package, bug-fixes, and introduction of new methods. Issues are tracked at Git­Hub (as of today none is open). So far the package had >118,000 downloads. Therefore, it is extremely un­like­ly that bugs were not detected given its large user base.
Validation of any software (yes, of SAS as well…) lies in the hands of the user.59 60 61 62
Execute the script test_ABEL.R which can be found in the /tests sub-directory of the package to reproduce tables given in the literature.63 You will notice some discrepancies: The authors employed only 10,000 simu­la­tions – which is not sufficient for a stable result (see below). Furthermore, they reported the minimum sample size which gives at least the target power, whereas sampleN.scABEL always rounds up to give balanced sequences.

• Q: Shall we throw away our sample size tables?
A: Not at all. File them in your archives to collect dust. Maybe in the future you will be asked by an agency how you arrived at a sample size. But: Don’t use them any more. What you should not do (and hopefully haven’t done before): Interpolate. Power and therefore, the sample size depends in a highly nonlinear fashion on the five conditions listed above, which makes interpolation of values given in a table a nontrivial job.

• Q: Which of the methods should we use in our daily practice?
A: sampleN.scABEL()/power.scABEL() for speed reasons. Only for the partial replicate designs and the – rare – case of CVwT > CVwR, use sampleN.scABEL.sdsims()/power.scABEL.sdsims() instead.

• Q: I fail to understand your example about dropouts. We finish the study with 28 eligible subjects as desired. Why is the dropout-rate ~18% and not the anticipated 15%?
A: That’s due to rounding up the calculated adjusted sample size (32.94…) to the next even number (34).
If you manage it to dose fractional subjects (I can’t), your dropout rate would indeed equal the anticipated one: 100 × (1 – 28/32.94…) = 15%.

• Q: Do we have to worry about unbalanced sequences?
A: sampleN.scABEL()/sampleN.scABEL.sdsims() will always give the total number of subjects for balanced sequences.
If you are interested in – irrelevant – post hoc power,64 give the sample size as a vector, i.e., power.scABEL(..., n = c(foo, bar, baz), where foo, bar, and baz are the number of subjects per sequence.

• Q: The default number of simulations in the sample size estimation is 100,000. Why?
A: We found that with this number the simulations are stable. For the background see another article. Of course, you can give a larger number in the argument nsims. However, you shouldn’t decrease the number.

• Q: How reliable are the results?
A: As stated above an exact method doesn’t – and will never – exist. We can only compare the empiric power of the ABE-component to the exact one obtained by power.TOST(). For an example see ‘Post hoc Power’ in the section about Dropouts.

A: I checked all URLs in July 2024. Contrary to us mere mortals who have to maintain a version control of documents, agen­cies don’t care. They change the structure of their websites (worst are the ones of the AN­VI­SA and the WHO), don’t establish auto­ma­tic re­di­rects, rename or even delete files… Quod licet Iovi, non licet bovi
If you discover an error, please drop me a note at [email protected]..

• Q: I still have questions. How to proceed?
A: The preferred method is to register at the BEBA Forum and post your question in the category  or   (please read the Forum’s Policy first).
You can contact me at [email protected]. Be warned – I will charge you for anything beyond most basic questions.

### Abbreviations

Abbreviation Meaning
(A)BE (Average) Bioequivalence
ABEL Average Bioequivalence with Expanding Limits
ASEAN Association of Southeast Asian Nations
CI $$\small{(1-2\,\alpha)}$$ Confidence Interval
CL Confidence Limit
$$\small{CV_\text{b}}$$ Between-subject Coefficient of Variation
$$\small{CV_\text{w}}$$ (Pooled) within-subject Coefficient of Variation
$$\small{CV_\text{wR},CV_\text{wT}}$$ Observed within-subject Coefficient of Variation of the Reference and Test product
EAC East African Community
EEA European Economic Area
EEU Eurasian Economic Union
EMA European Medicines Agency
EU European Union
FDA (U.S.) Food and Drug Administration
GCC Gulf Cooperation Council
$$\small{H_0}$$ Null hypothesis (inequivalence)
$$\small{H_1}$$ Alternative hypothesis (equivalence)
HVD(P) Highly Variable Drug (Product)
IBE Individual Bioequivalence
ICH International Council for Harmonisation
$$\small{k}$$ Regulatory constant (0.76)
NTID Narrow Therapeutic Index Drug
PE Point Estimate
$$\small{\text{R}}$$ Reference product
RSABE Reference-Scaled Average Bioequivalence (FDA and China’s CDE)
SABE Scaled Average Bioequivalence (ABEL and RSABE)
$$\small{s_\text{bR}^2,s_\text{bT}^2}$$ Observed between-subject variance of the Reference and Test product
$$\small{s_\text{wR},s_\text{wT}}$$ Observed within-subject standard deviation of the Reference and Test product
$$\small{s_\text{wR}^2,s_\text{wT}^2}$$ Observed within-subject variance of the Reference and Test product
$$\small{\text{T}}$$ Test product
TIE Type I Error (patient’s risk)
TIIE Type II Error (producer’s risk)
T/R T/R-ratio (observed, PE) or assumed in sample size estimation
$$\small{uc}$$ Upper cap of expansion in ABEL
WHO World Health Organization
$$\small{\alpha}$$ Nominal level of the test, probability of Type I Error (patient’s risk)
$$\small{\beta}$$ Probability of Type II Error (producer’s risk)
$$\small{\mu_\text{T}/\mu_\text{R},\theta_0}$$ True T/R-ratio
$$\small{\pi}$$ (Prospective) power, where $$\small{\pi=1-\beta}$$
$$\small{\sigma_\text{wR}}$$ True within-subject standard deviation of the Reference product
$$\small{\theta_1,\theta_2}$$ Fixed lower and upper limits of the acceptance range (generally 80.00 – 125.00%)
$$\small{\theta_{\text{s}_1},\theta_{\text{s}_2}}$$ Expanded lower and upper limits of the acceptance range $$\small{=100\exp(k\,\cdot s_\text{wR})}$$; in the guidelines denoted as $$\small{\left\{L,U\right\}}$$

Helmut Schütz 2024
R and PowerTOST GPL 3.0, klippy MIT, pandoc GPL 2.0.
1st version March 23, 2021. Rendered July 27, 2024 11:59 CEST by rmarkdown via pandoc in 0.95 seconds.

### Footnotes and References

I tried to give online-resources as far as possible. Others were published before the Internet was developed. I have them on yellowed or even faint thermal paper of (yes!) FAX-machines.

1. Labes D, Schütz H, Lang B. PowerTOST: Power and Sample Size for (Bio)Equivalence Studies. Package version 1.5.6. 2024-03-18. CRAN.↩︎

2. Schütz H. Reference-Scaled Average Bioequivalence. 2024-02-29. CRAN.↩︎

3. Labes D, Schütz H, Lang B. Package ‘PowerTOST’. March 18, 2024. CRAN.↩︎

4. Some gastric resistant formulations of diclofenac are HVDPs, practically all topical formulations are HVDPs, whereas diclofenac itself is not a HVD ($$\small{CV_\text{w}}$$ of a solution ~8%).↩︎

5. An inglorious counterexample: In the approval of dabigatran (the first univalent direct thrombin (IIa) in­hi­bi­tor) the originator withheld information about severe bleeding events. Although it is highly variable, it is considered an NTID by the FDA and reference-scaling not acceptable. The study has to performed in a four period replicate design in order to allow a comparison of $$\small{s_\text{wT}}$$ with $$\small{s_\text{wR}}$$.↩︎

6. Tóthfalusi L, Endrényi L, García-Arieta A. Evaluation of bioequivalence for highly variable drugs with scaled average bioequivalence. Clin Pharmacokinet. 2009; 48(11): 725–43. doi:10.2165/11318040-000000000-00000.↩︎

7. ABEL is one variant of SABE. Another is Reference-Scaled Average Bioequivalence (RSABE), which is recommended by the FDA and China’s CDE. It is covered in another article.↩︎

8. GCC. Executive Directorate Of Benefits And Risks Evaluation. The GCC Guidelines for Bioequivalence. Ver­sion 2.4. 30/3/2016. ↩︎

9. Final Concept Paper. M13: Bioequivalence for Immediate-Release Solid Oral Dosage Forms. Geneva. 19 June 2020. Endorsed 10 July 2020. Online.↩︎

10. EMA, CHMP. Guideline on the Inves­ti­ga­tion of Bioequivalence. London. 20 January 2010. Online.↩︎

11. WHO, Essential Medicines and Health Products: Multisource (generic) pharmaceutical products: guidelines on registration requirements to establish interchangeability. WHO Technical Report Series, No. 1003, Annex 6. Geneva. 28 April 2017. Online↩︎

12. WHO/PQT: medicines. Guidance Document: Application of reference-scaled criteria for AUC in bioequivalence studies conducted for submission to PQT/MED. Geneva. 02 July 2021. Online.↩︎

13. Australian Government, Department of Health and Aged Care, TGA. International scientific guideline: Guideline on the investigation of bioequivalence. Canberra. Effective 16 June 2011. Online.↩︎

14. Shohin LE, Rozhdestvenkiy DA, Medvedev VYu, Komarow TN, Grebenkin DYu. Russia, Belarus & Ka­zakh­stan. In: Kanfer I, editor. Bioequivalence Requirements in Various Global Jurisdictions. Charm: Springer; 2017. pp. 215–6.↩︎

15. EAC, Medicines and Food Safety Unit. Compendium of Medicines Evaluation and Regi­stra­tion for Medicine Re­gu­la­tion Harmonization in the East African Community, Part III: EAC Guidelines on The­ra­peutic Equivalence Re­qui­re­ments. Online.↩︎

16. ASEAN States Pharmaceutical Product Working Group. ASEAN Guideline for the Conduct of Bioequivalence Stu­dies. Vientiane. March 2015. Online.↩︎

17. Shohin LE, Rozhdestvenkiy DA, Medvedev VYu, Komarow TN, Grebenkin DYu. Russia, Belarus & Ka­zakh­stan. In: Kanfer I, editor. Bioequivalence Requirements in Various Global Jurisdictions. Charm: Springer; 2017. p. 207.↩︎

18. Eurasian Economic Commission. Regulations Conducting Bioequivalence Studies within the Framework of the Eur­asi­an Economic Union. 3 November 2016, amended 15 February 2023. Online. [Russian]↩︎

19. Ministry of Health and Population, The Specialized Scientific Committee for Evaluation of Bioavailability & Bioequivalence Studies. Egyptian Guideline For Conducting Bioequivalence Studies for Marketing Authorization of Generic Products. Cairo. February 2017. ↩︎

20. New Zealand Medicines and Medical Devices Safety Authority. Guideline on the Regulation of Therapeutic Pro­ducts in New Zealand. Part 6: Bioequivalence of medicines. Wellington. February 2018. Online.↩︎

21. Departamento Agencia Nacional de Medicamentos. Instituto de Salud Pública de Chile. Guia para La rea­li­za­ción de estudios de biodisponibilidad comparativa en formas farmacéuticas sólidas de administración oral y acción sistémica. Santiago. December 2018. [Spanish]↩︎

22. ANVISA. Dispõe sobre os critérios para a condução de estudos de biodisponibilidade relativa/bioequivalência (BD/BE) e estudos farmacocinéticos. Brasilia. 17 August 2022. Online. [Portuguese]↩︎

23. Executive Board of the Health Ministers’ Council for GCC States. The GCC Guidelines for Bioequivalence. Ver­sion 3.1. 10 August 2022. Online.↩︎

24. Secretaría de Salud. Disposiciones para los Estudios de Bioequivalencia para Fármacos Altamente Variables. Ciudad de México. 24 January 2023. Online. [Spanish]↩︎

25. Health Canada. Guidance Document. Comparative Bioavailability Standards: Formulations Used for Systemic Ef­fects. Ottawa. 08 June 2018. Online.↩︎

26. EMA, CHMP. Guideline on the pharmacokinetic and clinical evaluation of modified release dosage forms. London. 20 November 2014. Online.↩︎

27. Midha KK, Rawson MJ, Hubbard JW. Prescribability and switchability of highly variable drugs and drug products. J Contr Rel. 1999; 62(1-2): 33–40. ↩︎

28. Labes D, Schütz H. Inflation of Type I Error in the Evaluation of Scaled Average Bioequivalence, and a Method for its Control. Pharm Res. 2016: 33(11); 2805–14. doi:10.1007/s11095-016-2006-1.↩︎

29. Schütz H, Labes D, Wolfsegger MJ. Critical Remarks on Reference-Scaled Average Bioequivalence. J Pharm Phar­ma­ceut Sci. 2022; 25: 285–296. doi:10.18433/jpps32892. ↩︎

30. Schütz H, Tomashevskiy M, Labes D. replicateBE: Average Bioequivalence with Expanding Limits (ABEL). Pack­age version 1.1.3. 2022-05-02. CRAN.↩︎

31. Karalis V, Symillides M, Macheras P. On the leveling-off properties of the new bioequivalence limits for highly variable drugs of the EMA guideline. Europ J Pharm Sci. 2011; 44: 497–505. doi:10.1016/j.ejps.2011.09.008.↩︎

32. Commission of the European Communities, CPMP Working Party on Efficacy of Medicinal Products. Note for Guidance: Investigation of Bioavailability and Bio­equi­va­lence. Appendix III: Technical Aspects of Bio­equi­va­lence Sta­tistics. Brussels. December 1991.↩︎

33. Commission of the European Communities, CPMP Working Party on Efficacy of Medicinal Products. Note for Guidance: Investigation of Bioavailability and Bio­equi­va­lence. June 1992.↩︎

34. Blume H, Mutschler E, editors. Bioäquivalenz. Qualitätsbewertung wirkstoffgleicher Fertig­arzneimittel. Frank­furt / Main: Govi-Verlag; 6. Ergänzungslieferung 1996. [German]↩︎

35. EMA, CHMP EWP, Therapeutic Subgroup on Pharmacokinetics. Questions & Answers on the Bioavailability and Bioequivalence Guideline. London. 27 July 2006. Online.↩︎

36. MCC. Registration of Medicines. Biostudies. Pretoria. June 2015. Online.↩︎

37. League of Arab States, Higher Technical Committee for Arab Pharmaceutical Industry. Harmonised Arab Guide­line on Bioequivalence of Generic Pharmaceutical Products. Cairo. March 2014. Online.↩︎

38. Shohin LE, Rozhdestvenkiy DA, Medvedev VYu, Komarow TN, Grebenkin DYu. Russia, Belarus & Ka­zakh­stan. In: Kanfer I, editor. Bioequivalence Requirements in Various Global Jurisdictions. Charm: Springer; 2017. p. 222.↩︎

39. Of note, for NTIDs Health Canada’s limits are 90.0 – 112.0% and not 90.00 – 111.11% like in other jurisdictions. Guess the reason.↩︎

40. That’s contrary to ABE, where $$\small{CV_\text{w}}$$ is an assumption as well.↩︎

41. Senn S. Guernsey McPearson’s Drug Development Dictionary. 21 April 2020. Online.↩︎

42. Hoenig JM, Heisey DM. The Abuse of Power: The Pervasive Fallacy of Power Calculations for Data Analysis. Am Stat. 2001; 55(1): 19–24. doi:10.1198/000313001300339897. ↩︎

43. There is no statistical method to ‘correct’ for unequal carryover. It can only be avoided by design, i.e., a sufficiently long washout between periods. According to the guidelines subjects with pre-dose concentrations > 5% of their Cmax can by excluded from the comparison if stated in the protocol.↩︎

44. This is not always the case in ABEL. This issue is elaborated in another article.↩︎

45. Senn S. Statistical Issues in Drug Development. Chichester: John Wiley; 2nd ed. 2007.↩︎

46. Zhang P. A Simple Formula for Sample Size Calculation in Equivalence Studies. J Biopharm Stat. 2003; 13(3): 529–38. doi:10.1081/BIP-120022772.↩︎

47. Don’t be tempted to assume a ‘better’ T/R-ratio – even if based on a pilot or a previous study. It is a natural property of HVDPs that the T/R-ratio varies between studies. Don’t be overly optimistic!↩︎

48. Doyle AC. The Adventures of Sherlock Holmes. A Scandal in Bohemia. 1892. p. 3.↩︎

49. Schütz H. Sample Size Estimation in Bioequivalence. Evaluation. 2020-10-23. BEBA Forum. Vienna. 2020-10-23. Online.↩︎

50. Lenth RV. Two Sample-Size Practices that I Don’t Recommend. October 24, 2000. Online.↩︎

51. The only exception is design = "2x2x3" (the full replicate with sequences TRT|RTR). Then the first element is for sequence TRT and the second for RTR.↩︎

52. Quoting my late father: »If you believe, go to church.«↩︎

53. Berger RL, Hsu JC. Bioequivalence Trials, Intersection-Union Tests and Equivalence Confidence Sets. Stat Sci. 1996; 11(4): 283–302. JSTOR:2246021.↩︎

54. Zeng A. The TOST confidence intervals and the coverage probabilities with R simulation. March 14, 2014. Online.↩︎

55. The R Foundation for Statistical Computing. A Guidance Document for the Use of R in Regulated Clinical Trial En­vi­ron­ments. Vienna. October 18, 2021. Online.↩︎

56. The R Foundation for Statistical Computing. R: Software Development Life Cycle. A Description of R’s De­ve­lop­ment, Testing, Release and Maintenance Processes. Vienna. October 18, 2021. Online.↩︎

57. The R Foundation. R News. 2024-06-14. Online.↩︎

58. Bugzilla. R bug tracking system. Online.↩︎

59. FDA. General Principles of Software Validation; Final Guidance for Industry and FDA Staff. Rockville. January 11, 2002.

60. FDA. Statistical Software Clarifying Statement. May 6, 2015. Online.↩︎

61. WHO. Guidance for organizations performing in vivo bioequivalence studies. Technical Report Series No. 996, Annex 9. Section 4. Geneva. May 2016. Online.↩︎

62. EMA, GCP IWG. Guideline on computerised systems and electronic data in clinical trials. Amsterdam. 9 March 2023. Online.↩︎

63. Tóthfalusi L, Endrényi L. Sample Sizes for Designing Bioequivalence Studies for Highly Variable Drugs. J Pharm Phar­ma­col Sci. 2012; 15(1): 73–84. doi:10.18433/j3z88f.  Open Access.↩︎

64. User ‘BEQool’. ABEL is a framework (decision scheme). BEBA Forum. Vienna. 2024-01-30. Online.↩︎