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 version 4.1.3 (2022-03-10) and later. All scripts were run on a Xeon E3-1245v3 @ 3.40GHz (1/4 cores) 16GB RAM with R 4.3.1 on Win­dows 7.

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.RSABE() 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.

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).

previous section ↩︎

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/widened 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 observed¹⁵ \(\small{CV_\text{wR}}\) (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.

Of note, it is extremely unlikely that all assumptions will be exactly realized in a particular study. Hence, calculating retrospective (a.k.a. post hoc, a posteriori) power is not only futile but plain nonsense.¹⁷

Since generally the within-subject variability \(\small{CV_\text{w}}\) is smaller than the between-subject variability \(\small{CV_\text{b}}\), crossover studies are so popular. Of note, there is no relationship between \(\small{CV_\text{w}}\) and \(\small{CV_\text{b}}\). An example are drugs which are subjected to polymorphic metabolism. For these drugs \(\small{CV_\text{w}\ll CV_\text{b}}\).

Furthermore, the drugs’ within-subject variability may be unequal (i.e., \(\small{s_{\text{wT}}^{2}\neq s_{\text{wR}}^{2}}\)). For details see the section Heteroscedasticity.

It is a prerequisite that no carryover from one period to the next exists. Only then the comparison of treatments will be unbiased.¹⁸ Carryover is elaborated in another article.

Power is a hypersurface depending on \(\small{\theta_0}\), \(\small{n}\), and \(\small{CV}\). Don’t try to imagine four-dimensional space – it may fry your brain! We can only show spatial projections at a certain \(\small{CV\text{-}}\)value.

Fig. 3 Spatial projection of power for the ‘implied limits’ \(\small{\theta_{1,2}=0.6816-1.4671,\,\log_{e}\theta_{1,2}=\mp 0.38329}\), \(\small{n=18-32}\), and \(\small{CV_\text{wT}=CV_\text{wR}=0.45}\). 4-period full replicate design for the FDA’s RSABE.

Inspecting the power-surface for a given \(\small{n}\), we see the maximum at \(\small{\theta_0=1\;(\log_{e}\theta_0=0)}\). Due to the symmetry in \(\small{\log_{e}}\)-scale we require the same sample size for any given \(\small{\theta_0}\) and \(\small{1/\theta_0}\). Furthermore, at the ‘implied limits’ of the acceptance range (\(\small{\theta_1=0.6816,\theta_2=1.4671}\)) power \(\small{\leq 0.05}\), demonstrating that the Type I Error is controlled.¹⁹
For any given \(\small{\theta_0}\) power increases with \(\small{n}\) (asympotically approaching \(\small{1}\)).
Intersections with the horizontal power-planes (here \(\small{0.7,\,0.8,\,0.9}\)) show the U-shaped dependency of \(\small{n}\) on \(\small{\theta_0}\).

Note that the power-surface is ‘flatter’ than in the corresponding example of ABE.

previous section ↩︎

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).

Since theta0 = 0.90,²² targetpower = 0.80, and regulator = "FDA" 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.RSABE(CV = 0.45, 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) = 0.45; CVw(R) = 0.45
# True ratio = 0.9
# ABE limits / PE constraints = 0.8 ... 1.25 
# Regulatory settings: FDA 
# 
# Sample size
#  n    power
# 24   0.82450

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.RSABE(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] 24
x[['Achieved power']]
# [1] 0.8245

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          24         0.8245

With 24 subjects (twelve per sequence) we achieve the power we desire.

What happens if we have one dropout?

power.RSABE(CV = 0.45, design = "2x2x4",
            n = x[["Sample size"]] - 1)

# Unbalanced design. n(i)=12/11 assumed.

# [1] 0.8105

Still above 0.80 we desire. However, with two dropouts (22 eligible subjects) we would slightly fall short (0.7951).

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.

previous section ↩︎

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, believing²⁷ 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)
signif(ci, 4)

# lower CL upper CL 
#   0.3223   0.7629

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 24 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 scale more, we gain power. However, at large values the point estimate constraint cuts in and 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,
                          ci[["upper CL"]]),
                   power = NA)
for (i in 1:nrow(comp)) {
  comp$power[i] <- power.RSABE(CV = comp$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 0.79246
#  0.4500 0.87287
#  0.7629 0.81122

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.RSABE(CV = df$l[i], design = "2x2x4",
                            details = FALSE,
                            print = FALSE)[["Sample size"]]
  df[i, 6] <- sampleN.RSABE(CV = 0.45, design = "2x2x4",
                            details = FALSE,
                            print = FALSE)[["Sample size"]]
  df[i, 7] <- sampleN.RSABE(CV = df$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    30   24   32
#       18 0.45   0.3282   0.7300    30   24   26
#       24 0.45   0.3415   0.6685    28   24   24
#       30 0.45   0.3509   0.6334    28   24   24

Small pilot studies are practically useless. One leading generic company has an internal rule to perform pilot studies of HVD(P)s 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

Exlore 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$power[i] <- power.RSABE(CV = comp1$CV[i],
                                theta0 = theta0,
                                design = "2x2x4",
                                n = n)
}
comp1$power <- signif(comp1$power, 5)
for (i in 1:2) {
  comp2$power[i] <- power.RSABE(CV = CV,
                                theta0 = comp2$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 0.87287
#  0.4275 0.86914
#  theta0   power
#   0.900 0.87287
#   0.855 0.71029

    

Interlude

Fig. 4 Power curves for n = 28 (2×2×4 design).

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.

<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 higer 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.RSABE(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$power[i] <- power.RSABE(CV = CV,
                                theta0 = comp1$theta0[i],
                                design = design, n = n)
}
n       <- sampleN.RSABE(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$power[i] <- power.RSABE(CV = CV,
                                theta0 = comp2$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 24 0.8245
#  0.45 0.9091 FALSE 24 0.8492
#  0.45 1.1000 FALSE 24 0.8499
#  0.45 1.1110 FALSE 24 0.8242
#    CV theta0  base  n  power
#  0.45 0.9000 FALSE 22 0.7951
#  0.45 0.9091 FALSE 22 0.8199
#  0.45 1.1000  TRUE 22 0.8212
#  0.45 1.1110 FALSE 22 0.7954

</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,
                     regulator = "FDA",
                     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] <- "relative change"
print(pa, plotit = FALSE)
print(comp, row.names = FALSE)

# Sample size plan scABE (FDA/RSABE)
#  Design alpha CVwT CVwR theta0 theta1 theta2 Sample size
#   2x2x4  0.05 0.45 0.45    0.9    0.8   1.25          24
#  Achieved power Target power
#          0.8245          0.8
# 
# Power analysis
# CV, theta0 and number of subjects leading to min. acceptable power of ~0.7:
#  CV= 1.1132, theta0= 0.8647
#  n = 18 (power= 0.7141)
# 
#  parameter relative change
#         CV        +147.37%
#     theta0          -3.92%
#          n         -25.00%

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

Heteroscedasticity

I recommend to perform pilot studies in one of the fully replicated designs. If you are concerned about dropouts or the bioanalytical method requires large sample volumes, opt for one the 2-sequence 3-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 – hopefully – 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 of the pivotal study (scaling is based on \(\small{CV_\text{wR}}\) but the 90% CI on the – pooled – \(\small{s_\text{w}^{2}}\)).

\[\eqalign{\tag{7} s_\text{wT}^{2}&=\log_{e}(CV_\text{wT}^{2}+1)\\ s_\text{wR}^{2}&=\log_{e}(CV_\text{wR}^{2}+1)\\ s_\text{w}^{2}&=\left(s_\text{wT}^{2}+s_\text{wR}^{2}\right)/2\\ CV_\text{w}&=\sqrt{\exp(s_\text{w}^{2})-1}}\] It must be mentioned that \(\text{(7)}\) is only correct if the study was balanced (i.e., with an equal number of eligible subjects in each sequence). Otherwise, variances have to be weighed by the degrees of freedom.

Say, we performed two pilot studies.
In the partial replicate we estimated the \(\small{CV_\text{w}}\) with 0.45. We have to assume homoscedasticity (\(\small{CV_\text{wT}=CV_\text{wR}}\)), which might be wrong.
In the full replicate we estimated \(\small{CV_\text{wT}}\) with 0.414 and \(\small{CV_\text{wR}}\) with 0.484. Note that the \(\small{CV_\text{w}}\) is 0.45 as well. How will that impact the sample size 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.RSABE(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 24 0.82450
#      TRT|RTR 0.414 0.484 0.45 20 0.80146

Since bioanalytics drives study costs to a great extent, we may safe ~17%.

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 guidances 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?)²⁸ 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.RSABE(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 33 0.82802
#      TRT|RTR 0.414 0.484 0.45 27 0.81239

Again, a smaller sample size is possible.

previous section ↩︎

Multiple Endpoints

For demonstrating bioequivalence for the FDA and China’s CDE the pharmacokinetic metrics C_max, AUC_0–t, and AUC_0–∞ are mandatory.

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.^{29 30} We design the study always for the worst case combination, i.e., based on the PK metric requiring the largest sample size. Let’s explore that with different CVs and T/R-ratios.

metrics <- c("Cmax", "AUC")
CV      <- c(0.45, 0.30)
theta0  <- c(0.90, 0.925)
design  <- "2x2x4"
df      <- data.frame(metric = metrics, CV = CV, swR = CV2se(CV),
                      theta0 = theta0, n = NA, power = NA)
df[1, 5:6] <- sampleN.RSABE(CV = CV[1], theta0 = theta0[1],
                            design = design, details = FALSE,
                            print = FALSE)[8:9]
df[2, 5:6] <- sampleN.RSABE(CV = CV[2], theta0 = theta0[2],
                            design = design, details = FALSE,
                            print = FALSE)[8:9]
df$swR <- signif(df$swR, 5)
txt      <- paste0("Sample size based on ",
                   df$metric[df$n == max(df$n)], ".\n")
print(df, row.names = FALSE)
cat(txt)

#  metric   CV     swR theta0  n   power
#    Cmax 0.45 0.42942  0.900 24 0.82450
#     AUC 0.30 0.29356  0.925 22 0.81108
# Sample size based on Cmax.

As usual the ‘worse’ PK metric drives the sample size. Although we could scale for C_max (\(\small{s_\text{wR}\,0.42942\ge0.294}\)) but not for AUC (\(\small{s_\text{wR}\,0.29356<0.294}\)); this advantage is counteracted by its ‘worse’ T/R-ratio (0.900 instead of 0.925). Power is extremely sensitive to the T/R-ratio (see Fig. 4).
Consequently, with 24 subjects the study will be slightly ‘overpowered’ (~0.839) for AUC.

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 C_max giving still the target power?

opt <- function(x) {
  power.RSABE(theta0 = x, CV = df$CV[1],
              design = design,
              n = df$n[2]) - target
}
metrics <- c("Cmax", "AUC")
CV      <- c(0.45, 0.30)
theta0  <- 0.90
target  <- 0.80
design  <- "2x2x4"
df      <- data.frame(metric = metrics, CV = CV, swR = CV2se(CV),
                      theta0 = theta0, n = NA, power = NA)
df[1, 5:6] <- sampleN.RSABE(CV = CV[1], theta0 = theta0,
                            design = design, details = FALSE,
                            print = FALSE)[8:9]
df[2, 5:6] <- sampleN.RSABE(CV = CV[2], theta0 = theta0,
                            design = design, details = FALSE,
                            print = FALSE)[8:9]
df$swR   <- signif(df$swR, 5)
df$power <- signif(df$power, 5)
if (theta0[1] < 1) {
  res <- uniroot(opt, tol = 1e-8,
                 interval = c(0.80 + 1e-4, theta0))
} else {
  res <- uniroot(opt, tol = 1e-8,
                 interval = c(theta0, 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   CV     swR theta0  n   power
#    Cmax 0.45 0.42942    0.9 24 0.82450
#     AUC 0.30 0.29356    0.9 32 0.81666
# Target power for Cmax and sample size 32
# achieved for theta0 0.8661 or 1.1546.

That means, although we assumed for C_max the same T/R-ratio as for AUC, with the sample size of 32 required AUC, for C_max it can be as low as ~0.866 or as high as ~1.155, which is an interesting side-effect.

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