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Examples in this article were generated with 4.2.0 by the packages PowerTOST,¹ simcrossover,² and adjustalpha.³

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

Basics about the patient’s risk (Type I Error) and multiplicity – requiring no or only limited statistical expertise.

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

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

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

If you are not a neRd or statistics afficionado, skipping is recommended. Suggested for experts but might be confusing for others.

Click to show / hide R code.

Abbreviation	Meaning
\(\small{\alpha}\)	Nominal level of the test, probability of Type I Error (patient’s risk)
(A)BE	(Average) Bioequivalence
ABEL	Average Bioequivalence with Expanding Limits
\(\small{CV_\textrm{w}}\)	Within-subject Coefficient of Variation
\(\small{CV_\textrm{wT}}\), \(\small{CV_\textrm{wR}}\)	Within-subject Coefficient of Variation of the Test and Reference treatment
\(\small{\Delta}\)	Clinically relevant difference
\(\small{\widehat{\Delta}_\textrm{r}}\)	Realized clinically relevant difference
\(\small{H_0}\)	Null hypothesis
\(\small{H_1}\)	Alternative hypothesis (also \(\small{H_\textrm{a}}\))
\(\small{\mu_\textrm{T}/\mu_\textrm{R}}\)	True T/R-ratio
RSABE	Reference-scaled Average Bioequivalence
\(\small{s_\textrm{wR}}\), \(\small{s_\textrm{wT}}\)	Observed within-subject standard deviation of the Reference and Test product
\(\small{s_\textrm{wR}^2}\), \(\small{s_\textrm{wT}^2}\)	Observed within-subject variance of the Reference and Test product
\(\small{\sigma_\textrm{wR}}\)	True within-subject standard deviation of the Reference product
SABE	Scaled Average Bioequivalence
\(\small{\theta_0}\)	True T/R-ratio
\(\small{\theta_1,\;\theta_2}\)	Fixed lower and upper limits of the acceptance range in ABE
\(\small{\theta_{\textrm{s}_1},\;\theta_{\textrm{s}_2}}\)	Scaled lower and upper limits of the acceptance range
TIE	Type I Error (patient’s risk)
\(\small{uc}\)	Upper cap of expansion in ABEL

Introduction

What is the patient’s risk in Scaled Average Bioequivalence (SABE)?

For details about inferential statistics and hypotheses in equivalence see another article. See also the articles about RSABE and ABEL for the regulatory frameworks’ applicable conditions.

The conventional confidence interval inclusion approach of ABE is based on \[\begin{matrix}\tag{1} \theta_1=1-\Delta,\theta_2=\left(1-\Delta\right)^{-1}\\ H_0:\;\frac{\mu_\textrm{T}}{\mu_\textrm{R}}\ni\left\{\theta_1,\,\theta_2\right\}\;vs\;H_1:\;\theta_1<\frac{\mu_\textrm{T}}{\mu_\textrm{R}}<\theta_2, \end{matrix}\] where \(\small{\Delta}\) is the clinically relevant difference (commonly 20%), \(\small{\theta_1}\) and \(\small{\theta_2}\) are the fixed lower and upper limits of the acceptance range, \(\small{H_0}\) is the null hypothesis of inequivalence, and \(\small{H_1}\) is the alternative hypothesis. \(\small{\mu_\textrm{T}}\) and \(\small{\mu_\textrm{R}}\) are the geometric least squares means of \(\small{\textrm{T}}\) and \(\small{\textrm{R}}\), respectively.

However, for drugs with high variability the common approach for ABE requires extreme sample sizes. In a study for \(\small{CV_\textrm{w}=60\%}\) and \(\small{\theta_0=0.90}\) targeted at 90% power one would need 382 subjects in a 2×2×2 crossover design.

“For wide-therapeutic index highly variable drugs we should not have to study an excessive number of patients to prove that two equivalent products meet preset (one size fits all ) statistical criteria.
This is because, by definition, highly variable approved drugs must have a wide therapeutic index, otherwise there would have been significant safety issues and lack of efficacy during Phase 3.
Highly variable narrow therapeutic index drugs are dropped in Phase 2 since it is not possible to prove either efficacy or safety.
Leslie Z. Benet (2004)⁴

Hence, in Scaled Average Bioequivalence (SABE) the confidence interval inclusion approach \(\small{(1)}\) is modified to \[H_0:\;\frac{\mu_\textrm{T}}{\mu_\textrm{R}}\Big{/}\sigma_\textrm{wR}\ni\left\{\theta_{\textrm{s}_1},\,\theta_{\textrm{s}_2}\right\}\;vs\;H_1:\;\theta_{\textrm{s}_1}<\frac{\mu_\textrm{T}}{\mu_\textrm{R}}\Big{/}\sigma_\textrm{wR}<\theta_{\textrm{s}_2},\tag{2}\] where \(\small{\sigma_\textrm{wR}}\) is the standard deviation of the reference. The scaled limits \(\small{\left\{\theta_{\textrm{s}_1},\,\theta_{\textrm{s}_2}\right\}}\) of the acceptance range depend on conditions given by the agency.

\(\small{(2)}\) is formulated in the – unknown – population parameters. Both RSABE and ABEL are frameworks, where the scaled limits \(\small{\left\{\theta_{\textrm{s}_1},\,\theta_{\textrm{s}_2}\right\}}\) depend on the observed \(\small{s_\textrm{wR}}\).

In ABEL there is an ‘upper cap’ of scaling (\(\small{uc=50\%}\) limiting the expansion to \(\small{\left\{70.00-142.86\%\right\}}\), except for Health Canada, where \(\small{uc\approx 57.382\%}\) limiting the expansion to \(\small{\left\{67.7-150\%\right\}}\)).
Furthermore, in all jurisdictions additionally the point estimate (\(\small{PE}\)) has to lie within \(\small{80.00-125.00\%}\).

Since it is extremely unlikely that \(\small{s_\textrm{wR}\equiv \sigma_\textrm{wR}}\), a drug / drug product may be falsely classified as highly variable. Due to the scaled limits the chance of passing increases compared to the correct ABE.
Such a misclassificaton translates into an inflated Type I Error (increased patient’s risk).

Contrary to ABE, where the fixed limits are based on a pre-specified clinically not relevant difference \(\small{\Delta}\), it is not surprising that the Type I Error may be inflated, since in SABE the clinically not relevant difference \(\small{\Delta}\) is unknown beforehand and the null hypothesis generated post hoc based on the \(\small{CV_{\textrm{wR}}}\) observed in the study. Consequently, the regulatory goalposts become random variables and each study sets its own rules, awarding ones with high variability.

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

Although in a particular study the realized clinically not relevant difference \(\small{\widehat{\Delta}_\textrm{r}}\) can be recalculated \[\widehat{\Delta}_\textrm{r}=100\,(1-\theta_{\textrm{s}_1})\tag{3}\] but without access to the study report \(\small{\theta_{\textrm{s}_1}}\) is unknown to physicians, pharmacists, and patients alike. Hence, this is an unsatisfactory situation. We put the cart before the horse.
Regrettably, apart from one anecdotal report,¹⁰ regulatory agencies are paying little attention to the potentially increased consumer risk.

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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 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 4.0.5 (2021-03-31) and later.
All scripts were run on a Core i5-8265U @ 1.60GHz (1/4 cores) with R 4.2.0 on Windows 11 build 22000.

library(PowerTOST)          # attach it to run the examples
packs <- c("simcrossover", "adjustalpha")
if (sum(packs %in% rownames(installed.packages())) == 2) {
  lapply(packs, library, character.only = TRUE)
  adjustalpha.avail = TRUE  # use the package
} else {
  adjustalpha.avail = FALSE # use hard-coded results
}

Except for Health Canada and the FDA (where a mixed-effects model is required) the recommended evaluation by an ANOVA¹¹ assumes homoscedasticity (\(\small{s_{\textrm{wT}}^{2}\equiv s_{\textrm{wR}}^{2}}\)), which is – more often than not – wrong.

For a quick overview of the regulatory limits use the function scABEL() – for once in percent according to the guidelines.

df1 <- data.frame(regulator = "FDA", method = "RSABE",
                  CV = c(30, 50), L = NA_real_, U = NA_real_,
                  cap = c("lower", "  -  "))
for (i in 1:2) {
  df1[i, 4:5] <- sprintf("%.2f%%", 100*scABEL(df1$CV[i]/100,
                         regulator = "FDA"))
}
df1$CV <- sprintf("%.3f%%", df1$CV)
df2 <- data.frame(regulator = "EMA", method = "ABEL",
                  CV = c(30, 50),
                  L = NA_real_, U = NA_real_,
                  cap = c("lower", "upper"))
df3 <- data.frame(regulator = "HC", method = "ABEL",
                  CV = c(30, 57.382),
                  L = NA_real_, U = NA_real_,
                  cap = c("lower", "upper"))
for (i in 1:2) {
  df2[i, 4:5] <- sprintf("%.2f%%", 100*scABEL(df2$CV[i]/100))
  df3[i, 4:5] <- sprintf("%.1f%%", 100*scABEL(df3$CV[i]/100,
                         regulator = "HC"))
}
df2$CV <- sprintf("%.3f%%", df2$CV)
df3$CV <- sprintf("%.3f%%", df3$CV)
if (packageVersion("PowerTOST") >= "1.5.3") {
  df4 <- data.frame(regulator = "GCC", method = "ABEL",
                    CV = c(30, 50), L = NA_real_, U = NA_real_,
                    cap = c("lower", "  -  "))
  for (i in 1:2) {
    df4[i, 4:5] <- sprintf("%.2f%%", 100*scABEL(df4$CV[i]/100,
                           regulator = "GCC"))
  }
  df4$CV <- sprintf("%.3f%%", df4$CV)
}
if (packageVersion("PowerTOST") >= "1.5.3") {
  print(rbind(df1, df2, df3, df4), row.names = FALSE)
} else {
  print(rbind(df1, df2, df3), row.names = FALSE)
}

#  regulator method      CV      L       U   cap
#        FDA  RSABE 30.000% 80.00% 125.00% lower
#        FDA  RSABE 50.000% 65.60% 152.45%   -  
#        EMA   ABEL 30.000% 80.00% 125.00% lower
#        EMA   ABEL 50.000% 69.84% 143.19% upper
#         HC   ABEL 30.000%  80.0%  125.0% lower
#         HC   ABEL 57.382%  66.7%  150.0% upper
#        GCC   ABEL 30.000% 80.00% 125.00% lower
#        GCC   ABEL 50.000% 75.00% 133.33%   -

Note that these are the ‘implied limits’ of RSABE. The FDA assesses the Type I Error based on the limits of the ‘desired consumer risk model’. More about that later.

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Type I Error

The empiric Type I Error (TIE) can be obtained by simulating under the null hypothesis, i.e., setting \(\small{\theta_0}\) to one of the scaled limits \(\small{\left\{\theta_{\textrm{s}_1},\,\theta_{\textrm{s}_2}\right\}}\).

Numerous authors observed an inflated TIE.¹³ ¹⁴ ¹⁵ ¹⁶ ¹⁷ ¹⁸ ¹⁹ ²⁰ ²¹ ²² ²³ ²⁴ ²⁵ ²⁶
In the following by ‘critical region’ I’m referring to the range of \(\small{CV_\textrm{wR}}\)-values where an inflated TIE is obtained in simulations.

Let’s explore an example: A four-period full replicate design in 24 subjects, \(\small{CV_{\textrm{wR}}=0.20-0.60}\). We perform one million simulations at the respective upper boundary of the equivalence range \(\small{\theta_{\textrm{s}_2}}\) (due to the symmetry in log-scale similar results are obtained for the lower boundary \(\small{\theta_{\textrm{s}_1}}\)). Because of the positive skewness of both \(\small{CV_{\textrm{wR}}}\) and \(\small{\theta_0}\), the upper boundary represents generally the worst case scenario.

The standard error of the empiric TIE is calculated according to \(\small{SE=\sqrt{0.5\cdot TIE_\textrm{emp}/10^6}}\). Starred values denote a significant inflation of the TIE based on the one-sided binomial test (> 0.05036).

CV     <- sort(c(seq(0.20, 0.50, 0.05),
                 se2CV(c(0.25, 0.294)), 0.60))
n      <- 24
design <- "2x2x4"
nsims  <- 1e6
sign.i <- binom.test(0.05*nsims, nsims, alternative = "less")$conf.int[2]
reg    <- c("FDA", "EMA", "HC")
if (packageVersion("PowerTOST") >= "1.5.3") reg <- c(reg, "GCC")
res    <- data.frame(regulator = rep(reg, each = length(CV)),
                     method = c(rep("RSABE", length(CV)),
                                rep("ABEL", (length(reg)-1)*length(CV))),
                     CVwR = CV, theta2 = NA, TIE.emp = NA,
                     SE = NA, infl = "")
for (i in 1:nrow(res)) {
  res$theta2[i] <- scABEL(res$CVwR[i],
                          regulator = res$regulator[i])[["upper"]]
  if (res$regulator[i] == "FDA") {
    res$TIE.emp[i] <- power.RSABE(CV = res$CVwR[i], design = design,
                                  theta0 = res$theta2[i], n = n,
                                  nsims = nsims)
  } else {
    res$TIE.emp[i] <- power.scABEL(CV = res$CVwR[i], design = design,
                                   theta0 = res$theta2[i], n = n,
                                   nsims = nsims,
                                   regulator = res$regulator[i])
  }
  res$theta2[i] <- signif(res$theta2[i], 5)
  res$SE[i] <- sprintf("%.5f", sqrt(0.5 * res$TIE[i] / nsims))
  if (res$TIE[i] > sign.i) res$infl[i] <- "*"
}
res$CVwR    <- signif(res$CVwR, 5)
res$TIE.emp <- round(res$TIE.emp, 5)
print(res, row.names = FALSE)

#  regulator method    CVwR theta2 TIE.emp      SE infl
#        FDA  RSABE 0.20000 1.2500 0.05023 0.00016     
#        FDA  RSABE 0.25000 1.2500 0.06329 0.00018    *
#        FDA  RSABE 0.25396 1.2500 0.06634 0.00018    *
#        FDA  RSABE 0.30000 1.2500 0.13351 0.00026    *
#        FDA  RSABE 0.30047 1.3001 0.04782 0.00015     
#        FDA  RSABE 0.35000 1.3545 0.04098 0.00014     
#        FDA  RSABE 0.40000 1.4104 0.03071 0.00012     
#        FDA  RSABE 0.45000 1.4671 0.02139 0.00010     
#        FDA  RSABE 0.50000 1.5245 0.01455 0.00009     
#        FDA  RSABE 0.60000 1.6404 0.00707 0.00006     
#        EMA   ABEL 0.20000 1.2500 0.04994 0.00016     
#        EMA   ABEL 0.25000 1.2500 0.05224 0.00016    *
#        EMA   ABEL 0.25396 1.2500 0.05312 0.00016    *
#        EMA   ABEL 0.30000 1.2500 0.08040 0.00020    *
#        EMA   ABEL 0.30047 1.2504 0.08014 0.00020    *
#        EMA   ABEL 0.35000 1.2948 0.06527 0.00018    *
#        EMA   ABEL 0.40000 1.3402 0.05944 0.00017    *
#        EMA   ABEL 0.45000 1.3859 0.04780 0.00015     
#        EMA   ABEL 0.50000 1.4319 0.03289 0.00013     
#        EMA   ABEL 0.60000 1.4319 0.04490 0.00015     
#         HC   ABEL 0.20000 1.2500 0.05012 0.00016     
#         HC   ABEL 0.25000 1.2500 0.05385 0.00016    *
#         HC   ABEL 0.25396 1.2500 0.05490 0.00017    *
#         HC   ABEL 0.30000 1.2500 0.08414 0.00021    *
#         HC   ABEL 0.30047 1.2504 0.08396 0.00020    *
#         HC   ABEL 0.35000 1.2948 0.06869 0.00019    *
#         HC   ABEL 0.40000 1.3402 0.06150 0.00018    *
#         HC   ABEL 0.45000 1.3859 0.05283 0.00016    *
#         HC   ABEL 0.50000 1.4319 0.04299 0.00015     
#         HC   ABEL 0.60000 1.5000 0.03326 0.00013     
#        GCC   ABEL 0.20000 1.2500 0.05026 0.00016     
#        GCC   ABEL 0.25000 1.2500 0.07342 0.00019    *
#        GCC   ABEL 0.25396 1.2500 0.07838 0.00020    *
#        GCC   ABEL 0.30000 1.3333 0.02200 0.00010     
#        GCC   ABEL 0.30047 1.3333 0.02219 0.00011     
#        GCC   ABEL 0.35000 1.3333 0.03865 0.00014     
#        GCC   ABEL 0.40000 1.3333 0.04608 0.00015     
#        GCC   ABEL 0.45000 1.3333 0.04836 0.00016     
#        GCC   ABEL 0.50000 1.3333 0.04896 0.00016     
#        GCC   ABEL 0.60000 1.3333 0.04884 0.00016

We see a huge inflation of the TIE in RSABE and the GCC’s variant of ABEL up to the switching \(\small{C_\textrm{wR}}\) of 30%. Then the TIE is controlled due the inherent conservatism of the TOST and the PE-constraint.
Note the discontinuity of the GCC’s method because for any \(\small{C_\textrm{wR}>30\%}\) the limits are directly widened to 0.7500–1.3333.

ABEL for the EMA and Health Canada shows less inflation. However, it reaches beyond the switching \(\small{C_\textrm{wR}}\) of 30%. Due to the more liberal upper cap of scaling (≈57.4% instead of 50%) the inflation for Health Canada is slightly more pronounced than for the EMA. In either case, for large variabilities (> 45–50%) the TIE is controlled.

The TIE depends on the sample size as well. Simulations like above but with \(\small{n=24-48}\) at \(\small{CV_{\textrm{wR}}=0.30}\) (\(\small{\theta_{\textrm{s}_2}=1.25}\)); ABE for comparative purposes.

CV     <- 0.30
n      <- seq(24, 48, 12)
design <- "2x2x4"
nsims  <- 1e6
sign.i <- binom.test(0.05*nsims, nsims, alternative = "less")$conf.int[2]
reg    <- c("all", "FDA", "EMA", "HC")
if (packageVersion("PowerTOST") >= "1.5.3") reg <- c(reg, "GCC")
res    <- data.frame(regulator = rep(reg, each = length(n)),
                     method = c(rep("ABE", length(n)),
                                rep("RSABE", length(n)),
                                rep("ABEL", (length(reg)-2)*length(n))),
                     n = n, TIE.emp = NA, SE = NA, infl = "")
for (i in 1:nrow(res)) {
  if (res$regulator[i] == "all") {
    theta1 <- 1.25
  } else {
    theta2 <- scABEL(CV, regulator = res$regulator[i])[["upper"]]
  }
  if (res$regulator[i] == "FDA") {
    res$TIE.emp[i] <- power.RSABE(CV = CV, design = design,
                                  theta0 = theta2, n = res$n[i],
                                  nsims = nsims)
  } else {
    if (res$regulator[i] == "all") {
      res$TIE.emp[i] <- power.TOST(CV = CV, design = design,
                                   theta0 = 1.25, n = res$n[i])
    } else {
      res$TIE.emp[i] <- power.scABEL(CV = CV, design = design,
                                     theta0 = theta2, n = res$n[i],
                                     nsims = nsims,
                                     regulator = res$regulator[i])
    }
  }
  res$SE[i] <- sprintf("%.5f", sqrt(0.5 * res$TIE[i] / nsims))
  if (res$TIE[i] > sign.i) res$infl[i] <- "*"
}
res$TIE.emp <- round(res$TIE.emp, 5)
print(res, row.names = FALSE)

#  regulator method  n TIE.emp      SE infl
#        all    ABE 24 0.05000 0.00016     
#        all    ABE 36 0.05000 0.00016     
#        all    ABE 48 0.05000 0.00016     
#        FDA  RSABE 24 0.13351 0.00026    *
#        FDA  RSABE 36 0.15360 0.00028    *
#        FDA  RSABE 48 0.17076 0.00029    *
#        EMA   ABEL 24 0.08040 0.00020    *
#        EMA   ABEL 36 0.08192 0.00020    *
#        EMA   ABEL 48 0.08232 0.00020    *
#         HC   ABEL 24 0.08414 0.00021    *
#         HC   ABEL 36 0.08461 0.00021    *
#         HC   ABEL 48 0.08465 0.00021    *
#        GCC   ABEL 24 0.14932 0.00027    *
#        GCC   ABEL 36 0.19307 0.00031    *
#        GCC   ABEL 48 0.23244 0.00034    *

Fig. 2 TIE in 4-period full replicate designs.
a ABE,  b ABEL (EMA),
c ABEL (Health Canada),  d ABEL (GCC),
e RSABE (‘implied limits’),  f RSABE (‘desired consumer risk model’)

Of course, in ABE the TIE is always controlled, i.e., the TIE never exceeds nominal \(\small{\alpha}\). Since TOST is not a most powerful test,²⁷ for high \(\small{C_\textrm{wR}}\) together with relatively low sample sizes, it becomes conservative.
Again, RSABE and the GCC’s variant of ABEL behave similarly and show a substanial dependency on the sample size.
The two other ABEL methods are less sensitive to the sample size in the critical region.

Combining what we have seen so far for RSABE and the EMA’s ABEL (note the different scales of the z-axis):

Fig. 3 TIE of RSABE in 4-period full replicate designs (n 20–120).

Fig. 4 TIE of the EMA’s ABEL in 4-period full replicate designs (n 20–120).

The FDA assessed the Type I Error not at the ‘implied limits’ – like all other authors – but based on the limits of the so-called ‘desired consumer risk model’.²⁸ ²⁹ \[\begin{array}{ll}\tag{4} \theta_0=1.25 & \textrm{if}\; s_\textrm{wR}<0.294,\\ \theta_0=\exp(\frac{\log_{e}1.25}{0.25}\cdot s_\textrm{wR}) & \textrm{otherwise}. \end{array}\] Let’s explore that in four-period full replicate studies with a T/R-ratio of 0.90 powered at 80%.

CV  <- CV  <- sort(c(seq(0.2, 0.4, 0.05), se2CV(0.25), 0.272645))
res <- data.frame(n = NA, CVwR = CV, swR = CV2se(CV),
                  impl.L = NA, impl.U = NA, impl.TIE = NA,
                  des.L = NA, des.U = NA, des.TIE = NA)
for (i in 1:nrow(res)) {
  res$n[i] <- sampleN.RSABE(CV = CV[i], design = "2x2x4",
                           details = FALSE,
                           print = FALSE)[["Sample size"]]
  res[i, 4:5] <- scABEL(CV = CV[i], regulator = "FDA")
  if (CV2se(CV[i]) <= 0.25) {
    res[i, 7:8] <- c(0.80, 1.25)
  } else {
    res[i, 7:8] <- exp(c(-1, +1)*(log(1.25)/0.25)*CV2se(CV[i]))
  }
  res[i, 6]     <- power.RSABE(CV = CV[i], design = "2x2x4",
                               theta0 = res[i, 5], n = res$n[i],
                               nsims = 1e6)
  res[i, 9]     <- power.RSABE(CV = CV[i], design = "2x2x4",
                               theta0 = res[i, 8], n = res$n[i],
                               nsims = 1e6)
}
res[, 2:3] <- round(res[, 2:3], 3)
res[, c(4:5, 7:8)] <- round(res[, c(4:5, 7:8)], 4)
res[, c(6, 9)] <- round(res[, c(6, 9)], 5)
print(res, row.names = FALSE)

#   n  CVwR   swR impl.L impl.U impl.TIE  des.L  des.U des.TIE
#  20 0.200 0.198 0.8000 1.2500  0.05047 0.8000 1.2500 0.05047
#  28 0.250 0.246 0.8000 1.2500  0.06195 0.8000 1.2500 0.06195
#  30 0.254 0.250 0.8000 1.2500  0.06467 0.8000 1.2500 0.06467
#  32 0.273 0.268 0.8000 1.2500  0.08783 0.7874 1.2700 0.05000
#  32 0.300 0.294 0.8000 1.2500  0.14714 0.7695 1.2996 0.04611
#  28 0.350 0.340 0.7383 1.3545  0.03923 0.7383 1.3545 0.03923
#  24 0.400 0.385 0.7090 1.4104  0.03071 0.7090 1.4104 0.03071

Consequently, the TIE is apparently lower (though still significantly inflated). Note that the discontinuity disappears and the location of the maximum TIE shifts from \(\small{CV_\textrm{wR}=30\%\,(s_\textrm{wR}\sim0.294)}\) to the switching variability \(\small{CV_\textrm{wR}\sim 25.4\%\,(s_\textrm{wR}=0.25)}\).

Fig. 5 TIE in a 4-period full replicate design (n 32).
Colored line ‘implied limits’, black line ‘desired consumer risk model’.

In my opinion the ‘desired consumer risk model’ is nothing more than a mathematical prestidigitation as drug products are not approved according its conditions but the ones of the guidances.³⁰ ³¹ Some eminent statisticians³² call it just a hocus pocus…

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Remedies?

Ad hoc

Bonferroni

Bonferroni’s adjustment suggests itself. \[\alpha_\textrm{adj}=\alpha / k \tag{5}\] Whilst seemingly obvious, it is problematic for numerous reasons.

It is not known beforehand how many decisions and tests k will be performed in the regulatory frameworks and hence, any adjustment arbitrary.
- RSABE
  - s_wR ≥ 0.294
    - If the upper critical bound is ≤ 0, assess the PE (within the constraints 80.00–125.00%).
  - s_wR < 0.294
    - Assess the 90% CI for ABE.
- ABEL
  - CV_wR > 30%
    - If CV_wR ≤ 50% use s_wR for expanding the limits or the upper cap, otherwise.
    - If the 90% CI is within the expanded limits, assess the PE (within the constraints 80.00–125.00%).
  - CV_wR ≤ 30%
    - Assess the 90% CI for ABE.
The tests are not independent.
In simulations it was shown that the approach is overly conservative if an inflated TIE is unlikely (say, if CV_wR > 45% in ABEL and if CV_wR < 30% in RSABE).

CVwR   <- 0.35
design <- "2x2x4"
theta0 <- 0.90
tmp    <- sampleN.scABEL(CV = CVwR, theta0 = theta0,
                         design = design, details = FALSE,
                         print = FALSE)
cat("Design        ", design,
    "\nMethod         ABEL (EMA)",
    "\nCVwR          ", CVwR,
    "\ntheta0        ", theta0,
    "\nSample size   ", tmp[["Sample size"]],
    "\nAchieved power", tmp[["Achieved power"]], "\n\n")
scABEL.ad(alpha.pre = 0.05 / 2, CV = CVwR, theta0 = theta0,
          design = design, n = tmp[["Sample size"]],
          details = TRUE)

# Design         2x2x4 
# Method         ABEL (EMA) 
# CVwR           0.35 
# theta0         0.9 
# Sample size    34 
# Achieved power 0.81184 
# 
# +++++++++++ scaled (widened) ABEL ++++++++++++
#          iteratively adjusted alpha
#    (simulations based on ANOVA evaluation)
# ----------------------------------------------
# Study design: 2x2x4 (4 period full replicate)
# log-transformed data (multiplicative model)
# 1,000,000 studies in each iteration simulated.
# 
# CVwR 0.35, CVwT 0.35, n(i) 17|17 (N 34)
# Nominal alpha                 : 0.05, pre-specified alpha 0.025
# True ratio                    : 0.9000 
# Regulatory settings           : EMA (ABEL)
# Switching CVwR                : 0.3 
# Regulatory constant           : 0.76 
# Expanded limits               : 0.7723 ... 1.2948
# Upper scaling cap             : CVwR > 0.5 
# PE constraints                : 0.8000 ... 1.2500
# Empiric TIE for alpha 0.0250  : 0.03614 (rel. change of risk: -27.7%)
# Power for theta0 0.9000       : 0.722
# TIE = nominal alpha; the chosen pre-specified alpha is justified.

With \(\small{\alpha_{\textrm{adj}}=0.05/2}\) the TIE is controlled. However, the sample size has to be increased (in this example by ~24%) in order to maintain the target power.

In some extreme cases it leads still to an inflated TIE:

CVwR   <- 0.30
design <- "2x2x3"
theta0 <- 0.85
Bonf   <- 0.05 / 2
n      <- sampleN.scABEL(alpha = Bonf, CV = CVwR, theta0 = theta0,
                         design = design, print = FALSE,
                         details = FALSE)[["Sample size"]]
scABEL.ad(alpha.pre = Bonf, CV = CVwR, theta0 = theta0,
          design = design, n = n)

# +++++++++++ scaled (widened) ABEL ++++++++++++
#          iteratively adjusted alpha
#    (simulations based on ANOVA evaluation)
# ----------------------------------------------
# Study design: 2x2x3 (3 period full replicate)
# log-transformed data (multiplicative model)
# 1,000,000 studies in each iteration simulated.
# 
# CVwR 0.3, CVwT 0.3, n(i) 120|120 (N 240)
# Nominal alpha                 : 0.05, pre-specified alpha 0.025
# True ratio                    : 0.8500 
# Regulatory settings           : EMA (ABE)
# Switching CVwR                : 0.3
# BE limits                     : 0.8000 ... 1.2500
# Upper scaling cap             : CVwR > 0.5 
# PE constraints                : 0.8000 ... 1.2500
# Empiric TIE for alpha 0.0250  : 0.05243
# Power for theta0 0.8500       : 0.800
# Iteratively adjusted alpha    : 0.02356
# Empiric TIE for adjusted alpha: 0.05000
# Power for theta0 0.8500       : 0.793

top of section ↩︎ previous section ↩︎

Muñoz et al.

Muñoz et al.¹⁸ proposed to apply RSABE but according to the EMA’s conditions (regulatory constant 0.760, upper cap of scaling 50%).

# Cave: Long runtime
CV     <- 0.30
n      <- sampleN.scABEL(CV = CV, design = "2x2x4",
                         details = FALSE,
                         print = FALSE)[["Sample size"]]
CV     <- sort(c(se2CV(0.294),
                 seq(0.15, 0.65, 0.005)))
comp   <- data.frame(method = rep(c("EMA (2010)",
                                    "Muñoz et al. (2016)"),
                                  each = length(CV)),
                     CV = CV, TIE = NA)
theta2 <- rep(scABEL(CV = CV, regulator = "EMA")[, "upper"], 2)
for (i in 1:nrow(comp)) {
  if (comp$method[i] == "EMA (2010)") {
    comp$TIE[i] <- power.scABEL(CV = comp$CV[i], n = n,
                                design = "2x2x4",
                                theta0 = theta2[i],
                                nsims = 1e6)
  } else {
    comp$TIE[i] <- power.RSABE(CV = comp$CV[i], n = n,
                               design = "2x2x4",
                               theta0 = theta2[i],
                               regulator = "EMA",
                               nsims = 1e6)
  }
}
dev.new(width = 4.5, height = 4.5)
op   <- par(no.readonly = TRUE)
par(mar = c(4, 3.9, 0.1, 0.1), cex.axis = 0.9)
ylim <- c(0, max(comp$TIE))
xlim <- c(0.2, 0.6)
plot(CV, comp$TIE[comp$method == "EMA (2010)"], type = "n",
     xlab = expression(italic(CV)[wR]), xlim = xlim, ylim = ylim,
     ylab = "empiric Type I Error", axes = FALSE)
grid()
abline(v = c(0.3, 0.5), lty = 2)
abline(h = 0.05)
axis(1, at = pretty(CV),
     labels = sprintf("%i%%", as.integer(100*pretty(CV))))
axis(2, las = 1)
legend("bottomleft",
       legend = c("EMA (2010)",
                  expression("Muñoz "*italic("et al")*". (2016)")),
       col = c("#FF000080", "#FF00FF80"),
       lwd = 3, cex = 0.9, box.lty = 0, bg = "white", inset = 0.02)
box()
lines(CV, comp$TIE[comp$method == "EMA (2010)"],
      lwd = 3, col = "#FF000080")
lines(CV, comp$TIE[comp$method == "Muñoz et al. (2016)"],
      lwd = 3, col = "#FF00FF80")
par(op)

Fig. 6 TIE of the approaches in a 4-period full replicate design (n 34).

We see less inflation of the TIE (its maximum decreases from 0.0816 to 0.0684) and the critical region is substantially narrower. Whilst this is an improvement over the original method, it is not considered further.

top of section ↩︎ previous section ↩︎

Labes and Schütz

Labes and Schütz¹⁹ proposed to iteratively adjust \(\small{\alpha}\) based on the observed \(\small{CV_\textrm{wR}}\) if an inflated TIE is found in simulations.

Fig. 7 TIE of the EMA’s ABEL in 4-period full replicate designs (n 20–120)
when assessed with iteratively adjusted α.

CVwR   <- 0.35
design <- "2x2x4"
theta0 <- 0.90
tmp    <- sampleN.scABEL(CV = CVwR, theta0 = theta0,
                         design = design, details = FALSE,
                         print = FALSE)
cat("Design        ", design,
    "\nMethod         ABEL (EMA)",
    "\nCVwR          ", CVwR,
    "\ntheta0        ", theta0,
    "\nSample size   ", tmp[["Sample size"]],
    "\nAchieved power", tmp[["Achieved power"]], "\n\n")
scABEL.ad(CV = CVwR, theta0 = theta0,
          design = design, n = tmp[["Sample size"]],
          details = TRUE)

# Design         2x2x4 
# Method         ABEL (EMA) 
# CVwR           0.35 
# theta0         0.9 
# Sample size    34 
# Achieved power 0.81184 
# 
# +++++++++++ scaled (widened) ABEL ++++++++++++
#          iteratively adjusted alpha
#    (simulations based on ANOVA evaluation)
# ----------------------------------------------
# Study design: 2x2x4 (4 period full replicate)
# log-transformed data (multiplicative model)
# 1,000,000 studies in each iteration simulated.
# 
# CVwR 0.35, CVwT 0.35, n(i) 17|17 (N 34)
# Nominal alpha                 : 0.05 
# True ratio                    : 0.9000 
# Regulatory settings           : EMA (ABEL)
# Switching CVwR                : 0.3 
# Regulatory constant           : 0.76 
# Expanded limits               : 0.7723 ... 1.2948
# Upper scaling cap             : CVwR > 0.5 
# PE constraints                : 0.8000 ... 1.2500
# Empiric TIE for alpha 0.0500  : 0.06557 (rel. change of risk: +31.1%)
# Power for theta0 0.9000       : 0.812
# Iteratively adjusted alpha    : 0.03630
# Empiric TIE for adjusted alpha: 0.05000
# Power for theta0 0.9000       : 0.773 (rel. impact: -4.81%) 
# 
# Runtime    : 6.76 seconds
# Simulations: 7,100,000 (6 iterations)

With \(\small{\alpha_{\textrm{adj}}=0.0363}\) the TIE is controlled. However, we loose some power. We can counteract that in designing the study by means of the function sampleN.scABEL.ad().

CVwR   <- 0.35
design <- "2x2x4"
theta0 <- 0.90
sampleN.scABEL.ad(CV = CVwR, theta0 = theta0,
                  design = design)

# 
# +++++++++++ scaled (widened) ABEL ++++++++++++
#             Sample size estimation
#         for 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.
# 
# Assumed CVwR 0.35, CVwT 0.35
# Nominal alpha      : 0.05 
# True ratio         : 0.9000 
# Target power       : 0.8 
# Regulatory settings: EMA (ABEL)
# Switching CVwR     : 0.3 
# Regulatory constant: 0.76 
# Expanded limits    : 0.7723 ... 1.2948
# Upper scaling cap  : CVwR > 0.5 
# PE constraints     : 0.8000 ... 1.2500
# n  38,   adj. alpha: 0.03610 (power 0.8100), TIE: 0.05000

We need a ~12% larger sample size.

Similarly for RSABE, where we expect an inflated TIE for \(\small{CV_\textrm{wR}<30\%}\). This time spiced with heteroscedasticity (\(\small{CV_\textrm{wR}>CV_\textrm{wT}}\)).

CVw    <- 0.25
CVw    <- signif(CVp2CV(0.25, ratio = 0.80), 4)
design <- "2x2x4"
theta0 <- 0.90
tmp    <- sampleN.RSABE(CV = CVw, theta0 = theta0,
                        design = design, details = FALSE,
                        print = FALSE)
cat("Design        ", design,
    "\nMethod         RSABE (FDA)",
    "\nCVw (R, T)    ", paste(sprintf("%.4f", rev(CVw)), collapse = ", "),
    "\ntheta0        ", theta0,
    "\nSample size   ", tmp[["Sample size"]],
    "\nAchieved power", tmp[["Achieved power"]], "\n\n")
scABEL.ad(CV = CVw, theta0 = theta0,
          design = design, n = tmp[["Sample size"]],
          regulator = "FDA", details = FALSE)

# Design         2x2x4 
# Method         RSABE (FDA) 
# CVw (R, T)     0.2640, 0.2353 
# theta0         0.9 
# Sample size    28 
# Achieved power 0.81882 
# 
# +++++++++++ scaled (widened) ABEL ++++++++++++
#          iteratively adjusted alpha
# (simulations based on intra-subject contrasts)
# ----------------------------------------------
# Study design: 2x2x4 (4 period full replicate)
# log-transformed data (multiplicative model)
# 1,000,000 studies in each iteration simulated.
# 
# CVwR 0.264, CVwT 0.2353, n(i) 14|14 (N 28)
# Nominal alpha                 : 0.05 
# True ratio                    : 0.9000 
# Regulatory settings           : FDA (ABE)
# Switching CVwR                : 0.3
# BE limits                     : 0.8000 ... 1.2500
# PE constraints                : 0.8000 ... 1.2500
# Empiric TIE for alpha 0.0500  : 0.07697
# Power for theta0 0.9000       : 0.819
# Iteratively adjusted alpha    : 0.03053
# Empiric TIE for adjusted alpha: 0.05000
# Power for theta0 0.9000       : 0.747

The method has to be stated in the protocol. No problems are expected from authorities. Even if an assessor doesn’t believe [sic] in an inflated TIE, there are no reasons to object using an apparently more conservative α.
State not only the adjusted α (which is based on an assumed CV_wR and the planned sample size) but make clear that in the study it will be recalculated based on the observed CV_wR and actual sample size.

top of section ↩︎ previous section ↩︎

Molins et al.

Molins et al.²² argued that the true \(\small{\sigma_\textrm{wR}}\) is unknown, i.e. questioned the assumption of \(\small{s_\textrm{wR}=\sigma_\textrm{wR}}\) by Labes and Schütz.¹⁹ Hence, the authors proposed to adjust \(\small{\alpha}\) always for the worst case \(\small{CV_\textrm{wR}=30\%}\), where the largest inflation of the TIE is expected in ABEL.

CVwR   <- 0.50
design <- "2x2x4"
theta0 <- 0.90
tmp    <- sampleN.scABEL(CV = CVwR, theta0 = theta0,
                         design = design, details = FALSE,
                         print = FALSE)
ad     <- scABEL.ad(CV = 0.30, # worst case
                    n = tmp$n,
                    design = design,
                    theta0 = theta0,
                    print = FALSE)
power  <- power.scABEL(alpha = ad$alpha.adj,
                       CV = CVwR,
                       n = tmp$n,
                       design = design,
                       theta0 = theta0)
cat("Design        ", design,
    "\nMethod         ABEL (Molins et al. 2017)",
    "\nCVwR          ", CVwR,
    "\ntheta0        ", theta0,
    "\nSample size   ", tmp[["Sample size"]],
    "\nAdjusted alpha", ad$alpha.adj, "(based on CVwR 0.30)",
    "\nEmpiric TIE   ", ad$TIE.adj,
    "\nAchieved power", power)

# Design         2x2x4 
# Method         ABEL (Molins et al. 2017) 
# CVwR           0.5 
# theta0         0.9 
# Sample size    28 
# Adjusted alpha 0.028865 (based on CVwR 0.30) 
# Empiric TIE    0.05 
# Achieved power 0.74063

Of course, such a conservatism comes with a price. The sample size has to be increased even if an inflated TIE is unlikely (say, for \(\small{CV_\textrm{wR}\geq 45\%}\)).

The method has to be stated in the protocol. No problems are expected from authorities. Even if an assessor doesn’t believe [sic] in an inflated TIE, there are no reasons to object using an apparently more conservative α.
State not only the adjusted α (which is based on the planned sample size) but make clear that in the study it will be recalculated based on the actual sample size. If there are dropouts, slightly less adjustment will be required.

Ocaña and Muñoz

Ocaña and Muñoz²⁴ proposed a method to adjust \(\small{\alpha}\) both for the EMA’s and the FDA’s approaches, which allows also to take heteroscedasticity into account. Like the method of Muñoz et al.²⁰ it applies Howe’s approximation. For the example one gets an adjusted \(\small{\alpha}\) of 0.032407, which is less conservative than the one of Molins et al.²² (0.028865). However, similarly it has to be applied irrespective of the observed \(\small{CV_\textrm{wR}}\) and hence, leads to loss in power.

The R package adjustalpha can be obtained from the authors.

Comparison

Let’s compare the original ABEL and the ad hoc remedies.

\(\small{CV_\textrm{wR}=24\%}\), where we expect only a slight inflation of the TIE.

CVwR   <- 0.24
design <- "2x2x4"
theta0 <- 0.90
tmp    <- sampleN.scABEL(CV = CVwR, theta0 = theta0,
                         design = design, details = FALSE,
                         print = FALSE)
info   <- paste("Design     ", design,
                "\nMethod      ABEL (EMA)",
                "\nCVwR       ", CVwR,
                "\ntheta0     ", theta0,
                "\nSample size", tmp[["Sample size"]], "\n")
comp   <- data.frame(method = c("EMA (2010)",
                                "Bonferroni (1936)",
                                "Muñoz et al (2016)",
                                "Labes and Schütz (2016)",
                                "Molins et al. (2017)",
                                "Ocaña and Muñoz (2019)"),
                     adj = " \u2013 ", alpha = 0.05,
                     TIE.emp = NA, power = NA)
for (i in 1:nrow(comp)) {
  if (comp$method[i] == "EMA (2010)") {
    res <- scABEL.ad(CV = CVwR, n = tmp$n,
                     theta0 = theta0,
                     design = design,
                     print = FALSE)
    comp$TIE.emp[i] <- res$TIE.unadj
    comp$power[i]   <- res$pwr.unadj
  }
  if (comp$method[i] == "Bonferroni (1936)") {
    res <- scABEL.ad(alpha.pre = 0.05 / 2,
                     CV = CVwR, n = tmp$n,
                     theta0 = theta0,
                     design = design,
                     print = FALSE)
    comp$adj[i]     <- "yes"
    comp$alpha[i]   <- res$alpha.pre
    comp$TIE.emp[i] <- res$TIE.unadj
    comp$power[i]   <- res$pwr.unadj
  }
  if (comp$method[i] == "Muñoz et al (2016)") {
    comp$adj[i]     <- "no "
    comp$TIE.emp[i] <- power.RSABE(CV = CVwR,
                                   n = tmp$n,
                                   design = design,
                                   theta0 = scABEL(CV = CVwR)[["upper"]],
                                   regulator = "EMA")
    comp$power[i]   <- power.RSABE(CV = CVwR,
                                   n = tmp$n,
                                   design = design,
                                   theta0 = theta0,
                                   regulator = "EMA")
  }
  if (comp$method[i] == "Labes and Schütz (2016)") {
    res <- scABEL.ad(CV = CVwR, n = tmp$n,
                     theta0 = theta0,
                     design = design,
                     print = FALSE)
    if (is.na(res$alpha.adj)) {
      comp$adj[i]     <- "no "
      comp$alpha[i]   <- res$alpha
      comp$TIE.emp[i] <- res$TIE.unadj
      comp$power[i]   <- res$pwr.unadj
    } else {
      comp$adj[i]     <- "yes"
      comp$alpha[i]   <- res$alpha.adj
      comp$TIE.emp[i] <- res$TIE.adj
      comp$power[i]   <- res$pwr.adj
    }
  }
  if (comp$method[i] == "Molins et al. (2017)") {
    res <- scABEL.ad(CV = 0.30, n = tmp$n,
                     theta0 = theta0,
                     design = design,
                     print = FALSE)
    comp$adj[i]     <- "yes"
    comp$alpha[i]   <- res$alpha.adj
    comp$TIE.emp[i] <- power.scABEL(alpha = res$alpha.adj,
                                    CV = CVwR,
                                    n = tmp$n,
                                    design = design,
                                    theta0 = scABEL(CV = CVwR)[["upper"]])
    comp$power[i]   <- power.scABEL(alpha = res$alpha.adj,
                                    CV = CVwR,
                                    n = tmp$n,
                                    design = design,
                                    theta0 = theta0)
  }
  if (comp$method[i] == "Ocaña and Muñoz (2019)") {
    if (adjustalpha.avail) {
      comp$alpha[i] <- adjAlpha(n = tmp$n, beMethod = "HoweEMA",
                                design =  "TRTR-RTRT", nsim = 1e6,
                                seed = 123456)
    } else {
      comp$alpha[i] <- 0.03255385 # hard-coded
    }
    if (comp$alpha[i] < 0.05) comp$adj[i] <- "yes"
    comp$TIE.emp[i] <- power.RSABE(alpha = comp$alpha[i],
                                   CV = CVwR,
                                   n = tmp$n,
                                   design = design,
                                   theta0 = scABEL(CV = CVwR)[["upper"]],
                                   regulator = "EMA")
    comp$power[i]   <- power.RSABE(alpha = comp$alpha[i],
                                   CV = CVwR,
                                   n = tmp$n,
                                   design = design,
                                   theta0 = theta0,
                                   regulator = "EMA")
  }
}
comp[, 3:5] <- signif(comp[, 3:5], 5)
cat(info)
print(comp, row.names = FALSE)

# Design      2x2x4 
# Method      ABEL (EMA) 
# CVwR        0.24 
# theta0      0.9 
# Sample size 26 
#                   method adj    alpha  TIE.emp   power
#               EMA (2010)  –  0.050000 0.050538 0.81092
#        Bonferroni (1936) yes 0.025000 0.025016 0.71157
#       Muñoz et al (2016) no  0.050000 0.049970 0.79460
#  Labes and Schütz (2016) yes 0.049510 0.050000 0.80957
#     Molins et al. (2017) yes 0.029135 0.029191 0.73476
#   Ocaña and Muñoz (2019) yes 0.032554 0.032290 0.72699

\(\small{CV_\textrm{wR}=30\%}\), where we expect the maximum TIE.

CVwR   <- 0.30
design <- "2x2x4"
theta0 <- 0.90
tmp    <- sampleN.scABEL(CV = CVwR, theta0 = theta0,
                         design = design, details = FALSE,
                         print = FALSE)
info   <- paste("Design     ", design,
                "\nMethod      ABEL (EMA)",
                "\nCVwR       ", CVwR,
                "\ntheta0     ", theta0,
                "\nSample size", tmp[["Sample size"]], "\n")
comp   <- data.frame(method = c("EMA (2010)",
                                "Bonferroni (1936)",
                                "Muñoz et al (2016)",
                                "Labes and Schütz (2016)",
                                "Molins et al. (2017)",
                                "Ocaña and Muñoz (2019)"),
                     adj = " \u2013 ", alpha = 0.05,
                     TIE.emp = NA, power = NA)
for (i in 1:nrow(comp)) {
  if (comp$method[i] == "EMA (2010)") {
    res <- scABEL.ad(CV = CVwR, n = tmp$n,
                     theta0 = theta0,
                     design = design,
                     print = FALSE)
    comp$TIE.emp[i] <- res$TIE.unadj
    comp$power[i]   <- res$pwr.unadj
  }
  if (comp$method[i] == "Bonferroni (1936)") {
    res <- scABEL.ad(alpha.pre = 0.05 / 2,
                     CV = CVwR, n = tmp$n,
                     theta0 = theta0,
                     design = design,
                     print = FALSE)
    comp$adj[i]     <- "yes"
    comp$alpha[i]   <- res$alpha.pre
    comp$TIE.emp[i] <- res$TIE.unadj
    comp$power[i]   <- res$pwr.unadj
  }
  if (comp$method[i] == "Muñoz et al (2016)") {
    comp$adj[i]     <- "no "
    comp$TIE.emp[i] <- power.RSABE(CV = CVwR,
                                   n = tmp$n,
                                   design = design,
                                   theta0 = scABEL(CV = CVwR)[["upper"]],
                                   regulator = "EMA")
    comp$power[i]   <- power.RSABE(CV = CVwR,
                                   n = tmp$n,
                                   design = design,
                                   theta0 = theta0,
                                   regulator = "EMA")
  }
  if (comp$method[i] == "Labes and Schütz (2016)") {
    res <- scABEL.ad(CV = CVwR, n = tmp$n,
                     theta0 = theta0,
                     design = design,
                     print = FALSE)
    if (is.na(res$alpha.adj)) {
      comp$adj[i]     <- "no "
      comp$alpha[i]   <- res$alpha
      comp$TIE.emp[i] <- res$TIE.unadj
      comp$power[i]   <- res$pwr.unadj
    } else {
      comp$adj[i]     <- "yes"
      comp$alpha[i]   <- res$alpha.adj
      comp$TIE.emp[i] <- res$TIE.adj
      comp$power[i]   <- res$pwr.adj
    }
  }
  if (comp$method[i] == "Molins et al. (2017)") {
    res <- scABEL.ad(CV = 0.30, n = tmp$n,
                     theta0 = theta0,
                     design = design,
                     print = FALSE)
    comp$adj[i]     <- "yes"
    comp$alpha[i]   <- res$alpha.adj
    comp$TIE.emp[i] <- power.scABEL(alpha = res$alpha.adj,
                                    CV = CVwR,
                                    n = tmp$n,
                                    design = design,
                                    theta0 = scABEL(CV = CVwR)[["upper"]])
    comp$power[i]   <- power.scABEL(alpha = res$alpha.adj,
                                    CV = CVwR,
                                    n = tmp$n,
                                    design = design,
                                    theta0 = theta0)
  }
  if (comp$method[i] == "Ocaña and Muñoz (2019)") {
    if (adjustalpha.avail) {
      comp$alpha[i] <- adjAlpha(n = tmp$n, beMethod = "HoweEMA",
                                design =  "TRTR-RTRT", nsim = 1e6,
                                seed = 123456)
    } else {
      comp$alpha[i] <- 0.03225796 # hard-coded
    }
    if (comp$alpha[i] < 0.05) comp$adj[i] <- "yes"
    comp$TIE.emp[i] <- power.RSABE(alpha = comp$alpha[i],
                                   CV = CVwR,
                                   n = tmp$n,
                                   design = design,
                                   theta0 = scABEL(CV=CVwR)[["upper"]],
                                   regulator = "EMA")
    comp$power[i]   <- power.RSABE(alpha = comp$alpha[i],
                                   CV = CVwR,
                                   n = tmp$n,
                                   design = design,
                                   theta0 = theta0,
                                   regulator = "EMA")
  }
}
comp[, 3:5] <- signif(comp[, 3:5], 5)
cat(info)
print(comp, row.names = FALSE)

# Design      2x2x4 
# Method      ABEL (EMA) 
# CVwR        0.3 
# theta0      0.9 
# Sample size 34 
#                   method adj    alpha  TIE.emp   power
#               EMA (2010)  –  0.050000 0.081626 0.80281
#        Bonferroni (1936) yes 0.025000 0.044427 0.70537
#       Muñoz et al (2016) no  0.050000 0.068360 0.76872
#  Labes and Schütz (2016) yes 0.028572 0.050000 0.72506
#     Molins et al. (2017) yes 0.028572 0.050000 0.72506
#   Ocaña and Muñoz (2019) yes 0.032258 0.044960 0.69600

\(\small{CV_\textrm{wR}=50\%}\), where we expect no inflation of the TIE.

CVwR   <- 0.50
design <- "2x2x4"
theta0 <- 0.90
tmp    <- sampleN.scABEL(CV = CVwR, theta0 = theta0,
                         design = design, details = FALSE,
                         print = FALSE)
info   <- paste("Design     ", design,
                "\nMethod      ABEL (EMA)",
                "\nCVwR       ", CVwR,
                "\ntheta0     ", theta0,
                "\nSample size", tmp[["Sample size"]], "\n")
comp   <- data.frame(method = c("EMA (2010)",
                                "Bonferroni (1936)",
                                "Muñoz et al (2016)",
                                "Labes and Schütz (2016)",
                                "Molins et al. (2017)",
                                "Ocaña and Muñoz (2019)"),
                     adj = " \u2013 ", alpha = 0.05,
                     TIE.emp = NA, power = NA)
for (i in 1:nrow(comp)) {
  if (comp$method[i] == "EMA (2010)") {
    res <- scABEL.ad(CV = CVwR, n = tmp$n,
                     theta0 = theta0,
                     design = design,
                     print = FALSE)
    comp$TIE.emp[i] <- res$TIE.unadj
    comp$power[i]   <- res$pwr.unadj
  }
  if (comp$method[i] == "Bonferroni (1936)") {
    res <- scABEL.ad(alpha.pre = 0.05 / 2,
                     CV = CVwR, n = tmp$n,
                     theta0 = theta0,
                     design = design,
                     print = FALSE)
    comp$adj[i]     <- "yes"
    comp$alpha[i]   <- res$alpha.pre
    comp$TIE.emp[i] <- res$TIE.unadj
    comp$power[i]   <- res$pwr.unadj
  }
  if (comp$method[i] == "Muñoz et al (2016)") {
    comp$adj[i]     <- "no "
    comp$TIE.emp[i] <- power.RSABE(CV = CVwR,
                                   n = tmp$n,
                                   design = design,
                                   theta0 = scABEL(CV = CVwR)[["upper"]],
                                   regulator = "EMA")
    comp$power[i]   <- power.RSABE(CV = CVwR,
                                   n = tmp$n,
                                   design = design,
                                   theta0 = theta0,
                                   regulator = "EMA")
  }
  if (comp$method[i] == "Labes and Schütz (2016)") {
    res <- scABEL.ad(CV = CVwR, n = tmp$n,
                     theta0 = theta0,
                     design = design,
                     print = FALSE)
    if (is.na(res$alpha.adj)) {
      comp$adj[i]     <- "no "
      comp$alpha[i]   <- res$alpha
      comp$TIE.emp[i] <- res$TIE.unadj
      comp$power[i]   <- res$pwr.unadj
    } else {
      comp$adj[i]     <- "yes"
      comp$alpha[i]   <- res$alpha.adj
      comp$TIE.emp[i] <- res$TIE.adj
      comp$power[i]   <- res$pwr.adj
    }
  }
  if (comp$method[i] == "Molins et al. (2017)") {
    res <- scABEL.ad(CV = 0.30, n = tmp$n,
                     theta0 = theta0,
                     design = design,
                     print = FALSE)
    comp$adj[i]     <- "yes"
    comp$alpha[i]   <- res$alpha.adj
    comp$TIE.emp[i] <- power.scABEL(alpha = res$alpha.adj,
                                    CV = CVwR,
                                    n = tmp$n,
                                    design = design,
                                    theta0 = scABEL(CV = CVwR)[["upper"]])
    comp$power[i]   <- power.scABEL(alpha = res$alpha.adj,
                                    CV = CVwR,
                                    n = tmp$n,
                                    design = design,
                                    theta0 = theta0)
  }
  if (comp$method[i] == "Ocaña and Muñoz (2019)") {
    if (adjustalpha.avail) {
      comp$alpha[i] <- adjAlpha(n = tmp$n, beMethod = "HoweEMA",
                                design =  "TRTR-RTRT", nsim = 1e6,
                                seed = 123456)
    } else {
      comp$alpha[i] <- 0.03255385 # hard-coded
    }
    if (comp$alpha[i] < 0.05) comp$adj[i] <- "yes"
    comp$TIE.emp[i] <- power.RSABE(alpha = comp$alpha[i],
                                   CV = CVwR,
                                   n = tmp$n,
                                   design = design,
                                   theta0 = scABEL(CV = CVwR)[["upper"]],
                                   regulator = "EMA")
    comp$power[i]   <- power.RSABE(alpha = comp$alpha[i],
                                   CV = CVwR,
                                   n = tmp$n,
                                   design = design,
                                   theta0 = theta0,
                                   regulator = "EMA")
  }
}
comp[, 3:5] <- signif(comp[, 3:5], 5)
cat(info); print(comp, row.names = FALSE)

# Design      2x2x4 
# Method      ABEL (EMA) 
# CVwR        0.5 
# theta0      0.9 
# Sample size 28 
#                   method adj    alpha  TIE.emp   power
#               EMA (2010)  –  0.050000 0.033171 0.81428
#        Bonferroni (1936) yes 0.025000 0.015605 0.72004
#       Muñoz et al (2016) no  0.050000 0.029200 0.77748
#  Labes and Schütz (2016) no  0.050000 0.033171 0.81428
#     Molins et al. (2017) yes 0.028865 0.018291 0.74063
#   Ocaña and Muñoz (2019) yes 0.032407 0.018870 0.70922

top of section ↩︎ previous section ↩︎

Pros and Cons

Bonferroni is conservative and requires large sample sizes in order to maintain the target power.
I suggest to call scABEL.ad(alpha.pre = 0.025, ...) beforehand to be protected against the rare cases where more adjustment will be required.
The approach of Muñoz et al.¹⁸ leads still to an inflated TIE – although less than in the original ABEL and its critical region is narrower. It would require to change the statistical model in all regulatory guidelines currently recommending ABEL.
According to Labes and Schütz¹⁹ one has to adjust α only if an inflated TIE is expected. Such cases require a larger sample size than the original ABEL.
However, the method follows the plug-in principle, i.e., assumes that the observed CV_wR is the true one.
The approach of Molins et al.²² is more conservative and will require a larger sample size (particularly if no inflation of the TIE is expected).
The approach of Ocaña and Muñoz²⁴ controls the TIE but shows a loss in power similar to the one of Molins et al.²² Like the approach of Muñoz _et al.¹⁸, it would require to change the statistical model in all regulatory guidelines currently recommending ABEL.

Say, we want to perform a study in a four-period full replicate design for ABEL, assume \(\small{CV_\textrm{wR}=CV_\textrm{wT}=0.35}\), a T/R-ratio of 0.90 targeted at 0.80. How will the approaches perform in terms of controlling the TIE and maintaining power?
Note that I did not consider the approach of Muñoz et al.¹⁸ because it does not control the TIE.

# Cave: Very long runtime
CV.plan <- 0.35
# theta0 = 0.90 and targetpower = 0.80 are defaults
n    <- sampleN.scABEL(CV = CV.plan, design = "2x2x4",
                       details = FALSE,
                       print = FALSE)[["Sample size"]]
CV   <- sort(c(se2CV(0.294),
               seq(0.15, 0.65, 0.005)))
comp <- data.frame(method = rep(c("EMA (2010)",
                                  "Bonferroni (1936)",
                                  "Labes and Schütz (2016)",
                                  "Molins et al. (2017)",
                                  "Ocaña and Muñoz (2019)"),
                                  each = length(CV)),
                   CV = CV, alpha = 0.05, TIE = NA, power = NA)
if ("adjustalpha" %in% rownames(installed.packages())) {
  library(adjustalpha)
  adjustalpha.avail = TRUE
} else {
  adjustalpha.avail = FALSE
}
for (i in 1:nrow(comp)) {
  if (comp$method[i] == "EMA (2010)") {
    res <- scABEL.ad(CV = comp$CV[i], n = n,
                     design = "2x2x4", print = FALSE)
    comp$TIE[i]   <- res$TIE.unadj
    comp$power[i] <- res$pwr.unadj
  }
  if (comp$method[i] == "Bonferroni (1936)") {
    res <- scABEL.ad(alpha.pre = 0.05 / 2,
                     CV = comp$CV[i], n = n,
                     design = "2x2x4", print = FALSE)
    comp$alpha[i] <- res$alpha.pre
    comp$TIE[i]   <- res$TIE.unadj
    comp$power[i] <- res$pwr.unadj
  }
  if (comp$method[i] == "Labes and Schütz (2016)") {
    res <- scABEL.ad(CV = comp$CV[i], n = n,
                     design = "2x2x4", print = FALSE)
    if (is.na(res$alpha.adj)) {
      comp$alpha[i] <- res$alpha
      comp$TIE[i]   <- res$TIE.unadj
      comp$power[i] <- res$pwr.unadj
    } else {
      comp$alpha[i] <- res$alpha.adj
      comp$TIE[i]   <- res$TIE.adj
      comp$power[i] <- res$pwr.adj
    }
  }
  if (comp$method[i] == "Molins et al. (2017)") {
    res <- scABEL.ad(CV = 0.30, n = n,
                     design = "2x2x4", print = FALSE)
    comp$alpha[i] <- res$alpha.adj
    theta2        <- scABEL(CV = comp$CV[i])[["upper"]]
    comp$TIE[i]   <- power.scABEL(alpha = res$alpha.adj,
                                  CV = comp$CV[i], n = n,
                                  design = "2x2x4",
                                  theta0 = theta2)
    comp$power[i] <- power.scABEL(alpha = res$alpha.adj,
                                  CV = comp$CV[i], n = n,
                                  design = "2x2x4")
  }
  if (comp$method[i] == "Ocaña and Muñoz (2019)") {
    if (adjustalpha.avail) {
      comp$alpha[i] <- adjAlpha(n = n, beMethod = "HoweEMA",
                                design =  "TRTR-RTRT", nsim = 1e6,
                                seed = 123456)
    } else {
      comp$alpha[i] <- 0.03225796 # hard-coded
    }
    theta2        <- scABEL(CV = comp$CV[i])[["upper"]]
    comp$TIE[i]   <- power.RSABE(alpha = comp$alpha[i],
                                 CV = comp$CV[i], n = n,
                                 design = "2x2x4",
                                 theta0 = theta2,
                                 regulator = "EMA")
    comp$power[i] <- power.RSABE(alpha = comp$alpha[i],
                                 CV = comp$CV[i], n = n,
                                 design = "2x2x4",
                                 regulator = "EMA")
  }
}
col <- c("#FF000080", "#00800080", "#FF00FF80",
         "#0000FF80", "#A9A9A980")
### TIE ###
dev.new(width = 4.5, height = 4.5, record = TRUE)
op   <- par(no.readonly = TRUE)
par(mar = c(4, 3.9, 0.1, 0.1), cex.axis = 0.9)
ylim <- c(0, max(comp$TIE))
xlim <- c(0.2, 0.6)
plot(CV, comp$TIE[comp$method == "EMA (2010)"], type = "n",
     xlab = expression(italic(CV)[wR]), xlim = xlim, ylim = ylim,
     ylab = "empiric Type I Error", axes = FALSE)
grid()
abline(v = c(0.3, 0.5), lty = 2)
abline(h = 0.05)
axis(1, at = pretty(CV),
     labels = sprintf("%i%%", as.integer(100*pretty(CV))))
axis(2, las = 1)
legend("bottomleft", cex = 0.9, box.lty = 0, bg = "white", col = col,
       legend = c("EMA (2010)", "Bonferroni (1936)",
                  "Labes and Schütz (2016)",
                  expression("Molins "*italic("et al")*". (2017)"),
                  "Ocaña and Muñoz (2019)"), lwd = 3, inset = 0.01)
box()
lines(CV, comp$TIE[comp$method == "EMA (2010)"],
      lwd = 3, col = col[1])
lines(CV, comp$TIE[comp$method == "Bonferroni (1936)"],
      lwd = 3, col = col[2])
lines(CV, comp$TIE[comp$method == "Labes and Schütz (2016)"],
      lwd = 3, col = col[3])
lines(CV, comp$TIE[comp$method == "Molins et al. (2017)"],
      lwd = 3, col = col[4])
lines(CV, comp$TIE[comp$method == "Ocaña and Muñoz (2019)"],
      lwd = 3, col = col[5])

### power ###
ylim <- range(comp$power)
xlim <- c(0.2, 0.6)
plot(CV, comp$power[comp$method == "EMA (2010)"], type = "n",
     xlab = expression(italic(CV)[wR]), xlim = xlim, ylim = ylim,
     ylab = "empiric power", axes = FALSE)
grid()
abline(v = c(0.3, 0.5), lty = 2)
abline(h = 0.80)
axis(1, at = pretty(CV),
     labels = sprintf("%i%%", as.integer(100*pretty(CV))))
axis(2, las = 1)
legend("topright", cex = 0.9, box.lty = 0, bg = "white", col = col,
       legend = c("EMA (2010)", "Bonferroni (1936)",
                  "Labes and Schütz (2016)",
                  expression("Molins "*italic("et al")*". (2017)"),
                  "Ocaña and Muñoz (2019)"), lwd = 3, inset = 0.01)
box()
lines(CV, comp$power[comp$method == "EMA (2010)"],
      lwd = 3, col = col[1])
points(CV.plan, comp$power[comp$method == "EMA (2010)" &
                           comp$CV == CV.plan],
       pch = 19, cex=1.35, col = col[1])
lines(CV, comp$power[comp$method == "Bonferroni (1936)"],
      lwd = 3, col = col[2])
points(CV.plan, comp$power[comp$method == "Bonferroni (1936)" &
                           comp$CV == CV.plan],
       pch = 19, cex=1.35, col = col[2])
lines(CV, comp$power[comp$method == "Labes and Schütz (2016)"],
      lwd = 3, col = col[3])
points(CV.plan, comp$power[comp$method == "Labes and Schütz (2016)" &
                           comp$CV == CV.plan],
       pch = 19, cex=1.35, col = col[3])
lines(CV, comp$power[comp$method == "Molins et al. (2017)"],
      lwd = 3, col = col[4])
points(CV.plan, comp$power[comp$method == "Molins et al. (2017)" &
                           comp$CV == CV.plan],
       pch = 19, cex=1.35, col = col[4])
lines(CV, comp$power[comp$method == "Ocaña and Muñoz (2019)"],
      lwd = 3, col = col[5])
points(CV.plan, comp$power[comp$method == "Ocaña and Muñoz (2019)" &
                           comp$CV == CV.plan],
       pch = 19, cex=1.35, col = col[5])
par(op)

Fig. 8 TIE of the approaches in a 4-period full replicate design (n 34).

As expected, we see an inflated TIE with the original ABEL in the critical region and full control with Bonferroni. The method of Labes and Schütz¹⁹ adjusts \(\small{\alpha}\) only in the critical region. The methods of Molins et al.²² and Ocaña and Muñoz²⁴ control the TIE as well.

Fig. 9 Power of the approaches in a 4-period full replicate design (n 34).

Since we planned for ABEL, we will achieve a power of 0.812 if the assumed \(\small{CV_\textrm{wR}}\) is realized in the study.
With Bonferroni power will be 0.722, with the method of Labes and Schütz¹⁹ 0.773, with the one of Molins et al.²² 0.740, and with the one of Ocaña and Muñoz²⁴ 0.695.

Which sample sizes will we need and which sample size penalty will we face?

CV   <- 0.35
comp <- data.frame(method = rep(c("EMA (2010)",
                                  "Bonferroni (1936)",
                                  "Labes and Schütz (2016)",
                                  "Molins et al. (2017)",
                                  "Ocaña and Muñoz (2019)"),
                                  each = length(CV)),
                   alpha = 0.05, n = NA_integer_, power = NA_real_)
for (i in 1:nrow(comp)) {
  if (comp$method[i] == "EMA (2010)") {
    comp[i, 3:4] <- sampleN.scABEL(CV = CV,
                                   design = "2x2x4",
                                   details = FALSE,
                                   print = FALSE)[8:9]
  }
  if (comp$method[i] == "Bonferroni (1936)") {
    comp[i, 2]   <- 0.025
    comp[i, 3:4] <- sampleN.scABEL(alpha = 0.025,
                                   CV = CV,
                                   design = "2x2x4",
                                   details = FALSE,
                                   print = FALSE)[8:9]
  }
  if (comp$method[i] == "Labes and Schütz (2016)") {
    res          <- sampleN.scABEL.ad(CV = CV,
                                      design = "2x2x4",
                                      details = FALSE,
                                      print = FALSE)
    comp[i, 2]   <- res$alpha.adj
    comp[i, 3:4] <- res[c(12, 14)]
  }
  if (comp$method[i] == "Molins et al. (2017)") {
    res        <- scABEL.ad(CV = 0.30, n = comp[1, 3],
                            design = "2x2x4",
                            details = FALSE,
                            print = FALSE)
    comp[i, 2] <- res$alpha.adj
    comp[i, 3] <- sampleN.scABEL.ad(alpha.pre = res$alpha.adj,
                                    CV = CV,
                                    design = "2x2x4",
                                    details = FALSE,
                                    print = FALSE)[["Sample size"]]
    comp[i, 4] <- power.scABEL(alpha = res$alpha.adj,
                               CV = CV, n = comp[i, 3],
                               design = "2x2x4")
  }
  if (comp$method[i] == "Ocaña and Muñoz (2019)") {
    if (adjustalpha.avail) {
      comp[i, 2] <- adjAlpha(n = comp[1, 3], beMethod = "HoweEMA",
                             design =  "TRTR-RTRT", nsim = 1e6,
                             seed = 123456)
    } else {
      comp[i, 2] <- 0.03225796 # hard-coded
    }
    res        <- sampleN.RSABE(alpha = comp[i, 2],
                                CV = CV, design = "2x2x4",
                                regulator = "EMA",
                                details = FALSE,
                                print = FALSE)
    comp[i, 3] <- res[["Sample size"]]
    comp[i, 4] <- res[["Achieved power"]]
  }
}
comp$alpha   <- signif(comp$alpha, 5)
comp$power   <- signif(comp$power, 3)
comp$penalty <- sprintf("%+.1f%%", 100 * (comp$n - comp$n[1]) / comp$n[1])
print(comp, row.names = FALSE)

#                   method    alpha  n power penalty
#               EMA (2010) 0.050000 34 0.812   +0.0%
#        Bonferroni (1936) 0.025000 42 0.803  +23.5%
#  Labes and Schütz (2016) 0.036100 38 0.810  +11.8%
#     Molins et al. (2017) 0.028572 40 0.802  +17.6%
#   Ocaña and Muñoz (2019) 0.032258 46 0.813  +35.3%

top of section ↩︎ previous section ↩︎

‘Exact’

Tóthfalusi and Endrényi^{20 21} proposed an ‘exact’ procedure with a bias correction,³³ ³⁴ where the EMA’s regulatory constant \(\small{k=0.760}\) is kept but the upper cap of scaling at \(\small{CV_\textrm{wR}=50\%}\) and the PE-constraint are abandoned.

# 'Pure' EMA settings (without upper cap of expansion
# and PE constraint)
reg           <- reg_const("EMA")
reg$CVcap     <- Inf
reg$pe_constr <- FALSE
reg$name      <- "pure EMA"
# Cave: Subject data simulations = extremely long runtime
CV     <- 0.30
n      <- sampleN.scABEL(CV = CV, design = "2x2x4",
                         details = FALSE,
                         print = FALSE)[["Sample size"]]
CV     <- sort(c(se2CV(0.294),
                 seq(0.15, 0.65, 0.005)))
comp   <- data.frame(method = rep(c("EMA (2010)",
                                    "exact"),
                                  each = length(CV)),
                     CV = CV, TIE = NA)
theta2 <- c(scABEL(CV = CV, regulator = "EMA")[, "upper"],
            scABEL(CV = CV, regulator = reg)[, "upper"])
for (i in 1:nrow(comp)) {
  if (comp$method[i] == "EMA (2010)") {
    comp$TIE[i] <- power.scABEL(CV = comp$CV[i], n = n,
                                design = "2x2x4",
                                theta0 = theta2[i],
                                nsims = 1e6)
  }
  if (comp$method[i] == "exact") {
    comp$TIE[i] <- power.RSABE2L.sdsims(CV = comp$CV[i], n = n,
                                        design = "2x2x4",
                                        theta0 = theta2[i],
                                        regulator = reg,
                                        SABE_test = "exact",
                                        nsims = 1e6,
                                        progress = FALSE)
  }
  setTxtProgressBar(pb, i/nrow(comp))
}
dev.new(width = 4.5, height = 4.5, record = TRUE)
op   <- par(no.readonly = TRUE)
par(mar = c(4, 3.9, 0.1, 0.1), cex.axis = 0.9)
ylim <- c(0, max(comp$TIE))
xlim <- c(0.2, 0.6)
plot(CV, comp$TIE[comp$method == "EMA (2010)"], type = "n",
     xlab = expression(italic(CV)[wR]), xlim = xlim, ylim = ylim,
     ylab = "empiric Type I Error", axes = FALSE)
grid()
abline(v = c(0.3, 0.5), lty = 2)
abline(h = 0.05)
axis(1, at = pretty(CV),
     labels = sprintf("%i%%", as.integer(100*pretty(CV))))
axis(2, las = 1)
legend("bottomleft",
       legend = c("EMA (2010)",
                  "Tóthfalusi & Endrényi (2017)"),
       col = c("#FF000080", "#FF00FF80"),
       lwd = 3, cex = 0.9, box.lty = 0, bg = "white", inset = 0.02)
box()
lines(CV, comp$TIE[comp$method == "EMA (2010)"],
      lwd = 3, col = "#FF000080")
lines(CV, comp$TIE[comp$method == "exact"],
      lwd = 3, col = "#FF00FF80")
par(op)

Fig. 10 TIE of the approaches in a 4-period full replicate design (n 34).

With a maximum TIE of 0.0671 it performs better than the original ABEL (0.0816) and similarly to Muñoz et al.¹⁸ (0.0684). Furthermore, its critical region is slightly narrower (\(\small{CV_\textrm{wR}}\) up to 33.7% vs 35.7%).

The authors discussed a combination with the iterative adjustment proposed by Labes and Schütz¹⁹ in the critical region, which would provide a smaller sample size penalty.

# Extremely (!) long runtime due to subject data
# simulations and potential iterative alpha-adjustment
limits <- function(CV, regulator) {
  thetas        <- scABEL(CV = CV, regulator = regulator)
  names(thetas) <- c("lower", "upper")
  return(thetas)
}
check.TIE <- function(alpha = 0.05, CV, n, design,
                      regulator) {
  thetas <- limits(CV = CV, regulator = regulator)
  power.RSABE2L.sdsims(CV = CV, n = n,
                       theta0 = theta2,
                       design = design,
                       regulator = regulator,
                       SABE_test = "exact",
                       nsims = 1e6,
                       progress = FALSE)
}
check.power <- function(alpha = 0.05, CV, n, design,
                        theta0, regulator) {
  thetas <- limits(CV = CV, regulator = regulator)
  power.RSABE2L.sdsims(CV = CV, n = n,
                       theta0 = theta0,
                       design = design,
                       regulator = regulator,
                       SABE_test = "exact")
}
opt <- function(x) {
  power.RSABE2L.sdsims(alpha = x, CV = CV, n = n,
                       theta0 = theta2, design = design,
                       regulator = reg,
                       SABE_test = "exact", nsims = 1e6,
                       progress = FALSE) - alpha
}
# 'Pure' EMA settings (without upper cap of expansion
# and PE constraint)
reg           <- reg_const("EMA")
reg$CVcap     <- Inf
reg$pe_constr <- FALSE
reg$name      <- "pure EMA"
alpha         <- 0.05
CV            <- 0.35
theta0        <- 0.90
design        <- "2x2x4"
target        <- 0.80
theta2        <- scABEL(CV = CV, regulator = reg)[["upper"]]
step          <- as.integer(substr(design, 3, 3))
alpha.adj     <- NA
iter          <- 0
# start with Muñoz' approach
n             <- sampleN.RSABE(CV = CV, design = design,
                               theta0 = theta0,
                               targetpower = target,
                               regulator = "EMA",
                               details = FALSE,
                               print = FALSE)[["Sample size"]]
TIE           <- check.TIE(CV = CV, n = n, design = design,
                           regulator = reg)
power         <- check.power(CV = CV, n = n, design = design,
                             theta0 = theta0,
                             regulator = reg)
res           <- data.frame(iter = iter, n = n,
                            alpha.adj = NA,
                            TIE = TIE, power = power)
if (TIE > alpha) {# inflated TIE with first guess n
   iter      <- iter + 1
   theta2    <- limits(CV = CV,
                       regulator = regulator)[["upper"]]
   x         <- uniroot(opt, interval = c(0.025, alpha),
                        tol = 1e-8)
   alpha.adj <- x$root
   TIE       <- x$f.root + alpha
   power     <- check.power(alpha = alpha.adj,
                            CV = CV, n = n,
                            design = design,
                            theta0 = theta0,
                            regulator = reg)
   res[iter+1, ] <- c(iter, n, alpha.adj, TIE, power)
} else {          # TIE OK, try a lower sample size
  repeat {
    iter  <- iter + 1
    n     <- n - step
    power <- check.power(CV = CV, n = n,
                         design = design,
                         theta0 = theta0,
                         regulator = reg)
    TIE   <- check.TIE(CV = CV, n = n,
                       design = design,
                       regulator = reg)
    if (TIE > alpha) {
      x         <- uniroot(opt, interval = c(0.025, alpha),
                           tol = 1e-8)
      alpha.adj <- x$root
      TIE       <- x$f.root + alpha
    }
    res[iter+1, ] <- c(iter, n, alpha.adj, TIE, power)
    if (power >= target) {
      break
    }
  }
}
est  <- max(which(res$n == min(res$n) &
                  res$power >= target))
txt1 <- paste("Sample size   ", res$n[est],
              "\nAchieved power", res$power[est])
txt2 <- paste("\nEmpiric TIE   ", res$TIE[est], "\n")
if (is.na(res$alpha[est])) {
  cat(paste(txt1, "\nAlpha         ", alpha), txt2)
} else {
  cat(paste(txt1, "\nAdjusted alpha",
            signif(res$alpha.adj[est], 4),
            txt2))
}

In the example above we would need no \(\small{\alpha}\)-adjustment because the \(\small{CV_\textrm{wR}}\) is above the critical region. With a sample size of 36 (TIE 0.0486) we need two subjects less than by the approach of Labes and Schütz.¹⁹

Its implementation would require a major revision of the regulatory approach. Although statistically appealing, it will currently not be accepted by regulatory agencies.

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Acknowledgment

Jordi Ocaña (Department of Genetics, Microbiology and Statistics, Universitat de Barcelona, Spain) and Joel Muñoz (Faculty of Physical and Mathematical Sciences, Department of Statistics, Universidad de Concepción, Chile) for fruitful discussions and providing the packages simcrossover and adjustalpha.

Licenses

Helmut Schütz 2022
R, PowerTOST, simcrossover, and adjustalpha GPL 3.0, pandoc GPL 2.0.
1^st version May 04, 2021. Rendered April 28, 2022 09:42 CEST by rmarkdown via pandoc in 1.14 seconds.

Footnotes and References

Labes D, Schütz H, Lang B. PowerTOST: Power and Sample Size for (Bio)Equivalence Studies. Package version 1.5.4. 2022-02-21. CRAN ↩︎
Ocaña J. simcrossover: Data Generation from a Crossover Design as a data.frame. Package version 0.2.0. 2018-05-26.↩︎
Ocaña J. adjustalpha: Significance level adjustment in Scaled Average Bioequivalence. Package version 0.9.0. 2018-10-02.↩︎
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Commission of the EC. Note for Guidance. Investigation of Bioavailability and Bioequivalence. Appendix III: Technical Aspects of Bioequivalence Statistics. Brussels. December 1991.↩︎
Commission of the EC. Note for Guidance. Investigation of Bioavailability and Bioequivalence. Appendix III: Technical Aspects of Bioequivalence Statistics. 3CC15a. Brussels. June 1992. Download.↩︎
EMA, CHMP Efficacy Working Party, Therapeutic Subgroup on Pharmacokinetics (EWP-PK). Questions & Answers on the Bioavailability and Bioequivalence Guideline. EMEA/CHMP/EWP/40326/2006. London. 27 July 2006. Online.↩︎
Benet L. Why Highly Variable Drugs are Safer. Presentation at the FDA Advisory Committee for Pharmaceutical Science. Rockville. 06 October, 2006. Internet Archive.↩︎
Schütz H. ABEL: Type I Error. Vienna: Bioequivalence and Bioavailability Forum; 28 September 2020.↩︎
EMA. Clinical pharmacology and pharmacokinetics: questions and answers. 3.1 Which statistical method for the analysis of a bioequivalence study does the Agency recommend? Annex I. EMA/582648/2016. London. 21 September 2016. Online.↩︎
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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.↩︎
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Tóthfalusi L, Endrényi L. Algorithms for Evaluating Reference Scaled Average Bioequivalence: Power, Bias, and Consumer Risk. Stat Med. 2017; 36(27): 4378–90. doi:10.1002/sim.7440.↩︎
Molins E, Cobo E, Ocaña J. Two-Stage Designs Versus European Scaled Average Designs in Bioequivalence Studies for Highly Variable Drugs: Which to Choose? Stat Med. 2017; 36(30): 4777–88. doi:10.1002/sim.7452.↩︎
Endrényi L, Tóthfalusi L. Bioequivalence for highly variable drugs: regulatory agreements, disagreements, and harmonization. J Pharmacokin Pharmacodyn. 2019; 46(2): 117–26. doi:10.1007/s10928-019-09623-w.↩︎
Ocaña J, Muñoz J. Controlling type I error in the reference-scaled bioequivalence evaluation of highly variable drugs. Pharm Stat. 2019; 18(5): 583–99. doi:10.1002/pst.1950.↩︎
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Schütz H, Labes D, Wolfsegger MJ. Critical remarks on reference-scaled average bioequivalence. Manuscript in preparation. 2022.↩︎
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FDA, OGD. Draft Guidance on Progesterone. Rockville. Recommended April 2010, Revised February 2011. Download.↩︎
FDA, CDER. Bioequivalence Studies With Pharmacokinetic Endpoints for Drugs Submitted Under an ANDA. Rockville. August 2021. Download.↩︎
Detlew Labes and László Endrényi, just to name some.↩︎
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The authors may forgive me using quotation marks. Hedges’ correction is an approximation.↩︎

Inflation of the Type I Error in Scaled Average Bioequivalence

Helmut Schütz

April 28, 2022

Introduction

Preliminaries

Type I Error

Remedies?

Ad hoc

Bonferroni

Muñoz et al.

Labes and Schütz

Molins et al.

Ocaña and Muñoz

Comparison

Pros and Cons

‘Exact’