Introduction

Can we plan a study intended for Scaled Average Bioequivalence based on the within-subject CV of a Crossover Design?

 You could, although it would not be a particularly good idea. 

Regardless whether aiming at Average Bioequivalence with Expanding Limits (ABEL)^{2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17} or Reference-Scaled Average Bioequivalence (RSABE),^{18 19} we are lacking the most inportant information, namely the variability of the reference product.

Examples

Say, we had prior information about some CVs. We design 2×2×2 studies based on them, assuming a T/R-ratio of 0.95 and target 80% power. Let’s further assume that the outcome was exactly like assumed.

The \(\small{\alpha}\) confidence interval of the within-subject CV is obtained via the \(\small{\chi^2}\)-distribution of its error variance \(\small{\sigma_\text{w}^2}\) with \(\small{n-2}\) degrees of freedom, where \(\small{n}\) is the total number of subjects in the study. \[\eqalign{\tag{1} s_\text{w}^2&=\log_{e}(CV_\text{w}^2+1)\\ L=\frac{(n-1)\,s_\text{w}^2}{\chi_{\alpha/2,\,n-2}^{2}}&\leq\sigma_\text{w}^2\leq\frac{(n-1)\,s_\text{w}^2}{\chi_{1-\alpha/2,\,n-2}^{2}}=U\\ \left\{lower\;CL,\;upper\;CL\right\}&=\left\{\sqrt{\exp(L)-1},\sqrt{\exp(U)-1}\right\} }\]

library(PowerTOST)
# CVw in a nonreplicated crossover
CV    <- seq(0.30, 0.45, 0.05)
# sample size for theta0 = 0.95, targetpower = 0.8, design = "2x2"
n     <- numeric()
for (i in seq_along(CV)) {
  n[i] <- sampleN.TOST(CV = CV[i], print = FALSE)[["Sample size"]]
}
# confidence limits of CVw
ci <- data.frame(n = n, CV = CV, lo = NA_real_, hi = NA_real_)
for (i in seq_along(CV)) {
  ci[i, 3:4] <- CVCL(CV = CV[i], df = n[i] - 2,
                     side = "2-sided", alpha = 0.05)
}
names(ci)[3:4] <- c("lower CL", "upper CL")
print(signif(ci, 5), row.names = FALSE)

#   n   CV lower CL upper CL
#  40 0.30  0.24340  0.39229
#  52 0.35  0.29028  0.44216
#  66 0.40  0.33761  0.49227
#  82 0.45  0.38520  0.54267

Already this should flash a warning light. The CV is only an estimate and the true value might well be lower or higher. 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 another study.

If we would estimate the sample size for SABE based on the CV observed in a 2×2×2 crossover, it might fire back. If the CV will be higher, we gain power (more scaling) but that’s not economical. On the other hand, if it will be lower, we loose power (less scaling).

A high CV observed in a 2×2×2 crossover is nothing more than a hint of a highly variable drug / drug product (in jurisdictions applying Average Bioequivalence with Expanding Limits \(\small{CV_\text{wR}>30\%}\) and in ones applying Re­fe­rence-Scaled Average Bioequivalence \(\small{s_\text{wR}\geq0.294}\)).

The within-subject CV observed in a nonreplicated study is pooled from the – inaccessible – within-subject CV of the test and reference products. \[\eqalign{s_\text{wT}^2&=\log_{e}(CV_\text{wT}^2+1)\\ s_\text{wR}^2&=\log_{e}(CV_\text{wR}^2+1)\\ s_\text{w}^2&=(s_\text{wT}^2+s_\text{wR}^2)/2\\ CV_\text{w}&=\sqrt{\exp(s_\text{w}^2)-1}}\tag{2}\]

There is an infinite number of combinations of \(\small{CV_\text{wT}\textsf{-}}\) and \(\small{CV_\text{wR}\textsf{-}}\)values giving the same pooled \(\small{CV_\text{w}}\). We simply don’t – and can’t – know the variance components.

Leaves the pupil – rightly – dazed and confused.

As shown in articles about ABEL and RSABE, it is a good idea not only to have an estimate of \(\small{CV_\text{wR}}\) (required for SABE) but also of \(\small{CV_\text{wT}}\). Whereas the degree of scaling depends only on \(\small{CV_\text{wR}}\), the 90% confidence interval is based on \(\small{CV_\text{w}}\).

In ABE we have to assume homoscedasticity (\(\small{s_\text{wT}^2\equiv s_\text{wR}^2}\)). However, this assumption might be wrong. Since biopharmaceutical technology improves, quite often \(\small{s_\text{wT}^2< s_\text{wR}^2}\).

Let’s compare \(\small{s_\text{wT}^2\neq s_\text{wR}^2}\) (variance-ratios of 0.5 to 2) with \(\small{s_\text{wT}^2\equiv s_\text{wR}^2}\) and estimate sample sizes for an assumed T/R-ratio of 0.90 and target 80% power in a 2-sequence 4-period full replicate design.

# variance ratios of CVwT and CVwR
ratio <- sort(unique(c(seq(0.5, 1, 0.25), 1 / seq(0.5, 1, 0.25))))
# sample sizes for ABEL and RSABE: theta0 = 0.95, targetpower = 0.8,
# 2-sequence 4-period full replicate design
df    <- data.frame(CV = rep(CV, each = length(ratio)),
                    ratio = ratio, CVwT = NA_real_, CVwR = NA_real_,
                    n.ABEL = NA_integer_, n.RSABE = NA_integer_)
for (i in 1:nrow(df)) {
  df[i, 3:4]    <- CVp2CV(CV = df$CV[i], ratio = df$ratio[i])
  df$n.ABEL[i]  <- sampleN.scABEL(CV = as.numeric(c(df[i, 3:4])),
                                  design = "2x2x4", details = FALSE,
                                  print = FALSE)[["Sample size"]]
  df$n.RSABE[i] <- sampleN.RSABE(CV = as.numeric(c(df[i, 3:4])),
                                 design = "2x2x4", details = FALSE,
                                 print = FALSE)[["Sample size"]]
}
print(signif(df, 5), row.names = FALSE)

#    CV  ratio    CVwT    CVwR n.ABEL n.RSABE
#  0.30 0.5000 0.24318 0.34895     26      22
#  0.30 0.7500 0.27688 0.32172     30      26
#  0.30 1.0000 0.30000 0.30000     34      32
#  0.30 1.3333 0.32172 0.27688     38      36
#  0.30 2.0000 0.34895 0.24318     40      40
#  0.35 0.5000 0.28299 0.40814     24      20
#  0.35 0.7500 0.32268 0.37575     28      24
#  0.35 1.0000 0.35000 0.35000     34      28
#  0.35 1.3333 0.37575 0.32268     40      34
#  0.35 2.0000 0.40814 0.28299     50      48
#  0.40 0.5000 0.32250 0.46780     22      20
#  0.40 0.7500 0.36833 0.42995     26      22
#  0.40 1.0000 0.40000 0.40000     30      24
#  0.40 1.3333 0.42995 0.36833     38      30
#  0.40 2.0000 0.46780 0.32250     52      42
#  0.45 0.5000 0.36168 0.52795     22      18
#  0.45 0.7500 0.41381 0.48435     24      20
#  0.45 1.0000 0.45000 0.45000     28      24
#  0.45 1.3333 0.48435 0.41381     34      26
#  0.45 2.0000 0.52795 0.36168     48      36

As usual, sample sizes for ABEL are larger than the ones for RSABE (see also this article).

If \(\small{s_\text{wT}^2< s_\text{wR}^2}\) we get an incentive compared to the sample size assuming \(\small{s_\text{wT}^2\equiv s_\text{wR}^2}\). However, if \(\small{s_\text{wT}^2> s_\text{wR}^2}\) we would require substantially more subjects.

The FDA^{18 20 21 22} and China’s CDE²³ require in their RSABE approaches for Narrow Therapeutic Index Drugs also a comparison of \(\small{s_\text{wT}}\) with \(\small{s_\text{wR}}\).
Assumed T/R-ratio of 0.975, target power 80% in a two-sequence four-period full replicate design.

# CVw in a nonreplicated crossover
CV    <- seq(0.05, 0.25, 0.05)
# variance ratios of CVwT and CVwR
ratio <- sort(unique(c(seq(0.4, 1, 0.2), 1 / seq(0.4, 1, 0.2))))
# sample sizes for RSABE/NTID: theta0 = 0.975, targetpower = 0.8,
# 2-sequence 4-period full replicate design (default acc. to the guidances)
df    <- data.frame(CV = rep(CV, each = length(ratio)),
                    ratio = ratio, CVwT = NA_real_, CVwR = NA_real_,
                    n = NA_integer_)
for (i in 1:nrow(df)) {
  df[i, 3:4] <- CVp2CV(CV = df$CV[i], ratio = df$ratio[i])
  df$n[i]    <- sampleN.NTID(CV = as.numeric(c(df[i, 3:4])),
                             details = FALSE, print = FALSE)[["Sample size"]]
}
print(signif(df, 5), row.names = FALSE)

#    CV  ratio     CVwT     CVwR  n
#  0.05 0.4000 0.037786 0.059777 20
#  0.05 0.6000 0.043295 0.055910 24
#  0.05 0.8000 0.047137 0.052708 28
#  0.05 1.0000 0.050000 0.050000 32
#  0.05 1.2500 0.052708 0.047137 40
#  0.05 1.6667 0.055910 0.043295 54
#  0.05 2.5000 0.059777 0.037786 98
#  0.10 0.4000 0.075512 0.119650 12
#  0.10 0.6000 0.086549 0.111870 14
#  0.10 0.8000 0.094255 0.105440 16
#  0.10 1.0000 0.100000 0.100000 18
#  0.10 1.2500 0.105440 0.094255 20
#  0.10 1.6667 0.111870 0.086549 24
#  0.10 2.5000 0.119650 0.075512 40
#  0.15 0.4000 0.113120 0.179710 12
#  0.15 0.6000 0.129720 0.167940 12
#  0.15 0.8000 0.141330 0.158210 14
#  0.15 1.0000 0.150000 0.150000 16
#  0.15 1.2500 0.158210 0.141330 18
#  0.15 1.6667 0.167940 0.129720 22
#  0.15 2.5000 0.179710 0.113120 36
#  0.20 0.4000 0.150550 0.240060 12
#  0.20 0.6000 0.172780 0.224160 12
#  0.20 0.8000 0.188360 0.211050 14
#  0.20 1.0000 0.200000 0.200000 16
#  0.20 1.2500 0.211050 0.188360 18
#  0.20 1.6667 0.224160 0.172780 22
#  0.20 2.5000 0.240060 0.150550 36
#  0.25 0.4000 0.187750 0.300780 14
#  0.25 0.6000 0.215680 0.280580 14
#  0.25 0.8000 0.235300 0.263970 16
#  0.25 1.0000 0.250000 0.250000 16
#  0.25 1.2500 0.263970 0.235300 18
#  0.25 1.6667 0.280580 0.215680 22
#  0.25 2.5000 0.300780 0.187750 34

This has a massive impact on the sample size if the Test is more variable than the Reference, not only due to the degree of scaling but also the variance-ratio test. Here it would be grossly negligent to base the sample size on the result of a 2×2×2 study.

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Conclusion

Don’t be tempted to estimate the sample size for SABE based on the CV observed in a crossover study. Instead, perform a reasonably sized²⁴ pilot study in a replicate design.

I strongly recommend one of the full replicate designs²⁵ because in the – unfortunately popular – partial replicate (TRR|RTR|RRT) we don’t get an estimate of \(\small{s_\text{wT}^2}\) and hence, have to assume \(\small{s_\text{wT}^2\equiv s_\text{wR}^2}\). Such an assumption might be – and likely is – wrong.

For RSABE of NTIDs a pilot study in a full replicate design is unavoidable.

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Licenses

Helmut Schütz 2022
R GPL 3.0, pandoc GPL 2.0.
1^st version June 16, 2022. Rendered June 21, 2022 17:35 CEST by rmarkdown via pandoc in 0.17 seconds.

Footnotes and References