Bioequivalence and Bioavailability Forum

Dear Junhao,

❝ ❝ Don't invest too much in understanding or verifying this table.

❝ ❝ It shows sample sizes for the so-called "individual bioequivalence" in which the SbF (subject by formulation) interaction, your σ_D, plays a prominent role ...

Here I have to correct myself. Table 1 in the Appendix of the mentioned FDA guidance actually shows sample sizes for ABE.

Here some code to reproduce this table:
library(PowerTOST)
# function to calculate the 'pooled' CV if the individual component of the variability are given
# Wang/Chow 2002
# "ON Statistical Power for Average Bioequivalence Testing Under Replicated Crossover Designs"
# J. BIOPHARM. STAT. Vol. 12, No. 3, pp. 295–309, 2002
CVm <- function(sD, swT, swR=swT, design){
  if (design=="2x2"){
    s2diff <- (sD^2 + swT^2 + swR^2)/2
  }
  else if (design=="2x2x4"){
    s2diff <- sD^2 + (swT^2 + swR^2)/2
  } else stop("Design not implemented.")
  mse2CV(s2diff)
}

# gimme the numbers
delta <- 0.05 # this is measured in the log domain, corresponds to theta0 = 0.9512294
# assuming swT=swR as in Table 1
swT   <- 0.5
sD    <- 0.15
targetpower <- 0.8

design <- "2x2"
CV <- CVm(sD, swT, design=design)
sampleN.TOST(CV=CV, theta0=exp(-delta), targetpower=targetpower, design=design)

design <- "2x2x4"
CV <- CVm(sD, swT, design=design)
sampleN.TOST(CV=CV, theta0=exp(-delta), targetpower=targetpower, design=design)

Note that we need a 'pooled' variability for ABE only. Which we can get from a non-replicated design also.
Only in case of a full replicate design we may have informations about the components. But here it depends on the statistical model if a σ_D plays a role.
In case of a partial replicate design you can't get reasonable estimates of the variability components in all cases. Here a model with a SbF is overspecified and may lead to estimation problems like non-convergence of the REML procedure.

See f.i.
Endrenyi et al.
"Properties of the estimated variance component for subject-by-formulation interaction in studies of individual bioequivalence"
Statist. Med. 2000; 19:2867–2878
to find out that σ_D=0 is a reasonable assumption.

❝ But when we assume different between-subject variation for T and R, σ_D is never gonna be 0, and I right?

Correct. If your statistical model of the between-subject variability incorporates a SbF interaction. Otherwise this component is absorbed in the estimates of the within-subject variabilities σ_wT and σ_wR, respectively.

❝ ... but in the RSABE context, we are assuming same between-subject variation for T and R so that σ_D could be 0?

Also correct. But beside the CI component of T-R of RSABE, where we again have to use a pooled variability, there is no influence of σ_D.