Bioequivalence and Bioavailability Forum

Dear Helmut, Dear Zizou,

thank you for pointing out this interesting case!
I tried to write some code to see what could we expect with given variabilities from full replicate dataset.
May be it is not so fair since I'm sampling from given data, not from CV directly.
By the way some clues to the question 'what if'

library(dplyr)
library(ggplot2)
alpha <- 0.05
nsims <- 1000

Sample2from4 <- function(Data, datacol = "data", nsims = 1000, alpha = 0.05) {
  # columns should be: subject, sequence, period, treatment; data column could be changed in function args
  ow    <- options()
  options(contrasts = c("contr.treatment", "contr.poly"), digits = 12)
  Data <- as.data.frame(Data)
  Data$subject <- as.factor(Data$subject)
  Data$sequence <- as.factor(Data$sequence)
  Data$period <- as.numeric(Data$period)
  Data$treatment <- as.factor(Data$treatment)
  Data$data <- as.numeric(Data[, match(datacol, colnames(Data))])
  SamplingResults <-
    data.frame(
      PE = numeric(0),
      lowerCI = numeric(0),
      upperCI = numeric(0),
      CV = numeric(0))
  for (i in 1:nsims) {
    # sampling N          1   2  3   4
    # periods to include 12 34  14  23
    Data2periods <-
      Data %>%
      group_by(subject) %>%
      mutate(periodsample = sample(1:4, 1)) %>%
      filter((periodsample == 1 & (period == 1 | period == 2)) |
               (periodsample == 2 & (period == 3 | period == 4)) |
               (periodsample == 3 & (period == 1 | period == 4)) |
               (periodsample == 4 & (period == 2 | period == 3))
      ) %>%
      mutate(periodrank = rank(period))
    Data2periods$period <- as.factor(Data2periods$periodrank)
    mod <- lm(log(data) ~ sequence + treatment + period + subject %in% sequence,
         data = Data2periods)
    PE <- as.numeric(exp(coef(mod)["treatmentT"]))
    CI <- exp(confint(mod, "treatmentT", level = 1 - 2 * alpha))
    mse <- summary(mod)$sigma ^ 2
    CV <- sqrt(exp(mse) - 1)
    SamplingResults <-
      rbind(SamplingResults,
            data.frame(
              PE = PE,
              lowerCI = CI[1],
              upperCI = CI[2],
              CV = CV))
  }
  options(ow) # restore options
  return(SamplingResults)
}

plotSample2from4 <- function(SamplingResults, Data, datacol = "data"){
  ow    <- options()
  options(contrasts = c("contr.treatment", "contr.poly"), digits = 12)
  Data <- as.data.frame(Data)
  Data$subject <- as.factor(Data$subject)
  Data$sequence <- as.factor(Data$sequence)
  Data$period <- as.factor(Data$period)
  Data$treatment <- as.factor(Data$treatment)
  Data$data <- as.numeric(Data[, match(datacol, colnames(Data))])
  Data.modR <- lm(log(data) ~ sequence + subject%in%sequence + period,
                  data=Data[Data$treatment=="R",])
  Data.CVwR  <- sqrt(exp(summary(Data.modR)$sigma ^ 2) - 1)
  Data.modT <- lm(log(data) ~ sequence + subject%in%sequence + period,
                  data=Data[Data$treatment=="T",])
  Data.CVwT  <- sqrt(exp(summary(Data.modT)$sigma ^ 2) - 1)
  Data.CV <- cbind.data.frame(CV = c(Data.CVwR, Data.CVwT), CVw = c("CVwR", "CVwT") )
  MoreThanCVwR <- length(SamplingResults$CV[SamplingResults$CV<Data.CVwR]) / length(SamplingResults$CV)
  MoreThanCVwT <- length(SamplingResults$CV[SamplingResults$CV<Data.CVwT]) / length(SamplingResults$CV)
  Data.CV <- cbind.data.frame(Data.CV, MoreThanCVw = c(MoreThanCVwR, MoreThanCVwT))
  ribbon <- ggplot_build(ggplot() + geom_density(data = SamplingResults, aes(x = CV), colour = "<CVw>"))$data[[1]]
  ribbon$colour[ribbon$x<Data.CVwR & ribbon$x < Data.CVwT ] <- "<CVwR & <CVwT"
  ribbon$colour[ribbon$x>Data.CVwR & ribbon$x > Data.CVwT ] <- ">CVwR & >CVwT"
  title <- paste0("CVwR = ", sprintf("%.2f%%", Data.CVwR*100),
                  ", CVwT = ", sprintf("%.2f%%", Data.CVwT*100),
                  "; \n2X2 Study CV > CVwR in ", sprintf("%.2f%%", 100-MoreThanCVwR*100),
                  "; 2X2 Study CV > CVwT in ", sprintf("%.2f%%", 100-MoreThanCVwT*100))
  densityplot <-
    ggplot() +
    geom_ribbon(data = ribbon, aes(x = x, ymin = 0, ymax = y, fill = colour), alpha = .4) +
    geom_vline(data = Data.CV,  aes(xintercept = CV, color = CVw, linetype = CVw)) +
    theme_bw() +
    ggtitle(title)+
    labs(x="CV", y = "density")
  print(densityplot)
  return(Data.CV)
  options(ow) # restore options
}

now using the dataset from Helmut's post above:
SamplingResults <- Sample2from4(Data = Data, datacol = "data", nsims = 1000)
plotSample2from4(Data = Data, SamplingResults = SamplingResults, datacol = "data")

Amazing! CVwR and CVwT are far away from mean/median!

What about other examples? EMA full replicate:
SamplingResults <- Sample2from4(Data = ds01, datacol = "PK", nsims = 1000)
plotSample2from4(Data = ds01, SamplingResults = SamplingResults, datacol = "PK")


Looks reasonable. Somewhere between CVwR and CVwT. By the way I would not go with pooled CV for 2X2 knowing this...