Bioequivalence and Bioavailability Forum

❝ I wouldn't know about FDA and EMA, but I think that they'd accept as well.

❝ So it is not that easy to recruit these women, usually taking a lot of extra time in recruitment and screening

❝ I wouldn't know about FDA and EMA, but I think that they'd accept as well.

❝ While intuitively, you might think that the geometric mean of a set containing 0 should be 0 (since any product involving 0 is 0), the formal definition and intended purpose of the geometric mean don't align well with this interpretation.

❝ Imagine 2 datasets:

❝ c(1,1,1,1,0)

❝ c(0,0,0,0,1)

❝ Then both will have the same geometric mean value. It loses information about the other values and doesn't accurately reflect the "typical value" or "average rate" that the geometric mean aims to represent.

library(psych)    # contains geometric.mean()
library(EnvStats) # contains geoMean()
gm <- function(x, print = TRUE) {
  x   <- x[!is.na(x)]
  pos <- sign(x) == 1
  msg <- NULL
  if (sum(TRUE - pos) >= 1) {
    if (sum(TRUE - pos) == 1) {
      msg <- paste0("[1 non-positive value]")
    } else {
      msg <- paste0("[", sum(TRUE - pos), " non-positive values]")
    }
    res <- NA
  } else {
    res <- prod(x)^(1 / length(x))
  }
  if (print) {
    cat("gm()            :", res, msg, "\n")
  } else {
    return(res)
  }
}
gm1 <- function(x, print = TRUE) {
  x   <- x[!is.na(x)]
  neg <- sign(x) == -1
  msg <- NULL
  if (sum(neg) >=1) {
    msg <- paste0("[", sum(neg), " negative(s) excluded]")
    x   <- x[!neg]
  }
  res <- prod(x)^(1 / length(x))
  if (print) {
    cat("gm1()           :", res, msg, "\n")
  } else {
    return(res)
  }
}
gm2 <- function(x, print = TRUE) {
  x   <- x[!is.na(x)]
  pos <- sign(x) == 1
  msg <- NULL
  if (sum(TRUE - pos) >= 1) {
    if (sum(TRUE - pos) == 1) {
      msg <- paste0("[1 non-positive excluded]")
    } else {
      msg <- paste0("[", sum(TRUE - pos), " non-positives excluded]")
    }
    x <- x[pos]
  }
  res <- prod(x)^(1 / length(x))
  if (print) {
    cat("gm2()           :", res, msg, "\n")
  } else {
    return(res)
  }
}
gm3 <- function(x, print = TRUE) {
  res <- geometric.mean(x)
  if (print) {
    cat("geometric.mean():", res, "\n")
  } else {
    return(res)
  }
}
gm4 <- function(x, print = TRUE) {
  res <- geoMean(x)
  if (print) {
    cat("geoMean()       :", res, "\n")
  } else {
    return(res)
  }
}
am <- function(x, print = TRUE) {
  res <- mean(x, rm = TRUE)
  if (print) {
    cat("mean()          :", res, "\n")
  } else {
    return(res)
  }
}

set1 <- c(1, 1, 1, 1, 0)
set2 <- c(0, 0, 0, 0, 1)

gm(set1); gm1(set1); gm3(set1); gm2(set1); gm4(set1); am(set1)

gm()            : NA [1 non-positive value]
gm1()           : 0
geometric.mean(): 0
gm2()           : 1 [1 non-positive excluded]
geoMean()       : NA
Warning message:
In geoMean(x) : Non-positive values in 'x'
mean()          : 0.8

gm(set2); gm1(set2); gm3(set2); gm2(set2); gm4(set2); am(set2)

gm()            : NA [4 non-positive values]
gm1()           : 0
geometric.mean(): 0
gm2()           : 1 [4 non-positives excluded]
geoMean()       : NA
Warning message:
In geoMean(x) : Non-positive values in 'x'
mean()          : 0.2

❝ Regarding geometric mean: kind of tricky taking into account the information above. The median is better, but why don't you want to follow the method mentioned in BE GL at the first place (arithmetic mean)? Yes, it is bad, but at least not worse than others

Multiple baseline endogenous concentrations should be measured from each subject in the time period before administration of the study drug. The time-averaged baseline [… is] subtracted from post-dose concentrations for those subjects in an appropriate manner consistent with the PK properties of the drug. For the time-averaged method, either the mean or median value may be used.

1. Netz H. Formeln der Mathematik. München: Hanser; 6^th ed. 1986. p. 18.
   
2. Nolan D, Speed T. Stat Labs. Mathematical Statistics Through Applications. New York: Springer; 2001. p. 68.
   
3. Sachs S, Hedderich J. Angewandte Statistik. Methodensammlung mit R. Berlin: Springer; 12^th ed. 2006. p. 76.

❝ Note that equally spaced calibrators are mandatory in most accredited methods in environmental analysis.

❝ The exponentially increasing spacing of calibrators likely goes back to the dark ages where weighted regression was not available (pocket calculator, Excel?)

❝ Ahem. What to you mean by nonsense – apart from the word “terminal”?

❝ Agree. See this stupid example (slides 46–49) from the early days of LC/MS-MS (no stable isotope IS and hit by the matrix effect). Slide 49 compares results of the first 12 (of 24) subjects obtained by the lousy LC/MS-MS method with stable isotope IS GC/MS. Does it matter? No.