Ratio/product of two lognormals [NCA / SHAM]

posted by Helmut Homepage – Vienna, Austria, 2021-10-03 22:29 (960 d 13:44 ago) – Posting: # 22616
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Dear Detlew,

❝ ❝ ❝ what do you think is a reasonable assumption about the distribution of the metric AUC*k?

❝ ❝ Since both are lognormal, their ratio should be lognormal as well. I trust here Martin; will meet him in the evening and ask again.


Answer: The difference of two normal distributions is normal distributed. The ratio and product of two lognormal distributions are lognormal distributed.
Furthermore, the unit of AUC·k is mass/volume.

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n   <- 2501
x   <- seq(0.01, 2.5, length.out = n)
a   <- dlnorm(x = x, meanlog = log(1),    sdlog = sqrt(log(0.15^2 + 1)))
b   <- dlnorm(x = x, meanlog = log(0.95), sdlog = sqrt(log(0.30^2 + 1)))
c   <- a / b
d   <- a * b
clr <- c("#FF808080", "#8080FF80", "#FF80FF80")
dev.new(width = 4.5, height = 4.5, record = TRUE)
op <- par(no.readonly = TRUE)
par(mar = c(4, 0, 0, 0), cex.axis = 0.9)
plot(x = x, y = a, type = "n", xlab = expression(mu[T]/mu[R]),
     ylab = "", ylim = range(a, b, c), axes = FALSE)
axis(1)
polygon(x = c(x, rev(x)), y = c(rep(0, n), rev(a)), col = clr[1], border = NA)
polygon(x = c(x, rev(x)), y = c(rep(0, n), rev(b)), col = clr[2], border = NA)
polygon(x = c(x, rev(x)), y = c(rep(0, n), rev(c)), col = clr[3], border = NA)
legend("topright", lwd = 5, seg.len = 2, col = clr, cex = 0.8,
       legend = c(expression(italic(a)*" = dlnorm(log(1), CV 0.15)"),
                  expression(italic(b)*" = dlnorm(log(0.95), CV 0.30)"),
                  expression(italic(c==a/b))))
plot(x = x, y = a, type = "n", xlab = expression(mu[T]/mu[R]),
     ylab = "", ylim = range(a, b, d), axes = FALSE)
axis(1)
polygon(x = c(x, rev(x)), y = c(rep(0, n), rev(a)), col = clr[1], border = NA)
polygon(x = c(x, rev(x)), y = c(rep(0, n), rev(b)), col = clr[2], border = NA)
polygon(x = c(x, rev(x)), y = c(rep(0, n), rev(d)), col = clr[3], border = NA)
legend("topright", lwd = 5, seg.len = 2, col = clr, cex = 0.8,
       legend = c(expression(italic(a)*" = dlnorm(log(1), CV 0.15)"),
                  expression(italic(b)*" = dlnorm(log(0.95), CV 0.30)"),
                  expression(italic(d==a %.% b))))
par(op)


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