R script demonstrating how spurious correlations between two variables can be introduced by common normalization factor

spurious_correlation.R

# Create 3 random vectors of length 50
# We use human / mouse gene symbols to illustrate that this
# reflects commonly used procedures in biomedical science,
# e.g. the analysis of qPCR results with a single 
# reference gene.
# See MIQE guidelines on how to normalize properly:
# https://www.ncbi.nlm.nih.gov/pubmed/19246619
Il6 <- rnorm(mean = 10, n = 50)
Il1b <- rnorm(mean = 10, n = 50)
Gapdh <- rnorm(mean = 10, sd = 3, n = 50)

# random numbers are not correlated with each other
panel.cor <- function(x, y, digits = 2, ...) {
  usr <- par("usr"); on.exit(par(usr))
  par(usr = c(0, 1, 0, 1))
  r <- cor(x, y)
  p <- cor.test(x, y)$p.value
  txt <- sprintf("R = %.02f\np = %.02f", r, p)
  text(0.5, 0.5, txt, cex = 2)
}
pairs(cbind(Il6, Il1b, Gapdh), upper.panel = panel.cor)

# But when Il6 and Il1b are 'normalized' to Gapdh,
# strong correlations are induced
plot(Il6/Gapdh, Il1b/Gapdh)
  title(main = sprintf(
  "R = %.02f p = %s", 
  cor(Il6/Gapdh, Il1b/Gapdh),
  signif(cor.test(Il6/Gapdh, Il1b/Gapdh)$p.value)))
abline(0, 1)