Going through the Wiki article on the Phi coefficient, I've noticed that for paired binary data "a Pearson correlation coefficient estimated for two binary variables will return the phi coefficient".
Upon running a quick simulation I found this to not be the case. However, it appears that the phi coefficient does approximate the pearson's correlation coefficient.
x <- c(1, 1, 0, 0, 1, 0, 1, 1, 1) y <- c(1, 1, 0, 0, 0, 0, 1, 1, 1) cor(x,y) sqrt(chisq.test(table(x,y))$statistic/length(x)) # phi x <- rep(x, 1000) y <- rep(y, 1000) sqrt(chisq.test(table(x,y))$statistic/length(x)) # phi # it now DOES approximates the pearsons correlation. cor(x,y)
But it is not apparent to me why (mathematically) this is the case.