Remove outliers from correlation coefficient calculation

Question

Assume we have two numeric vectors x and y. The Pearson correlation coefficient between x and y is given by

Answer 1

Using method = "spearman" in cor will be robust to contamination and is easy to implement since it only involves replacing cor(x, y) with cor(x, y, method = "spearman").

Repeating Prasad's analysis but using Spearman correlations instead we find that the Spearman correlation is indeed robust to the contamination here, recovering the underlying zero correlation:

set.seed(1)

# x and y are uncorrelated
x <- rnorm(1000)
y <- rnorm(1000)
cor(x,y)
## [1] 0.006401211

# add contamination -- now cor says they are highly correlated
x <- c(x, 500)
y <- c(y, 500)
cor(x, y)
## [1] 0.995741

# but with method = "spearman" contamination is removed & they are shown to be uncorrelated
cor(x, y, method = "spearman")
## [1] -0.007270813