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

This may have been already obvious to the OP, but just to make sure... You have to be careful because trying to maxmimize correlation may actually tend to include outliers. (@Gavin touched on this point in his answer/comments.) I would be first removing outliers, then calculating a correlation. More generally, we want to be calculating a correlation that is robust to outliers (and there are many such methods in R).

Just to illustrate this dramatically, let's create two vectors x and y that are uncorrelated:

set.seed(1)
x <- rnorm(1000)
y <- rnorm(1000)
> cor(x,y)
[1] 0.006401211

Now let's add an outlier point (500,500):

x <- c(x, 500)
y <- c(y, 500)

Now the correlation of any subset that includes the outlier point will be close to 100%, and the correlation of any sufficiently large subset that excludes the outlier will be close to zero. In particular,

> cor(x,y)
[1] 0.995741

If you want to estimate a "true" correlation that is not sensitive to outliers, you might try the robust package:

require(robust)
> covRob(cbind(x,y), corr = TRUE)
Call:
covRob(data = cbind(x, y), corr = TRUE)

Robust Estimate of Correlation: 
            x           y
x  1.00000000 -0.02594260
y -0.02594260  1.00000000

You can play around with parameters of covRob to decide how to trim the data. UPDATE: There is also the rlm (robust linear regression) in the MASS package.