Some results on a Winsorized correlation coefficient

Abstract

Recent investigations indicate that psychometric measures often have distributions with very heavy tails, and that outliers are quite common. It has long been known that in terms of both power and Type I errors, even slight departures from normality can have serious consequences, and it is fairly evident that there are problems when using conventional measures of effect size, as is briefly illustrated in this paper. Similar problems arise when dealing with the usual correlation coefficient. Robust correlation coefficients have already been proposed that reflect the linear relationship between two random variables, but many have the unfortunate property of not always being equal to 0 under independence. One exception is the Winsorized correlation. The primary goal in this paper is to suggest a simple method for testing the hypothesis that the Winsorized correlation is equal to zero. A minor goal is to describe a formal definition of the Winsorized correlation coefficient in terms of Winsorized expected values.