Measuring the Relationship of Bivariate Data Using Hodges-Lehman Estimator

Abstract

The relationship of bivariate data ordinarily measured using correlation coefficient. The most commonly used correlation coefficient is the Pearson correlation coefficient. This coefficient is well-known as the best coefficient for interval or ratio bivariate data with a linear relationship. Even though this coefficient is good under the mentioned condition, it also becomes very sensitive to a small departure from linearity. Usually, this is because of the existence of an outlier. For that reason, this paper provides new robust correlation coefficients which combine the elements of nonparametric technique from the Hodges Lehmann estimator and the parametric technique based on the Pearson correlation coefficient. This paper also introduces different scale estimators such as median and median absolute deviation (MADn) and denoted by rHL(med) and rHL(MADn) respectively. The performance of the proposed correlation coefficients is measured by the coefficient values and these values are also being compared to the Pearson correlation coefficient and several existing robust correlation coefficients. The results show that the Pearson correlation coefficient (r) with no doubt is very good under perfect data condition, but with only 10% outliers, it not only give poor correlation value but turns the direction of the relationship to negative. While the rHL(med) and rHL(MADn) offer the highest coefficient values and these values are robust to the existence of outliers by up to 30%. With very good performance under all data conditions yet simple in the calculation, the rHL(med) and rHL(MADn) is considered a good alternative to the r when need to deal with outliers. Keywords: correlation coefficient; Hodges Lehmann; median; median absolute deviation (MADn)