The Weighted Rank Correlation Coefficient $r_{W2}$

Abstract

A new weighted rank correlation coefficient, $r_{W2}$, has been introduced in Pinto da Costa, Weighted Correlation, 2011, [74] and applied in a bioinformatics context in Pinto da Costa et al., IEEE/ACM Trans Comput Biol Bioinf 8(1):246–252, 2011, [73]. This coefficient is the second of its series, following the coefficient $r _{W}$ introduced in Pinto da Costa et al., Nonlinear Estimation and Classification, MSRI, 2001 [61], Pinto da Costa and Soares, Australian New Zealand J Stat 47(4):515–529, 2005, [63], Soares et al., JOCLAD 2001: VII Jornadas de Classificação e Análise de Dados, Porto, 2001, [93], which was motivated by a machine learning problem concerning the recommendation of learning algorithms. These coefficients were inspired by Spearman’s rank correlation coefficient, $r_S$. Nevertheless, unlike Spearman’s, which treats all ranks equally, these coefficients weigh the distance between two ranks using a linear function of those ranks in the case of $r_W$ and a quadratic function in the case of $r_{W2}$. In both cases, these functions give more importance to top ranks than lower ones, although $r_{W2}$ has some advantages over $r_W$ as we will see. In some of the applications of weighted correlation, ties can happen naturally; nevertheless, the existing coefficients tend to ignore this situation. We give here the expression of $r_{W2}$ in the case of ties. We present also some simulations in order to compare the three coefficients $r_{W2}$, $r_W$, and $r_S$.