Element weighted Kemeny distance for ranking data

Preference data are a particular type of ranking data that arise when n individuals express their preferences over a finite set of items. Within this framework, the main issue concerns the aggregation of the preferences to identify a compromise or a “consensus”, defined as the closest ranking (i.e. with the minimum distance or maximum correlation) to the whole set of preferences.  Many approaches have been proposed, but they are not sensitive to the importance of items: i.e.  changing the rank of a highly-relevant element should result in a higher penalty than changing the rank of a negligible one. The goal of this paper is to investigate the consensus between rankings taking into account the importance of items (element weights).  For this purpose, we present:  i) an element weighted rank correlation coefficient tau_ew as an extension of the Emond and Mason’s tau, and ii) an element weighted rank distance d_ew as an extension of the Kemeny distance d. The one-to-one correspondence between the weighted distance and the rank correlation coefficient is analytically proved. Moreover, a procedure to obtain the consensus ranking among n individuals is described and its performance is studied both by simulation and by the application to real datasets.