Weighting Distance Matrices Using Rank Correlations

In a number of applications of multivariate analysis, the data matrix is not fully observed. Instead a set of distance matrices on the same entities is available. A reasonable strategy to construct a global distance matrix is to compute a weighted average of the partial distance matrices, provided that an appropriate system of weights can be defined. The Distatis method developed by Abdi et al. (2005) is a three-step procedure for computing the global distance matrix. An important aspect of that procedure is the computation of the vector correlation coefficient (RV) to measure the similarity between partial distance matrices. The RV coefficient is based on the Pearson product moment correlation coeffcient, which is highly prone to the effects of outliers. We are convinced that, in many measurable phenomena, the relationships between distances are far more likely to be ordinal than interval in nature, and it is therefore preferable to adopt an approach appropriate to ordinal data. The goal of our paper is to revise the system of weights of the Distatis procedure substituting the conventional Pearson coefficient with rank correlations that are less affected by errors of measurement, perturbation or presence of outliers in the data. In the light of our findings on real and simulated data sets, we recommend the use of a speci c coefficient of rank correlation to replace, where necessary, the conventional vector correlation.