Ordinal principal component analysis theory and an application

Abstract

In this study we investigate the problem of ordering multivariate data. We propose the use of the so-called (first) ordinal principal component for this purpose. The ordinal principal component is defined as a new ordinal variable which orders the sample observations in such a way that the sum of the squares of the (rank) correlation coefficients between the original variables and the ordinal principal component is maximal. The definition provides a direct way to estimate a rank order for multivariate observations and it is also applicable to varibles measured only on ordinal scales. It is consistent with the usual definition of the principal component transformation in the sense that the sum of the (weighted) squares of the correlation coefficients between the original variables and the principal component is also maximal. Because the correlation coefficients (Spearman's and Kendall's rank correlation coefficients) can be defined for ordinal variables as well, the ordinal principal component can be defined without using any cardinal information. Our approach was applied to a problem of ordering 19 hypermarkets in Finland, when a set of various performance indicators determines conflicting rank orders for those hypermarkets.