A topological approach of multiple correspondence analysis

Abstract

ABSTRACT Topological multiple correspondence analysis (TMCA) studies a group of categorical variables defined on the same set of individuals. It is a topological method of data analysis that consists of exploring, analyzing, and representing the associations between several qualitative variables in the context of multiple correspondence analysis (MCA). It compares and classifies proximity measures to select the best one according to the data under consideration, then analyzes, interprets, and visualizes with graphic representations, the possible associations between several categorical variables relating to the known problem of MCA. Based on the notion of neighborhood graphs, some of these proximity measures are more-or-less equivalent. A topological equivalence index between two measures is defined and statistically tested according to the degree of description of the associations between the modalities of these qualitative variables. We compare proximity measures and propose a topological criterion for choosing the best association measure, adapted to the data considered, from among some of the most widely used proximity measures for categorical data. The principle of the proposed approach is illustrated using a real dataset with conventional proximity measures for binary variables from the literature. The first step is to find the proximity measure that can best be adapted to the data; the second step is to use this measure to perform the TMCA.