Singly and doubly ordered cumulative correspondence analysis.

The classical approach to correspondence analysis (CA) is designed to allow its user to a graphically summarize the association between two or more categorical variables that form a contingency table. Despite its popularity and utility, the classical approach does not take in consideration the structure of ordered variables. One way to performing CA when the variables have an ordered structure is to consider the Taguchi’s statistic (Taguchi, 1974). Beh, D’Ambra, Simonetti (2010) demonstrated the applicability of considering this statistic which takes into account the ordered structure by considering the cumulative sum of cell frequencies across the variable. Thus, the statistic is defined by summing the chi-squared statistic for each I × 2 contingency table obtained by aggregating the column categories 1 to j and aggregating the column categories (j+1) to J. For this reason, the Taguchi’s statistic is also referred to as cumulative chi-squared statistic (Nair; 1987). Cuadras (2002) proposes an approach to correspondence analysis based on double cumulative frequencies. However, it does not decompose any known index. In this paper we explore a generalization of Taguchi’s statistic which takes into account the presence of two ordinal categorical variables by considering their cumulative sum of cell frequencies. This generalization is analogous to the doubly cumulative chi-squared statistic which is constructed by summing the chi-squared statistic for each 2×2 sub-table formed by pooling adjacent rows and columns of the original contingency table; see Hirotsu (1986). We illustrate this approach to CA using a partition of the statistic proposed by Hirotsu. Its application presents some interesting properties and allows the analyst to represent the variations of row and column categories rather than the categories on the space generated by cumulative frequencies.