Vol.:(0123456789) AStA Advances in Statistical Analysis https://doi.org/10.1007/s10182-020-00382-5 1 3 ORIGINAL PAPER Confdence regions and other tools for an extension of correspondence analysis based on cumulative frequencies Antonello D’Ambra 1  · Pietro Amenta 2  · Eric J. Beh 3 Received: 29 February 2020 / Accepted: 12 October 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020 Abstract Over the past 50 years, correspondence analysis (CA) has increasingly been used by data analysts to examine the association structure of categorical variables that are cross-classifed to form a contingency table. However, the literature has paid little attention to the case where the variables are ordinal. Indeed, Pearson’s chi- squared statistic X 2 can perform badly in studying the association between ordinal categorical variables (Agresti in An introduction to categorical data analysis, Wiley, Hoboken, 1996; Barlow et al. in Statistical inference under order restrictions, Wiley, New York, 1972). Taguchi’s (Nair in Technometrics 28(4):283–291, 1986; Nair in J Am Stat Assoc 82:283–291, 1987) and Hirotsu’s (Biometrika 73: 165–173, 1986) statistics have been introduced in the literature as simple alternatives to Pearson’s index for contingency tables with ordered categorical variables. Taguchi’s statistic takes into account the presence of an ordinal categorical variable by considering the cumulative sum of the cell frequencies across the variable. An extension of corre- spondence analysis using a decomposition of Taguchi’s statistic has been introduced to accommodate this feature of the variables. This considers the impact of difer- ences between adjacent ordered categories on the association between row and col- umn categories. Therefore, the main aim of this paper is to introduce a confdence region for each of the ordered categories so that one may determine the statistical signifcance of a category with respect to the null hypothesis of independence. We highlight that the construction of these circles has not been considered in the litera- ture for this approach to CA. We also introduce a suitable decomposition of Tagu- chi’s statistic to test the statistical signifcance of each column category. Keywords Contingency table · Chi-squared statistic · Single cumulative chi-squared statistic · Confdence circle * Pietro Amenta amenta@unisannio.it Extended author information available on the last page of the article