Single Ordinal Correspondence Analysis with External Information

Several non-iterative procedures for performing correspondence analysis with external information are proposed in literature. The interpretation of the multidimensional representation of the row and column categories may be greatly simplified if additional information about the row and column structure are incorporated. In this paper, a new combined approach to impose external information (as linear constraints) in analyzing a contingency table which can be of ordinal nature, is showed. Linear constraints are imposed using the polynomial approach to correspondence analysis. The classical approach to correspondence analysis decomposes the Pearson chi-squared statistic into singular values by partitioning the matrix of Pearson contingencies using singular value decomposition. The polynomial approach to correspondence analysis decomposes the same statistic by partitioning the matrix of Pearson contingencies using orthogonal polynomials rather than singular value decomposition. An alternative approach to partitioning the Pearson chi-squared statistic for a two-way contingency table is essentially to combine the approach of orthogonal polynomials for the ordered columns and singular vectors for the unordered rows. With this mixed approach, the researcher can determine the statistically significant source of variation (location, dispersion and higher order components) of the ordered columns along a particular axis using the simple correspondence analysis. Main aim of this paper is then to introduce external information in this last approach. In our proposal, external information, for instance to take into account not-equally spaced categories, are then included directly on suitable matrices which reflect the most important components overcoming the problem to impose linear constraints based on subjective decisions.