Single Ordinal Correspondence Analysis with External Information

Several non-iterative procedures for performing correspondence analysis with external information have been 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 an 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 any statistically significant sources of variation (location, dispersion and higher order components) of the ordered columns along the a particular axis using the simple correspondence analysis. Main aim of the present study is to introduce external information to this approach. In our proposal external information, such as taking into account that categories are not equally spaced is then included directly on suitable matrices which reflect the most important components. This approach allows for one to overcome the problem of having to impose linear constraints at the variables based on subjective decisions.