Diagnostics for regression dependence in tables re-ordered by the dominant correspondence analysis solution

Correspondence analysis is an exploratory technique for analyzing the interaction in a contingency table. Tables with meaningful orders of the rows and columns may be analyzed using a model-based correspondence analysis that incorporates order constraints. However, if there exists a permutation of the rows and columns of the contingency table so that the rows are regression dependent on the columns and, vice versa, the columns are regression dependent on the rows, then both implied orders are reflected in the first dimension of the unconstrained correspondence analysis [Schriever, B.F., 1983. Scaling of order dependent categorical variables with correspondence analysis. International Statistical Review 51, 225–238]. Thus, using unconstrained correspondence analysis, we may still find that the data fit an ordinal stochastic model. Fit measures are formulated that may be used to verify whether the re-ordered contingency table is regression dependent in either the rows or columns. Using several data examples, it is shown that the fit indices may complement the usual geometric interpretation of the unconstrained correspondence analysis solution in low-dimensional space.