Multivariate co-inertia analysis for qualitative data by partial least squares

In this paper we propose a generalization of the covariance criterion, which is at the heart of the Partial Least Squares (PLS) approach (Wold, 1966), studying the regression problem between a response and a set of ordinal predictor variables. From a practical point of view, the use of a complete ordering of predictors can be considered as a way to extract more interpretable information. Under the constraint of the complete order different techniques have been proposed in literature, specifically within the context %%@ of optimal scaling, under the constraint of complete order only one solution satisfies %%@ both the constraint and the criterion of optimal scaling (Nishisato 198?, Nishisato and Arri 1975). In this paper the first axis maximizing the covariance between the variables is re-computed by a rank regression in order to preserve the original ordering of predictor categories, while the remaining axes are calculated by PLS or SIMPLS (Tenenhaus 1999) approach.