Ordinal Non‐Symmetric Correspondence Analysis for Four‐Way Data

Summary Non‐symmetric correspondence analysis (NSCA) is a powerful analytical technique designed to facilitate the exploration, interpretation and visualisation of the asymmetric association among nominal or ordinal categorical variables. We extend the theory of three‐way ordinal NSCA to multiway data; in particular, we consider the case of four‐way data using four‐variate moment decomposition ( FMD ). The present study focuses on ordinal NSCA, specifically tailored for a completely ordered contingency table. Ordinal NSCA visualises the asymmetric association among the ordered categories of the variables through the decomposition of the centred predictor profile array using orthogonal polynomials. The traditional method of ordinal NSCA for three‐way data utilises the Kronecker product for decomposing the centred column–tube profile array. We propose the use of tensor operations to perform FMD instead of the Kronecker product. We summarise some interesting tensor properties. Furthermore, we develop an algorithm to perform four‐way ordinal NSCA using a tensorial approach. The advantage of the proposed algorithm is that it can be easily generalised to a multiway case. We explore the association between the variables through an interactively coded predictor isometric biplot. An R function is developed to perform ordinal NSCA on four‐way data, and its implementation is demonstrated through a numerical example.