Multidimensional Association Models

Abstract

This article develops several multidimensional multilinear association models for sociologists and other social science researchers to analyze the relationship between categorical variables in multiway cross-classification tables. The proposed multilinear approach not only provides satisfactory fit by conventional standards in the illustrative examples but also offers better understanding of the complex relationship between variables. This study highlights the relationship between two alternative decompositions in the multilinear framework—the PARAFAC/CANDECOMP and the Tucker 3-mode methods to decompose log-linear parameters—as well as the relationship between the multilinear approach and the log-multiplicative association models developed by Goodman and others. In addition, the author discusses empirical strategies to determine whether some or all cross-dimensional and other identifying restrictions can be relaxed in certain restricted models and to account for the proper degrees of freedom for these models.