Jointness and Duality in Algebraic Approaches to Dichotomous Data

A number of algebraic approaches to dichotomous data have attracted interest in the social sciences. Despite their underlying connections, these different approaches have different assumptions and realms of applicability. In particular, some approaches (which may be termed explanatory) assume a certain a process that generates the data, while others (which may be termed interpretive) give a parsimonious representation of data patterns irrespective of a generative process. The former may also be termed one-sided in that they privilege one dimension of a conventional data matrix, while the latter are dual, in that they do not do this. In certain cases, one-sided analyses can be combined to produce “joint†analyses, which differ from dual analyses. A consideration of these differences casts light on the information retrieved by the various methods.