Contingency Table Analysis

A contingency table is a table of counts, produced by cross-classifying items (e.g., persons, products, institutions) according to two or more categorical variables (attributes). Contingency tables group item-specific data in cells without losing information of the cross-classified variables, thus providing a summary presentation and a first descriptive analysis, aiming at examining the underlying associations among the classification variables. Contingency tables commonly occur in applications of psychological, educational, and social sciences. The information of a contingency table is further summarized by measures of association and analyzed by appropriate models.

The decision for the appropriate type of measures and models for a specific contingency table analysis relies on the nature of the study, the hypothesis under investigation, and the measurement scale of the classification variables (nominal or ordinal). Studies may target measuring and analyzing the effect of a set of explanatory variables on one or more response variables or may treat variables in a symmetric manner investigating the underlying association and interaction structures, in which case all variables are considered as response variables. Measures of association are [Page 288]distinguished to directional and nondirectional, depending on whether a distinction between explanatory and response variables exists or not. Furthermore, measures for nominal variables refer only to the strength of the association, whereas for ordinal variables provide information also for its direction (positive or negative). The development of association measures was pioneering, the contribution of Leo A. Goodman and William H. Kruskal.

The standard models used for contingency tables analysis are the loglinear models that are applicable when all the classification variables are responses and interest lies on the underlying structure of dependencies (or conditional independencies) among them. When the focus is on modeling the effect of a set of categorical explanatory variables on a categorical response, then appropriate are the logit models, that is, logistic regression models, having all explanatory variables categorical. Both loglinear and logit models belong to the family of generalized lineal models (GLMs), introduced by John A. Nelder and Robert W. M. Wedderburn in 1972. The family of GLMs provides a powerful and flexible framework, since it unifies a broad set of models through the choice for the distribution of the random component and the link function. This way, procedures of statistical inference, model fit, and model selection are also unified and easily implemented in statistical software. GLMs facilitated the analysis of contingency tables and stimulated the further development of modeling options.

The analysis of high-dimensional sparse contingency tables is an interesting area rooting back in the 1980s and the pioneering work of Stephen E. Fienberg. He explored the role of maximum likelihood estimates (MLEs) and loglinear model selection and identified challenges related to statistical learning problems and actual applications such as analysis of social media data or mobility tables in social networks.