Uncovering interactions in multivariate contingency tables: a multi-level modelling exploratory approach

Much quantitative behavioural social science – a great deal of it exploratory in nature – involves the analysis of multivariate contingency tables, usually deploying logistic binomial and multinomial regression models with no exploration of interaction effects, despite arguments that this should be a crucial element of the analysis. This paper builds on suggestions that the search for interaction effects should employ multi-level modelling strategies and outlines a procedure for modelling patterns in data sets with small numbers of observations in many, if not all, of their multivariate contingency table cells; all expected cells must be non-zero. The procedure produces precision-weighted estimates of the observed:expected rates for each and every cell, together with associated Bayesian credible intervals, and is illustrated using a large survey data set relating voting (and abstaining)at the 2015 UK general election to age, sex and educational qualifications. Crucially while fine detail can be explored in the analysis unreliable rates for particular subgroups are automatically down-weighted to what is happening generally. The identification of reliable differential rates then allows a simpler hybrid model that captures the main trends to be fitted and interpreted.