On the index of dissimilarity for lack of fit in log-linear and log-multiplicative models

The index of dissimilarity, often denoted by Delta, is commonly used, especially in social science and with large datasets, to describe the lack of fit of models for categorical data. The definition and sampling properties of the index for general loglinear and log-multiplicative models are investigated. It is argued that in some applications a standardized version of the index is appropriate for interpretation. A simple, approximate variance formula is derived for the index, whether standardized or not. A simple bias reduction formula is also given. The accuracy of these formulae and of confidence intervals based upon them is investigated in a simulation study based on large-scale social mobility data.