Ranking group-level outcomes with multilevel models: An information-theoretic measure of statistical separation

Social epidemiologists frequently aim to quantify how social, spatial, or organizational contexts shape individual outcomes, an aim commonly addressed through the use of multilevel models. These models readily estimate the magnitude of group-level differences, but it is more difficult to formalize the uncertainty in the relative ordering of predicted group-level outcomes, which is often considered qualitatively in practice. We propose an entropy-based coefficient, grounded in information theory, which quantifies the stability of predicted group-level rankings. This metric, termed the separation statistic (S), integrates both the magnitude of group-level differences and their statistical uncertainty, providing a principled summary of how well groups are separated in terms of their predicted outcomes. Our motivation is drawn from Multilevel Analysis of Individual Heterogeneity and Discriminatory Accuracy (MAIHDA), a widely used approach in social epidemiology for assessing group-level heterogeneity with multilevel models. We show how the separation statistic can be applied to group-level predictions derived from MAIHDA models and is compatible with both Bayesian and frequentist estimation approaches. The metric can be computed globally, across all groups, or locally, within specific subsets of interest. We demonstrate its utility using applied examples from intersectional MAIHDA and provide accompanying code to facilitate its use in future studies. By quantifying the stability of group-level predictions, the separation statistic offers a broadly applicable tool for describing certainty in the relative ordering of outcomes from multilevel models. Importantly, it should be interpreted as a descriptive measure of ranking uncertainty rather than a prescriptive target, with its limitations carefully considered in applied settings.