Extending the Median Odds Ratio (MOR), the Interval Odds Ratio (IOR), and the Proportion of Opposed Odds Ratios (POOR) for use with 3-level multilevel logistic regression models

Multilevel data occur frequently in health services, population and public health, and epidemiologic research (e.g., patients nested within hospitals). Multilevel logistic regression models allow one to account for the clustering of individuals within higher-level units when estimating the association of individual-level and cluster-level characteristics with individual-level binary outcomes. Odds ratios for cluster-level variables generated from a multilevel logistic regression model have an interpretation that is conditional on the random effects, compared to the marginal interpretation of odds ratios from logistic regression models that do not incorporate cluster-specific random effects. The interval odds ratio (IOR) and the proportion of opposed odds ratios (POOR) are two summary measures that have been proposed for quantifying the magnitude and heterogeneity of the association of cluster-level variables with individual-level binary outcomes. The median odds ratio (MOR) is a measure of the between-cluster heterogeneity of a binary outcome. The IOR, the POOR, and the MOR were developed for use with two-level multilevel logistic regression models (e.g., patients nested within hospitals). In applied research, data with a three-level multilevel structure are common (e.g., hospitalized patients cared for by the attending physician who in turn practice within hospitals). We mathematically derived extensions of the MOR, the IOR, and the POOR for use with three-level multilevel logistic regression models. We then illustrated the application and interpretation of the MOR, the IOR, and the POOR by analyzing mortality in patients hospitalized with acute myocardial infarction, where patients are nested within physicians who in turn are nested within hospitals.