Measures of clustering and heterogeneity in multilevel Poisson regression analyses of rates/count data

Multilevel data occur frequently in many research areas like health services research and epidemiology. A suitable way to analyze such data is through the use of multilevel regression models. These models incorporate cluster-specific random effects that allow one to partition the total variation in the outcome into between-cluster variation and between-individual variation. The magnitude of the effect of clustering provides a measure of the general contextual effect. When outcomes are binary or time-to-event in nature, the general contextual effect can be quantified by measures of heterogeneity like the median odds ratio or the median hazard ratio, respectively, which can be calculated from a multilevel regression model. Outcomes that are integer counts denoting the number of times that an event occurred are common in epidemiological and medical research. The median (incidence) rate ratio in multilevel Poisson regression for counts that corresponds to the median odds ratio or median hazard ratio for binary or time-to-event outcomes respectively is relatively unknown and is rarely used. The median rate ratio is the median relative change in the rate of the occurrence of the event when comparing identical subjects from 2 randomly selected different clusters that are ordered by rate. We also describe how the variance partition coefficient, which denotes the proportion of the variation in the outcome that is attributable to between-cluster differences, can be computed with count outcomes. We illustrate the application and interpretation of these measures in a case study analyzing the rate of hospital readmission in patients discharged from hospital with a diagnosis of heart failure.