The median hazard ratio: a useful measure of variance and general contextual effects in multilevel survival analysis.

Multilevel data occurs 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 (MLRM). MLRM incorporate cluster-specific random effects which allow one to partition the total individual variance into between-cluster variation and between-individual variation. Statistically, MLRM account for the dependency of the data within clusters and provide correct estimates of uncertainty around regression coefficients. Substantively, the magnitude of the effect of clustering provides a measure of the General Contextual Effect (GCE). When outcomes are binary, the GCE can also be quantified by measures of heterogeneity like the Median Odds Ratio (MOR) calculated from a multilevel logistic regression model. Time-to-event outcomes within a multilevel structure occur commonly in epidemiological and medical research. However, the Median Hazard Ratio (MHR) that corresponds to the MOR in multilevel (i.e., 'frailty') Cox proportional hazards regression is rarely used. Analogously to the MOR, the MHR is the median relative change in the hazard of the occurrence of the outcome when comparing identical subjects from two randomly selected different clusters that are ordered by risk. We illustrate the application and interpretation of the MHR in a case study analyzing the hazard of mortality in patients hospitalized for acute myocardial infarction at hospitals in Ontario, Canada. We provide R code for computing the MHR. The MHR is a useful and intuitive measure for expressing cluster heterogeneity in the outcome and, thereby, estimating general contextual effects in multilevel survival analysis. © 2016 The Authors. Statistics in Medicine published by John Wiley & Sons Ltd.