Modelling volume-outcome relationships in health care

Despite the ongoing strong interest in associations between quality of care and the volume of health care providers, a unified statistical framework for analyzing them is missing, and many studies suffer from poor statistical modelling choices. We propose a flexible, additive mixed model for studying volume-outcome associations in health care that takes into account individual patient characteristics as well as provider-specific effects through a multi-level approach. More specifically, we treat volume as a continuous variable, and its effect on the considered outcome is modelled as a smooth function. We take account of different case-mixes by including patient-specific risk factors and of clustering on the provider level through random intercepts. This strategy enables us to extract a smooth volume effect as well as volume-independent provider effects. These two quantities can be compared directly in terms of their magnitude, which gives insight into the sources of variability of quality of care. Based on a causal DAG, we derive conditions under which the volume-effect can be interpreted as a causal effect. The paper provides confidence sets for each of the estimated quantities relying on joint estimation of all effects and parameters. Our approach is illustrated through simulation studies and an application to German health care data. Keywords: health care quality measurement, volume-outcome analysis, minimum provider volume, additive regression models, random intercept