  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
