Simultaneous autoregressive (SAR) model

Simultaneous autoregressive (SAR) models are useful for accommodating various forms of dependence among data that have discrete support in a space of interest. These models are often specified hierarchically as mixed-effects regression models with first-moment structure controlled by a conventional linear regression term and second-moment structure induced by correlated random effects. In their general form, SAR models resemble conditional autoregressive (CAR) models, and can be made equivalent but are often parameterized differently. Importantly, SAR models can be specified by simultaneously regressing a discrete spatial process on itself. Thus, they allow one to construct statistical models for processes with directional graphical properties that pertain to data generating mechanisms. Most commonly SAR models have been used to account for structure among data with areal spatial support in applications involving ecology, epidemiology, sociology, and environmental science.