ABSTRACT
We are often interested in making inferences on many parameters corresponding to different “units,” for example, different patients, geographical areas, schools, hospitals, etc. The main goal is often to quantify the degree of similarity across units so that we can make predictions about “new” units. Suppose, for example, that a specific operation is carried out in a number of hospitals, labelled A, B, C, D, etc. Further suppose that the observed mortality rates for that operation in hospitals A, B, and C are 0.1, 0.19, and 0.14, respectively. What would you predict for hospital D? What information did you use to come up with that prediction? Most people would predict a value that is similar, in some sense, to the other values. We tend to recognise that it is unlikely that all hospitals have the same underlying mortality rate, due to employing different surgeons and having different catchment areas, for example, but we also tend to assume that knowing something about the other hospitals tells us at least something about the one of interest. Our natural inclination, therefore, is towards an assumption somewhere between the unit-specific parameters being identical and being entirely independent.
