Notation

We consider two-level hierarchical data structures and follow the notation of Bryk and Rauden-bush4 who developed HLM/2L. For details of three-level models see Bryk and Raudenbush,4 Gatsonis et al.5 and Skene and Wakefield.6

Similar to our example (the PORT study, restricting attention to one site) consider an application involving a two-stage sampling strategy. In the simplest two-stage sampling strategy a simple random sample of primary sampling units (PSUs) is selected and then, within each PSU, a simple random sample of secondary sampling units (SSUs) is selected. The PSUs could reflect hospitals, physicians, clinics or some other entity and the SSUs could reflect physicians (within hospitals) patients (within physician practices or within clinics) or some other entity. In the PORT study, the PSU was the physician and the SSU was the patient at each site. In two-level hierarchical analyses, observations are classified as level 1 (within) or level 2 (between) units. In the PORT study, patients represent the level 1 units and physicians represent the level 2 units.

We use J to denote the number of level 2 units (in the PORT study, physicians) and within each of they = 1, ... ,J level 2 units (physician practices) there are rij level 1 units (in the PORT study, there are rij patients in the jth physician practice). The data do not have to be balanced (that is, it is not necessary that n} = nk for j # k). Figure 1 displays the data structure for the two-level hierarchical model.

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