Separability in the fixed part of multilevel models

Separability in ordinary regression is achieved by partitioning the set of explanatory variables into mutually orthogonal subsets. The coefficient vector of each subset is separate: its estimate depends only on the response and on the explanatory variable scores of the subset. The feasibility of formulating multilevel models with subsets of separate parameters in the fixed part is discussed. Generic sufficient conditions for separability and a series of rules and examples are provided. The search for instances of separability rests on an analysis of the covariance matrix of the multilevel model. Its structure, in terms of its spectral decomposition, explains the role of within-cluster centered and orthogonalized variables in multilevel models.