Multilevel dimensionality-reduction methods

When data sets are multilevel (group nesting or repeated measures), different sources of variations must be identified. In the framework of unsupervised analyses, multilevel simultaneous component analysis (MSCA) has recently been proposed as the most satisfactory option for analyzing multilevel data. MSCA estimates submodels for the different levels in data and thereby separates the “within”-subject and “between”-subject variations in the variables. Following the principles of MSCA and the strategy of decomposing the available data matrix into orthogonal blocks, and taking into account the between- and the within data structures, we generalize, in a multilevel perspective, multivariate models in which a matrix of response variables can be used to guide the projections (formed by responses predicted by explanatory variables or by a limited number of their combinations/composites) into choices of meaningful directions. To this end, the current paper proposes the multilevel version of the multivariate regression model and dimensionality-reduction methods (used to predict responses with fewer linear composites of explanatory variables). The principle findings of the study are that the minimization of the loss functions related to multivariate regression, principal-component regression, reduced-rank regression, and canonical-correlation regression are equivalent to the separate minimization of the sum of two separate loss functions corresponding to the between and within structures, under some constraints. The paper closes with a case study of an application focusing on the relationships between mental health severity and the intensity of care in the Lombardy region mental health system.