A Novel Identification Approach to Bayesian Factor Analysis with Sparse Loadings Matrices

Abstract

Sparse factor analysis comprises aspects of exploratory and confirmatory factor analysis, seeking to establish a parsimonious structure in the loadings matrix of the model. This task is related to the issue of determining the number of factors required for model representation, the question of which variables are useful and which ones can be excluded from the analysis, and the problem whether some variables are driven by a subset of all factors only. Whereas sparsity analysis focuses mainly on the third of these questions, it can provide helpful hints to tackle the first two questions as well. I use multivariate highest posterior density (HPD) intervals calculated for the posterior densities derived from the weighted orthogonal Procrustes (WOP) ex-post identification approach to find a sparse loadings structure. In a simulation study, this method is used to identify different sparse structures, including those with excess variables, and to determine the number of factors in the model, where all three tasks are well achieved. Eventually, I apply the approach on a data set of intelligence test results to determine the number of factors, the required variables and the sparsity structure, where it yields results not only well-comprehensible, but also very similar to those found in former studies analyzing the data set.