Factor Analysis Procedures Revisited from the Comprehensive Model with Unique Factors Decomposed into Specific Factors and Errors

Factor analysis (FA) procedures can be classified into three types (Adachi in WIREs Comput Stat https://onlinelibrary.wiley.com/doi/abs/10.1002/wics.1458 , 2019): latent variable FA (LVFA), matrix decomposition FA (MDFA), and its variant (Stegeman in Comput Stat Data Anal 99: 189–203, 2016) named completely decomposed FA (CDFA) through the theorems proved in this paper. We revisit those procedures from the Comprehensive FA (CompFA) model, in which a multivariate observation is decomposed into common factor, specific factor, and error parts. These three parts are separated in MDFA and CDFA, while the specific factor and error parts are not separated, but their sum, called a unique factor, is considered in LVFA. We show that the assumptions in the CompFA model are satisfied by the CDFA solution, but not completely by the MDFA one. Then, how the CompFA model parameters are estimated in the FA procedures is examined. The study shows that all parameters can be recovered well in CDFA, while the sum of the parameters for the specific factor and error parts is approximated by the LVFA estimate of the unique factor parameter and by the MDFA estimate of the specific factor parameter. More detailed results are given through our subdivision of the CompFA model according to whether the error part is uncorrelated among variables or not.