Factor analysis: Latent variable, matrix decomposition, and constrained uniqueness formulations

Abstract

Factor analysis (FA) is a time‐honored multivariate analysis procedure for exploring the factors that underlie observed multiple variables to explain their variations. According to how the factors are treated, the formulations of FA can be classified into three approaches. One of them can be called a latent variable (LV) formulation, while the remaining two can be named matrix decomposition (MD) ones. The factors are regarded as random latent variables in the LV formulation, but treated as fixed parameter matrices in the MD ones. Here, the latter MD formulations are subdivided into the one simply called MD and the constrained uniqueness (CU) formulation with a special constraint. As compared to the LV formulation which is classic, MD and CU ones are recently established. Thus, it is useful to review the formulations now. This article is categorized under: Statistical and Graphical Methods of Data Analysis > Multivariate Analysis

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

“…These three equations imply that the common factors, specific factors, and errors are mutually decomposed completely, thus the name CDFA. This name is considered more suitable than another name, constrained uniqueness FA, used in Adachi (2019), as the CompFA model has not been considered in the latter naming. Further, the theorems show that Stegeman's (2016) procedure in Sect.…”

“…This implies that the p variables in x have one-to-one correspondences to the p unique factors in u: its jth element uniquely affects the jth one of x with the jth diagonal element of being a coefficient. We refer to the FA procedure based on (1) as latent variable FA (LVFA) following Adachi (2019), in order to distinguish it from a procedure to be introduced later.…”

“…However, a recently proposed FA procedure, which is called matrix decomposition FA (MDFA) to distinguish it from LVFA (Adachi, 2019;Adachi & Trendafilov, 2018), can be viewed as a parameter estimation procedure for the CompFA model. In more exactness, MDFA can be modeled as a nonrandom matrix version of (3):…”

“…Bertisimas, Cohenhaver, and Mazumder's (2017) algorithm is used for the first MRFA step in CDFA, and Stegeman's (2016) algorithm is used for the remaining steps in CDFA. The above LS-LVFA algorithm can give an improper solution including a negative unique variance; such a solution cannot be given by the algorithms for the other procedures (Adachi, 2013(Adachi, , 2019.…”

Two-Stage Procrustes Rotation with Sparse Target Matrix and Least Squares Criterion with Regularization and Generalized Weighting

“…A similar but different formulation of factor analysis was recently presented [4] [5], and the following indeterminacy on which the proposer procedure is based on also holds in another formulation.…”