<b>SOME CONTRIBUTIONS TO DATA-FITTING FACTOR </b><b>ANALYSIS WITH EMPIRICAL COMPARISONS TO </b><b>COVARIANCE-FITTING FACTOR ANALYSIS </b>

Abstract

A data-fitting factor analysis (FA) procedure was recently presented, which is very different from the prevailing covariance-fitting FA. In the former procedure, common and unique factor scores are modeled as fixed unknown parameters, and an unweighted least squares (ULS) function, which is not scale invariant, is minimized for fitting the model to a data matrix. The main purpose of this paper is to settle four remaining problems with data-fitting FA. First, we present a weighted least squares (WLS) procedure which can be scale invariant, and include the above ULS procedure as a special case according to the choice of weights. Second, we prove that the WLS loss function can be minimized, even if raw data are unknown and only their sample covariance matrix is available, despite being a data-fitting approach. Third, we propose an estimator of factor scores that cannot be uniquely determined. Fourth, we empirically compare this data-fitting FA procedure with covariance-fitting FA with respect to recovery of parameter matrices.

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

“…The norm of the undetermined part is constant (Adachi, ) with || n 1/2 K ⊥ L ⊥ ′|| = ( nm ) 1/2 and that part is orthogonal to the determined part with ( KL ′)′( K ⊥ L ⊥ ′) = O p × m . Those facts allow us to depict (34) as the cone in Figure , where matrices are depicted as arrows (Adachi & Trendafilov, ).…”

“…As (36) makes (32) feasible, the MDFA algorithm for obtaining B from S can be listed as in Figure 5. Table 2 shows the MDFA, LVFA, and CUFA solutions for the correlation matrix S obtained from the 62 × 6 baseball data matrix in Adachi (Adachi, 2012). The CUFA solution is mentioned in Section 5, but not here.…”

“…It is the sum of the determined part n 1/2 KL 0 = XBLΔ −1 L 0 and the undetermined part n 1/2 K ⊥ L ⊥ 0 , which shows the factor indeterminacy in MDFA. The norm of the undetermined part is constant (Adachi, 2012) with ||n 1/2 K ⊥ L ⊥ 0 || = (nm) 1/2 and that part is orthogonal to the determined part with (KL 0 ) 0 (K ⊥ L ⊥ 0 ) = O p × m . Those facts allow us to depict (34) as the cone in Figure 6, where matrices are depicted as arrows (Adachi & Trendafilov, 2018).…”

Sparsest factor analysis for clustering variables: a matrix decomposition approach

“…As detailed in Adachi (2012), the MDFA loss function (4) is not scale-free as MLFA in (2) (e.g., Harman, 1976). Thus, the MDFA and MDFA-based SSFA solutions for covariances are essentially different from those for the correlations obtained from the same data set.…”

“…It is empirically known that MDFA and MLFA provide almost equivalent solutions (Adachi, 2012(Adachi, , 2014.…”