Generalized Structured Component Analysis

What is Generalized Structured Component Analysis (GSCA)

GSCA is a component based structural equation model method and can be used as PLS Path Modeling.

This method introduced by Hwang and Takane (2011), allows to optimize a global function using an algorithm called Alternating Least Square algorithm (ALS).

GSCA lies in the tradition of component analysis. It substitutes components for factors as in PLS. Unlike PLS, however, GSCA offers a global least squares optimization criterion, which is consistently minimized to obtain the estimates of model parameters. GSCA is thus equipped with an overall measure of model fit while fully maintaining all the advantages of PLS (e.g., less restricted distributional assumptions, no improper solutions, and unique component score estimates). In addition, GSCA handles more diverse path analyses, compared to PLS.

Let Z denote an N by J matrix of observed variables. Assume that Z is columnwise centered and scaled to unit variance. Then, the model for GSCA may be expressed as

ZV = ZWA + E,

P = GA + E,    (1)

where P = ZV, and G = ZW. In (1), P is an N by T matrix of all endogenous observed and composite variables, G is an N by D matrix of all exogenous observed and composite variables, V is a J by T matrix of component weights associated with the endogenous variables, W is a J by D matrix of component weights for the exogenous variables, A is a D by T supermatrix consisting of a matrix of component loadings relating components to their observed variables, denoted by C, in addition to a matrix of path coefficients between components, denoted by B, that is, A = [C, B], and E is a matrix of residuals.

We estimate the unknown parameters V,W, and A in such a way that the sum of squares of the residuals, E = ZV - ZWA = P- GA, is as small as possible. This amounts to minimizing

f = SS(ZV - ZWA)

                    = SS(P- GA),                (2)

with respect to V, W, and A, where SS(X) = trace(X’X). The components in P and/or G are subject to normalization for identification purposes.

We cannot solve (2) in an analytic way since V, W, and A can comprise zero or any fixed elements. Instead, we develop an alternating least squares (ALS) algorithm (de Leeuw, Young, & Takane, 1976) to minimize (2). In general, ALS can be viewed as a special type of the FP algorithm where the fixed point is a stationary (accumulation) point of a function to be optimized.

The proposed ALS algorithm consists of two steps: In the first step, A is updated for fixed V and W. In the second step, V and W are updated for fixed A. (Hwang and Takane, 2004)