Abstract
Factor analysis is a well-known method for describing the covariance structure among a set of manifest variables through a limited number of unobserved factors. When the observed variables are collected at various occasions on the same statistical units, the data have a three-way structure and standard factor analysis may fail. To overcome these limitations, three-way models, such as the Parafac model, can be adopted. It is often seen as an extension of principal component analysis able to discover unique latent components. The structural version, i.e., as a reparameterization of the covariance matrix, has been also formulated but rarely investigated. In this article, such a formulation is studied by discussing under what conditions factor uniqueness is preserved. It is shown that, under mild conditions, such a property holds even if the specific factors are assumed to be within-variable, or within-occasion, correlated and the model is modified to become scale invariant.
Highlights
Factor analysis (FA) (Bartholomew, Knott, & Moustaki, 2011) is a well-known method explaining the relationships among a set of manifest variables, observed on a sample of statistical units, in terms of a limited number of latent variables
FA deals with two-way two-mode data, where the modes are the entities of the data matrix, i.e., statistical units and manifest variables, and the ways are the indexes of the elements of X, i.e., i = 1, . . . , I and j = 1, . . . , J
As a matter of fact, Carroll & Chang (1970) proposed a canonical decomposition equivalent to Parafac, named Candecomp; as an extension of PCA in multidimensional scaling, Kroonenberg & De Leeuw (1980) named three-mode principal component analysis a method based on the ordinary least squares (OLS) estimation of Tucker3
Summary
PSYCHOMETRIKA the so-called array, or tensor, usually denoted by X of order (I × J × K ). The FA model fitted to XA discovers latent variables ignoring that the same manifest variables are replicated across the occasions For this reason, the standard FA model has been extended in order to take into account and exploit the increasing complexity of three-way three-mode data. As a matter of fact, Carroll & Chang (1970) proposed a canonical decomposition equivalent to Parafac, named Candecomp; as an extension of PCA in multidimensional scaling, Kroonenberg & De Leeuw (1980) named three-mode principal component analysis a method based on the ordinary least squares (OLS) estimation of Tucker3 For these reasons, hereinafter such models will be said to follow a component-based approach. The effectiveness of the proposal is illustrated by a real-life example
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