Abstract
Hierarchical models are often considered to measure latent concepts defining nested sets of manifest variables. Therefore, by supposing a hierarchical relationship among manifest variables, the general latent concept can be represented by a tree structure where each internal node represents a specific order of abstraction for the latent concept measured. In this paper, we propose a new latent factor model called second-order disjoint factor analysis in order to model an unknown hierarchical structure of the manifest variables with two orders. This is a second-order factor analysis, which—respect to the second-order confirmatory factor analysis—is exploratory, nested and estimated simultaneously by maximum likelihood method. Each subset of manifest variables is modeled to be internally consistent and reliable, that is, manifest variables related to a factor measure “consistently” a unique theoretical construct. This feature implies that manifest variables are positively correlated with the related factor and, therefore, the associated factor loadings are constrained to be nonnegative. A cyclic block coordinate descent algorithm is proposed to maximize the likelihood. We present a simulation study that investigates the ability to get reliable factors. Furthermore, the new model is applied to identify the underlying factors of well-being showing the characteristics of the new methodology. A final discussion completes the paper.
Highlights
Factor analysis (FA, Anderson & Rubin, 1956; Horst, 1965) is one of the most used models to reconstruct manifest variables (MVs) through a set of latent variables
We extend disjoint factor analysis (DFA), which is not appropriate to identify the hierarchical structure of factors since it assumes that factors are orthogonal, that is, factors are not mutually related and do not share common information that could be summarized by the general factor
We conducted an additional study to assess the performance of the 2O-DFA called Second-Order Nonnegative Disjoint Factor Analysis (2ON-DFA) in detecting a simple structure model (SSM), with respect to EFA followed by rotation, where each variable was assigned to the factor with the highest loading resulting from EFA followed by a rotation method
Summary
Factor analysis (FA, Anderson & Rubin, 1956; Horst, 1965) is one of the most used models to reconstruct manifest variables (MVs) through a set of latent variables. It is worth remarking that many hierarchical extensions of FA were already proposed throughout the years (Le Dien & Pages, 2003; Schmid & Leiman, 1957; Thompson, 1951; Wherry, 1959; 1975; 1984) All of these hierarchical extensions were developed as sequential analysis, at times not even guaranteeing to obtain a simple structure, i.e., the partition in H classes of variables where common relations in each class are represented by a single factor. The ARI between the partition found by EFA followed by an oblique rotation method and the generated one resulted equal to 0.78; our methodology, applied to the same dataset, was able to perfectly detect the generated blocks (i.e., ARI = 1) To extend this result to a reasonable number of examples, we generated 200 samples with the same four-block structure.
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