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Disjoint factor analysis with cross-loadings

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  • Maurizio Vichi

    (Sapienza University of Rome)

Abstract

Disjoint factor analysis (DFA) is a new latent factor model that we propose here to identify factors that relate to disjoint subsets of variables, thus simplifying the loading matrix structure. Similarly to exploratory factor analysis (EFA), the DFA does not hypothesize prior information on the number of factors and on the relevant relations between variables and factors. In DFA the population variance–covariance structure is hypothesized block diagonal after the proper permutation of variables and estimated by Maximum Likelihood, using an Coordinate Descent type algorithm. Inference on parameters on the number of factors and to confirm the hypothesized simple structure are provided. Properties such as scale equivariance, uniqueness, optimal simplification of loadings are satisfied by DFA. Relevant cross-loadings are also estimated in case they are detected from the best DFA solution. DFA has also the option to constrain a variable to load on a pre-specified factor so that the researcher can assume, a priori, some relations between variables and loadings. A simulation study shows performances of DFA and an application to optimally identify the dimensions of well-being is used to illustrate characteristics of the new methodology. A final discussion concludes the paper.

Suggested Citation

  • Maurizio Vichi, 2017. "Disjoint factor analysis with cross-loadings," Advances in Data Analysis and Classification, Springer;German Classification Society - Gesellschaft für Klassifikation (GfKl);Japanese Classification Society (JCS);Classification and Data Analysis Group of the Italian Statistical Society (CLADAG);International Federation of Classification Societies (IFCS), vol. 11(3), pages 563-591, September.
    • Handle: RePEc:spr:advdac:v:11:y:2017:i:3:d:10.1007_s11634-016-0263-9
    DOI: 10.1007/s11634-016-0263-9
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    References listed on IDEAS

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    Cited by:

    1. Cavicchia, Carlo & Sarnacchiaro, Pasquale & Vichi, Maurizio, 2021. "A composite indicator for the waste management in the EU via Hierarchical Disjoint Non-Negative Factor Analysis," Socio-Economic Planning Sciences, Elsevier, vol. 73(C).
    2. Carlo Cavicchia & Maurizio Vichi, 2021. "Statistical Model-Based Composite Indicators for Tracking Coherent Policy Conclusions," Social Indicators Research: An International and Interdisciplinary Journal for Quality-of-Life Measurement, Springer, vol. 156(2), pages 449-479, August.
    3. Carlo Cavicchia & Maurizio Vichi, 2022. "Second-Order Disjoint Factor Analysis," Psychometrika, Springer;The Psychometric Society, vol. 87(1), pages 289-309, March.
    4. Naoto Yamashita, 2023. "Principal component analysis constrained by layered simple structures," Advances in Data Analysis and Classification, Springer;German Classification Society - Gesellschaft für Klassifikation (GfKl);Japanese Classification Society (JCS);Classification and Data Analysis Group of the Italian Statistical Society (CLADAG);International Federation of Classification Societies (IFCS), vol. 17(2), pages 347-367, June.
    5. Adelaide Freitas & Eloísa Macedo & Maurizio Vichi, 2021. "An empirical comparison of two approaches for CDPCA in high-dimensional data," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 30(3), pages 1007-1031, September.
    6. Carlo Cavicchia & Maurizio Vichi & Giorgia Zaccaria, 2023. "Hierarchical disjoint principal component analysis," AStA Advances in Statistical Analysis, Springer;German Statistical Society, vol. 107(3), pages 537-574, September.

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