IDEAS home Printed from https://ideas.repec.org/a/spr/alstar/v107y2023i3d10.1007_s10182-022-00458-4.html
   My bibliography  Save this article

Hierarchical disjoint principal component analysis

Author

Listed:
  • Carlo Cavicchia

    (Erasmus University Rotterdam)

  • Maurizio Vichi

    (University of Rome La Sapienza)

  • Giorgia Zaccaria

    (University of Rome La Sapienza)

Abstract

Dimension reduction, by means of Principal Component Analysis (PCA), is often employed to obtain a reduced set of components preserving the largest possible part of the total variance of the observed variables. Several methodologies have been proposed either to improve the interpretation of PCA results (e.g., by means of orthogonal, oblique rotations, shrinkage methods), or to model oblique components or factors with a hierarchical structure, such as in Bi-factor and High-Order Factor analyses. In this paper, we propose a new methodology, called Hierarchical Disjoint Principal Component Analysis (HierDPCA), that aims at building a hierarchy of disjoint principal components of maximum variance associated with disjoint groups of observed variables, from Q up to a unique, general one. HierDPCA also allows choosing the type of the relationship among disjoint principal components of two sequential levels, from the lowest upwards, by testing the component correlation per level and changing from a reflective to a formative approach when this correlation turns out to be not statistically significant. The methodology is formulated in a semi-parametric least-squares framework and a coordinate descent algorithm is proposed to estimate the model parameters. A simulation study and two real applications are illustrated to highlight the empirical properties of the proposed methodology.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:alstar:v:107:y:2023:i:3:d:10.1007_s10182-022-00458-4
    DOI: 10.1007/s10182-022-00458-4
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10182-022-00458-4
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s10182-022-00458-4?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Yiu-Fai Yung & David Thissen & Lori McLeod, 1999. "On the relationship between the higher-order factor model and the hierarchical factor model," Psychometrika, Springer;The Psychometric Society, vol. 64(2), pages 113-128, June.
    2. 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.
    3. Matteo Mazziotta & Adriano Pareto, 2019. "Use and Misuse of PCA for Measuring Well-Being," Social Indicators Research: An International and Interdisciplinary Journal for Quality-of-Life Measurement, Springer, vol. 142(2), pages 451-476, April.
    4. S. K. Vines, 2000. "Simple principal components," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 49(4), pages 441-451.
    5. Vichi, Maurizio & Saporta, Gilbert, 2009. "Clustering and disjoint principal component analysis," Computational Statistics & Data Analysis, Elsevier, vol. 53(8), pages 3194-3208, June.
    6. Jarvis, Cheryl Burke & MacKenzie, Scott B & Podsakoff, Philip M, 2003. "A Critical Review of Construct Indicators and Measurement Model Misspecification in Marketing and Consumer Research," Journal of Consumer Research, Journal of Consumer Research Inc., vol. 30(2), pages 199-218, September.
    7. Chen, Song Xi & Zhang, Li-Xin & Zhong, Ping-Shou, 2010. "Tests for High-Dimensional Covariance Matrices," Journal of the American Statistical Association, American Statistical Association, vol. 105(490), pages 810-819.
    8. Robert Wherry, 1959. "Hierarchical factor solutions without rotation," Psychometrika, Springer;The Psychometric Society, vol. 24(1), pages 45-51, March.
    9. Carlo Cavicchia & Maurizio Vichi & Giorgia Zaccaria, 2020. "The ultrametric correlation matrix for modelling hierarchical latent concepts," 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. 14(4), pages 837-853, December.
    10. Lee Cronbach, 1951. "Coefficient alpha and the internal structure of tests," Psychometrika, Springer;The Psychometric Society, vol. 16(3), pages 297-334, September.
    11. Lawrence Hubert & Phipps Arabie, 1985. "Comparing partitions," Journal of Classification, Springer;The Classification Society, vol. 2(1), pages 193-218, December.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Carlo Cavicchia & Maurizio Vichi, 2022. "Second-Order Disjoint Factor Analysis," Psychometrika, Springer;The Psychometric Society, vol. 87(1), pages 289-309, March.
    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 & Giorgia Zaccaria, 2020. "The ultrametric correlation matrix for modelling hierarchical latent concepts," 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. 14(4), pages 837-853, December.
    4. 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.
    5. John S. Hill & Myung-Su Chae & Jinseo Park, 2012. "The Effects of Geography and Infrastructure on Economic Development and International Business Involvement," Journal of Infrastructure Development, India Development Foundation, vol. 4(2), pages 91-113, December.
    6. Matthijs J. Warrens, 2021. "Kappa coefficients for dichotomous-nominal classifications," 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. 15(1), pages 193-208, March.
    7. Sönke Albers & Lutz Hildebrandt, 2006. "Methodische Probleme bei der Erfolgsfaktorenforschung — Messfehler, formative versus reflektive Indikatoren und die Wahl des Strukturgleichungs-Modells," Schmalenbach Journal of Business Research, Springer, vol. 58(1), pages 2-33, February.
    8. Sajtos, Laszlo & Magyar, Bertalan, 2016. "Auxiliary theories as translation mechanisms for measurement model specification," Journal of Business Research, Elsevier, vol. 69(8), pages 3186-3191.
    9. Georg Fassott, 2006. "Operationalisierung latenter Variablen in Strukturgleichungsmodellen: Eine Standortbestimmung," Schmalenbach Journal of Business Research, Springer, vol. 58(1), pages 67-88, February.
    10. Ohiomah, Alhassan & Andreev, Pavel & Benyoucef, Morad & Hood, David, 2019. "The role of lead management systems in inside sales performance," Journal of Business Research, Elsevier, vol. 102(C), pages 163-177.
    11. Huete-Alcocer, Nuria & Hernandez-Rojas, Ricardo David, 2022. "Do SARS-CoV-2 safety measures affect visitors experience of traditional gastronomy, destination image and loyalty to a World Heritage City?," Journal of Retailing and Consumer Services, Elsevier, vol. 69(C).
    12. Leibao Zhang & Liming Sheng & Wenyu Zhang & Shuai Zhang, 2020. "Do Personal Norms Predict Citizens’ Acceptance of Green Transport Policies in China," Sustainability, MDPI, vol. 12(12), pages 1-16, June.
    13. Nitzl, Christian, 2016. "The use of partial least squares structural equation modelling (PLS-SEM) in management accounting research: Directions for future theory development," Journal of Accounting Literature, Elsevier, vol. 37(C), pages 19-35.
    14. Eggers, Fabian & Niemand, Thomas & Kraus, Sascha & Breier, Matthias, 2020. "Developing a scale for entrepreneurial marketing: Revealing its inner frame and prediction of performance," Journal of Business Research, Elsevier, vol. 113(C), pages 72-82.
    15. Carlo Cavicchia & Maurizio Vichi & Giorgia Zaccaria, 2022. "Gaussian mixture model with an extended ultrametric covariance structure," 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. 16(2), pages 399-427, June.
    16. Matheus Pereira Libório & Oseias da Silva Martinuci & Sandro Laudares & Renata de Mello Lyrio & Alexei Manso Correa Machado & Patrícia Bernardes & Petr Ekel, 2020. "Measuring Intra-Urban Inequality with Structural Equation Modeling: A Theory-Grounded Indicator," Sustainability, MDPI, vol. 12(20), pages 1-18, October.
    17. Markus Eckey & Utz Schäffer, 2006. "Kontrolle von Mehrheitsbeteiligungen in börsennotierten Management-Holdings," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 17(3), pages 251-280, November.
    18. 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).
    19. José Fernando Romero Cañizares & Purificación Vicente Galindo & Yannis Phillis & Evangelos Grigoroudis, 2022. "Graphical sustainability analysis using disjoint biplots," Operational Research, Springer, vol. 22(2), pages 1575-1596, April.
    20. Guillard, Valérie, 2009. "La tendance de certains consommateurs à tout garder," Economics Thesis from University Paris Dauphine, Paris Dauphine University, number 123456789/5480 edited by Pinson, Christian.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:alstar:v:107:y:2023:i:3:d:10.1007_s10182-022-00458-4. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.