IDEAS home Printed from https://ideas.repec.org/a/spr/psycho/v83y2018i4d10.1007_s11336-017-9599-0.html
   My bibliography  Save this article

Direct Schmid–Leiman Transformations and Rank-Deficient Loadings Matrices

Author

Listed:
  • Niels G. Waller

    (University of Minnesota)

Abstract

The Schmid–Leiman (S–L; Psychometrika 22: 53–61, 1957) transformation is a popular method for conducting exploratory bifactor analysis that has been used in hundreds of studies of individual differences variables. To perform a two-level S–L transformation, it is generally believed that two separate factor analyses are required: a first-level analysis in which k obliquely rotated factors are extracted from an observed-variable correlation matrix, and a second-level analysis in which a general factor is extracted from the correlations of the first-level factors. In this article, I demonstrate that the S–L loadings matrix is necessarily rank deficient. I then show how this feature of the S–L transformation can be used to obtain a direct S–L solution from an unrotated first-level factor structure. Next, I reanalyze two examples from Mansolf and Reise (Multivar Behav Res 51: 698–717, 2016) to illustrate the utility of ‘best-fitting’ S–L rotations when gauging the ability of hierarchical factor models to recover known bifactor structures. Finally, I show how to compute direct bifactor solutions for non-hierarchical bifactor structures. An online supplement includes R code to reproduce all of the analyses that are reported in the article.

Suggested Citation

  • Niels G. Waller, 2018. "Direct Schmid–Leiman Transformations and Rank-Deficient Loadings Matrices," Psychometrika, Springer;The Psychometric Society, vol. 83(4), pages 858-870, December.
    • Handle: RePEc:spr:psycho:v:83:y:2018:i:4:d:10.1007_s11336-017-9599-0
    DOI: 10.1007/s11336-017-9599-0
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s11336-017-9599-0
    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/s11336-017-9599-0?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. Karl Holzinger & Frances Swineford, 1937. "The Bi-factor method," Psychometrika, Springer;The Psychometric Society, vol. 2(1), pages 41-54, March.
    3. David Grayson & Herbert Marsh, 1994. "Identification with deficient rank loading matrices in confirmatory factor analysis: Multitrait-multimethod models," Psychometrika, Springer;The Psychometric Society, vol. 59(1), pages 121-134, March.
    4. Ledyard Tucker, 1940. "The role of correlated factors in factor analysis," Psychometrika, Springer;The Psychometric Society, vol. 5(2), pages 141-152, June.
    5. Carl Eckart & Gale Young, 1936. "The approximation of one matrix by another of lower rank," Psychometrika, Springer;The Psychometric Society, vol. 1(3), pages 211-218, September.
    6. Norman Cliff, 1966. "Orthogonal rotation to congruence," Psychometrika, Springer;The Psychometric Society, vol. 31(1), pages 33-42, March.
    7. Harry Harman & Wayne Jones, 1966. "Factor analysis by minimizing residuals (minres)," Psychometrika, Springer;The Psychometric Society, vol. 31(3), pages 351-368, September.
    8. John Schmid & John Leiman, 1957. "The development of hierarchical factor solutions," Psychometrika, Springer;The Psychometric Society, vol. 22(1), pages 53-61, March.
    9. R. Jennrich & P. Sampson, 1966. "Rotation for simple loadings," Psychometrika, Springer;The Psychometric Society, vol. 31(3), pages 313-323, September.
    10. Michael Browne, 1968. "A comparison of factor analytic techniques," Psychometrika, Springer;The Psychometric Society, vol. 33(3), pages 267-334, September.
    11. Robert Jennrich & Peter Bentler, 2011. "Exploratory Bi-Factor Analysis," Psychometrika, Springer;The Psychometric Society, vol. 76(4), pages 537-549, October.
    12. Robert Jennrich & Peter Bentler, 2012. "Exploratory Bi-factor Analysis: The Oblique Case," Psychometrika, Springer;The Psychometric Society, vol. 77(3), pages 442-454, July.
    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 & 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.
    2. Michela Gnaldi, 2017. "A multidimensional IRT approach for dimensionality assessment of standardised students’ tests in mathematics," Quality & Quantity: International Journal of Methodology, Springer, vol. 51(3), pages 1167-1182, May.
    3. Robert Jennrich & Peter Bentler, 2012. "Exploratory Bi-factor Analysis: The Oblique Case," Psychometrika, Springer;The Psychometric Society, vol. 77(3), pages 442-454, July.
    4. Li Cai, 2015. "Lord–Wingersky Algorithm Version 2.0 for Hierarchical Item Factor Models with Applications in Test Scoring, Scale Alignment, and Model Fit Testing," Psychometrika, Springer;The Psychometric Society, vol. 80(2), pages 535-559, June.
    5. Stanley Mulaik, 1986. "Factor analysis and Psychometrika: Major developments," Psychometrika, Springer;The Psychometric Society, vol. 51(1), pages 23-33, March.
    6. Robert Jennrich & Peter Bentler, 2011. "Exploratory Bi-Factor Analysis," Psychometrika, Springer;The Psychometric Society, vol. 76(4), pages 537-549, October.
    7. Henk Kiers, 1994. "Simplimax: Oblique rotation to an optimal target with simple structure," Psychometrika, Springer;The Psychometric Society, vol. 59(4), pages 567-579, December.
    8. Stegeman, Alwin, 2016. "A new method for simultaneous estimation of the factor model parameters, factor scores, and unique parts," Computational Statistics & Data Analysis, Elsevier, vol. 99(C), pages 189-203.
    9. Jos Berge & Henk Kiers, 1993. "An alternating least squares method for the weighted approximation of a symmetric matrix," Psychometrika, Springer;The Psychometric Society, vol. 58(1), pages 115-118, March.
    10. Kan, Kees-Jan & van der Maas, Han L.J. & Levine, Stephen Z., 2019. "Extending psychometric network analysis: Empirical evidence against g in favor of mutualism?," Intelligence, Elsevier, vol. 73(C), pages 52-62.
    11. Robert Boik, 1996. "An efficient algorithm for joint correspondence analysis," Psychometrika, Springer;The Psychometric Society, vol. 61(2), pages 255-269, June.
    12. Harold Bechtoldt, 1961. "An empirical study of the factor analysis stability hypothesis," Psychometrika, Springer;The Psychometric Society, vol. 26(4), pages 405-432, December.
    13. M. Corballis & R. Traub, 1970. "Longitudinal factor analysis," Psychometrika, Springer;The Psychometric Society, vol. 35(1), pages 79-98, March.
    14. Kohei Adachi, 2022. "Factor Analysis Procedures Revisited from the Comprehensive Model with Unique Factors Decomposed into Specific Factors and Errors," Psychometrika, Springer;The Psychometric Society, vol. 87(3), pages 967-991, September.
    15. Charley Xia & Sarah J. Pickett & David C. M. Liewald & Alexander Weiss & Gavin Hudson & W. David Hill, 2023. "The contributions of mitochondrial and nuclear mitochondrial genetic variation to neuroticism," Nature Communications, Nature, vol. 14(1), pages 1-14, December.
    16. Minjeong Jeon & Frank Rijmen & Sophia Rabe-Hesketh, 2018. "CFA Models with a General Factor and Multiple Sets of Secondary Factors," Psychometrika, Springer;The Psychometric Society, vol. 83(4), pages 785-808, December.
    17. Henk Kiers, 1997. "Weighted least squares fitting using ordinary least squares algorithms," Psychometrika, Springer;The Psychometric Society, vol. 62(2), pages 251-266, June.
    18. Henk Kiers & Yoshio Takane & Jos Berge, 1996. "The analysis of multitrait-multimethod matrices via constrained components analysis," Psychometrika, Springer;The Psychometric Society, vol. 61(4), pages 601-628, December.
    19. Farid Anvari & Stephan Billinger & Pantelis P. Analytis & Vithor Rosa Franco & Davide Marchiori, 2024. "Testing the convergent validity, domain generality, and temporal stability of selected measures of people’s tendency to explore," Nature Communications, Nature, vol. 15(1), pages 1-23, December.
    20. Frank Rijmen & Minjeong Jeon & Matthias von Davier & Sophia Rabe-Hesketh, 2014. "A Third-Order Item Response Theory Model for Modeling the Effects of Domains and Subdomains in Large-Scale Educational Assessment Surveys," Journal of Educational and Behavioral Statistics, , vol. 39(4), pages 235-256, August.

    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:psycho:v:83:y:2018:i:4:d:10.1007_s11336-017-9599-0. 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.