Skip to main content
SpringerLink
Log in

Disjoint factor analysis with cross-loadings

  • Regular Article
  • Published:
Advances in Data Analysis and Classification Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others

Notes

  1. For matrix V columns and rows will be denoted by the \(\mathbf{v}_{.h}\) and \(\mathbf{v}_j\)., respectively.

  2. The OECD defines regions as the first tier of sub-national government (for example states in the United States, provinces in Canada, “régions” in France, “regioni” in Italy).

  3. The Cronbach’s Alpha measures an internal consistency: \(\upalpha \ge 0.9, excellent; 0.7 \le \upalpha < 0.9\), good; \(0.6 \le \upalpha < 0.7\), acceptable; \(0.5 \le \upalpha < 0.6\) poor; \(\upalpha < 0.5\), Unacceptable.

  4. Note that in 50 random starts the same optimal solution was found 41 times.

  5. Note that in 50 random starts the optimal solution was found 40 times.

References

  • Abdi H (2003) Factor Rotations in Factor Analyses. In: Lewis-Beck M, Bryman A, Futing T (eds) Encyclopedia of Social Sciences Research Methods, pp 792–795. Sage, Thousand Oaks (CA)

  • Amstrong JS, Soelberg P (1968) On the interpretation of factor analysis. Psychol Bull 70(5):361–364

    Article  Google Scholar 

  • Akaike H (1973) Information theory and an extension of the maximum likelihood principle. In: Petrov BN, Csáki F (eds) 2nd International Symposium on Information Theory, Tsahkadsor, Armenia, USSR. Akadémiai Kiadó, Budapest, pp 267–281

  • Anderson TW, Rubin H (1956) Statistical inferences in factor analysis. In: Proceedings of the Third Symposium on Mathematical Statistics and Probability, vol V, pp 111–150

  • Aoyagi H, Ogunseitan OA (2015) Toxic releases and risk disparity: a spatiotemporal model of industrial ecology and social empowerment. Int J Environ Res Public Health 12:6300–6318

    Article  Google Scholar 

  • Bartlett MS (1937) The statistical conception of mental factors. Br J Psychol 28:97–104

    Google Scholar 

  • Bartlett MS (1954) A note on the multiplying factors for various \(\chi \)2 approximations. J R Stat Soc Ser B (Methodol) 16(2):296–298

    MATH  Google Scholar 

  • Boer D, Abubakar A (2014) Music listening in families and peer groups: benefits for young people’s social cohesion and emotional well-being across four cultures. Frontiers Psychol 5(392):1–15

    Google Scholar 

  • Costello AB, Osborne JW (2005) Best practices in exploratory factor analysis: four recommendations for getting the most from your analysis. Pract Assess Res Eval. http://pareonline.net/getvn.asp?v=10&n=7

  • d’Aspremont A, El Ghaoui L, Jordan MI, Lanckriet GRG (2007) A direct formulation for sparse PCA using semidefinite programming. SIAM Rev 49(3):434–448

    Article  MathSciNet  MATH  Google Scholar 

  • Elliot AJ, McGregor HA (2001) A 2 \(\times \) 2 achievement goal Framework. J Pers Soc Psychol 80(3):501–519

    Article  Google Scholar 

  • Grbovic M, Dance CR, Vucetic S (2012) Sparse principal component analysis with constraints. In: Proceedings of the twenty-sixth AAAI conference on artificial intelligence. https://www.aaai.org/ocs/index.php/AAAI/AAAI12/paper/viewFile/5109/5503

  • Horst P (1965) Factor analysis of data matrices. Holt, Rinehart and Winston, New York

    MATH  Google Scholar 

  • Hubert L, Arabie P (1985) Comparing partitions. J Classif 2:193–218

    Article  MATH  Google Scholar 

  • Joereskog KG (1969) A general approach to confirmatory maximum-likelihood factor analysis. Psychometrika 34:183–202

    Article  Google Scholar 

  • Joereskog KG (1978) Structural analysis of covariance and correlation matrices. Psychometrika 43:443–477

    Article  MathSciNet  Google Scholar 

  • Joereskog KG (1979) A general approach to confirmatory maximum likelihood factor analysis, with addendum. In: Joereskog KG, Sorbom D, Magidson J (eds) Advances in factor analysis and structural equation models, pp 40–43

  • Joereskog KG, Sorbom D (1979) Advances in factor analysis and equation models. Abt Books, Cambridge

    Google Scholar 

  • Joereskog KG, Sorbom D (1981) LISREL V: Analysis of linear structural relationships by the method of maximum likelihood [Computer program]. Scientific Software, Mooresville

    Google Scholar 

  • Jolliffe IT, Trendafilov NT, Uddin M (2003) A modified principal component technique based on the LASSO. J Comput Graph Stat 12(3):531–547

    Article  MathSciNet  Google Scholar 

  • Kaiser HF (1958) The varimax criterion for analytic rotation in factor analysis. Psychometrika 23:187–200

    Article  MATH  Google Scholar 

  • Kim EJ (2013) Accessing factor structure and construct validity of the successful aging inventory. J Korean Acad Nurs 43(4):568–578

  • Kline P (2000) The handbook of psychological testing (2nd edn.). Routledge, London, p 13

  • Leiva-Bianchi MC, Araneda AC (2013) Validation of the Davidson Trauma Scale in its original and a new shorter version in people exposed to the F-27 earthquake in Chile. Eur J Psychotraumatology 4:21239

    Article  Google Scholar 

  • Masaya I, Satomi N, Daisuke F, Mitsunori M, Kim Y, Shear MK, Angela G, Wall MM (2012) Brief measure for screening complicated grief: reliability and discriminant validity. PLoS One 7(2):e31209

    Article  Google Scholar 

  • Morikawa M, Okada T, Ando M, Aleksic B, Kunimoto S, Nakamura Y, Kubota C,, Uno Y, Tamaji A, Hayakawa N, Furumura K, Shiino T, Morita T, Ishikawa N, Ohoka H, Usui H, Banno N, Murase S, Goto S, Kanai A, Masuda T, Ozaki1 N (2015) Relationship between social support during pregnancy and postpartum depressive state: a prospective cohort study. Nature 5:10520

  • Mulaik SA (1972) The foundations of factor analysis. McGraw Hill, New York

    MATH  Google Scholar 

  • Park Y-M, Kwon HS, Lim SY, Lee JH, Kim SR, Yoon KH, Cha BY, Son HY, Park YG, Kim DS, Meng K, Le WC (2006) Clustering characteristics of risk variables of metabolic syndrome in Korean rural populations. J Korean Diabetes Assoc 30:177–189

  • Raftery AE (1993) Bayesian model selection in structural equation models. In: Bollen KA, Long JS (eds) Testing structural equation models. sage, Newbury Park, pp 163–180

    Google Scholar 

  • Rosman A, Maggie G, Lee S, Butler S, Schwartzy A (2013) DOSPERT\(+\)M: A survey of medical risk attitudes in the United States. Judgm Decis Mak 8(4):470–481

    Google Scholar 

  • Sherin RJ (1966) A matrix formulation of Kaiser’s varimax criterion. Psychometrika 31(4):535–538

    Article  MathSciNet  MATH  Google Scholar 

  • Shen H, Huang JZ (2008) Sparse principal component analysis via regularized low rank matrix approximation. J Multivar Anal 99:1015–1034

    Article  MathSciNet  MATH  Google Scholar 

  • Spearman C (1904) General intelligence, objectively determined and measured. Am J Psychol 15:201–293

  • Steiger JH, Lind J (1980) Statistically based tests for the number of common factors. Paper presented at the annual meeting of the Psychometric Society, Iowa City

  • Steiger JH (1989) EzPATH: A supplementary module for SYSTAT and SYSGRAPH [Computer program]. SYSTAT Inc, Evanston

    Google Scholar 

  • Stiglitz JE, Sen A, Fitoussi JP (2009) Report by the Commission on the Measurement of Economic Performance and Social Progress. Insee. http://www.insee.fr/fr/publications-et-services/dossiers_web/stiglitz/doc-commission/RAPPORT_anglais.pdf

  • Schwarz G (1978) Estimating the dimension of a model. Ann Stat 6(2):461–464

    Article  MathSciNet  MATH  Google Scholar 

  • ten Berge JMF (1984) A joint treatment of varimax rotation and the problem of diagonalizing symmetric matrices simultaneously in the least-squares sense. Psychometrika 49(3):347–358

    Article  MathSciNet  MATH  Google Scholar 

  • ten Berge JMF, Knol DL, Kiers HAL (1988) A treatment of the orthomax rotation family in terms of diagonalization, and a re-examination of a singular value approach to varimax rotation. Comput Stat Q 3:207–217

    MATH  Google Scholar 

  • Thomson GH (1935) The definition and measurement of g (general intelligence). J Educ Psychol 26:241–262

    Article  Google Scholar 

  • Ullman JB (2006) Structural equation modeling: reviewing the basics and moving forward. J Personal Assess 87(1):35–50

  • Vichi M, Saporta G (2009) Clustering and disjoint principal component analysis. Comput Stat Data Anal 53(8):3194–3208

  • Yang B, Fu X, Sidiropoulos ND (2015) Joint factor analysis and latent clustering. In: Proceedings of IEEE 6th International Workshop on Computational Advances in Multi-Sensor Adaptive Processing (CAMSAP2015), pp 173–176

  • Zangwill WI (1969) Nonlinear programming: a unified approach. Prentice-Hall, Englewood Cliffs

    MATH  Google Scholar 

  • Zou H, Hastie T, Tibshirani R (2006) Sparse principal component analysis. J Comput Graph Stat 15:262–286

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Statistical Sciences, Sapienza University of Rome, P.le Aldo Moro 5, 00185, Rome, Italy

    Maurizio Vichi

Authors
  1. Maurizio Vichi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Maurizio Vichi.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Vichi, M. Disjoint factor analysis with cross-loadings. Adv Data Anal Classif 11, 563–591 (2017). https://doi.org/10.1007/s11634-016-0263-9

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11634-016-0263-9

Keywords

Mathematics Subject Classification

Search

Navigation