×

A new method for simultaneous estimation of the factor model parameters, factor scores, and unique parts. (English) Zbl 1468.62181

Summary: In the common factor model the observed data is conceptually split into a common covariance producing part and an uncorrelated unique part. The common factor model is fitted to the data itself and a new method is introduced for the simultaneous estimation of loadings, unique variances, factor scores, and unique parts. The method is based on Minimum Rank Factor Analysis and allows for the percentage of explained common variance to be computed. Taking into account factor indeterminacy, an explicit description of the complete class of solutions for the factor scores and unique parts is given. The method is evaluated in a simulation study and fitted to a dataset in the literature.

MSC:

62-08 Computational methods for problems pertaining to statistics
62H25 Factor analysis and principal components; correspondence analysis
62P15 Applications of statistics to psychology
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Bartholomew, D. J., Posterior analysis of the factor model, Psychometrika, 34, 93-99, (1981) · Zbl 0462.62084
[2] Browne, M. W., On oblique procrustes rotation, Psychometrika, 32, 125-132, (1967)
[3] Browne, M. W., An overview of analytic rotation in exploratory factor analysis, Multivariate Behav. Res., 36, 111-150, (2001)
[4] De Leeuw, J., Least squares optimal scaling of partially observed linear systems, (van Montfort, K.; Oud, J.; Satorra, A., Recent Developments on Structural Equation Models: Theory and Applications, (2004), Kluwer Academic Publishers Dordrecht), 121-134 · Zbl 05196650
[5] De Leeuw, J., 2008. Factor analysis as matrix decomposition. Preprint series: Department of Statistics, University of California, Los Angeles. Available online at: http://www.stat.ucla.edu/ deleeuw/janspubs/2008/notes/deleeuw_U_08g.pdf.
[6] Eckart, C.; Young, G., The approximation of one matrix by another of lower rank, Psychometrika, 1, 211-218, (1936) · JFM 62.1075.02
[7] Grice, J. W., Computing and evaluating factor scores, Psychol. Methods, 6, 430-450, (2001)
[8] Grice, J. W.; Krohn, E. J.; Logerquist, S., Cross-validation of the WISC-III factor structure in two samples of children with learning disabilities, J. Psychoeduc. Assess., 17, 236-248, (1999)
[9] Grung, B.; Manne, R., Missing values in principal component analysis, Chemometr. Intell. Lab. Syst., 42, 125-139, (1998)
[10] Guttman, L., The determinacy of factor score matrices with implications for five other basic problems of common factor theory, British J. Statist. Psych., 8, 65-81, (1955)
[11] Harman, H. H.; Jones, W. H., Factor analysis by minimizing residuals (MINRES), Psychometrika, 31, 351-368, (1966)
[12] Hoshino, T.; Shigemasu, K., Standard errors of estimated latent variable scores with estimated structural parameters, Appl. Psychol. Meas., 32, 181-189, (2008)
[13] Jöreskog, K. G., Some contributions to maximum likelihood factor analysis, Psychometrika, 32, 443-482, (1967) · Zbl 0183.24603
[14] Lin, T.-I.; Wu, P. H.; McLachlan, G. J.; Lee, S. X., A robust factor analysis model using the restricted skew-\(t\) distribution, TEST, 24, 510-531, (2015) · Zbl 1327.62344
[15] McDonald, R. P., The simultaneous estimation of factor loadings and scores, Br. J. Math. Stat. Psychol., 32, 212-228, (1979) · Zbl 0439.62038
[16] Maraun, M. D., Metaphor taken as math: indeterminacy in the factor analysis model, with comments and reply, Multivariate Behav. Res., 31, 517-689, (1996)
[17] Montanari, A.; Viroli, C., A skew-normal factor model for the analysis of student satisfaction towards university courses, J. Appl. Stat., 37, 473-487, (2010)
[18] Mulaik, S. A., Foundations of factor analysis, (2010), Chapman & Hall FL, USA · Zbl 1188.62185
[19] Ng, S., Constructing common factors from continuous and categorical data, Econometric Rev., 34, 1141-1171, (2015)
[20] Schafer, J. L.; Graham, J. W., Missing data: our view of the state of the art, Psychol. Methods, 7, 147-177, (2002)
[21] Schönemann, P. H., A generalized solution of the orthogonal procrustes problem, Psychometrika, 31, 1-10, (1966) · Zbl 0147.19401
[22] Schönemann, P. H.; Wang, M.-M., Some new results on factor indeterminacy, Psychometrika, 37, 61-91, (1972) · Zbl 0236.92005
[23] Shapiro, A.; Ten Berge, J. M.F., Statistical inference of minimum rank factor analysis, Psychometrika, 67, 79-94, (2002) · Zbl 1297.62137
[24] Skrondal, A.; Rabe-Hesketh, S., Generalized latent variable modeling, (2004), Chapman & Hall Boca Raton, USA · Zbl 1097.62001
[25] Smits, I. A.M.; Timmerman, M. E.; Stegeman, A., Modelling non-normal data: the relationship between the skew-normal factor model and the quadratic factor model, British J. Math. Statist. Psych., (2015), in press. Available online via
[26] Sočan, G., The incremental value of minimum rank factor analysis, (2003), University of Groningen The Netherlands, (PhD Thesis)
[27] Spearman, C., General intelligence objectively determined and measured, Am. J. Psychol., 15, 201-293, (1904)
[28] Steiger, J. H., Factor indeterminacy in the 1930’s and the 1970’s, some interesting parallels, Psychometrika, 44, 157-167, (1979)
[29] Stewart, G. W., The efficient generation of random orthogonal matrices with an application to condition estimators, SIAM J. Numer. Anal., 17, 403-409, (1980) · Zbl 0443.65027
[30] Ten Berge, J. M.F., Some recent developments in factor analysis and the search for proper communalities, (Rizzi, A.; Vichi, M.; Bock, H. H., Advances in Data Science and Classification, (1998), Springer Heidelberg, Germany)
[31] Ten Berge, J. M.F.; Kiers, H. A.L., A numerical approach to the approximate and the exact minimum rank of a covariance matrix, Psychometrika, 56, 309-315, (1991) · Zbl 0850.62462
[32] Ten Berge, J. M.F.; Nevels, K., A general solution to mosier’s oblique procrustes problem, Psychometrika, 42, 593-600, (1977) · Zbl 0387.62086
[33] Thurstone, L. L., The vectors of mind, (1935), University of Chicago Press Chicago · JFM 62.1380.06
[34] Trendafilov, N. T.; Unkel, S., Exploratory factor analysis of data matrices with more variables than observations, J. Comput. Graph. Statist., 20, 874-891, (2011)
[35] Unkel, S.; Trendafilov, N. T., Simultaneous parameter estimation in exploratory factor analysis: an expository review, Internat. Statist. Rev., 78, 363-382, (2010) · Zbl 1284.62361
[36] Unkel, S.; Trendafilov, N. T., Zig-zag exploratory factor analysis with more variables than observations, Comput. Statist., 28, 107-125, (2013) · Zbl 1305.65081
[37] Wechsler, D., WISC-III manual, (1991), Psychological Corporation San Antonio, TX
[38] Wilson, E., On hierarchical correlation systems, Proc. Natl. Acad. Sci., 14, 283-291, (1928) · JFM 54.0558.05
[39] Zhang, G., Estimating standard errors in exploratory factor analysis, Multivariate Behav. Res., 49, 339-353, (2014)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.