Robert Kapłon

(Polish) PDF


The goal of factor analysis is to reduce the redundancy among variables by using smaller number of factors that are treated as constructs or latent variables. Unfortunately, if we face with data heterogeneity, the estimates of a single set of means, factor loadings and specific variances may be misleading.One way of accounting for unobserved heterogeneity is to include another latent variable in a factor analysis model. As a consequence, the observations in a samples are assumed to arise from two or more subpopulations that are mixed in unknown proportions. Since putting some restrictions on parameters such as factor loadings and specific variancesone can get more parsimonious models. Therefore, the purpose of this paper is to present the eight factor analysis models. Methods of optimization to derive the maximum likelihood estimates based on EM algorithm as well as model selection procedure are considered. Proposed approach is illustrated by using a set of data referring to preferences.


[1] Akaike H. (1973), Information theory and anextension of the maximum likelihoodprinciple, [w:] Petrov B.N., Csaki F. (Eds.), Second international symposium on information theory (pp.). Budapest: AcademiaiKiado, s. 267-281.

[2] Andrews L., Currim I.S. (2003), A Comparison of segment retention criteria for finite mixture logit models, Journal of Marketing Research, 40(2), s. 235-243.

[3] Banfield, J.D.,Raftery, A.E. (1993). Model-based Gaussian and non-Gaussian clustering, Biometrics, 49, s. 803-821.

[4] Bozdogan, H. (1987), Model selection and Akaike’s information criterion (AIC): The general theory and its analytical extensions. Psychometrika, 52, s. 345-370.

[5] Celeux G., Govaert G. (1995), Gaussian Parsimonious Clustering Models, Pattern Recognition, 28(5), s. 781-793.

[6] Hair J.F., Black W.C., Babin B.J., Anderson R.E., (2010), Multivariate Data Analysis, Prentice Hall, New York.

[7] Kapłon R. (2004), Estymacja parametrów modelu czynnikowego wykorzystującego klasy ukryte, [w:] Jajuga K.,Walesiak M. „Klasyfikacja i analiza danych – teoria i zastosowania”. Taksonomia nr 11, Prace Naukowe Akademii Ekonomicznej we Wrocławiu, s. 204-211.

[8] Kapłon R. (2007), O liczbie klas w modelu analizy czynnikowej z dwoma zmiennymi ukrytymi, [w:] Jajuga K.,Walesiak M. „Klasyfikacja i analiza danych – teoria i zastosowania”. Taksonomia nr 14, Prace Naukowe Akademii Ekonomicznej we Wrocławiu, s. 253-260.

[9] Kass R. E., Raftery A. E. (1995), Bayes factors,Journal of the American Statistical Association, 90, s. 773-795.

[10] Luenberger D.G., Ye Y. (2008), Linear and Nonlinear Programming, Springer.

[11] McLachlan G.J., Krishnan T. (1997), The EM Algorithm and Extensions. New York: Wiley.

[12] McLachlan G.J., Peel. D. (2000), Finite Mixture Models, New York: Wiley.

[13] McLachlan, G.J, Basford K. (1988),Mixturemodels. New York: Marcel Dekker.

[14] McLachlan, G.J. (1987), On bootstrapping the likelihood ratio test statistic for the number of components in a normal mixture,Journal of the Royal Statistical Society Series C (Applied Statistics), 36, s. 318-324.

[15] R Development Core Team (2011), R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria. ISBN 3-900051-07-0, URL

[16] Walesiak M. (1996), Metody analizy danych marketingowych, Wydawnictwo Naukowe PWN, Warszawa.

Back to top
© 2019–2022 Copyright by Statistics Poland, some rights reserved. Creative Commons Attribution-ShareAlike 4.0 International Public License (CC BY-SA 4.0) Creative Commons — Attribution-ShareAlike 4.0 International — CC BY-SA 4.0