Hostname: page-component-745bb68f8f-kw2vx Total loading time: 0 Render date: 2025-01-08T16:22:00.687Z Has data issue: false hasContentIssue false
  • English
  • Français

Factor Analysis with (mixed) Observed and Latent Variables in the Exponential Family

Published online by Cambridge University Press:  01 January 2025

Michel Wedel  and
Wagner A. Kamakura
Michel Wedel*
Affiliation:
Universites of Groningen and Michigan
Wagner A. Kamakura
Affiliation:
Duke University
*
Requests for reprints should be sent to: Michel Wedel, Faculty of Economics, University of Groningen, PO Box 800, 9700 AV Groningen, THE NETHERLANDS. E-Mail: M.Wedel@eco.rug.nl, or Wedel@umich.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

We develop a general approach to factor analysis that involves observed and latent variables that are assumed to be distributed in the exponential family. This gives rise to a number of factor models not considered previously and enables the study of latent variables in an integrated methodological framework, rather than as a collection of seemingly unrelated special cases. The framework accommodates a great variety of different measurement scales and accommodates cases where different latent variables have different distributions. The models are estimated with the method of simulated likelihood, which allows for higher dimensional factor solutions to be estimated than heretofore. The models are illustrated on synthetic data. We investigate their performance when the distribution of the latent variables is mis-specified and when part of the observations are missing. We study the properties of the simulation estimators relative to maximum likelihood estimation with numerical integration. We provide an empirical application to the analysis of attitudes.

Keywords

factor modelsimulated likelihoodlatent variable model
Type
Articles
Information
Psychometrika , Volume 66 , Issue 4 , December 2001 , pp. 515 - 530
Copyright
Copyright © 2001 The Psychometric Society

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Akaike, H. (1987). Factor analysis and AIC. Psychometrika, 52, 317–332.CrossRefGoogle Scholar
Anderson, E.B. (1980). Discrete statistical models with social science applications. New York, NY: North Holland.Google Scholar
Anderson, T.W., & Rubin, H. (1956). Statistical inference in factor analysis. Proceedings of the third Berkeley Symposium in Mathematical Statistics and Probability, 5, 111–150.Google Scholar
Ansari, A., & Jedidi, K. (2000). Bayesian factor analysis for multilevel binary observations. Psychometrika, 475–496.CrossRefGoogle Scholar
Arminger, G., & Kusters, U. (1988). Latent trait models with indicators of mixed measurement level. In Langeheine, R., & Rost, J. (Eds.), Latent trait and latent class models (pp. 51–71). New York, NY: Plenum.CrossRefGoogle Scholar
Bartholomew, D.J. (1988). The sensitivity of latent trait analysis to choice of prior distribution. British Journal of Mathematical and Statistical Psychology, 41, 101–107.CrossRefGoogle Scholar
Bartholomew, D.J., & Knott, M. (1999). Latent variable models and factor analysis 2nd. ed., New York, NY: Edward Arnold.Google Scholar
Bozdogan, H. (1987). Model selection and Akaike's information criterion (AIC): The general theory and its analytical extensions. Psychometrika, 52, 345–370.CrossRefGoogle Scholar
Fahrmeier, L., & Tutz, G. (1991). Multivariate statistical modeling based on generalized linear models. New York, NY: Springer-Verlag.Google Scholar
Geweke, J.M. (1988). Anthitetic acceleration of Monte Carlo integration in Bayesian inference. Journal of Econometrics, 57, 1317–1339.CrossRefGoogle Scholar
Geweke, J., Keane, M., & Runkle, D. (1994). Alternative computational approaches to inference in the multinomial probit model. Review of Economics and Statistics, 76(4), 609–632.CrossRefGoogle Scholar
Gill, R.D. (1977). Consistency of maximum likelihood estimator of the factor analysis model when the observations are not multivariate normal. In Bara, J.R., Brodeau, F., Romier, G., & van Custem, B. (Eds.), Recent developments in statistics (pp. 437–440). Amsterdam: North Holland.Google Scholar
Gouriéroux, C., & Monfort, A. (1997). Simulation based econometric methods. New York, NY: Oxford University Press.CrossRefGoogle Scholar
Haberman, S.J. (1982). Analysis of dispersion of multinomial responses. Journal of the American Statistical Association, 77, 568–580.CrossRefGoogle Scholar
Kamakura, W.A., & Wedel, M. (2000). Factor analysis and missing data. Journal of Marketing Research, 37, 490–498.CrossRefGoogle Scholar
Lee, L.F. (1995). Asymptotic bias in simulated maximum likelihood estimation of discrete choice models. Econometric Theory, 11, 437–483.CrossRefGoogle Scholar
Lee, L.F. (1997). Simulated maximum likelihood estimation of dynamic discrete choice statistical models: Some Monte Carlo results. Journal of Econometrics, 82, 1–35.CrossRefGoogle Scholar
Lee, L.F. (1999). Statistical inference with simulated likelihood functions. Econometric Theory, 15, 337–351.CrossRefGoogle Scholar
Little, R.J.A., & Rubin, D.B. (1987). Statistical analysis with missing data. New York, NY: Wiley.Google Scholar
McFadden, D. (1989). A method of simulated moments for estimation of discrete response models without numerical integration. Econometrika, 57, 995–1026.CrossRefGoogle Scholar
McKay, M.D., Conover, W.J., & Beckman, R.J. (1979). A Comparison of three methods for selecting values of input variables in the analysis of output from a computer code. Technometrics, 21, 239–245.Google Scholar
Moustaki, I. (2001). A latent trait and latent class model for mixed observed variables. British Journal of Mathematical and Statistical Psychology, 49, 313–334.CrossRefGoogle Scholar
Moustaki, I., & Knott, M. (2000). Generalized latent trait models. Psychometrika, 65, 391–411.CrossRefGoogle Scholar
Mulaik, S.A. (1972). The foundations of factor analysis. New York, NY: McGraw Hill.Google Scholar
Muthén, B.O. (1984). A general structural model with dichotomous, ordered categorical and continuous latent variable indicators. Psychometrika, 49, 115–132.CrossRefGoogle Scholar
Owen, A. (1992). Orthogonal arrays for computer experiments, integration and visualization. Statistica Sinica, 2, 439–452.Google Scholar
Sammel, M.D., Ryan, L.M., & Legler, J.M. (1997). Latent variable models for mixed discrete and continuous outcomes. Journal of the Royal Statistical Society, Series B, 59(3), 667–678.CrossRefGoogle Scholar
Sándor, Z., & András, P. (2000). Alternative sampling methods for estimating multivariate normal probabilities. Groningen, Netherlands: University of Groningenm Faculty of Economics.Google Scholar
Scales, L.E. (1985). Introduction to non-linear optimization. London: Macmillan.CrossRefGoogle Scholar
Seong, T.J. (1990). Sensitivity of marginal maximum likelihood estimation of item and ability parameters to the characteristics of the prior ability distributions. Applied Psychological Measurement, 14, 299–311.CrossRefGoogle Scholar
Stern, S. (1997). Simulation-based estimation. Journal of Economic Literature, 35, 2006–2039.Google Scholar
Tang, B. (1993). Orthogonal array-based hypercubes. Journal of the American Statistical Association, 88, 1392–1397.CrossRefGoogle Scholar