Psychometrika

, Volume 54, Issue 2, pp 181–202 | Cite as

Factor and ideal point analysis for interpersonally incomparable data

  • Henry E. Brady
Article

Abstract

Interpersonally incomparable responses pose a significant problem for survey researchers. If the manifest responses of individuals differ from their underlying true responses by monotonic transformations which vary from person to person, then the covariances of the manifest responses tools such as factor analysis may yield incorrect results. Satisfactory results for interpersonally incomparable ordinal responses can be obtained by assuming that rankings are based upon a set of multivariate normal latent variables which satisfy the factor or ideal point models of choice. Two statistical methods based upon these assumptions are described; their statistical properties are explored; and their computational feasibility is demonstrated in some simulations. We conclude that is possible to develop methods for factor and ideal point analysis of interpersonally incomparable ordinal data.

Key words

interpersonally incomparable responses rankings factor analysis ideal-point analysis ordinal data 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Amemiya, Takeshi. (1973). Regression analysis when the dependent variable is truncated normal.Econometrica, 41, 997–1016.Google Scholar
  2. Anderosn, T. W., & Rubin, R. (1956). Statistical inference in factor analysis.Proceedings of the third Berkeley symposium on mathematical statistics and probability, Volume V.Google Scholar
  3. Arrow, K. (1951).Social choice and individual values. New York: John Wiley.Google Scholar
  4. Bartholomew, D. J. (1980). Factor analysis for categorical data (with discussion).Journal of the Royal Statistical Society, Series B,42, 293–321.Google Scholar
  5. Bartholomew, D. J. (1983). Latent variable models for ordered categorical data.Journal of Econometrics, 22, 229–243.Google Scholar
  6. Billingsly, P. (1979).Probability and measure. New York: John Wiley.Google Scholar
  7. Bock, R. Darrell, & Aitkin, M. (1981). Marginal maximum likelihood estimation of item parateters: Application of an EM algorithm.Psychometrika, 46, 443–459.Google Scholar
  8. Bock, R. D., & Lieberman, M. (1970). Fitting a response model forn dichotomously scored items.Psychometrica, 35, 179–197.Google Scholar
  9. Browne, M. W. (1974). Generalized least squares estimators in the analysis of covariance structures.South African Statistical Journal, 8, 1–24.Google Scholar
  10. Browne, M. W. (1982). Covariance structures. In D. M. Hawkins (Ed.),Topics in applied multivariate analysis. Cambridge: Cambridge University Press.Google Scholar
  11. Christofferson, A. (1975). Factor analysis of dichotomized variables.Psychometrika, 40, 5–32.Google Scholar
  12. Dijkstra, T. (1983). Some comments on maximum likelihood and partial least squares methods.Journal of Econometrics.22, 67–90.Google Scholar
  13. Domowitz, I., & White, H. (1982). Misspecified models with dependent observations.Journal of Econometrics.20, 35–38.Google Scholar
  14. Enelow, J., & Hinich, M. (1984).The spatial theory of voting: An introduction. Cambridge University Press: Cambridge.Google Scholar
  15. Halff, H. M. (1976). Choice theories for differentially comparable alternatives.Journal of Mathematical Psychology, 14, 244–246.Google Scholar
  16. Harman, H. (1976).Modern Factor Analysis (3rd. ed.). Chicago: University of Chicago Press.Google Scholar
  17. Harshman, R. A., & Lundy, M. E. (1984). Data preprocessing and the extended PARAFAC model. In H. G. Law, C. W. Snyder, Jr., J. A. Hattie, & R. P. McDonald (Eds.),Research Methods for Multimode Data Analysis. New York: Praeger.Google Scholar
  18. Hoffman, K. (1975).Analysis in Euclidean Space. Englewood Cliffs, NJ: Prentice-Hall.Google Scholar
  19. Johnson, N. L., & Kotz, S. (1972).Distributions in statistics: Continuous multivariate distributions. New York: John Wiley.Google Scholar
  20. Jöreskog, K. G., & Sörbom, D. (1979).Advances in factor analysis and structural equation models. Cambridge, MA: Abt Books.Google Scholar
  21. Kelly, J. (1978).Arrow impossibility theorems. New York: Academic Press.Google Scholar
  22. Lawley, D. N., & Maxwell, A. E. (1971).Factor analysis as a Statistical Method. London: Butterworths. (Original work published in 1963).Google Scholar
  23. Lee, S.-Y., & Bentler, P. M. (1980). Some asymptotic properties of constrained generalized least squares estimation in covariance structure models.South African Statistical Journal, 14, 121–136.Google Scholar
  24. Lerman, S. R., & Manski, C. F. (1981). On the use of simulated frequencies to approximate choice probabilities. In C. F. Manski & D. McFadden (Eds.),Structural analysis of discrete data with econometric applications, Cambridge, MA: MIT Press.Google Scholar
  25. Luce, R. D., & Suppes, P. (1965). Preference, utility and subjective probability. In R. D. Luce, R. R. Bush, & E. Galanter (Eds.),Handbook of mathematical psychology, Volume III. New York: John Wiley.Google Scholar
  26. Manski, C. F., & McFadden, D. (1981). Alternative estimators and sample designs for discrete choice analysis. In C. F. Manski & D. McFadden (Eds.),Structural analysis of discrete data with econometric applications. Cambridge, MA: MIT Press.Google Scholar
  27. Muthén, B. (1978). Contributions to factor analysis of dichotomous variables.Psychometrika, 43, 551–560.Google Scholar
  28. Muthén, B. (1983). Latent variable structural equation modeling with categorical data.Journal of Econometrics, 22, 43–65.Google Scholar
  29. Niemi, R. G., & Weisberg, H. F. (1972).Probability models of collective decision-making. Columbus, OH: Merrill.Google Scholar
  30. Rao, C. R. (1973).Linear statistical inference and its applications (2nd ed.). New York: John Wiley.Google Scholar
  31. Rao, C. R., & Mitra, S. K. (1971).Generalized inverse of matrices and its applications. New York: John Wiley.Google Scholar
  32. Sen, A. (1970).Collective choice and social welfare. San Francisco: Holden-Day.Google Scholar
  33. Shapiro, A. (1983). Asymptotic distribution theory in the analysis of covariance structures (a unified approach).South African Statistical Journal, 17, 33–81.Google Scholar
  34. Shepard, R. N. (1962a) Analysis of proximities: Multidimensional scaling with an unknown distance function. I.Psychometrika, 27, 125–140.Google Scholar
  35. Shepard, R. N. (1962b). Analysis of proximities: Multidimensional scaling with an unknown distance function. II.Psychometrika, 27, 219–246.Google Scholar
  36. Torgerson, W. S. (1958).Theory and methods of scaling. New York: John Wiley & Sons.Google Scholar
  37. White, H. (1980). Nonlinear regression on cross-section data.Econometrica, 48, 721–746.Google Scholar
  38. White, H. (1982). Maximum likelihood estimation of misspecified models.Econometrica, 50, 1–25.Google Scholar

Copyright information

© The Psychometric Society 1989

Authors and Affiliations

  • Henry E. Brady
    • 1
  1. 1.Department of Political ScienceUniversity of ChicagoChicago

Personalised recommendations