A full-information maximum likelihood method for fitting a multidimensional latent variable model to a set of ordinal observed variables is discussed. This method is an implementation of a general class of models for ordinal variables, and for regression models with one ordinal dependent variable and all explanatory variables observed. Estimation of the model, scoring of persons on the latent dimensions, and the goodness-of-fit of the model are also discussed. The method is applied to an example dataset concerning attitudes toward technology.
Get full access to this article
View all access options for this article.
References
1.
Arminger, G. ,& Küsters, U. (1988). Latent trait models with indicators of mixed measurement level. In R. Langeheine and J. Rost (Eds.), Latent trait and latent class models. New York: Plenum.
2.
Arminger, G. , & Küsters, U. (1989). Construction principles for latent trait models. In C. Clogg (Ed.), Sociological methodology(Vol. 19, pp. 369–393). Washington DC: American Sociological Association.
3.
Bartholomew, D. J. (1980). Factor analysis for categorical data. Journal of the Royal Statistical Society, Series B, 42, 293–321.
4.
Bartholomew, D. J. (1981). Posterior analysis of the factor model. British Journal of Mathematical and Statistical Psychology, 34, 93–99.
5.
Bartholomew, D. J. (1983). Latent variable models for ordered categorical data. Journal of Econometrics, 22, 229–243.
6.
Bartholomew, D. J. (1984a). The foundations of factor analysis. Biometrika, 71, 221–232.
7.
Bartholomew, D. J. (1984b). Scaling binary data using a factor model. Journal of the Royal Statistical Society, Series B, 46, 120–123.
8.
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.
9.
Bartholomew, D. J. , & Knott, M. (1999). Latent variable models and factor analysis(2nd ed.). London: Griffin.
10.
Bartholomew, D. J. , & Tzamourani, P. (1999). The goodness-of-fit of latent trait models in attitude measurement. Sociological Methods and Research, 27, 525–546.
11.
Bock, R. D. , & Aitkin, M. (1981). Marginal maximum likelihood estimation of item parameters: Application of EM algorithm. Psychometrika, 46, 443–459.
12.
Collett, D. (1991). Modelling binary data. NewYork: Chapman and Hall.
13.
reskog, K. G. (1994). On the estimation of polychoric correlations and their asymptotic covariance matrix. Psychometrika, 59, 381–389.
14.
Lancaster, H. (1954). Traces and cumulants of quadratic forms in normal variables. Journal of the Royal Statistical Society, Series B, 16, 247–254.
15.
Lee, S.-Y. , Poon, W.-Y., & Bentler, P. (1992). Structural equation models with continuous and polytomous variables. Psychometrika, 57, 89–105.
16.
Lee, S.-Y. , Poon, W.-Y., & Bentler, P. (1995). A two-stage estimation of structural equation models with continuous and polytomous variables. British Journal of Mathematical and Statistical Psychology, 48, 339–358.
17.
McCullagh, P. (1980). Regression models for ordinal data. Journal of the Royal Statistical Society Series B, 42, 109–142.
18.
Moustaki, I. (1996). A latent trait and a latent class model for mixed observed variables. British Journal of Mathematical and Statistical Psychology, 49, 313–334.
19.
Moustaki, I. (1999). LATENT: A computer program for fitting a one- or two-factor latent variable model to binary, nominal, ordinal, normal, gamma and mixed observed items with missing values [Technical Report]. London: London School of Economics and Political Science, Statistics Department.
20.
Muraki, E. (1990). Fitting a polytomous item response model to Likert-type data. Applied Psychological Measurement, 14, 59–71.
21.
Muraki, E. , & Carlson, E. (1995). Full-information factor analysis for polytomous item responses. Applied Psychological Measurement, 19, 73–90.
22.
MuthÈn, B. (1984). A general structural equation model with dichotomous, ordered categorical and continuous latent variables indicators. Psychometrika, 49, 115–132.
23.
Samejima, F. (1969). Estimation of latent ability using a response pattern of graded scores.Psychometrika Monograph Supplement, 17.
24.
Shi, J.-Q. , & Lee, S.-Y. (1997). Estimation of factor scores with polytomous data by the EM algorithm. British Journal of Mathematical and Statistical Psychology, 50, 215–226.
25.
Stroud, A. , & Secrest, D. (1966). Gaussian quadrature formulas. Englewood Cliffs NJ: Prentice-Hall.
26.
Takane, Y. (1996). An item response model for multidimensional analysis of multiple-choice data. Behaviormetrika, 23, 153–167.
27.
Takane, Y. (1998). Choice model analysis of the “pick any/n” type of binary data. Japanese Psychological Research, 40, 31–39.
28.
Takane, Y. , & De Leeuw, J. (1987). On the relationship between item response theory and factor analysis of discretized variables. Psychometrika, 52, 393–408.