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Contributions: Abstract

Confirmatory factor analysis with ordinal variables: a simulation study

 Fan Wallentin University of Uppsala Sweden

When data is collected through questionnaires the variables are often ordinal. Observations on an ordinal variable are assumed to represent responses to a set of ordered categories such as a five-category Likert scale. It is common practice to treat such scores representing the ordered categories as if they have interval scale properties. Most people would therefore compute an ordinary covariance matrix or product-moment correlation matrix and use this to fit the model. Jöreskog (2002) says it is wrong to do so. He argues that ordinal variables do not have origins or units of measurements. Means, variances and covariances of ordinal variables therefore have no meaning. He suggests that polychoric correlations should be used with ordinal variables. On the other hand it could be argued that the theory of asymptotic robustness of Browne (1984, 1987) and Satorra (1993) developed for continuous variables, still holds even for ordinal variables treated as continuous despite their severe non-normality.

In this paper we simulate ordinal variables and a small confirmatory factor analysis model under normal and non-normal conditions and investigate how wrong is wrong. Three kinds of matrices can be computed from the ordinal data. The model can be fitted to the matrix by different methods, such as Unweighted Least Squares (ULS), Maximum Likelihood (ML) or Weighted Least Squares (WLS) using a weight matrix, which is the inverse of an estimated asymptotic covariance matrix (ACM). Two hybrid procedures, Robust Unweighted Least Squares (RULS) and Robust Maximum Likelihood (RML), are to use ULS and ML to estimate the parameters and the ACM matrix to obtain standard errors and chi-square values. Thus there are six procedures to be investigated, as each estimation method can be used with each matrix. For each procedure an issue is how large samples are required for it to work properly.