Ordinary Least Squares Estimation of Parameters in Exploratory Factor Analysis With Ordinal Data

Abstract

Exploratory factor analysis (EFA) is often conducted with ordinal data (e.g., items with 5-point responses) in the social and behavioral sciences. These ordinal variables are often treated as if they were continuous in practice. An alternative strategy is to assume that a normally distributed continuous variable underlies each ordinal variable. The EFA model is specified for these underlying continuous variables rather than the observed ordinal variables. Although these underlying continuous variables are not observed directly, their correlations can be estimated from the ordinal variables. These correlations are referred to as polychoric correlations. This article is concerned with ordinary least squares (OLS) estimation of parameters in EFA with polychoric correlations. Standard errors and confidence intervals for rotated factor loadings and factor correlations are presented. OLS estimates and the associated standard error estimates and confidence intervals are illustrated using personality trait ratings from 228 college students. Statistical properties of the proposed procedure are explored using a Monte Carlo study. The empirical illustration and the Monte Carlo study showed that (a) OLS estimation of EFA is feasible with large models, (b) point estimates of rotated factor loadings are unbiased, (c) point estimates of factor correlations are slightly negatively biased with small samples, and (d) standard error estimates and confidence intervals perform satisfactorily at moderately large samples.

Notes

¹The procedure provides standard error estimates for both unrotated factor loadings and rotated factor loadings and factor correlations. Any rotation method is allowed: the only required modifications are the rotation constraints c (θ) and their derivatives in A¹¹.

²We thank Shanhong Luo for making the data set available to us.

³R code implementing standard errors and confidence intervals for OLS rotated factor loadings and factor correlation of EFA with polychoric correlation matrices is downloadable from http://www.nd.edu/~gzhang3/Papers/EFAOrdinal/EFAOrdinal.html and is also available from Guangjian Zhang upon request.