This paper reports on a simulation study that evaluated the performance of five structural equation model test statistics appropriate for categorical data. Both Type I error rate and power were investigated. Different model sizes, sample sizes, numbers of categories, and threshold distributions were considered. Statistics associated with both the diagonally weighted least squares (cat-DWLS) estimator and with the unweighted least squares (cat-ULS) estimator were studied. Recent research suggests that cat-ULS parameter estimates and robust standard errors slightly outperform cat-DWLS estimates and robust standard errors (Forero, Maydeu-Olivares, & Gallardo-Pujol, 2009). The findings of the present research suggest that the mean- and variance-adjusted test statistic associated with the cat-ULS estimator performs best overall. A new version of this statistic now exists that does not require a degrees-of-freedom adjustment (Asparouhov & Muthén, 2010), and this statistic is recommended. Overall, the cat-ULS estimator is recommended over cat-DWLS, particularly in small to medium sample sizes.
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