Estimators for structural equation models of Likert scale data

Which estimation method is optimal for structural equation modeling (SEM) of Likert scale data? Conventional SEM assumes continuous measurement, and some SEM estimators assume a multivariate normal distribution, but Likert scale data are ordinal and do not necessarily resemble a discretized normal distribution. When treated as continuous, these data may yet be skewed due to item difficulty, choice of population, or various response biases. One can fit an SEM to a matrix of polychoric correlations, which estimate latent, continuous constructs underlying ordinally measured variables, but polychoric correlations also assume these latent factors are normally distributed. To what extent are these methods robust with continuous versus ordinal data and with varying degrees of skewness and kurtosis? To answer, I simulated 10,000 samples of multivariate normal data, each consisting of 500 observations of five strongly correlated variables. I transformed each consecutive sample to an incrementally greater degree to increase skew and kurtosis from approximately normal levels to extremes beyond six and 30, respectively. I then performed five confirmatory factor analyses on each sample using five different estimators: maximum likelihood (ML), weighted least squares (WLS), diagonally weighted least squares (DWLS), unweighted least squares (ULS), and generalized least squares (GLS). I compared results for continuous and discretized (ordinal) data, including loadings, error variances, fit statistics, and standard errors. I also noted frequencies of failures, which complicated calculation of polychoric correlations, and particularly plagued the WLS estimator. WLS estimation produced relatively biased loadings and error variance estimates. GLS also underestimated error variances. Neither estimator exhibited any unique advantage to offset these disadvantages. ML estimated parameters more accurately, but some fit statistics appeared biased by it, especially in the context of extreme nonnormality. Specifically, the chi squared goodness-of-fit test statistic and the root mean square error of approximation (RMSEA) began higher with ML-estimated SEMs of approximately normal data, and worsened sharply with greater nonnormality. The Tucker Lewis Index (TLI) and standardized root mean square residual (SRMR) also worsened more moderately with nonnormality when using ML estimation. GLS-estimated fit statistics shared ML’s sensitivity to nonnormality, and were even worse for the TLI and SRMR. Results generally favored ULS and DWLS estimators, which produced accurate parameter estimates, good and robust fit statistics, and small standard errors (SEs) for loadings. DWLS tended to produce smaller SEs than ULS when skewness was below three, but ULS SEs were more robust to nonnormality and smaller with extremely nonnormal data. ML SEs were larger for loadings, but smaller for error variance estimates, and fairly robust to nonnormality...

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