- The document compares different estimation methods (ML, WLS, ULS, DWLS, GLS) for confirmatory factor analysis with ordinal/Likert scale data. - WLS estimates deviated most from the target values, while ML, ULS, and DWLS estimates were roughly equal. WLS estimates were the least accurate. - For non-normal data, fit statistics from ML and GLS were biased, while ULS and DWLS were considered the most robust and liberal estimators.

1. Results
Population correlation = black dashed line. Estimator legend:
Exponentiation decreased correlations (blue). ML = green WLS = purple
Binning continuous data attenuates correlations (red); ULS = blue DWLS = red GLS = olive
gap widens with bigger exponents (dashed green).
Within-sample median rs and skews used for independence.
Polychoric correlations resist bias from binning and nonnormality Loading SEs based on polychoric correlations
Loadings based on polychoric correlations Loadings based on product-moment correlations
Item residuals based on polychoric correlations χ² fit statistics based on polychoric correlations
Summary
WLS loading estimates deviate most from target (dashed black, square root of median correlation)
Parameter estimates roughly equal for ML, ULS, & DWLS.
WLS estimates are least accurate. For GLS, only error variances are inaccurate.
No reasons to recommend weighted or generalized least squares.
Fit statistics are biased by nonnormality for ML and GLS.
ML and GLS have worse χ² and RMSEA even in normal data.
Nonnormality widens standard errors of loadings for DLWS & WLS
Liberal & robust (best?): ULS and DWLS.
Method
Simulated 10,000 multivariate normal datasets for 5-item CFAs with n = 500.
Correlations = .581 ± sampling error.
Continuous values plotted as vertically jittered triangles. Ordinal counts and bin
thresholds in red.
Exponentiated continuous values to 1–6th powers to increase skew & kurtosis across
samples:
Sample #10,000. Generated like Sample #1, but continuous values were raised to 6th
power. Continuous/ordinal skewness = 4.3/6.5, kurtosis = 35/58.
Compared results for continuous and binned (ordinal) data
Compared loadings, error variances, fit statistics, etc.
Nick Stauner
Case Western Reserve University
References
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Publications.
Contact: NickStauner@gmail.com
Limitations and Future Directions
Haven’t explored theoretical differences yet. Maximum likelihood might have more meaningfully defined standard errors.
Didn’t use scaled χ² or mean & variance adjustments to improve robustness of fit statistics or standard errors against nonnormality yet.
Future sensitivity analyses:
• Sample size (half-done), correlation strength, # of bins, model complexity
• SRMRs, WRMRs, and residual SEs for all estimators
• Interactions? Could test for them with generalized linear models…
Introduction
Maximum likelihood (ML) is a popular default estimator for SEM.
Alternatives in R’s lavaan:
• Weighted least squares (WLS)
• Diagonally weighted least squares (DWLS)
• Unweighted least squares (ULS)
• Generalized least squares (GLS)
Which estimator is optimal for SEM of Likert scale data?
Polychoric correlations assume a normally distributed latent
variable and estimate thresholds from ordinal data. How do
estimators work with these?