Weighted Least Squares Estimation

When you work with structural equation models (SEM) using ordinal survey data or variables that violate normality assumptions, choosing the wrong estimation method can invalidate your conclusions. Weighted least squares (WLS) estimation, and its variant WLSMV (weighted least squares mean and variance adjusted), provide robust alternatives to maximum likelihood for such data. Understanding when and how to apply these methods is essential for accurate model testing and credible research in the social, behavioral, and health sciences.

Foundations of Estimation in Structural Equation Modeling

Structural equation modeling is a versatile framework for testing complex relationships among observed and latent variables. The process requires an estimation method—a statistical procedure to find the parameter values (like factor loadings or regression paths) that make the model-implied covariance matrix best match the sample covariance matrix from your data. For decades, maximum likelihood (ML) estimation has been the default choice due to its desirable properties, including efficiency and the production of standard fit indices like the chi-square (${\chi }^2$) test, CFI, and RMSEA. ML estimation operates under the assumption that your observed variables are continuous and follow a multivariate normal distribution.

However, real-world data often defy these assumptions. Survey research routinely employs ordinal data from Likert scales (e.g., 1=Strongly Disagree to 5=Strongly Agree), which are categorical by nature. Other variables may be non-normally distributed, exhibiting skewness or kurtosis. When ML is applied to such data, it can produce distorted results: chi-square statistics become inflated, standard errors may be biased, and the overall model fit can appear worse than it truly is. This is where weighted least squares estimation becomes a critical tool in your analytical arsenal.

Why Weighted Least Squares for Categorical and Non-Normal Data?

The core strength of WLS estimation lies in its direct use of the asymptotic covariance matrix of the sample variances and covariances. While ML minimizes a function based on the assumption of normality, WLS minimizes a different fit function:

$$F_{WLS}=(s-\sigma (\theta ){)}^{\prime }{W}^{-1}(s-\sigma (\theta ))$$

Here, $s$ is a vector of the sample statistics (e.g., variances and covariances), $\sigma (\theta )$ is the vector of model-implied statistics, and $W$ is a weight matrix. This weight matrix is crucial—it is an estimate of the asymptotic covariance matrix of the sample statistics. By using this weight matrix, WLS accounts for the actual distribution and scale of the data, making it theoretically more appropriate for variables that are not multivariate normal.

For categorical data, you typically analyze a polychoric correlation matrix instead of a Pearson correlation matrix. Polychoric correlations estimate the linear relationship between assumed normally distributed continuous variables underlying your observed ordinal categories. WLS is then applied to this matrix. The method is "weighted" because it gives less influence to correlations that are estimated with greater sampling variability. This approach provides more accurate parameter estimates and model fit assessments when the normality assumption is violated. You should consider WLS when your data are ordinal or exhibit significant non-normality that transformations cannot adequately correct.

WLSMV: A Tailored Method for Ordinal Data

A standard WLS estimator can be challenging to use with smaller samples because the weight matrix $W$ becomes very large and unstable. WLSMV is a diagonally weighted least squares estimator with robust corrections that address this issue. It is specifically designed for ordinal data. The "mean and variance adjusted" part refers to how the method calculates the test statistic and standard errors: it uses a diagonal weight matrix (which is more stable) and then applies corrections to the chi-square statistic and standard errors to account for this simplification.

In practice, when you specify WLSMV in software like Mplus or the lavaan package in R, the analysis follows a clear workflow. First, the software computes polychoric correlations for your ordinal variables. Then, it uses the WLSMV estimator to fit your SEM model to this correlation matrix. The output includes robust fit indices like the Robust Comparative Fit Index (CFI) and Robust Root Mean Square Error of Approximation (RMSEA), which are corrected for the nature of the data. For graduate researchers analyzing survey data, WLSMV is often the recommended estimator because it balances statistical rigor with practical performance across a range of sample sizes.

Model Fit Indices and Parameter Interpretation with WLS

Shifting from ML to WLS estimation changes how you evaluate your model's fit. The traditional likelihood-based chi-square test is replaced with a robust chi-square test, often referred to as the Satorra-Bentler or mean-adjusted chi-square in the context of WLSMV. This test is better calibrated for non-normal data. Similarly, fit indices like the CFI and RMSEA are calculated using robust formulas. A common guideline is that a Robust CFI value above 0.95 and a Robust RMSEA value below 0.06 indicate good fit, but you should consult methodological literature specific to your field.

Interpreting parameter estimates also requires careful attention. The parameter estimates themselves (e.g., factor loadings, path coefficients) from WLS/WLSMV are generally interpreted in the same way as those from ML—they represent the strength and direction of relationships. However, the standard errors are computed differently, leading to potentially different significance levels. A path coefficient might be statistically significant under ML but not under WLSMV, or vice versa, due to the correction for non-normality. Therefore, you must ensure that the estimation method aligns with your data's properties to draw valid inferences about the significance of your model's paths.

Practical Application in Graduate Research

Consider a concrete scenario: you are a graduate student testing a theory of job satisfaction using a survey with 20 items, all measured on a 5-point ordinal scale from "Strongly Disagree" to "Strongly Agree." Your proposed model has three latent factors (e.g., Workload, Autonomy, Recognition) predicting a latent Satisfaction factor. Here is how you would apply WLS estimation:

1. Data Preparation: Screen your data for other issues like missingness. Confirm that the variables are correctly coded as ordinal.
2. Software Specification: In your SEM software, explicitly declare your variables as ordinal or categorical. This instructs the program to compute the polychoric correlation matrix.
3. Estimator Selection: Choose the WLSMV estimator for the analysis. In lavaan, this is specified as estimator = "WLSMV".
4. Model Evaluation: Examine the robust fit indices from your output. Do not compare these values directly to cut-offs established for ML-based indices without noting the difference.
5. Reporting: In your thesis or paper, clearly state, "The model was estimated using the WLSMV estimator due to the ordinal nature of the observed variables." Present the robust fit indices and note any differences from a potential ML analysis.

This approach ensures your methodology is defensible and that your conclusions about the structural relationships are built on a solid statistical foundation.

Common Pitfalls

1. Using WLS Unnecessarily with Normal, Continuous Data: WLS and WLSMV are computationally more intensive and less efficient than ML when the data actually meet the assumption of multivariate normality. If your data are continuous and normally distributed, ML is the preferred estimator. Applying WLS in this scenario can waste computational resources without benefit.

• Correction: Always conduct preliminary diagnostics to assess the scale (continuous vs. ordinal) and distribution (normality tests, skewness/kurtosis values) of your variables before selecting an estimator.

1. Ignoring Sample Size Requirements: While WLSMV is more robust than standard WLS for smaller samples, it still has requirements. Very small sample sizes (e.g., below 100) can lead to unstable polychoric correlation estimates and unreliable model results, regardless of the estimator.

• Correction: Be aware of the sample size recommendations for the WLSMV estimator in your specific model context. When in doubt, consult simulation studies or methodological advisors. Consider alternative methods or acknowledge limitations if your sample is very small.

1. Misinterpreting or Mismatching Fit Indices: A critical mistake is reporting the robust chi-square and fit indices but evaluating them against the traditional cut-offs for ML without recognition. Similarly, comparing fit indices from a WLSMV analysis directly to those from an ML analysis on the same data is invalid.

• Correction: Use the robust versions of fit indices (e.g., Robust CFI, Robust RMSEA) provided in the output. Reference methodological papers that discuss interpretation of these robust indices when justifying your model's fit.

1. Assuming WLS Corrects for All Data Problems: WLS methods address non-normality and categorical measurement, but they do not solve other fundamental issues like poor model specification, multicollinearity, or measurement error not accounted for in the model.

• Correction: Use WLS as part of a comprehensive, thoughtful modeling process. Ensure your model is theoretically sound, your measures are valid, and you have checked for other statistical assumptions relevant to SEM.

Summary

• Weighted least squares (WLS) estimation, particularly the WLSMV variant, is the recommended approach for fitting structural equation models with ordinal data or variables that significantly depart from multivariate normality.
• These methods work by applying a weight matrix to a polychoric correlation matrix, yielding more accurate parameter estimates and model fit statistics than standard maximum likelihood estimation under these conditions.
• When using WLS/WLSMV, you must interpret the robust model fit indices (like Robust CFI and Robust RMSEA) provided in the output, not the standard ML-based indices.
• Avoid using WLS methods for continuous, normally distributed data, as maximum likelihood remains more efficient and appropriate in that scenario.
• Always report your choice of estimator transparently in your research, linking it directly to the properties of your data to bolster the credibility of your findings.