验证性因子分析：ML与MLM估计器的选择——基于小样本纵向数据

Great question—this is a common (and important) dilemma when working with small, non-normal longitudinal Likert-scale data for CFA, and you’re already on the right track referencing Brown’s Confirmatory Factor Analysis for Applied Research (a definitive resource for this topic). Here’s a breakdown of the choice tailored to your specific scenario:

Core Recommendation: Prioritize Robust ML (MLM) #

Given your dataset’s characteristics—small sample size (n < 130), 5-point Likert-scale data (inherently non-normal/ordinal), and 4-wave longitudinal structure—MLM is the clear preferred choice over traditional Maximum Likelihood (ML). Here’s why:

• Addresses non-normality bias head-on: Traditional ML assumes multivariate normality, which is rarely met with Likert-scale data (even 5-point scales often have skewed distributions or floor/ceiling effects). As Brown outlines, non-normality distorts ML’s chi-square test statistics, standard errors, and model fit indices (e.g., inflating Type I error rates for model comparison). MLM uses sandwich-style robust standard errors and a scaled chi-square statistic (e.g., Satorra-Bentler correction) to adjust for these biases, producing more reliable parameter estimates and fit assessments.

• Performs better with small samples: While robust estimators are often associated with large samples, MLM has been shown to outperform traditional ML even in smaller samples (n ≥ 50, which your dataset meets). For n < 130, ML’s unadjusted standard errors are likely to be underestimated, leading you to overestimate the significance of factor loadings or model differences. MLM’s correction mitigates this issue, making results more trustworthy.

• Adapts to longitudinal panel structure: Your 4-wave data introduces potential dependencies between repeated measures and possible heteroscedasticity across time points. Traditional ML assumes independent observations and homogeneous variance, which doesn’t hold in most longitudinal designs. MLM’s robust standard errors automatically account for these complex variance-covariance structures without requiring additional model adjustments (e.g., specifying residual correlations manually).

When Might You Consider Traditional ML? #

Only in edge cases where:

• Your data is surprisingly close to multivariate normal (unlikely with Likert scales, but you can confirm via Shapiro-Wilk tests or Mardia’s multivariate kurtosis).
• Your analysis software doesn’t support robust ML estimators (rare today—tools like Lavaan, Mplus, and AMOS all include MLM options).

Practical Implementation Tip #

If you’re using Lavaan (a popular open-source tool for CFA), specify the MLM estimator with this code:

fit <- cfa(model = your_cfa_model, data = your_data, estimator = "mlm")
summary(fit, fit.measures = TRUE, robust = TRUE)

In Mplus, use:

ESTIMATOR = MLM;

Be sure to report the scaled chi-square, robust standard errors, and adjusted fit indices (e.g., robust CFI/TLI) from the MLM output, rather than the unadjusted ML metrics.

内容的提问来源于stack exchange，提问作者Amonet