Structural equation modeling with robust covariances

Structural equation modeling with robust covariances (DOI: https://doi.org/10.1111/0081-1750.00052) was written by Ke-Hai Yuan and Peter M. Bentler in 1998. It was published in Sociological Methodology (vol. 28).

The authors examine properties of estimators for the population covariance matrix (Σ(θ), whose components are notated σ(θ)). The sample covariance matrix S is sensitive to outliers, is inefficient when the tails of the distribution are fat, and is biased when the sample is not normally distributed.

They contrast this to a robust sample covariance (S_n, whose components are notated s_n). Robust estimators have a bounded influence function (IF). (For an estimator T based on a population distribution F, an IF maps a proportion of the sample not actually drawn from F to a change in measured T. A bounded IF means that an improperly specified distribution can only bias the measurement so much.) Many robust estimators converge to a scaled version of the population parameters rather than the parameters themselves. The authors notate this scalar as α.

Convergence to a scaled version of the population covariance matrix does not matter for SEM as long as the model Σ(θ) is invariant under a constant scaling factor (ICSF). Mathematically, this means that for any α there is a parameter vector θ* such that Σ(θ*) = α Σ(θ).

It remains true that θ* depends on α. The authors explore the consequences for this on the most common SEM patterns: setting all factor covariances to 1 and setting a single factor loading to 1. Even so, they discern that the following relations are unaffected:

• testing if some factor loadings are 0
• testing relations (e.g., equality) between factor loadings
• magnitudes of the coefficients of reliability

The key property needed for a robust sample covariance matrix in SEM is convergence to a normal distribution characterized by a mean vector of zero and a covariance matrix of Γ.

This is where the paper loses me... don't understand why this convergence is important, don't understand the convergence itself even (what is the curly L), don't understand what M- and S-estimators are...

The authors propose a new robust sample covariance estimator. Under some conditions, it is equivalent to the Satorra-Bentler adjustment.

Reading notes

Another one to try again with.

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