Corrections to Test Statistics and Standard Errors in Covariance Structure Analysis
Corrections to Test Statistics and Standard Errors in Covariance Structure Analysis was written by Albert Satorra and Peter M. Bentler in 1994. It was published in Latent variables analysis: Applications for developmental research (eds. Alexander von Eye and Clifford C. Clogg). The chapter can be accessed online.
The authors explore three chi-square goodness-of-fit statistics (T1, T2, and T3) used in covariance structure analysis. In particular, T3 does not rely upon normality. It instead involves estimating a matrix C with Cn. (What is C though? Hard to say. The authors are using a completely different notation as compared to Browne. I think it's a covariance matrix calculated through GLS.)
The authors notate population moments as σ and sample moments as s. They also notate parameters as θ with a corresponding parameter space Θ. They assume that σ = σ(θ), i.e., the population moments can be modeled; that σ(θ) is differentiable on θ; and that the true population value θ0 is interior to Θ. Lastly, they assume that √n(s-σ) converges to a normal distribution (with mean of 0 and asymptotic covariance matrix Γ) as n increases.
Under normality conditions (e.g., σ = σ0 = σ(θ0)), the three goodness-of-fit statistics are asymptotically equivalent. That is, as n increases, the difference between estimators tends toward 0.
There's then something about estimating Γ with Γn, and then a Un matrix, and... I'm lost.
The authors promote the use of T̅ = c-1T where c = trace(UnΓn/r). T̅ better follows the chi-squared distribution in all cases. This correction factor is generally applicable to any goodness-of-fit statistic T.
The authors also calculate a method for correcting standard errors, based on the information matrix calculated above.
Reading notes
Probably need to try again with this paper at some point.