= 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 [[https://archive.org/details/latentvariablesa0000unse/page/398/mode/2up|online]].

The authors explore three [[Statistics/PearsonsChiSquaredTest#Test_for_Goodness-of-fit|chi-square goodness-of-fit statistics]] (''T,,1,,'', ''T,,2,,'', and ''T,,3,,'') used in [[Statistics/StructuralEquationModeling|covariance structure analysis]]. In particular, ''T,,3,,'' does not rely upon [[Analysis/NormalDistribution|normality]]. It instead involves estimating a matrix ''C'' with ''C,,n,,''. (What is ''C'' though? Hard to say. The authors are using a completely different notation as compared to [[AsymptoticallyDistributionFreeMethodsForTheAnalysisOfCovarianceStructures|Browne]]. I ''think'' it's a [[Statistics/Covariance#Matrix|covariance matrix]] calculated through [[Statistics/GeneralizedLeastSquares|GLS]].)

The authors notate population [[Statistics/Moments|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 ''θ,,0,,'' 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 = σ,,0,, = σ(θ,,0,,)''), 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 ''Γ,,n,,'', and then a ''U,,n,,'' matrix, and... I'm lost.

The authors promote the use of ''T̅ = c^-1^T'' where ''c = trace(U,,n,,Γ,,n,,/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.



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