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Asymptotic standard errors of the estimated asymptotic standard errors for parameter estimates in structural equation modelling are derived using the delta method with the assumption of multivariate normality for observed variables. The derivation covers the cases with and without restrictions on parameters. The result can be used to derive the asymptotic standard error of the z score (a parameter estimate divided by its estimated standard error), which is frequently substantially different from one. The case of standardized observed variables is dealt with as a typical example with restrictions on parameters. For actual covariance (correlation) structure models, the exploratory factor analysis model with factor rotation and the confirmatory factor analysis model are presented with numerical examples. Simulations are performed to assess the accuracy of our method for normally and non-normally distributed variables.
I. Introduction
In structural equation modelling, parameter estimates are subject to sampling errors even when a model is true, and hence their accuracy should be evaluated. The stability of the parameter estimates can be assessed by their corresponding standard errors. In typical software for structural equation modelling such as LISREL (Joreskog & Sorbom, 1996) and EQS (Bentler, 1989), the asymptotic standard errors of parameter estimates are reported. The standard errors can also be used as auxiliary goodness-of-fit indices or modification indices with corresponding parameter estimates Bollen, 1989, Chapter 7).
When the number of observations is large and some regularity conditions are satisfied, the maximum likelihood estimators of the parameters are normally distributed. In such cases, we obtain the oath quantiles of the parameter estimates from the distribution function of the standardized normal distribution. However, generally the standard errors of the parameter estimates depend on the parameters, which are usually unknown except in experimental situations. In practice, the standard errors are estimated by using the parameter estimates. Consequently, the estimated standard errors which involve the estimated parameters are also subject...