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I am trying to understand how parameter estimates in structural models are calculated. I have searched through various SEM textbooks, journal articles, etc. and have not found a good description of how the sample input data is used to calculate these values.

What I generally see and what I understand (I think) is that assuming multivariate normality, maximum likelihood (ML) is used on the sample data. ML analyzes the sample covariance matrix and compares it to the implied covariance matrix (created following model specification). ML outputs the model fit indices AND outputs the parameter estimate values, standard error values, a t-value (critical ratio) and p-value of the corresponding critical ratio.

While I can quite easily find the various formulas for the fit indices, I can not seem to find and formula that explains how the parameters estimates (or as AMOS refers to them, Regression Weights) are calculated.

Any information in regards to an equation, or maybe a better explanation for me to understand how these estimates are made I would really appreciate it!

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There isn't a close-form solution for SEMs, the model must be computed iteratively using numerical optimization algorithms. No doubt you ran into the ML formula during your search, which is one particular objective function that is being optimized (SEM has several). Essentially, if you plugged in any parameter estimates other than the ML estimates you would obtain a (log-)likelihood value that was less than the maximized (log-)likelihood. Numerical solvers do this laborious and high-dimensional search for you, and you would never want to do this by hand.

It might be helpful to look over open-source projects to understand this. I'd recommend inspecting the source code from the sem or lavaan packages in R if you are really interested in the how-to part of estimating these models.

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