How is goodness of fit estimated in SEM models?

Structural equation models allow for estimation of complex networks including latent and observed variables, and endogenous and exogenous factors. When an SEM is fit, the model results are summarized in terms of a number of fit indices. I assume these indices (and their $p$-values) are variations of goodness-of-fit-tests. An overview is given by Kenny here.

1. What measures is an SEM capable of generating predicted values for?
2. The site refers to a $\chi^2$ value but does not explain how it is generated. When there is no clear, single "Y" in the model, is every single variable in the model taken as an (expected-observed)?
3. How is it possible for an SEM to have a low R^2 but "perfect fit"? And what does perfect fit mean in this sense?
• I don't think I understand #1. – Patrick Malone Apr 13 '18 at 15:42
• @PatrickMalone I guess I had the assumption that SEM was capable of calculating or simulating predictions for at least one of the measures conditional on some subset (possibly empty?) of the other values? – AdamO Apr 13 '18 at 20:34
• In core SEM, the estimating process results in a model-implied variance-covariance matrix. As Jeremy noted, it can also result in model-implied means. So in the evaluation of fit, there's no distinction between independent/predictor/dependent/etc. A model will usually have those things, but the fit evaluation is whether the specified model can closely reproduce the data. Does that help? – Patrick Malone Apr 14 '18 at 0:25

1. SEM doesn't really give predicted values, in the sense of (say) regression. You can get predicted values for the latent variables, but these are non-deterministic - that is, there is more than one (an infinite number) or sets of predicted values. This is the same as factor analysis. What is predicted is the fitted covariance matrix.

2. $\chi^2$ is a measure of how closely the fitted covariance matrix matches the sample covariance matrix. Most commonly, maximum likelihood is used. In ML:

$F_{ML} = log|\Sigma| + tr(S \Sigma^{-1}) - log|S| - p$

Where S is the sample covariance matrix, $\Sigma$ is the fitted covariance matrix, and p is the number of variables in the model.

$\chi^2 = F_{ML}(N - 1)$

3. Yes, it's very possible. Perfect fit means that you reproduce the sample covariance matrix perfectly. Imagine a really simple model where $x$ predicts $y$. In the sample, $x$ and $y$ are unrelated. Your model predicts a zero relationship between $x$ and $y$. The sample covariance matrix will be perfectly matched by the fitted covariance matrix. $\chi^2$ will be zero, and fit will be perfect.

Note: I've been referring to the simpler case here where we're only modeling covariances - what's sometimes called the extended model can also include means. The principle is the same.

• Very illuminating, thank you. I have a question about the relevance of the sample covariance matrix. In categorical analyses, we can create a saturated model where the fitted values are the empirical proportions. That forms the basis of the Chi-square goodness of fit test. Is there a concept of an SEM model which results in the sample covariance matrix as a fitted matrix? If so, how is such a model constructed? – AdamO Apr 13 '18 at 17:27
• Yes, this is called the saturated model, and has perfect fit - chisquare is zero, df is zero. SEMs can be evuated against one another if the models are nested. The model can be thought of as being compared with the saturated model. – Jeremy Miles Apr 13 '18 at 20:07