Correlation residuals vs standardized residuals in SEM package in R

Question

I've been working with SEM package in R recently that I happened to read it's manual for the standardizedResiduals. In the manual,

Residuals are defined as S - C, where S is the sample covariance matrix of the observed variables and C is the model-reproduced covariance matrix. The standardized residual covariance for a pair of variables divides the residual covariance by the product of the sample standard deviations of the two variables, (s_{ij} - c_{ij})/(s_{ii}s_{jj})^{1/2}.

I can see from the formula that the standardized residuals computed in SEM package are standardized residual covariance that are taken to the correlation scale.

But, when I looked at the Residuals section in Kenneth A. Bollen (1989) to clear my understanding, I found another term, correlation residuals.

My question is then that what's the difference between correlation residuals and standardized residuals other than computation itself? When would I want to look at standardized residuals verses correlation residuals? I really appreciate your help to understand the output of SEM correctly.

Answer 1

Standardized residuals should have an approximately normal distribution, with mean 0. If they don't look normally distributed, you might have a problem. You expect (about) 5% of them to have absolute values > 2, so you can determine if you have some serious outliers. This can indicate a problem with your model, even if the overall model fit is good.

Correlation residuals are bounded by the range of the correlation - the absolute maximum is therefore 1 - (-1) = 2 (although this would be very unusual).