How to Use Normalized Residuals Heat Map in Structural Equation Models?

[#1]

I understand that residuals are the difference between sample covariance matrix and model implied covariance matrix. But I don't know how to use it. Can anyone tell me how can I use the normalized residuals heat map to identify the problem in my model and find potential ways (such as add or delete paths between variables) to improve model fit? Attached is the matrix and heat map of my model. Thanks!

[#2]

[#3 · Accepted solution]

Hi @lujc07, 

Great question! Normalized residuals (along with their heatmap) can be scanned visually to see if there is over- or under-fitting. Generally speaking, we want these values to be within about |2| units to reassure us that our model fits well. The heatmap facilitates one's understanding of whether the misfit (if present) is all over the place, or whether it's due to a misspecification within a smaller part of the model. In your example, it seems the latter is true; that is, there's misspecification involving the BFI, CrFungiPar, SrPhelnsec, and BcDO variables. Because these values are computed by subtracting the model-implied covariance matrix (MIC) from the sample covariance matrix (SCM), you could figure out if you're over- or under-fitting by looking at the signs of the residuals--if the SCM and MIC have all positive values, then large negative residuals point to over-fitting and large positive ones point to under-fitting. However, this process can get a bit more complicated if you have a mix of negative and positive values in the SCM and MIC. Indeed, if your interest lies on identifying which paths should be added, then your next step would entail using the Modification Indices option. This option will show the paths that will improve your model fit (they show up ranked by most to least improvement in fit). Please beware, however, that Modification Indices should be used carefully (i.e., in conjunction with domain expertise) because they could lead you astray otherwise. Here's a useful article that describes the potential issues.

HTH,

~Laura

[#4]

Hi Laura,

Thank you so much for your answer! The article is also very helpful. I read the article and it talked about the threshold of T value of parameter estimate and modification index. You mentioned that the normalized residuals should be within in |2|. Could you please provide me any article talking about it if available so that I could cite in my article? Thanks!

[#5]

You're very welcome! Bollen (1989) is the best reference for that:

Bollen, K. A. (1989). Structural equations with latent variables (Vol. 210). John Wiley & Sons.

Page 259 shows the formula for the normalized residuals, which is simply the residual divided by its asymptotic variance. This formulation gives normalized residuals a scale that's similar* to that of a standard normal variable, and it's why a value of around |2| is recommended as a guideline.

HTH,

~Laura

*A caveat is that the asymptotic variance in the formula has been shown to be a bit high (Joreskog & Sorbom, 1988), so the scale of normalized residuals isn't exactly like a std normal variable, but rather a bit smaller. Still, the |2| guideline is used.

[#6]

Thank you so much for the reference!