Bollen's normalized residuals PreviousNext
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 Leslie Rutkowski posted on Tuesday, March 06, 2018 - 7:05 am

I'm looking at this:

And comparing the normalized residuals to Bollen's (1989, p 259). Bollen's denominator is the model-implied values (hat(sigma)_ii or hat(sigma)_ij) whereas yours are the empirical values (s_ii or s_ij).

Is there literature that would explain when and why this changed?

 Tihomir Asparouhov posted on Tuesday, March 06, 2018 - 4:12 pm
There are several reasons for computing the normalized residual that way. The denominator interpretation is that it is Var(s_ij), not Var(sigma_ij). The Var(sigma_ij) is not expressed like that at all.

1) Var(s_ij) is computed by estimating the unrestricted model, so it shouldn't have anything to do with the null model that is being tested

2) That formula in the note applies only for the ML estimator and it doesn't hold for the MLR estimator (under non-normality or complex sampling). With the MLR estimator we actually estimate the unrestricted model and standard errors and use Var(s_ij) from that estimation. For consistency we use the same method for the ML estimation.

3) Similarly that formula does not apply to estimation with missing data. In that case again Mplus estimates the unrestricted model and standard errors and uses those standard errors.

4) Consider as an example the case where sigma_ij=0. The normalized residuals (as computed in Mplus) would match precisely the T-value for sigma_ij from the unrestricted model estimation and thus it is guaranteed to have standard normal distribution. If we use the formula from Bollen book this will not happen.

5) Using Mplus normalized residuals you can independently evaluate the fit for each residual. If you use Bollen's formula there is a risk that a misfit in sigma_ii will negatively impact the residual evaluation for sigma_ij.

6) Using Mplus normalized residual approach we got better performance for the standardized residuals.

Finally, note that in most applications the difference between the two approaches will be negligible. Note also that the normalized residuals do not in general have a standard normal distribution under the null hypothesis. The standardized residuals do, and if one is to look at the precise value it is better to look at the standardized residual. Mostly the normalized residuals should be used as a guidance for model modification.
 Leslie Rutkowski posted on Wednesday, March 07, 2018 - 6:04 am

Thanks for your response. I understand the deficiencies of the normalized approach. But I was confused because the technical note references Bollen's formula directly and I am trying to understand the discrepancy and the motivation for the change.

Then, is there evidence (e.g., a published paper or manuscript) that makes the case for this shift? I'd like to better understand what is meant by "better performance," as you mention above.

 Tihomir Asparouhov posted on Wednesday, March 07, 2018 - 7:57 am
The better performance in item 6) is regarding the standardized not the normalized residuals and it means that the negative variance problem occurs less often. There is no paper besides that note. There has never been a shift in what Mplus does - we have always used the approach that is documented in the note. The way I see it, the above argument is so overwhelming that the question of which is better hardly exist. Consider for example point 3). If there is a large amount of missing data in one variable, say 50%, it would be quite unreasonable to use N as the sample size that drives the denominator - since that doesn't represent the number of observations you have. The rest of the arguments above are equally important though.
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