# Confirmatory factor analysis: using ML vs MLM estimator

I have a relatively small dataset (n < 130) based on a questionnaire set up with 5-point likert type scale. Each respondent has filled out the questionnaire at 4 time points, i.e. a longitudinal (or panel) data study.

In papers and other theory that address Maximum Likelihood estimation in case of non-normal data, the authors usually recommend to use the robust ML estimator (MLM) (e.g. Brown, T.A. (2006) Confirmatory Factor Analysis for Applied Research, New York, NY: The Guilford Press).

In case of non-continuous data, e.g. ordinal data other estimators are recommended such as Diagonally Weighted Least Squares (DWLS).

Two questions:

1) Isn't it always a good idea to fit your models with both ML and MLM, even if you think the data is normally distributed? I was thinking of doing this myself and if the results differ a lot, ML is probably not a good approach.

2) In all these cases of whether or not the data is non-normal, non-continuous, etc. Do the authors refer to the outcome data type (i.e. the latent variable that you try to measure), the residuals or the manifest variables / indicators?

P.s. If your answer can be backed by a paper I would appreciate a reference to read the details.

EDIT I forgot to link a somewhat related forum topic: Exploratory Factor Analysis (EFA) followed by Confirmatory Factor Analysis (CFA)