To a lesser extent I am also interested in when one should use Unweighted Least Squares (ULS), and other less common methods.
I have been taught Maximum Likelihood (ML) as a default but have just done a CFA and model fit was better under Generalized Least Squares (GLS) than ML. Is that sufficient to indicate I should use GLS? If not, what would be sufficient to indicate I should use GLS?
Kline (2016) p256 writes:
The GLS method generally requires less computation time and computer memory, but this potential advantage is not very meaningful today, given fast processors and abundant memory in relatively inexpensive personal computers. In general, ML is preferred over both ULS and GLS.
He does not mention any other advantage for GLS, or explain why ML is generally preferred, and nor do any of the other textbooks I have consulted.
I don't have any particular problems with univariate or multivariate normality, but might that make a difference to my choice?
Kline, R. B. (2016). Principles and practice of structural equation modeling. Methodology in the social sciences. 4th edition. New York: Guilford Press.