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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.

user1205901 - Reinstate Monica
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  • I am not an expert on CFA, but I am wondering what do you mean that the "model fit was better"? How do you measure model fit? – amoeba Mar 23 '15 at 12:53
  • I used RMSEA and Chi Square as measures of fit. I did this analysis in SPSS AMOS. – user1205901 - Reinstate Monica Mar 23 '15 at 12:54
  • I am asking this because as far as know, chi square statistic is essentially a measure of likelihood of the data, so I don't understand how a *maximum likelihood* solution can have lower likelihood (higher p-value from chi square test) than any other... – amoeba Mar 23 '15 at 12:56
  • Thanks for the information. If that's the case I must have typoed the original question, and got a higher Chi Square with ML. – user1205901 - Reinstate Monica Mar 23 '15 at 13:15
  • I would be skeptical that one should always default to ML estimation. What if the distribution is not normal? OLS will still work, as I recall, but MLE will be on shaky grounds. – RegressForward May 05 '15 at 03:16
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    Though [this post](http://stats.stackexchange.com/q/50745/3277) is about Exploratory FA, not CFA, the ideas behind the ML/GLS algorithms are being shared, so you may glance at that post. The two algorithms are quite similar computationally, but ML does assumes multivariate normality. CFA test (chi-square etc.) also assume it. – ttnphns May 05 '15 at 13:46
  • Thanks! However, isn't it also the case that GLS assumes multivariate normality? The Kline book I cited earlier states on p176 that "Two methods for endogenous variables with multivariate normal distributions include generalized least squares (GLS) and unweighted least squares (ULS)." – user1205901 - Reinstate Monica May 05 '15 at 20:28
  • No, GLS (just as OLS) does not require normality (be it multivariate or univariate), at least asymptotically. In small samples it could be desirable, though. – Richard Hardy May 06 '15 at 06:41

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