I was wondering which is a better estimator to use for categorical data: ML or WLSMV. I saw on a discussion on the Mplus website that they recommend WLSMV for categorical data but didn't explain why. Does anyone know specifically why ML doesn't work as well?
Preferably, I am looking for a reference that compares these two estimation approaches, but have not been able to locate one after hours of searching.
Thank you for sharing your knowledge and experience!
In one medical research paper, Proitsi et al. (2009) write:
"The WLSMV is a robust estimator which does not assume normally distributed variables and provides the best option for modelling categorical or ordered data (Brown, 2006)".
For your convenience, I'm including the cited reference in the reference list below (I use APA format):
Brown, T. (2006). Confirmatory factor analysis for applied research. New York: Guildford.
Proitsi, P., Hamilton, G., Tsolaki, M., Lupton, M., Daniilidou, M., Hollingworth, P., ..., Powell, J. F. (2009, in press). A multiple indicators multiple causes (MIMIC) model of behavioural and psychological symptoms in dementia (BPSD). Neurobiology Aging. doi:10.1016/j.neurobiolaging.2009.03.005
I hope this is helpful and answers your question.
The most obvious reason for choosing one over the other would be the kind of fit indices you need. The WLSMV will give you CFI, TLI and RMSEA, which will help you evaluate the fit of a given model. If you need to compare non-nested models, you would need AIC and/or BIC, which aren't available with WLSMV and categorical data. The opposite is true of ML (again, only when dealing with categorical data).
I'm not sure why they recommend WLSMV on the Mplus website, but if you are comparing nested models, the WLSMV is probably the most convenient as it will allow you to both (1) evalute whether the models provide adequate fit to the data (e.g. CFI > .90 and RMSEA < .5), and (2) use a chi2 difference test to see which models provides the best fit out of a number of competing models.
Your question does not specifically reference factor analysis (FA) or structural equation modeling (SEM), though I will assume you are broadly interested in differences between estimators for continuous latent variable models of categorical data. Broadly speaking, there are three classes of modeling approaches$^1$. The first is the direct approach - a FA/SEM modeling approach, which treats categorical data as continuous and the most commonly used estimator for this approach is robust maximum likelihood (commonly referred to as MLR in FA/SEM packages such as lavaan). I assume such an approach is what you referenced in the question, because as @Jeremy Miles said, "ML for categorical data in SEM hasn't been around for all that long." The second approach, like the first, is a FA/SEM approach, but it can also be referred to as an item factor analysis (IFA) approach since it treats the data as categorical. The most common estimator used for this approach is some form of diagonally weighted least squares (DWLS). WLSMV falls under the DWLS umbrella, though it is not technically an estimator. DWLS is the estimator, and calling WLSMV in a software package (e.g., lavaan or Mplus) tells the program to report robust standard errors and to use a particular adjustment to the test statistic used to assess model fit. The final class of models are item response theory (IRT) models, some of which have analytic relationships to FA/SEM models of categorical data (Kamata & Bauer, 2008). The most common estimator for this class of models is ML, specifically marginal maximum likelihood (MML) via the Bock-Aitkin EM algorithm (Bock & Aitkin, 1981).
I have excluded many important aspects of the models and estimators mentioned above, though hopefully what I wrote will help you better understand my recommended articles! For articles looking at IFA models generally, I highly recommend Chen & Zhang (2021) and Wirth & Edwards (2007). These two articles primarily discuss the latter two classes of models, as their focus is on IFA models. Regarding articles concerned with the direct approach and FA/SEM models for categorical data, I recommend Flora & Curran (2004), Li (2016), and Robitzsch (2020).
$^1$ Note that there are several others, though they are seldom used in practice.
References
Bock, R. D., & Aitkin, M. (1981). Marginal maximum likelihood estimation of item parameters: Application of an EM algorithm. Psychometrika, 46(4), 443-459.
Chen, Y., & Zhang, S. (2021). Estimation methods for item factor analysis: An overview. Modern Statistical Methods for Health Research, 329-350.
Flora, D. B., & Curran, P. J. (2004). An empirical evaluation of alternative methods of estimation for confirmatory factor analysis with ordinal data. Psychological Methods, 9(4), 466.
Kamata, A., & Bauer, D. J. (2008). A note on the relation between factor analytic and item response theory models. Structural Equation Modeling: A Multidisciplinary Journal, 15(1), 136-153.
Li, C. H. (2016). Confirmatory factor analysis with ordinal data: Comparing robust maximum likelihood and diagonally weighted least squares. Behavior Research Methods, 48, 936-949.
Robitzsch, A. (2020, October). Why ordinal variables can (almost) always be treated as continuous variables: Clarifying assumptions of robust continuous and ordinal factor analysis estimation methods. In Frontiers in Education (Vol. 5, p. 589965). Frontiers Media SA.
Wirth, R. J., & Edwards, M. C. (2007). Item factor analysis: Current approaches and future directions. Psychological Methods, 12(1), 58.
