Is the ML chi-square ever robust to nonnormality? A cautionary note with missing data

Normal theory maximum likelihood (ML) is by far the most popular estimation and testing method used in structural equation modeling (SEM), and it is the default in most SEM programs. Even though this approach assumes multivariate normality of the data, its use can be justified on the grounds that it is fairly robust to the violations of the distributional assumptions under some conditions. In support of this claim, a large literature exists outlining conditions under which the ML chi-square retains its asymptotic distribution even under nonnormality. The most important of these conditions is specifying a model in which the exogenous variables presumed to be uncorrelated (e.g., factors and errors in a confirmatory factor analysis model) are also statistically independent. The goal of this article is to point out that these conditions were developed for complete data, and in fact no longer ensure robustness when the data are both nonnormal and incomplete. This lack of robustness is illustrated both mathematically and empirically. Violation becomes more severe when the data are highly nonnormal and when a higher proportion of data is missing. It is concluded that if the proportion of missing data is greater than about 10%, robustness of the ML chi-square with incomplete nonnormal data cannot be counted on, even if the necessary assumptions such as independence are made. Other approaches to model testing are to be preferred in this case.