Covariance Structure Model Fit Testing Under Missing Data: An Application of the Supplemented EM Algorithm

Abstract

We apply the Supplemented EM algorithm (Meng & Rubin, 1991) to address a chronic problem with the “two-stage” fitting of covariance structure models in the presence of ignorable missing data: the lack of an asymptotically chi-square distributed goodness-of-fit statistic. We show that the Supplemented EM algorithm provides a convenient computational procedure that leads to such a chi-square statistic, and we provide a SAS macro implementing this method. Our derivations are corroborated with results from a small simulation study. We also apply the proposed method to 2 empirical data sets: (a) confirmatory factor analysis of Mardia, Kent, & Bibby's 1979 Open-book Closed-book data and (b) conditional latent curve modeling of adolescent aggressive behavior as discussed by Curran (1997).

ACKNOWLEDGMENTS

We thank the editor and the reviewers for helpful comments and suggestions that led to an improved article. Li Cai also gratefully acknowledges financial support from the following funding agencies: National Science Foundation (# SES-0717941), the National Center for Research on Evaluation, Standards and Student Testing (CRESST) through an award from the U.S. Department of Education's Institute of Education Sciences (IES) (# R305A050004), and a pre-doctoral advanced quantitative methods training grant awarded to the UCLA Departments of Education and Psychology by IES. The views expressed in this paper are of the authors alone and do not reflect the views or policies of the funding agencies.

Notes

¹ We deliberately avoid considering such extended CSM models as nonlinear or mixture models because those models can only be identified from raw data.

² Loosely speaking, monotone missing means that the columns of the data set can be arranged such that if a case is missing in one of the columns, then all subsequent columns are missing. The reader should refer to Schafer (1997) for the precise definition of monotone missing data.