| Abstract: | Evaluating the fit of a structural equation model via bootstrap requires a transformation of the data so that the null hypothesis holds exactly in the sample. For complete data, such a transformation was proposed by Beran and Srivastava (1985) Beran, R. and Srivastava, M. S. 1985. Bootstrap tests and confidence regions for functions of a covariance matrix. The Annals of Statistics, 13: 95–115. [Crossref], [Web of Science ®] , [Google Scholar] for general covariance structure models and applied to structural equation modeling by Bollen and Stine (1992) Bollen, K. A. and Stine, R. A. 1992. Bootstrapping goodness-of-fit measures in structural equation models. Sociological Methods and Research, 21: 205–229. [Crossref], [Web of Science ®] , [Google Scholar]. An extension of this transformation to missing data was presented by Enders (2002) Enders, C. K. 2002. Applying the Bollen-Stine bootstrap for goodness-of-fit measures to structural equation models with missing data. Multivariate Behavioral Research, 37: 359–377. [Taylor &; Francis Online], [Web of Science ®] , [Google Scholar], but it is an approximate and not an exact solution, with the degree of approximation unknown. In this article, we provide several approaches to obtaining an exact solution. First, an explicit solution for the special case when the sample covariance matrix within each missing data pattern is invertible is given. Second, 2 iterative algorithms are described for obtaining an exact solution in the general case. We evaluate the rejection rates of the bootstrapped likelihood ratio statistic obtained via the new procedures in a Monte Carlo study. Our main finding is that model-based bootstrap with incomplete data performs quite well across a variety of distributional conditions, missing data mechanisms, and proportions of missing data. We illustrate our new procedures using empirical data on 26 cognitive ability measures in junior high students, published in Holzinger and Swineford (1939) Holzinger, K. J. and Swineford, F. 1939. A study in factor analysis: The stability of a bi-factor solution. Supplementary Educational Monographs, 48: 1–91. [Google Scholar]. |
