title: On the Model-Based Bootstrap With Missing Data: Obtaining a P-Value for a Test of Exact Fit.
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) for general covariance structure models and applied to structural equation modeling by Bollen and Stine (1992) . An extension of this transformation to missing data was presented by Enders (2002) , 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) .
authors.affiliation: a University of British Columbia.
authors.forename: Victoria
authors.initials: V
authors.lastname: Savalei
authors.affiliation: b University of Notre Dame.
authors.forename: Ke-Hai
authors.initials: KH
authors.lastname: Yuan