Bootstrapping Goodness-of-Fit Measures in Structural Equation Models

Assessing overall fit is a topic of keen interest to structural equation modelers, yet measuring goodness of fit has been hampered by several factors. First, the assumptions that underlie the chi-square tests of model fit often are violated. Second, many fit measures (e.g., Bentler and Bonett's [1980] normed fit index) have unknown statistical distributions so that hypothesis testing, confidence intervals, or comparisons of significant differences in these fit indices are not possible. Finally, modelers have little knowledge about the distribution and behavior of the fit measures for misspecified models or for nonnested models. Given this situation, bootstrapping techniques would appear to be an ideal means to tackle these problems. Indeed, Bentler's (1989) EQS 3.0 and Jöreskog and Sörbom's (forthcoming) LISREL 8 have bootstrap resampling options to bootstrap fit indices. In this article the authors (a) demonstrate that the usual bootstrapping methods will fail when applied to the original data, (b) explain why this occurs, and, (c) propose a modified bootstrap method for the chi-square test statistic for model fit. They include simulated and empirical examples to illustrate their results.