Bootstrap Tests for the Goodness of Fit in Factor Analysis

Asymptotic robustness studies have shown that normal theory based test statistic for the goodness of fit in factor analysis and related structural models retains its asymptotic chi-square distribution under the null hypothesis if the latent vector variables are independently distributed. The asymptotic test, however, may not be robust against the independence assumption, as suggested by a recent Monte Carlo study. A Monte Carlo experiment is conducted to compare the asymptotic and the bootstrap tests across 4 exploratory factor analysis models, 5 sample sizes and 6 distributional conditions; in some of these conditions the common factors and the unique factors are taken to be dependent. Results of a simulation study indicate that the asymptotic (bootstrap) test rejects (accepts) the null hypothesis too often than expected from nominal levels of tests when the common factors and the unique factors are mutually independent. When they are just uncorrelated the asymptotic test completely broke down, while the bootstrap test performed much better, though it rejected the null hypothesis too often.