Data-Matrix-Based Overall Goodness-of-Fit Testing for Confirmatory Factor Analysis: In-Sample Prediction Error for Observed Variables

Conventional χ²-based tests in confirmatory factor analysis assess overall fit through covariance reproduction and remain the standard inferential tools for overall goodness-of-fit testing. However, they do not directly quantify the magnitude of model misfit and may become unavailable when estimation fails to converge or produces improper solutions. We introduce and evaluate two data-matrix-based alternatives within structured factor analysis: the in-sample prediction error for observed variables (IPEo) test for a single model and the ΔIPEo test for nested models. IPEo quantifies the proportion of variability in the observed data matrix left unexplained by the hypothesized model, while its bootstrap p value provides an inferential decision. Monte Carlo simulations showed that the IPEo-based tests controlled Type I error, gained power with increasing sample size, and remained available when conventional χ²-based tests frequently failed. An empirical illustration further showed that the IPEo value and its p value provide complementary information about the magnitude and statistical significance of model misfit. These findings support IPEo-based testing as a complementary inferential framework for evaluating CFA model fit directly at the level of the observed data matrix.