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A full structural equation model (SEM) typically consists of both a measurement model (describing relationships between latent variables and observed scale items) and a structural model (describing relationships among latent variables). However, often researchers are primarily interested in testing hypotheses related to the structural model while treating the measurement model as a necessary but not primary focus of the overall model. In this case, researchers often wish to isolate and just evaluate the fit of the structural model. In our research, we examine a two-stage approach that can compute the chi-square statistic and fit indices for evaluating only the fit of the structural model in a full SEM. We call these the structural chi-square statistic and structural fit indices. For structural fit indices, we focused on the root mean square error of approximation (RMSEA), comparative fit index (CFI), and standardized root mean square residual (SRMR). We developed several new versions of the structural chi-square statistic, structural fit indices, and confidence intervals (CIs) of the structural fit indices. Through a simulation study, we demonstrated that several versions of our newly developed structural chi-square statistic yielded the nominal Type-I error rate; and the same versions of the structural fit indices exhibited low bias and their corresponding CIs had high coverage rates. Therefore, we recommend researchers use these versions of the structural chi-square test of fit alongside the structural fit indices when evaluating the fit of the structural model.
Zhang et al. (2024) studied this question.