Longitudinal measurement invariance (LMI) is a critical prerequisite to assessing change over time with intensive longitudinal data (ILD). For LMI testing with ILD, we propose cross-classified factor analysis (CCFA) to detect non-invariant item parameters and alignment optimization (AO) to detect non-invariant time points as a supplement to CCFA. In addition, we use a covariate in CCFA to identify a source of non-invariance. To evaluate the proposed models under unique features of ILD, such as autoregression (AR), we conducted a Monte Carlo simulation study. The results showed CCFA can be an excellent tool for ILD LMI testing regardless of simulation factors even when AR was misspecified and can identify a source of non-invariance using a covariate. AO can supplement CCFA to find non-invariant time points although AO requires a large number of persons. We provide detailed discussions and practical suggestions.

Model size has been shown to influence model fit measures in structural equation modeling, whereas its impact on measurement invariance testing using recommended cutoff criteria of fit measures is not clear. This study examined the impact of model size on the sensitivity of fit measures (CFI, RMSEA, AIC, BIC, SaBIC, LRT) to measurement noninvariance. Results suggested that CFI and RMSEA failed to detect metric noninvariance when model size was large (e.g., 6 factors with 8 items per factor). AIC and LRT were minimally affected by model size. Recommendations for empirical researchers were provided based on the findings of this simulation study.

Exploratory structural equation modeling (ESEM) allows for the estimation of all cross-loadings, which leads to the number of parameters estimated substantially greater than that in conventional SEM. This study examined the sensitivity of fit measures (CFI, RMSEA, AIC, BIC, SaBIC, LRT) to measurement noninvariance in ESEM. Results suggested that the sensitivity of the fit measures depended on the magnitude of noninvariance, model complexity, and sample size. The sensitivity of LRT to both metric and scalar noninvariance was high. Comparative fit index tended to filter out small noninvariance with large sample sizes. Root mean squared error of approximation failed to detect even a large measurement noninvariance when the model was complex and sample size was below 300 per group. Information criteria were more sensitive to scalar noninvariance than to metric noninvariance. Implications and recommendations based on the findings were discussed.

Examining how well a hypothesized model fits observed data is pivotal in Bayesian structural equation modeling (SEM). Recent efforts have aimed to formulate Bayesian SEM fit indices akin to those used in frequentist SEM. However, the assessment of these fit indices in Bayesian SEM has been limited in scope. This research entailed two simulation studies to explore the impact of various factors, including prior choices, model misspecification, model size, and sample size, on the sensitivity of fit indices. The first study delved into the prior sensitivity of Bayesian SEM fit indices (BRMSEA [Bayesian root mean square error of approximation], BCFI [Bayesian comparative fit index], BTLI [Bayesian Tucker–Lewis index], and PPp [posterior predictive p-value]) under latent factor and cross-loading misspecifications. The second study explored how alterations in model complexity influenced the prior sensitivity of these fit indices to model misspecifications. The findings indicate a robust performance of Bayesian fit indices under relatively severe model misspecification, suggesting a preference for highly informative priors when prior knowledge is certain. As model size increased, PPp exhibited a balanced performance between true and false positive rates, whereas BRMSEA was less reliable in the presence of latent factor misspecification. Implications for applied research are discussed.

Exploratory structural equation modeling (ESEM) allows items to cross-load on nontarget factors and provides flexibility in modeling multidimensional survey data. However, this flexibility comes at the cost of increased model complexity due to the estimation of a significantly greater number of parameters than conventional SEM. This study systematically evaluated the model fit indices in ESEM through a Monte Carlo simulation. Design factors included model complexity,atent factor correlations, magnitudes and percentages of major cross-loadings, and sample sizes. Results showed that both the number ofatent factors m and the item–factor ratio p/m had distinct impacts on rejection rates, particularly in underspecified models. Among the fit measures, ${\chi }^2$ tests and Mc demonstrated high power for detecting misspecification while, in general, elevating false positives. CFI and TLI provided a more balanced trade-off between false- and true-positive rates. When evaluating ESEM, it is important to consider multiple fit indices and justify the cutoff criteria across models with different factor structures.