The performance of Bayesian fit indices in approximate measurement invariance testing with many groups in cross-cultural research.

This simulation study examined the performance of commonly used fit indices in traditional Bayesian estimation, including posterior predictive p value (PPP), Bayesian information criterion (BIC), and deviance information criterion (DIC), as well as the Bayesian version of approximate fit indices, such as Bayesian comparative fit index (BCFI), Bayesian Tucker-Lewis Index (BTLI), and Bayesian root mean square error of approximation (BRMSEA), in testing approximate measurement invariance (AMI) using multigroup Bayesian structural equation modeling (MG-BSEM) in the context of cross-cultural research involving a large number of groups. Key design factors included the number of groups, group size, and the magnitude of AMI, operationalized as the variance of parameter differences across groups. In MG-BSEM, the prior variance controls the extent to which measurement parameters are allowed to differ across groups. Results showed that when the prior variance aligned with the true magnitude of AMI, the mean PPP value was closest to 0.50. BCFI, BTLI, and BRMSEA stabilized once the aligned prior variance was reached; further increases in prior variance did not yield additional improvements in model fit. BIC tended to classify models with smaller AMI as invariant and those with larger AMI as noninvariant but failed to identify the aligned prior variance in cases of moderate AMI. In contrast, DIC was highly sensitive and more effective in detecting the aligned prior variance.