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Latent variable modeling is an essential technique for understanding and measuring the relationships between latent constructs and observed scores accurately in social sciences such as education and psychology. Further, measurement invariance, a statistical property, is an important concern in latent variable modeling that assumes a latent construct is measured in the same way concerning covariates or background variables such that meaningful statistical comparisons and interpretations can be made without the confounding effect of those covariates. This dissertation includes three papers in examining or developing methods for measurement invariance via Monte Carlo simulations that can be used by applied researchers. (1) Paper 1 systematically examined how Bayesian model evaluation indices, namely the deviance information criterion (DIC), Bayesian information criterion (BIC), and the posterior predictive p value (PPP), performed in detecting violations of measurement invariance under varying sample sizes and priors, as compared to frequentist indices and tests, such as Akaike information criterion (AIC), root mean square error of approximation (RMSEA), McDonald’s noncentrality index (Mc) and likelihood ratio tests (LRTs). Using specificity (i.e., true negative rates to invariance) and sensitivity (i.e., true positive rates to noninvariance), results showed that the deviance information criterion (DIC) with a weakly informative prior in standard normal distribution N (0, 1) had higher specificity in detecting true invariance, especially when sample size n = 100, and kept competitive sensitivity in detecting noninvariant items when n = 200, compared to the frequentist approach. Therefore, DIC is a Bayesian alternative for investigating measurement invariance. (2) Paper 2 developed a novel use of the alignment optimization (AO) method that examined measurement invariance concerning a continuous violator by discretizing that violator into many groups (e.g., 10) and using the categorized variable as a group membership. Compared to multiple indicators, multiple causes (MIMIC) and MIMIC-interaction methods, the AO method was as competitive as the sequential likelihood ratio tests and Wald tests with Bonferroni corrections in detecting linear invariance violations, far better in detecting nonlinear quadratic violations, with classification accuracy (CA = .88) when the sample size n =1000 (i.e., 100 per group for 10 groups), and more convenient to implement than the MIMIC-interaction method that requires formulating and comparing a lot of models. (3) Paper 3 proposed using the structural equation model trees to detect measurement invariance concerning multiple observed violators and identified invariance testing procedure. Invariance evaluation under the SEM tree requires model comparisons at different invariance constraint levels. The LRTs or likelihood comparisons under the SEM tree had Type I error rates = .052 when n = 1000 and statistical power rates in .964–1.00 in detecting both linear and nonlinear quadratic intercept noninvariance when n = 1000. The split rates of true violators were in .928–1.00 across all replications when n = 1000, suggesting that it was very likely for a group membership for a tree node being a violator related to noninvariance, especially for a dichotomous violator, compared to a continuous noise covariate which could be a group membership due to accounting for factor mean differences. |
