Measurement invariance test concerns with whether the group membership is related to the attributes of a test. In the framework of Structural Equation Modeling (SEM), it is implemented by the multiple-group confirmatory factor analysis (CFA). Although this test has been widely applied in empirical studies, two research questions still need to be explored. One is how to appropriately choose the reference indicator (RI), the second is how to locate the non-invariant items. The challenge lies in appropriately choosing an invariant RI and accurately locating the non-invariant item parameters. In this dissertation, we used two well-designed simulation studies to answer these two aforementioned questions. In Study I, we compared three commonly-used methods, named MaxL, Minχ^2 and BSEM and evaluated their performance of reference indicator selection. In Study II, we employed two Bayesian methods, Bayes factor and Bayesian estimation to locate the positions of non-invariance. All methods were applied to empirical datasets. We found Minχ^2 and BSEM are superior to MaxL in correctly choosing the reference indicator. In addition we discovered Bayes factor can more accurately locate the non-invariant item parameters and distinguish the invariant items from the contaminated ones. Finally, the application of Cauchy prior will help to improve Bayes factor’s performance.