Exact and Bayesian Approximate Measurement Invariance
Cross-cultural research is concerned with comparisons of substantive variables across cultures, countries, or other types of socially relevant groups and contexts. This chapter introduces the case of Bayesian approximate measurement invariance as an alternative when exact measurement invariance fails. It presents an empirical example based on data from the International Social Survey Program to test the equivalence of a scale measuring the attitude toward granting citizenship rights to immigrants across several European countries. With approximate measurement invariance, all model parameters can be subject to a small degree of deviation from exact zero constraints, while the differences across groups are kept at a minimum. However, to the best of our knowledge, only a few applications exist to date that make use of Bayesian approximate measurement invariance. The alignment method can be used in conjunction with Bayesian approximate measurement invariance. Future research should also expand the range of possible prior distributions when testing measurement invariance issues.