Using product indicators in restricted factor analysis models to detect nonuniform measurement bias

When sample sizes are too small to support multiple-group models, an alternative method to evaluate measurement invariance is restricted factor analysis (RFA), which is statistically equivalent to the more common multiple-indicator multiple-cause (MIMIC) model. Although these methods traditionally were capable of detecting only uniform measurement bias, RFA can be extended with latent moderated structural equations (LMS) to assess nonuniform measurement bias. As LMS is implemented in limited structural equation modeling (SEM) computer programs (e.g., Mplus), we propose the use of the product indicator (PI) method in RFA models, which is available in any SEM software. Using simulated data, we illustrate how to apply this method to test for measurement bias, and we compare the conclusions with those reached using LMS in Mplus. Both methods obtain comparable results, indicating that the PI method is a viable alternative to LMS for researchers without access to SEM software featuring LMS.