Finally, measurement non-invariance of the craving criterion was tested across gender, race/ethnicity, and age. Race/ethnicity was operationalized as a five-level nominal variable (non-Hispanic White, non-Hispanic Black, Native American/Alaska Native, Asian/Pacific Islander, and Hispanic), and age was operationalized as a four-level ordinal variable (18–29, 30–44, 45–64, 65+). MPLUS uses CFA with covariates (Multiple Indicators Multiple Causes [MIMIC] model) to test item measurement non-invariance due to differential item functioning. We tested the direct effect of the covariate on the craving item that is unmediated through the underlying latent trait. An alpha of 0.05 was selected a priori. A significant direct effect means that for the same severity level, members of different covariate groups have different probabilities of endorsing the item; this is indicative of measurement non-invariance, as group membership should not be a determinant of criterion endorsement probability at equal levels of severity if the criterion is completely unbiased. The final MIMIC model included all significant indirect (latent trait regressed on covariate) and direct (item regressed on covariate) effects. A similar approach has been taken previously in IRT analyses of alcohol criteria
