Next, we examined a multigroup model, in which group differences were captured either by differences in criterion parameters (in the unconstrained model), or by mean differences in the multi-substance factor (in the constrained model). In the first multigroup model, the “unconstrained” model, thresholds (for the categorical observed criteria) and factor loadings were estimated freely within each group, while residual variances and factor means were fixed (at one and zero, respectively) in all groups. In the second multigroup model, the “constrained” model, thresholds (for the categorical observed criteria) and factor loadings were constrained equal across all groups, while residual variances and the multi-substance latent trait mean were fixed (at one and zero, respectively) only in the reference group (MTFS males) and freely estimated in all other groups. The comparison of these two models allowed us to evaluate measurement invariance between the different sex and sampling groups. Because only the means (and residual variances) vary among groups in the constrained model, it is more parsimonious, but assumes that the substance dependence criteria exhibit measurement invariance across groups. That is, the structure of
