Chi-square difference tests were used to test for overall DIF (another term for measurement invariance) of all the dependence and abuse items, excluding legal problems and including craving. We first constructed a model where all factor loadings and thresholds were allowed to vary across covariate categories (baseline model). We then compared the baseline model to a model where item thresholds and factor loadings were set equal across covariate categories (constrained model). If the chi-square difference test reaches significance (p < 0.05), then measurement non-invariance exists and the items behave differently in the two groups. DIF covariates included gender, age (≤ or >the median, 37 years), race/ethnicity (white vs. non-white), and the three combined psychiatric disorder variables (mood disorders, anxiety disorders, and personality disorders). Since we analysed DIF using six covariates, we used Bonferroni adjustment for multiple comparisons, considering a result significant at p ≤ 0.008. When significant overall DIF was identified, we explored which items contributed to DIF by identifying anchor items whose thresholds or factor loadings did not significantly differ by group in order to set the metric between
