In Mplus 5.1 (Muthén and Muthén 2007) one factor models were specified with the four CAGE items as indicators. First, a configural invariance model was fitted. In this model the factor loadings and thresholds were estimated for each group. Strong factorial invariance and strict factorial invariance models were tested in line with Slof-Op ‘t Landt et al. (2009). In the strong factorial invariance model, the factor loadings and thresholds were constrained to be equal over the 12 groups. The residual variances and means and variances of the latent factor were then estimated. In the strict factorial invariance model, the residual variances were also constrained to be equal over groups. This model tested if for all groups the same proportion of variance is explained by the latent factor. In an additional model, sex differences at the latent factor level were evaluated by constraining the factor means for males and females to be equal. Absence of sex differences at the latent factor level is not a requirement for measurement invariance, but makes the interpretation of results from the cross-sectional and longitudinal analyses
