This talk concerns measurement invariance analysis for situations with many groups or time points. A BSEM (Bayesian Structural Equation Modeling) approach is proposed for detecting non-invariance that is similar to modification indices with maximum-likelihood estimation, but unlike maximum-likelihood is applicable also for high-dimensional latent variable models for categorical variables. Under certain forms of non-invariance, BSEM gives proper comparisons of factor means and variances using only approximate measurement invariance and without relaxing the invariance specifications or deleting non-invariant items. To ensure correct estimation, a two-step Bayesian analysis procedure is proposed, where step 1 uses BSEM to identify non-invariant parameters and step 2 frees those parameters. An application involves PISA data with binary items measuring math achievement in 40 countries.

The goal of multiple group factor analysis is to study measurement invariance and also group differences in factor means and variances. This is typically carried out using confirmatory factor analysis with equality constraints. A radically different method is proposed which is particularly useful when there are many groups with many non-invariant parameters as is common in cross-cultural studies. The method has three steps. Step is a configural factor analysis with no across-group restrictions. Step 2 makes a rotation according to a simplicity criterion that favors few non-invariant measurement parameters. Step 3 adjusts the factor means and factor variances in line with the rotation. Applications include cross-cultural data from the ESS.