A comparison of regularization, alignment, and a traditional method for estimating structural relationships across multiple groups

Establishing the correct partial measurement invariance model is crucial for ensuring unbiased comparisons of relationships between latent variables across multiple groups. While traditional approaches rely on detecting non-invariant items followed by estimation of structural relationships, more recently, approaches that estimate latent parameters without prior knowledge of anchor items have been developed. Specifically, regularization and alignment are powerful approaches that can be used to estimate various multiple group structural models. This study compares a traditional sequential search based on multiple-group CFA (MGCFA) to alignment and regularization approaches (i.e., lasso, elastic net, and ridge), for estimating the correlation between latent variables across two groups without pre-specifying anchor items. In the simulation study, we varied the percentage and magnitude of non-invariance, sample sizes, number of indicators, and correlation value and evaluated the bias and efficiency of the methods. Results indicated that alignment and Bayesian alignment provided unbiased and efficient estimates across most conditions, while ridge and the traditional MGCFA procedure encountered more difficulty, particularly when the percentage or magnitude of non-invariance was large.