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 noninvariant 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 multiple group structural models. This study compares a traditional sequential search based on multiple-group CFA (MGCFA) to alignment, lasso, elastic net, and ridge regression for estimating the correlation and means between latent variables without pre-specifying anchor items. In the simulation study, we varied the percentage, magnitude, and pattern of noninvariance, sample size, number of indicators, and correlation value and evaluated the bias and efficiency of the methods in terms of the recovery of the factor correlation, means, and item parameters. Results indicated that elastic net led to less biased and more efficient estimates under some conditions, while MGCFA and alignment approaches provided more biased estimates, particularly when the proportion of noninvariance was large and the pattern of noninvariance was unbalanced. We provide recommendations for researchers estimating latent correlations and means under different levels of measurement invariance.