Adjusting for partial invariance in latent parameter estimation: Comparing forward specification search and approximate invariance methods.

Measurement invariance is the condition that an instrument measures a target construct in the same way across subgroups, settings, and time. In psychological measurement, usually only partial, but not full, invariance is achieved, which potentially biases subsequent parameter estimations and statistical inferences. Although existing literature shows that a correctly specified partial invariance model can remove such biases, it ignores the model uncertainty in the specification search step: flagging the wrong items may lead to additional bias and variability in subsequent inferences. On the other hand, several new approaches, including Bayesian approximate invariance and alignment optimization methods, have been proposed; these methods use an approximate invariance model to adjust for partial measurement invariance without the need to directly identify noninvariant items. However, there has been limited research on these methods in situations with a small number of groups. In this paper, we conducted three systematic simulation studies to compare five methods for adjusting partial invariance. While specification search performed reasonably well when the proportion of noninvariant parameters was no more than one-third, alignment optimization overall performed best across conditions in terms of efficiency of parameter estimates, confidence interval coverage, and type I error rates. In addition, the Bayesian version of alignment optimization performed best for estimating latent means and variances in small-sample and low-reliability conditions. We thus recommend the use of the alignment optimization methods for adjusting partial invariance when comparing latent constructs across a few groups.