An extension of the invariance alignment for scale linking

We examine the extension of the invariance alignment (IA) method originally proposed by Asparouhov and Muthén (2014). The generalized form of a loss function for the IA is discussed, and different forms of the loss function are evaluated using Monte Carlo studies and an empirical example using European Social Survey Data. We compare results obtained by the Mplus software (Muthén & Muthén, 1998-2017) with the R package sirt (Robitzsch, 2019). It is shown that different forms of loss functions that are implemented in the sirt package differ in their performance according to the recovery of group means. This suggests that the performance of IA heavily depends on the form of the loss functions, type of the data (mostly sample size), and type of invariance that could be encountered. The results show that the loss function proposed by Asparouhov and Muthén (2014) might not be optimal in all situations.