Implementation aspects in invariance alignment

In social sciences, multiple groups, such as countries, are frequently compared regarding a construct that is assessed using a number of items administered in a questionnaire. The corresponding scale is assessed with a unidimensional factor model involving a latent factor variable. To enable a comparison of the mean and standard deviation of the factor variable across groups, identification constraints on item intercepts and factor loadings must be imposed. Invariance alignment (IA) provides such a group comparison in the presence of partial invariance (i.e., a minority of item intercepts and factor loadings are allowed to differ across groups). IA is a linking procedure that separately fits a factor model in each group in the first step. In the second step, a linking of estimated item intercepts and factor loadings is conducted using a robust loss function 𝐿0.5. The present article discusses implementation alternatives in IA. It compares the default 𝐿0.5 loss function with 𝐿𝑝 with other values of the power p between 0 and 1. Moreover, the nondifferentiable 𝐿𝑝 loss functions are replaced with differentiable approximations in the estimation of IA that depend on a tuning parameter ε (such as, e.g., ε=0.01). The consequences of choosing different values of ε are discussed. Moreover, this article proposes the 𝐿0 loss function with a differentiable approximation for IA. Finally, it is demonstrated that the default linking function in IA introduces bias in estimated means and standard deviations if there is noninvariance in factor loadings. Therefore, an alternative linking function based on logarithmized factor loadings is examined for estimating factor means and standard deviations. The implementation alternatives are compared through three simulation studies. It turned out that the linking function for factor loadings in IA should be replaced by the alternative involving logarithmized factor loadings. Furthermore, the default 𝐿0.5 loss function is inferior to the newly proposed 𝐿0 loss function regarding the bias and root mean square error of factor means and standard deviations.