Implementation Aspects in Invariance Alignment

Abstract

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.

New Developments in Measurement Invariance Testing - An Overview and Comparison of EFA-based Approaches

“…Alignment aims at enabling a comparison of latent means across groups when full MI is not supported; that is, when there are some small differences in parameters across groups (Asparouhov & Muthén, 2014, 2023. This is done by first estimating a configural model, that is, a model where all parameters are estimated group-specifically (this corresponds to the model in equation ( 1)).…”

“…As a consequence, the alignment does not change the model fit when searching for optimal values of the factor means and variances. To resolve this unidentifiability and to arrive at the optimal (i.e., "most invariant") values, an alignment function F is minimized with respect to the factor means and variances (Asparouhov & Muthén, 2014, 2023:…”

“…It is chosen to be √ x 2 + ϵ, where ϵ is a small number, e.g., 0.001. This function is approximately the same as |x| with ϵ being added to ensure continuous differentiability (Asparouhov & Muthén, 2014;Robitzsch, 2023). Equation ( 5) is minimized when the majority of loadings and intercepts are invariant, and only a small number of parameters are (largely) non-invariant (i.e., the number of non-invariant parameters is minimized).…”

“…If one is willing to make this assumption, AESEM produces a model with a clear interpretation about (non-)invariance. Researchers are provided with approximate latent means which can be used to compare groups even if exact MI is not supported (Asparouhov & Muthén, 2014, 2023. This makes AESEM a powerful follow-up method for the other methods presented here and elsewhere because it provides a way of handling non-invariant measurement models.…”

Examining Differences of Invariance Alignment in the Mplus Software and the R Package Sirt

“…Nowadays, the IA method is frequently applied in social sciences for analyzing questionnaire data [9][10][11][12]. Unfortunately, most methodological developments of IA (but see [13][14][15] for exceptions) are strongly coupled to the popular but commercial (and closedsource) Mplus software [16]. Previous simulation studies for one-dimensional factor models investigated the case of continuous items [5,8,[17][18][19], dichotomous items [20,21], and polytomous items [14,22].…”