Abstract: In 2023, Asparouhov and Muthén proposed a new framework for structural equation modeling called the penalized structural equation modeling (PSEM). The penalized alignment method exemplifies the utilization of PSEM within the field of measurement invariance testing. The penalized alignment method inherits the benefits of multiple-group exploratory factor analysis, such as estimating cross-loadings, and the benefits of alignment optimization method which uses a component loss function allowing for a certain amount of noninvariant parameters to exist in the model. It also adopts advantages from Bayesian structural equation modeling, such as setting alignment prior distributions for model parameters and testing approximate measurement invariance.
At the same time, this method overcomes limitations of multiple-group exploratory factor analysis, including the need for strong invariance, and low tolerance for errors when using targeted rotation methods. It also addresses shortcomings of the alignment optimization method, such as the ease of mis-specifications when setting reference groups for latent factor means in multiple-factor models and its inability to be applied to multiple-indicators multiple-causes (MIMIC) models. Additionally, it resolves issues with Bayesian structural equation modeling related to measurement invariance testing, such as the requirement for parameter priors to follow a normal distribution, the need for sensitivity analysis, and slower computations. Due to these integrated features, the penalized alignment method has clear advantages over traditional measurement invariance testing methods and holds great promise for future applications.
To illustrate the application of penalized alignment method, a study on work values among college students is employed as an example to showcase the utilization of the penalized alignment method for conducting measurement invariance testing and multiple-group analysis. Multiple-group CFA, CFA-based penalized alignment, and ESEM-based penalized alignment models are used to fit the data.
The results show that the analysis using the CFA-based penalized alignment method outperforms that of traditional multiple-group CFA. Firstly, the CFA-based penalized alignment model holds weak measurement invariance, whereas the traditional multiple-group CFA using WLSMV estimation rejects the strong invariance model, and the weak invariance model cannot be identified. This leads to a significant discrepancy between the multiple-group CFA model and the real data. Secondly, the penalized alignment method not only yields approximate measurement invariance diagnosis for all model parameters but also estimates the latent factor mean parameters, allowing for direct multiple-group comparisons. In contrast, the chi-square likelihood-ratio test in traditional multiple-group CFA rejects the strong invariance assumption, precluding direct multiple-group comparisons of latent factor means.
The results also indicate that the CFA-based penalized alignment method and ESEM-based penalized alignment method are generally consistent, but the ESEM-based penalized alignment method appears to better reflect the real data. First, the ranking of latent factor means provided by the two models across the four institutional type groups is identical for both latent factors, but the absolute differences in the latent factor means among groups vary, and there are slight differences in the significance of pairwise Z-tests. Second, since the two models are not nested, the CFA-based penalized alignment model has a chi-square value of 1403.827 with 32 degrees of freedom, while the ESEM-based penalized alignment model has a chi-square value of 1219.664 with 16 degrees of freedom. The large difference in chi-square values suggests that the ESEM-based penalized alignment method may perform better. Third, regarding the results of approximate measurement invariance tests, the CFA-based penalized alignment model inherently ignores the estimation of cross-loadings, whereas the approximate measurement invariance test results from the ESEM-based penalized alignment method indicate that three of these ignored cross-loadings do not satisfy approximate measurement invariance. Estimating cross-loadings with ESEM-based penalized alignment model can better reflect the real data.
In summary, for the data from the work values scale of university students in this research example, using the ESEM-based penalized alignment model should be the better choice.

Key words: measurement invariance testing, penalized alignment, Bayesian structural equation modeling, exploratory structural equation model, alignment loss function prior