Table 5.

Comparison of the classical (exploratory) factor analysis, confirmatory factor analysis, multi-group confirmatory factor analysis and multi-group confirmatory factor analysis with alignment

	Classical (exploratory) factor analysis/confirmatory factor analysisa	Multi-group confirmatory factor analysis (MG-CFA)	Multi-group confirmatory factor analysis with alignment
Application	Only one population/group is analysed; To test whether measures of a construct are consistent with a researcher’s understanding of the nature of that construct; in other words, to test whether the data fit a hypothesized measurement model that is based on a theory and/or previous research	A few populations/groups are analysed; To test whether measures of a construct are consistent with a researcher’s understanding; To measure the construct in different populations	Multiple populations are analysed (<100); To test whether measures of a construct are consistent with a researcher’s understanding; To measure the construct in different populations
Model specification	$X_{ij}={\mathit{\tau }}_j+{\mathit{\lambda }}_j{\mathit{\eta }}_i+{\mathit{\epsilon }}_{ij}$ (1), where $X_{ij}$-observed scored X of variable j for person i; ${\mathit{\tau }}_j$-intercept of variable j; ${\mathit{\eta }}_i$-latent factor score of person i; ${\mathit{\lambda }}_j$-factor loading corresponding to variable j; ${\mathit{\epsilon }}_{ij}$-unique factor score or residual	$X_{ijg}={\mathit{\tau }}_{jg}+{\mathit{\lambda }}_{jg}{\mathit{\eta }}_{ig}+{\mathit{\epsilon }}_{ijg}$ (2), where $X_{ijg}$-observed scored X of variable j for person i in group g; ${\mathit{\tau }}_j$- intercept of variable j; ${\mathit{\eta }}_i$-latent factor score of person i; ${\mathit{\lambda }}_j$- factor loading corresponding to variable j; ${\mathit{\epsilon }}_{ij}$-unique factor score or residual	The same as Eq. (2) The same as for the MG-CFA
Interpretation	For each statement j, the Eq. (1) is to predict the item score X for an individual i as a linear combination of the statement-specific intercept ${\mathit{\tau }}_j$, the factor score ${\mathit{\eta }}_i$ times the statement-specific factor loading ${\mathit{\lambda }}_j$ and a random error component ${\mathit{\epsilon }}_{ij}$; The statement-specific intercept ${\mathit{\tau }}_j$ relates to the score of the statement j when the factor score of the latent variable $\mathit{\eta }$ equals 0. The statement-specific factor loading ${\mathit{\lambda }}_j$ indicates the change in the score of the statement j when the score of the latent variable $\mathit{\eta }$ of an individual i is changed by one unit	For each statement j, the Eq. (2) predicts the item score X for an individual i from the group g as a linear combination of the statement-specific intercept ${\mathit{\tau }}_{jg}$, the factor score ${\mathit{\eta }}_{ig}$ times the factor loading ${\mathit{\lambda }}_{jg}$ and a random error component ${\mathit{\epsilon }}_{ijg}$; The subscript g in the formula indicates that the parameters $\mathit{\tau }$ and $\mathit{\lambda }$ are allowed to vary across groups, i.e. depend on the group membership of the individual; In the MG-CFA the mean and variance of the latent variable $\mathit{\eta }$ are estimated and differ depending on the group membership	The same as for the MG-CFA
Measurement invariance issue	Because score of each statement is a function of only a latent factor $\mathit{\eta }$ and not of a group membership, it is silently assumed that the underlying construct is comparable across groups and it can be measured comparably across groups, i.e. it is characterised by full configural, weak and strong measurement invariance	Because score of each statement is a function of both a latent factor $\mathit{\eta }$ and a group membership in order to compare the construct across groups the measurement invariance feature of the model is needed; In order to ensure  1) configural measurement invariance, it is necessary to obtain well-fitted MG-CFA model with unconstrained factor loadings and intercepts; unfortunately in such a model factor, means and variances are not identified and are typically set to 0 and 1, respectively  2) weak measurement invariance, it is necessary to verify the condition ${\mathit{\lambda }}_1={\mathit{\lambda }}_2=\dots ={\mathit{\lambda }}_g$  3) strong measurement invariance, it is necessary to verify two conditions ${\mathit{\lambda }}_1={\mathit{\lambda }}_2=\dots ={\mathit{\lambda }}_g$ and ${\mathit{\tau }}_1={\mathit{\tau }}_2=\dots ={\mathit{\tau }}_g$	The alignment process aims to minimize the amount of non-invariance by  1) estimation of the configural model (i.e. model with the same number of factors and same pattern of zero factor loadings in all groups) in which loadings and intercepts are free across the groups, factor means are fixed at 0 and factor variances are fixed at 1;  2) alignment optimisation where the estimation of the model is conducted under the assumption that the number of measurement non-invariance parameters is as small as possible;  3) adjustment of the factor means and variances in line with the optimal alignment. The relationship between models in points 1 and 2 is similar to the relationship we observe in factor analysis between unrotated and rotated (which simplifies the loading matrix) solutions; specifically, the alignment aims at finding a solution with a few large non-invariant parameters and many invariant parameters rather than many medium-sized non-invariant parameters. In Mplus, two alignment optimisations are available: fixed and free. The fixed option assumes that the mean of the latent factor in the first analysed group is equal to 0. The free option, instead, assumes that all means of the latent factor are estimated freely. Whenever more than two groups are compared, free alignment provides more accurate estimates. However, in practical, applications often appear that free alignment may be poorly identified (warning obtained automatically in Mplus) and the application of the fixed alignment (with the mean of the latent factor in the group where it is the smallest fixed as 0) is suggested. Estimated model has the same fit as the configural model
Fit measures	CFI, TLI, RMSEA	CFI, TLI, RMSEA	CFI, TLI, RMSEA to assess the configural model
Software	Classical (exploratory) factor analysis: SPSS, Statistica, SAS, Mplus (the only one where the fit statistics are available) Confirmatory factor analysis: LISREL, EQS, SPSS-Amos, Statistica, SAS, Stata, Mplus	LISREL, EQS, SPSS-Amos, Stata, Mplus	Mplus

Note Based on the information presented by Muthen and Asparouhov (2013); Asparouhov and Muthén (2014) and Segeritz and Pant (2013)

^a Classical (exploratory) factor analysis and confirmatory factor analysis are perfect counterparts only for a one-factor model