The MM defines how the factors are measured by the items and MG-CFA is used to compare MMs across groups. Note that we estimate the MM per factor, which implies that we assume the factors to be independent in Step 1. Indicating an individual in Group g (g=1,…,G) by ng and gathering the responses on the Jq items measuring Factor q (q=1,…,Q) in the Vector xng, the MG-CFA model for Factor q is expressed as:

where τg is a Jq-dimensional vector of intercepts for Group g, λg is a Jq-dimensional vector of factor loadings (i.e., item-factor relations) for Group g, ηng denotes the latent variable score for the individual, and ϵng is a Jq-dimensional vector of residuals, with the diagonal of Θg containing the group-specific unique variances of the items. To set the scale of each factor, one can either set its variance to one or use the marker variable approach by fixing one loading (ideally, a strong and invariant loading) to one, for each group. In this paper, we adopt the marker variable approach to ensure that a one-unit change in the underlying factor has the same meaning across groups.

Since small differences in MM parameters are common across many groups and still allow for latent variable comparisons, we apply the assumption of approximate metric invariance (i.e., λg≈λ for all Groups g) instead of exact invariance (i.e., λg=λ) in Step 1 of MixMG-BSEM. This is accomplished by using MG-CFA with Bayesian estimation¹1

The possibility to use a different estimator in each step, such as Bayesian estimation for the MM and maximum likelihood for the SM (see also Zhao et al., 2025a) is an important advantage of the SAM approach.

Since Bayesian estimation can be computationally challenging, two measures are taken to lower the computation time of Step 1 of MixMG-BSEM: (1) the data are centered per group to remove the mean structure (i.e., τg=0 and αg=0), which is irrelevant to the comparison of structural relations, and (2) MG-BCFA is performed for each factor separately, which is in line with the “measurement blocks” approach in SAM (Rosseel & Loh, 2024) with one factor per measurement block. This approach lowers the number of parameters to be estimated and also enhances the model's robustness against MM misspecifications, such as a few unmodeled crossloadings.

In this paper, we assume that all factor loadings, except for the marker variable loadings, are approximately invariant, while the unique variances and factor variances are estimated as group-specific parameters (i.e., with the default non-informative priors per group). Note that it is harmless to specify exactly invariant loadings as approximately invariant since they will then be estimated as nearly identical across groups. Of course, in practice, combinations of exactly and approximately invariant loadings can be applied in Step 1 of MixMG-BSEM. Moreover, in theory, all combinations of exact invariance, approximate invariance and non-invariance can be used, but complex combinations may cause convergence problems.

To determine which parameters are (approximately) invariant or non-invariant, MI testing should be performed prior to using MixMG-BSEM. Note that, if exact invariance does not hold for a parameter, standard MG-CFA requires a tedious process of comparing group-specific parameter estimates to determine whether differences reflect non-invariance or approximate invariance. Instead, MG-BCFA allows to test the tenability of AMI directly by imposing small-variance priors on MM parameters and assessing model fit. Muthén and Asparouhov (2012) recommend starting with a very small variance (e.g., 0.001) and, if needed, the priors’ variances can be increased to reach a good model fit. In this way, MG-BCFA provides information on how large the parameter differences are (i.e., on the level of AMI). Model fit can be assessed using the posterior predictive p value (Gelman et al., 1996), but it is not very sensitive to the prior variances in case of large samples. Other fit measures include the Bayesian RMSEA (BRMSEA; Hoofs et al., 2018) and the Deviance Information Criterion (DIC; Spiegelhalter et al., 2002). The DIC balances model fit (i.e., the posterior mean deviance) and complexity (i.e., the effective number of parameters) in Bayesian models, with smaller values indicating a better balance. Regarding the prior selection in MG-BSEM, Kim et al. (2017) found that the DIC often selected models with smaller prior variances when the sample size was small and Pokropek et al. (2020) found that the DIC performed better as sample size increased, and recommended using the DIC with thresholds tailored to different sample sizes.

Once the marker variables and the approximately invariant loadings are confirmed by the MI testing, we obtain the specification of the MG-BCFA model that corresponds to the first step of MixMG-BSEM. In the next step, we need estimates of the factor scores and their uncertainty. To this end, the means and standard deviations of the posterior distributions of the individuals’ factor scores (i.e., estimated latent variable scores) are appended to the data file.

In a single-indicator approach, the factor scores are used as the “observed” proxy (or a single indicator) for the latent variable (Vermunt, 2025). Since factor scores are only estimates of the true latent variable scores, we apply Croon’s correction (2002) to the factor score covariances (covfg) to obtain unbiased estimates of the true latent variable covariances (covηg), here denoted as Φgs2:

where Λg^ corresponds to the Q×Q diagonal matrix of group-specific factor loadings (reflecting the reliability of the factor scores) and Θg^ is the Q×Q diagonal matrix of group-specific unique variances. These estimates correspond to the MM parameters of the single indicators of the factors (i.e., the factor scores) rather than the original, observed indicators. These MM parameters are derived from the posterior mean and standard deviation estimates for the factor scores, obtained from Step 1. For details, please refer to Equations (7–8) in the MixML-SEM paper (Zhao et al., 2025a).

In Step 3, MixMG-BSEM clusters the groups and estimates cluster-specific structural relations. The SM is thus conditional on the cluster membership, zgk, which denotes whether Group g belongs to Cluster k. Whereas the true cluster membership is assumed to be either 1 or 0, its estimation, z^gk, ranges from 0 to 1 and represents the probability of Group g belonging to Cluster k. The model-implied factor covariance matrix Φgk, given that zgk=1, is defined as:

where Βk contains the cluster-specific regression coefficients between latent variables, and Ψgk is the residual factor covariance matrix, which is specified as group-and-cluster-specific to ensure that clustering is driven only by the regressions Βk (for details, see Perez Alonso et al., 2024). The SM is estimated with maximum likelihood estimation using Φgs2 as input.

For the mixture clustering in MixMG-BSEM, it is assumed that the (true) latent variable scores ηng are sampled from a mixture of K multivariate normal distributions. Specifically, all latent variable scores of Group g, Hg, are assumed to be sampled from the same distribution:

where f is the population density function, υ represents the set of population parameters, and πk is the prior probability that Group g belongs to Cluster k. The scores in Hg are assumed to follow a normal distribution with αg as the factor mean (which is zero due to centering) and Φgk as the factor covariance matrix. The unknown parameters υ are estimated by maximizing the following log-likelihood function:

where Φgs2 is the group-specific factor covariance matrix from Step 2 (Equation 2), and Φgk is the group-and-cluster-specific factor covariance matrix from Step 3 (Equation 3). The maximum likelihood estimation is performed using the EM algorithm (Dempster et al., 1977). Specifically, in the E-step, the algorithm estimates the classification probabilities z^gk given the current parameter estimates. In the M-step, the algorithm estimates the unknown parameters 𝜐 given the classification probabilities obtained from the E-step. The E- and M-steps are iterated until convergence. A multi-start procedure is applied to mitigate convergence to local maxima, where the converged solution with the highest loglikelihood across the different starts is selected as the final result. For an in-depth explanation of the technical details of Step 3, readers are referred to Appendix A of the paper by Perez Alonso et al. (2024).

We evaluated the recovery of the group-specific factor loading estimates for each item j, using the Root Mean Squared Error (RMSE) across groups as follows:

where λgj is the true group-specific loading of the j-th item on the factor, and λ^gj is the corresponding estimate. Note that RMSEλj is also affected by the variance of the estimates, rather than just the bias.

When using MixMG-BSEM with the true prior variances for the loadings, the average RMSEλj across the four factors and all simulated data sets was 0.066, 0.091, 0.066, and 0.066, respectively, for the loadings of the second to the fifth item of each factor (Table 1, last row). Note that RMSEλ3 was larger due to the disregarded crossloadings on that item. This was also the only RMSEλj value that differed across the four factors. Specifically, the RMSEλ3 values were 0.093, 0.072, 0.098, and 0.101, for F1 to F4, respectively. It seems that the third loading for F2 is affected by the ignored crossloading the least, which may be explained by the fact that, unlike the other factors, F2 is involved in only one direct regression relation with the other factors⁴4

It has indirect relations with the other factors via the correlation between F1 and F2, but the expected value of this correlation is zero.

To illustrate the effect of the prior variances for the loadings, RMSEλ2 across different prior variances is shown in Figure 4. The diagonal of the plot represents cases where the prior variances were correctly specified, while the lower part shows cases where the priors were narrower than the true level of AMI. In general, applying too narrow or too wide priors resulted in larger RMSEλ2 values. Since the prior variance affected the loading recovery, we also evaluated prior selection using the Deviance Information Criterion (DIC), which balances model fit and complexity. When looking at the prior selection per loading, the correct selection rate was 59.2% across all loadings and all simulated data sets.⁵5

Similar results were found with the widely applicable information criterion (WAIC; Watanabe, 2010) and leave-one-out information criterion (LOOIC; Vehtari et al., 2017): WAIC: 58.2%; LOOIC: 57.6%.

To evaluate how often (Step 3 of) MixMG-BSEM converged to a local maximum, we compared the log-likelihood of the final best solution (out of 50 random starts) to the one obtained when starting from the true clustering, which is a proxy for the global maximum. If log⁡Lηwas more than 0.001 lower than the proxy, the solution was considered a local maximum. Overall, when applying the true priors, MixMG-BSEM ended up in a local maximum for 0.01% of the data sets, with all local maxima occurring in case of Ng=50 with K=4.

The Adjusted Rand Index (ARI; Hubert & Arabie, 1985) measures the similarity between two partitions while correcting for chance, with a value of one indicating perfect agreement and zero indicating the level of agreement between two random partitions. To compute the ARI, the modal clustering (i.e., assigning each group to the cluster with the highest classification probability) was compared to the true clustering. Additionally, the correct clustering rate (%CC) was computed as the percentage of correctly clustered groups.

When using the true priors, the average ARI across all simulated data was 0.644 and the %CC was 84.8%.⁶6

We further examined the distributions of the ARI using boxplots across relevant conditions (see Figure 1 in the Supplementary Material, S3; Zhao et al., 2025b). Under more challenging conditions (e.g., small sample sizes, low reliability, small regression effects), we observed larger variability. For example, with β=0.2 and Ng=50, the distributions were more right-skewed, indicating a majority of low ARI values. In contrast, under easier conditions, such as β=0.4 and Ng =200, the distributions became symmetric or slightly left-skewed, with less variability, indicating more consistent high ARI values. Therefore, we have added a table reporting the medians and MADs for ARI and %CC in the Supplementary Material (S4; Zhao et al., 2025b), to complement the means and SDs reported in the main text.

Across different prior variances, the ARI remained relatively stable. When the applied prior was too narrow or too large, the ARI slightly dropped. For example, for an AMI level of 0.1, it decreased from 0.635 when using the true prior variance to 0.608 when using a prior variance of 0.001. For an AMI level of 0.001, it decreased from 0.651 when using the true prior variance to 0.638 when using a prior variance of 0.1.

To evaluate the recovery of the regression parameters, we computed the MEβ and the RMSEβ per regression parameter (i.e., β1, β2, β3, and β4):

where β^k is the estimated regression coefficient in cluster k and βk is the corresponding true value. To get a better idea of the separate β coefficients in different clusters (i.e., without averaging across clusters), we also reported the bias for β1 in the Supplementary Material as an example (S2; Zhao et al., 2025b), with positive values indicating overestimation in most conditions. Similar trends were observed for β3 and β4. For β2, the bias values were the smallest and close to zero (slightly negative).

On average, MEβ was 0.037, -0.004, 0.039, and 0.031 (Table 3), and RMSEβ was 0.073, 0.054, 0.072 and 0.069 (Table 4) for β1, β2, β3, and β4, respectively.⁷7

We also checked the distributions of MEβ1, which were mostly symmetric (see Figure 2 in the Supplementary Material, S3; Zhao et al., 2025b). Thus, we retained the use of means and SDs and did not include a separate table for the medians and MADs for these outcomes.

We assessed the performance of MixMG-BSEM when the true number of clusters was known. We found that performing 50 random starts in Step 3 largely prevented local maxima. The recovery of clusters and regression parameters was good when the within-group sample size was at least 200 and/or in case of a larger cluster separation (i.e., β=0.4). Ignoring larger crossloadings (by estimating the MM per factor) resulted in more biased estimates for factor loadings and regression parameters, while cluster recovery was less affected. Although prior selection based on the DIC was not always optimal, the recovery of clusters and regression parameters was relatively stable across the priors.