Multivariate Behavioral Research[JOURNAL]
Sun
200 papers
Unique Contributions of Dynamic Affect Indicators - Beyond Static Variability.
Koslowski K, Holtmann J
Multivariate Behav Res · 2025 · PMID 40891493 · Publisher ↗
Indicators of affect dynamics (IADs) capture temporal dependencies and instability in affective trajectories over time. However, the relevance of IADs for the prediction of time-invariant outcomes (e.g., depressive sympt...
Indicators of affect dynamics (IADs) capture temporal dependencies and instability in affective trajectories over time. However, the relevance of IADs for the prediction of time-invariant outcomes (e.g., depressive symptoms) was recently challenged due to results suggesting low predictive utility beyond intraindividual means and variances. We argue that these results may in part be explained by mathematical redundancies between IADs and static variability as well as the chosen modeling strategy. In three extensive simulation studies we investigate the accuracy and power for detecting non-null relations between IADs and an outcome variable in different relevant settings, illustrating the effect of the length of a time series, the presence of missing values or measurement error, as well as of erroneously fixing innovation variances to be equal across persons. We show that, if uncertainty in individual IAD estimates is not accounted for, relations between IADs (i.e., autoregressive effects) and a time-invariant outcome are underestimated even in large samples and propose the use of a latent multilevel one-step approach. In an empirical application we illustrate that the different modeling approaches can lead to different substantive conclusions regarding the role of negative affect inertia in the prediction of depressive symptoms.
How to Get MAD: Generating Uniformly Sampled Correlation Matrices with a Fixed Mean Absolute Discrepancy.
Waller NG
Multivariate Behav Res · 2025 · PMID 40755439 · Publisher ↗
This article describes a simple and fast algorithm for generating uniformly sampled correlation matrices () with a fixed mean absolute discrepancy (MAD) relative to a target (population) The algorithm can be profitably...
This article describes a simple and fast algorithm for generating uniformly sampled correlation matrices () with a fixed mean absolute discrepancy (MAD) relative to a target (population) The algorithm can be profitably used in many settings including model robustness studies and stress testing of investment portfolios, or in dynamic model-fit analyses to generate matrices with a known degree of model-approximation error (as operationalized by the MAD). Using results from higher-dimensional geometry, I show that matrices with a fixed MAD lie in the intersection of two sets that represent: (a) an elliptope and (b) the surface of a cross-polytope. When = 3, these sets can be visualized as an elliptical tetrahedron and the surface of an octahedron. An online supplement includes code for implementing the algorithm and for reproducing all of the results in the article.
How to Capture Synchronization in Triads in One Single Measure: Development of the Measure and an Associated Significance Test.
Wang S, Carlier C, Verhees MWFT … +1 more , Ceulemans E
Multivariate Behav Res · 2025 · PMID 40613655 · Publisher ↗
Interpersonal synchronization is a concept often studied in psychology. Whereas most research focuses on dyads, triadic systems such as family triads warrant increased attention. A crucial challenge in taking a triadic v...
Interpersonal synchronization is a concept often studied in psychology. Whereas most research focuses on dyads, triadic systems such as family triads warrant increased attention. A crucial challenge in taking a triadic view on synchronization is how to quantify it, since a statistical measure that captures the level of triadic synchronization in one value, while discarding dyadic synchronization only, is lacking so far. The current paper therefore investigated three existing measures that show potential to capture triadic synchronization and proposes two novel ones. We also present a significance test that allows to investigate whether the observed triadic synchronization in a triad is stronger than can be expected by chance, while accounting for potential auto-dependence in the data. By means of a simulation study, we tested (1) how the measures react to different potential synchronization patterns; (2) the Type I error rate and the power of the significance test. The results showed that only one measure, i.e., the newly proposed adapted multiplication of pairwise correlations (), can effectively capture triadic synchronization, while discarding dyadic synchronization. We then applied the measure to intensive longitudinal data on attachment-related measures in families, showing that can detect meaningful triadic synchronization in empirical data.
Empirical Bayes Priors for MCMC Estimation of the Multivariate Social Relations Model.
Bhangale AM, Jorgensen TD
Multivariate Behav Res · 2025 · PMID 40600284 · Publisher ↗
The social relations model (SRM) is a linear random-effects model applied to examine dyadic round-robin data within social networks. Such data have a unique multilevel structure in that dyads are cross-classified within...
The social relations model (SRM) is a linear random-effects model applied to examine dyadic round-robin data within social networks. Such data have a unique multilevel structure in that dyads are cross-classified within individuals who may be nested within different social networks. The SRM decomposes perceptual or behavioral measures into multiple components: case-level random effects (in-coming and out-going effects) and dyad-level residuals (relationship effects), the associations among which are often of substantive interest. Multivariate SRM analyses are increasingly common, requiring more sophisticated estimation algorithms. This article evaluates Markov chain Monte Carlo (MCMC) estimation of multivariate-SRM parameters, compares MCMC to maximum-likelihood estimation, and introduces two methods to reduce bias in MCMC point estimates using empirical-Bayes priors. Four simulation studies are presented, two of which reveal dependency of small-group results on priors by manipulating location and precision hyperparameters, respectively. The third simulation study explores the impact of sampling more small groups on prior sensitivity. The fourth simulation study explores how Bayesian model averaging might compensate for underestimated variance due to empirical-Bayes priors. Finally, recommendations for future research are made and extensions of the SRM are discussed.
Disaggregating Associations of Between-Person Differences in Change over Time from Within-Person Associations in Longitudinal Data.
Hoffman L
Multivariate Behav Res · 2025 · PMID 40544333 · Publisher ↗
Longitudinal designs afford the opportunity to examine the many different ways in which variables can be related over time, which can be both a blessing and a curse. Much has been written about the need to distinguish be...
Longitudinal designs afford the opportunity to examine the many different ways in which variables can be related over time, which can be both a blessing and a curse. Much has been written about the need to distinguish between-person relations of individual mean differences from within-person relations of time-specific residuals for time-varying predictors. The present work expands on this topic by describing the need to further distinguish between-person relations among individual slopes for change over time. Using simulation methods, this problem is demonstrated within univariate longitudinal models (i.e., multilevel or mixed-effects models using observed predictors), as well as in multivariate longitudinal models (i.e., structural equation models using latent predictors). The discussion presents recommendations for practice, along with caveats and concerns regarding related longitudinal models for lead-lag effects.
How to Estimate Intraclass Correlation Coefficients for Interrater Reliability from Planned Incomplete Data.
Ten Hove D, Jorgensen TD, van der Ark LA
Multivariate Behav Res · 2025 · PMID 40524384 · Publisher ↗
The interrater reliability (IRR) of observational data is often estimated by means of intraclass correlation coefficients (ICCs), which are flexible IRR estimators that are based on the variance decomposition of scores o...
The interrater reliability (IRR) of observational data is often estimated by means of intraclass correlation coefficients (ICCs), which are flexible IRR estimators that are based on the variance decomposition of scores obtained by observations. ICCs are typically estimated using mean squares from an ANOVA model, the computation of which is not straightforward for incomplete data. However, many studies in behavioral research use planned missing observational designs, in which the raters partially vary across subjects. Planned missing designs result in incomplete data. Therefore, we simulated planned incomplete data and compared the computational accuracy (bias of point estimates, bias of variability estimates, root mean squared error, and coverage rates) and computational feasibility (convergence rates and estimation time) of three recently proposed estimation methods for ICCs: Markov chain Monte Carlo estimation of Bayesian hierarchical linear models, maximum likelihood estimation of random-effects models, and maximum likelihood estimation of common-factor models. Maximum likelihood estimation of random-effects models with Monte-Carlo confidence intervals is preferred based on all criteria. This article is accompanied by R code, which enables researchers to apply these estimation methods. A demonstration of the R code to a real-data set from an educational context is provided.
Estimating IRT Models Under Gaussian Mixture Modeling of Latent Traits: An Application of MSAEM Algorithm.
Cheng S, Meng X
Multivariate Behav Res · 2025 · PMID 40483708 · Publisher ↗
The assumption of a normal distribution for latent traits is a common practice in item response theory (IRT) models. Numerous studies have demonstrated that this assumption is often inadequate, impacting the accuracy of...
The assumption of a normal distribution for latent traits is a common practice in item response theory (IRT) models. Numerous studies have demonstrated that this assumption is often inadequate, impacting the accuracy of statistical inferences in IRT models. To mitigate this issue, Gaussian mixture modeling (GMM) for latent traits, known as GMM-IRT, has been proposed. Moreover, the GMM-IRT models can also serve as powerful tools for exploring the heterogeneity of latent traits. However, the computation of GMM-IRT model estimation encounters several challenges, impeding its widespread application. The purpose of this paper is to propose a reliable and robust computing method for GMM-IRT model estimation. Specifically, we develop a mixed stochastic approximation EM (MSAEM) algorithm for estimating the three-parameter normal ogive model with GMM for latent traits (GMM-3PNO). Crucially, the GMM-3PNO is augmented to be a complete data model within the exponential family, thereby substantially streamlining the computation of the MSAEM algorithm. Furthermore, the MSAEM algorithm adeptly avoid the label-switching issue, ensuring its convergence. Finally, simulation and empirical studies are conducted to validate the performance of the MSAEM algorithm and demonstrate the superiority of the GMM-IRT models.
Rnest: An R Package for the Next Eigenvalue Sufficiency Test for Factor Analysis.
Caron PO
Multivariate Behav Res · 2025 · PMID 40483704 · Publisher ↗
To address the challenge of determining the number of factors to retain in exploratory factor analysis, a plethora of techniques, called stopping rules, has been developed, compared and widely used among researchers. Des...
To address the challenge of determining the number of factors to retain in exploratory factor analysis, a plethora of techniques, called stopping rules, has been developed, compared and widely used among researchers. Despite no definitive solution to this key issue, the recent Next Eigenvalue Sequence Test (NEST) showed interesting properties, such as being theoretically grounded in the factor analysis framework, robustness to cross loadings, a low false positive rate, sensitive to small but true factors, and better accuracy and unbiased compared to traditional stopping rules. Despite these strengths, there is no existing software readily available for researcher. These considerations have led to the development of the R package Rnest. This paper introduces NEST, presents the functionality of the Rnest package, and illustrates its workflow using a reproducible data example. By providing a practical and reliable approach to factor retention, this package aims to encourage its widespread adoption among practitioners, psychometricians, and methodological researchers conducting exploratory factor analyses.
Causal Inference with Unobserved Confounding: Leveraging Negative Control Outcomes Using Lavaan.
Loh WW
Multivariate Behav Res · 2025 · PMID 40476472 · Publisher ↗
Causal conclusions about non-randomized treatments rest on the absence of unobserved confounding. While often made in practice, this fundamental yet empirically untestable assumption can rarely be definitively justified....
Causal conclusions about non-randomized treatments rest on the absence of unobserved confounding. While often made in practice, this fundamental yet empirically untestable assumption can rarely be definitively justified. In most realistic settings, the threat of unobserved confounding lurks. Can causal effects be unbiasedly estimated when unobserved confounding is present? In this tutorial, we introduce an approach from the causal inference and epidemiological literature that permits doing so: negative control outcomes. We explain what a negative control outcome is and how to leverage it to counteract bias due to unobserved confounding. Estimation using a negative control outcome is carried out using the Control Outcome Calibration Approach (COCA). To encourage the adoption of COCA in practice, we implement COCA using lavaan, a popular and free statistical modeling software in R. We illustrate COCA using two publicly available real-world datasets. COCA is practically elegant, straightforward to implement, and under certain assumptions about the potential outcomes, able to unbiasedly estimate causal effects even when unobserved confounding is present.
Ruling out Latent Time-Varying Confounders in Two-Variable Multi-Wave Studies.
Kenny DA, McCoach DB
Multivariate Behav Res · 2025 · PMID 40459159 · Publisher ↗
There has been considerable interest in estimating causal cross-lagged effects in two-variable, multi-wave designs. However, there does not currently exist a strategy for ruling out unmeasured time-varying covariates tha...
There has been considerable interest in estimating causal cross-lagged effects in two-variable, multi-wave designs. However, there does not currently exist a strategy for ruling out unmeasured time-varying covariates that may act as confounders. In this paper, we propose a new strategy for testing whether an unmeasured time-varying covariate explains all covariation between the two "causal" variables in the data. That model, called the (LTVC) model, can be tested with observations for two variables assessed across three or more measurement waves. If the LTVC model fits well, then a time-varying covariate can explain the covariance structure, which undermines the plausibility of causal cross-lagged effects. Although the LTVC model tends to be underpowered when causal cross-lagged effects are small, if testable stationarity constraints on the LTVC model are imposed, adequate power can be achieved. We illustrate the LTVC approach with three examples from the literature. Additionally, we introduce the LTVC-CLPM model, which is identified given strong stationarity constraints. Also considered are multivariate and multi-factor models, the inclusion of measured time-invariant covariates in model, measurement of the stability of the LTVC, and the lag-lead model. These methods allow researchers to probe the assumption that an unmeasured time-varying confounder is the source of all the covariation. Our methods help researchers to rule out certain forms of confounding in two-variable, multi-wave designs.
Modeling Cycles, Trends and Time-Varying Effects in Dynamic Structural Equation Models with Regression Splines.
Sørensen Ø, McCormick EM
Multivariate Behav Res · 2025 · PMID 40454572 · Publisher ↗
Intensive longitudinal data with a large number of timepoints per individual are becoming increasingly common. Such data allow going beyond the classical growth model situation and studying population effects and individ...
Intensive longitudinal data with a large number of timepoints per individual are becoming increasingly common. Such data allow going beyond the classical growth model situation and studying population effects and individual variability not only in trends over time but also in autoregressive effects, cross-lagged effects, and the noise term. Dynamic structural equation models (DSEMs) have become very popular for analyzing intensive longitudinal data. However, when the data contain trends, cycles, or time-varying predictors which have nonlinear effects on the outcome, DSEMs require the practitioner to specify the correct parametric form of the effects, which may be challenging in practice. In this paper, we show how to alleviate this issue by introducing regression splines which are able to flexibly learn the underlying function shapes. Our main contribution is thus a building block to the DSEM modeler's toolkit, and we discuss smoothing priors and hierarchical smooth terms using the special cases of two-level lag-1 autoregressive and vector autoregressive models as examples. We illustrate in simulation studies how ignoring nonlinear trends may lead to biased parameter estimates, and then show how to use the proposed framework to model weekly cycles and long-term trends in diary data on alcohol consumption and perceived stress.
Ideal Point or Dominance Process? Unfolding Tree Approaches to Likert Scale Data with Multi-Process Models.
Zeng B, Wen H, Jeon M
Multivariate Behav Res · 2025 · PMID 40420796 · Publisher ↗
This study introduces a new multi-process analytical framework based on the ideal point assumption for analyzing Likert scale data with three newly developed Unfolding Tree (UTree) models. Through simulations, we tested...
This study introduces a new multi-process analytical framework based on the ideal point assumption for analyzing Likert scale data with three newly developed Unfolding Tree (UTree) models. Through simulations, we tested the performance of proposed models and existing Item Response Tree (IRTree) models across various conditions. Subsequently, empirical data were utilized to analyze and compare the UTree models relative to IRTree models, exploring respondents' decision-making processes and underlying latent traits. Simulation results showed that fit indices could effectively discern the correct model underlying the data. When the correct model was employed, both IRTree and UTree accurately retrieved item and individual parameters, with the recovery precision improving as the number of items and sample size increased. Conversely, when an incorrect model was utilized, the mis-specified model consistently returned biased results in estimating individual parameters, which was pronounced when the respondents followed an ideal point response process. Empirical findings highlight that respondents' decisions align with the ideal point process rather than the dominance process. The respondents' choices of extreme response options are more driven by target traits than by extreme response style. Furthermore, evidence indicates the presence of two distinct but moderately correlated target traits throughout the different decision stages.
Missing Data Handling via EM and Multiple Imputation in Network Analysis using Glasso and Atan Regularization.
Nehler KJ, Schultze M
Multivariate Behav Res · 2025 · PMID 40415497 · Publisher ↗
The existing literature on missing data handling in psychological network analysis using cross-sectional data is currently limited to likelihood based approaches. In addition, there is a focus on convex regularization, w...
The existing literature on missing data handling in psychological network analysis using cross-sectional data is currently limited to likelihood based approaches. In addition, there is a focus on convex regularization, with the missing handling implemented using different calculations in model selection across various packages. Our work aims to contribute to the literature by implementing a missing data handling approach based on multiple imputation, specifically stacking the imputations, and evaluating it against direct and two-step EM methods. Standardized model selection across the multiple imputation and EM methods is ensured, and the comparative evaluation between the missing handling methods is performed separately for convex regularization (glasso) and nonconvex regularization (atan). Simulated conditions vary network size, number of observations, and amount of missingness. Evaluation criteria encompass edge set recovery, partial correlation bias, and correlation of network statistics. Overall, missing data handling approaches exhibit similar performance under many conditions. Using glasso with EBIC model selection, the two-step EM method performs best overall, closely followed by stacked multiple imputation. For atan regularization using BIC model selection, stacked multiple imputation proves most consistent across all conditions and evaluation criteria.
Bayesian Multilevel Latent Class Profile Analysis: Inference and Estimation for Exploring the Diverse Pathways to Academic Proficiency.
Lee J, McCoach DB, Harel O … +1 more , Chung H
Multivariate Behav Res · 2025 · PMID 40400340 · Publisher ↗
Multilevel latent class profile analysis (MLCPA) is a recently developed technique for understanding latent class dynamics in longitudinal studies; however, conventional maximum likelihood (ML) estimation may face challe...
Multilevel latent class profile analysis (MLCPA) is a recently developed technique for understanding latent class dynamics in longitudinal studies; however, conventional maximum likelihood (ML) estimation may face challenges, particularly with small sample sizes or boundary solutions. As an alternative method, we propose a Bayesian estimation for MLCPA by employing non-informative prior distributions. In addition, we shed light on the underflow problem, which denotes a phenomenon such that the logarithm of the likelihood is negative infinity due to the multilevel structure. We perform extensive numerical studies to compare the behaviors of the MLE and the Bayesian estimates and investigate the accuracies of approximated model selection criteria. The simulation study revealed that Bayesian estimates are preferred to ML estimates when the underlying latent classes are well-separated, while the ML estimates are preferred when the underlying latent classes overlap. Utilizing the Progress Monitoring and Reporting Network data, which includes longitudinal academic performance metrics, our analysis uncovers distinct pathways of latent classes for students, further differentiated by latent groups of schools. These findings shed light on the considerable variations in academic proficiency trajectories and thus may offer new perspectives on academic proficiency patterns, with important implications for policy development and targeted educational interventions.
Three Approaches to Testing for Statistical Suppression.
Muniz FB, MacKinnon DP
Multivariate Behav Res · 2025 · PMID 40395214 · Full text
Suppression effects are important for theoretical and applied research because these effects occur when there is an unexpected increase in an effect when it is adjusted for a third variable. This paper investigates three...
Suppression effects are important for theoretical and applied research because these effects occur when there is an unexpected increase in an effect when it is adjusted for a third variable. This paper investigates three approaches to testing for statistical suppression. The first test was proposed in 1978 and is based on the relationship between the zero-order and semi-partial correlations. The second test comes from a condition that is necessary for suppression proposed in 1997. The third test is an extension of the test for the inconsistent mediated effect. We derive standard errors for the Velicer, and Sharpe and Roberts tests, conduct a statistical simulation study, and apply all three tests to two real data sets and several published correlation matrices. In the simulation study, the test based on inconsistent mediation had the best properties overall. For the data examples, when raw data were available, we constructed bootstrap confidence intervals to assess significance, and for correlations, we compared each test statistic to the normal distribution to assess statistical significance. Each test gave consistent results when applied to the example data sets. Analytical work demonstrated conditions where each test gave conflicting results. The mediation test of suppression based on the sign of the product of the mediated effect and the direct effect had the best overall performance.
Accounting for Measurement Invariance Violations in Careless Responding Detection in Intensive Longitudinal Data: Exploratory vs. Partially Constrained Latent Markov Factor Analysis.
Vogelsmeier LVDE, Jongerling J, Ulitzsch E
Multivariate Behav Res · 2025 · PMID 40326463 · Publisher ↗
Intensive longitudinal data (ILD) collection methods like experience sampling methodology can place significant burdens on participants, potentially resulting in careless responding, such as random responding. Such behav...
Intensive longitudinal data (ILD) collection methods like experience sampling methodology can place significant burdens on participants, potentially resulting in careless responding, such as random responding. Such behavior can undermine the validity of any inferences drawn from the data if not properly identified and addressed. Recently, a confirmatory mixture model (here referred to as fully constrained latent Markov factor analysis, LMFA) has been introduced as a promising solution to detect careless responding in ILD. However, this method relies on the key assumption of measurement invariance of the attentive responses, which is easily violated due to shifts in how participants interpret items. If the assumption is violated, the ability of the fully constrained LMFA to accurately identify careless responding is compromised. In this study, we evaluated two more flexible variants of LMFA-fully exploratory LMFA and partially constrained LMFA-to distinguish between careless and attentive responding in the presence of non-invariant attentive responses. Simulation results indicated that the fully exploratory LMFA model is an effective tool for reliably detecting and interpreting different types of careless responding while accounting for violations of measurement invariance. Conversely, the partially constrained model struggled to accurately detect careless responses. We end by discussing potential reasons for this.
Estimating Latent State-Trait Models for Experience-Sampling Data in R with the Package: A Tutorial.
Norget J, Weiss A, Mayer A
Multivariate Behav Res · 2025 · PMID 40277228 · Publisher ↗
As the popularity of the experience-sampling methodology rises, there is a growing need for suitable analytical procedures. These studies often aim to separate fleeting situation-specific influences from more enduring on...
As the popularity of the experience-sampling methodology rises, there is a growing need for suitable analytical procedures. These studies often aim to separate fleeting situation-specific influences from more enduring ones. Latent state-trait (LST) models can make this differentiation. This tutorial discusses multiple-indicator wide-format LST models suitable for experience-sampling data. We outline second-order and first-order model specifications, their advantages and disadvantages, and make the assumptions of first-order specifications explicit. These LST models are very flexible, allowing for various different models and for testing invariance assumptions. However, their specification is tedious and error-prone. This tutorial introduces a new user-friendly browser app and R-function for experience sampling models in the R-package . Extending on existing models, the software also allows to add covariates, which can further explain the stable components. Throughout the tutorial, we answer exemplary research questions about well-being in everyday life with data from a five-day experience-sampling study. An autoregressive model with indicator-specific traits was most appropriate for the data and revealed relatively high consistency, implying that well-being depends more strongly on the person than the current situation. Of the Big Five, extraversion, emotional stability and agreeableness are predictive of trait well-being. We conclude with recommendations about model fit and comparisons.
Gaussian distributional structural equation models: A framework for modeling latent heteroscedasticity.
Fazio L, Bürkner PC
Multivariate Behav Res · 2025 · PMID 40241475 · Publisher ↗
Accounting for the complexity of psychological theories requires methods that can predict not only changes in the means of latent variables - such as personality factors, creativity, or intelligence - but also changes in...
Accounting for the complexity of psychological theories requires methods that can predict not only changes in the means of latent variables - such as personality factors, creativity, or intelligence - but also changes in their variances. Structural equation modeling (SEM) is the framework of choice for analyzing complex relationships among latent variables, but the modeling of latent variances as a function of other latent variables is a task that current methods only support to a limited extent. In this article, we develop a Bayesian framework for Gaussian distributional SEM, which broadens the scope of feasible models for latent heteroscedasticity. We use statistical simulation to validate our framework across four distinct model structures, in which we demonstrate that reliable statistical inferences can be achieved and that computation can be performed with sufficient efficiency for practical everyday use. We illustrate our framework's applicability in a real-world case study that addresses a substantive hypothesis from personality psychology.
Measurement invariance and confirmatory measurement modeling of a psychological flexibility questionnaire across Likert and Expanded response formats.
Hsu T, Hoffman L, Thomas EBK
Multivariate Behav Res · 2025 · PMID 40236375 · Full text
Abstract loading — click title to view on PubMed.
Regularized Variational Bayesian Approximations for Variable Selection in Extended Multiple-Indicators Multiple-Causes Models.
Jin Y, Chen J
Multivariate Behav Res · 2025 · PMID 40208625 · Publisher ↗
Variable selection in structural equation modeling has merged as a new concern in social and psychological studies. Researchers often aim to strike a balance between achieving predictive accuracy and fostering parsimonio...
Variable selection in structural equation modeling has merged as a new concern in social and psychological studies. Researchers often aim to strike a balance between achieving predictive accuracy and fostering parsimonious explanations by identifying the most informative variables. While recent developments in Bayesian regularization methods offer promising solutions to promote model sparsity with much fewer "active" variables, their computational burden due to reliance on the Markov chain Monte Carlo technique limits practical utility. In response, this study proposes a variational Bayesian expectation-maximum algorithm (VBEM) for variable selection to extend the multiple-indicators multiple-causes (MIMIC) model. On the basis of traditional MIMIC models, a partially confirmatory framework that operates within the exploratory-confirmatory continuum is introduced, allowing for the flexible incorporation of substantive knowledge and regularization into both measurement and structural parts while accounting for factor correlation. The proposed method demonstrated its flexibility, reliability, and efficiency on both simulated and real data.
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