Structural Equation Modeling: A Multidisciplinary Journal

Small sample sizes pose a severe threat to convergence and accuracy of between-group level parameter estimates in multilevel structural equation modeling (SEM). However, in certain situations, such as pilot studies or when populations are inherently small, increasing samples sizes is not feasible. As a remedy, we propose a two-stage regularized estimation approach designed for scenarios with both a small number of groups and small group sizes, and a low ICC. The method employs the wide format approach to multilevel SEM, where, at first, the sample covariance matrix is replaced by a shrinkage estimate, and then, this estimate is used to fit the SEM. By means of a simulation study, we evaluated the effectiveness of our two-stage approach. Our findings demonstrate that this method consistently ensures model convergence, provides more accurate between-level estimates, and even improves accuracy of within-level estimates in cases of very small group sizes.

The bifactor model is becoming a popular tool for modeling hierarchical constructs. However, the bifactor predictive model, which uses both the general factor and all group factors to predict a criterion, often encounters symptoms of empirical under-identification. The augmentation approach can effectively alleviate these issues, but it is not always feasible. There is a need for a more readily available approach. In the present study, we examined the extent to which the Structural After Measurement (SAM; Rosseel & Loh, 2022) approach can mitigate these statistical issues in bifactor predictive models. Monte Carlo simulations showed that, compared to the classic one-step Structural Equation Modeling (SEM) approach, SAM can effectively enhance the statistical performance of bifactor predictive models. This enhancement is evidenced by higher convergence rates, smaller RMSE, more accurate standard error estimates, better coverage rates, and improved control of Type I error rates, albeit at the cost of somewhat higher bias. Additionally, we demonstrated that combining the SAM approach with the augmentation approach resulted in the best performance across all simulated conditions. An empirical illustration was also provided to demonstrate the feasibility of the SAM approach.

Currently, dynamic structural equation modeling (DSEM) and residual DSEM (RDSEM) are commonly used in testing intensive longitudinal data (ILD). Researchers are interested in ILD mediation models, but their analyses are challenging. The present paper mathematically derived, empirically compared, and step-by-step demonstrated three types (i.e., 1-1-1, 2-1-1, and 2-2-1) of intensive longitudinal mediation (ILM) analyses based on DSEM and RDSEM models. Specifically, each ILM model was demonstrated with a simulated example and illustrated with the corresponding annotated Mplus codes. We compared two types of detrending methods in mediation analyses and showed that RDSEM was superior to DSEM because the latter included the time tj variable as a Level 1 predictor. Lastly, we extended ILM analyses based on DSEM and RDSEM to multilevel autoregressive mediation models, cross-classified DSEM, and intensive longitudinal moderated mediation models. KEYWORDS Dynamic structural equation modeling; intensive longitudinal data; mediation effect; moderated mediation model; residual dynamic structural equation modeling Analyses of mediation, particularly in longitudinal designs, are important in causation analyses. Recently, digital data devices (e.g., digital watch monitoring health) have made possible the collection of intensive longitudinal data (ILD) over a huge number of time points. Massive data improves construct ecological validity but is challenging in analyses. The present study mathematically examined and extended the common analytical strategies. Simulated data were then used to compare and demonstrate their proper use. Furthermore, our detailed Mplus programming codes provide step-by-step guidance to applied researchers on how ILD mediation models should be analyzed. 1. Causation and Mediation Analyses In psychological and behavioral studies, mediation analyses are important in investigating causal chains among the independent variable X, the mediator M, and the dependent variable Y (Baron & Kenny, 1986). The mediation effect of X on Y through M is often quantified and denoted by a  b, where a and b are the effects of X on M and the effect of M on Y controlling for X (MacKinnon, 2008), respectively. The part of the effect of X on Y that the mediation effect cannot explain is called the direct effect (denoted by c 0).

We present a bias-adjusted three-step estimation approach for multilevel latent class models (LC) with covariates. The proposed approach involves (1) fitting a single-level measurement model while ignoring the multilevel structure, (2) assigning units to latent classes, and (3) fitting the multilevel model with the covariates while controlling for measurement error introduced in the second step. Simulation studies and an empirical example show that the three-step method is a legitimate modeling option compared to the existing one-step and two-step methods.

This paper aims to advocate for a balanced approach to model fit evaluation in structural equation modeling (SEM). The ongoing debate surrounding chi-square test statistics and fit indices has been characterized by ambiguity and controversy. Despite the acknowledged limitations of relying solely on the chi-square test, its careful application can enhance its effectiveness in evaluating model fit and specification. To illustrate this point, we present three common scenarios relevant to social and behavioral science research using Monte Carlo simulations, where fit indices may inadequately address concerns regarding goodness-of-fit, while the chi-square statistic can offer valuable insights. Our recommendation is to report both the chi-square test and fit indices, prioritizing precise model specification to ensure the reliability of model fit indicators.

A two-level data set can be structured in either long format (LF) or wide format (WF), and both have corresponding SEM approaches for estimating multilevel models. Intuitively, one might expect these approaches to perform similarly. However, the two data formats yield data matrices with different numbers of columns and rows, and their cols:rows is related to the magnitude of eigenvalue bias in sample covariance matrices. Previous studies have shown similar performance for both approaches, but they were limited to settings where cols≪rows in both data formats. We conducted a Monte Carlo study to investigate whether varying cols:rows result in differing performances. Specifically, we examined the p:N (cols:rows) effect on convergence and estimation accuracy in multilevel settings. Our findings suggest that (1) the LF approach is more likely to achieve convergence, but for the models that converged in both, (2) the LF and WF approach yield similar estimation accuracy, which is related to (3) differential cols:rows effects in both approaches, and (4) smaller ICC values lead to less accurate between-group parameter estimates.

Multilevel factor mixture modeling (FMM) is a hybrid of multilevel confirmatory factor analysis (CFA) and multilevel latent class analysis (LCA). It allows researchers to examine population heterogeneity at the within level, between level, or both levels. This tutorial focuses on explicating the model specification of multilevel FMM that considers the conceptualization of multilevel constructs. Empirical data sets are used to demonstrate the applications of multilevel FMM for within-level constructs, between-level constructs, and within- and between-level constructs. Detailed model specifications of integrating latent classes into multilevel constructs are provided. For modeling the heterogeneity at the between level, parametric and nonparametric approaches are compared both conceptually and substantively using demonstration data. The interpretations of results using multilevel FMM are also provided. The tutorial is concluded with a discussion of some important aspects of applying multilevel FMM.

The use of structural equation models for causal inference from panel data is critiqued in the causal inference literature for unnecessarily relying on a large number of parametric assumptions, and alternative methods originating from the potential outcomes framework have been recommended, such as inverse probability weighting (IPW) estimation of marginal structural models (MSMs). To better understand this criticism, we describe three phases of causal research. We explain (differences in) the assumptions that are made throughout these phases for structural equation modeling (SEM) and IPW-MSM approaches using an empirical example. Second, using simulations we compare the finite sample performance of SEM and IPW-MSM for the estimation of time-varying exposure effects on an end-of-study outcome under violations of parametric assumptions. Although increased reliance on parametric assumptions does not always translate to increased bias (even under model misspecification), researchers are still well-advised to acquaint themselves with causal methods from the potential outcomes framework.

We investigated three different approaches for quantifying individual change and reporting it back to persons: (a) the common change score, which is obtained by first computing scale scores from two consecutive measurements and then subtract these scores from one another, (b) the ad-hoc approach, which is similar to the former approach but uses regression scores instead of scale scores, and (c) Kelley’s approach, which computes the change score directly from a regression. Specifically, we compared these approaches with one another with regard to the mean squared error (MSE), a measure of the accuracy of an estimator. Our findings indicated that the ad-hoc approach and Kelley’s approach provide change scores that can be more accurate in terms of the MSE than the common change score and that Kelley’s approach can be more accurate than the ad-hoc approach under certain conditions. Moreover, we present an example and provide a step by step guide to implement the approaches.

GSCA Pro is free, user-friendly software for generalized structured component analysis structural equation modeling (GSCA-SEM), which implements three statistical methods for estimating models with factors only, models with components only, and models with both factors and components. This tutorial aims to provide step-by-step illustrations of how to use the software to estimate such various models after briefly discussing model specification, estimation, and evaluation in GSCA-SEM.

This study examined the false positive (FP) rates and sensitivity of Bayesian fit indices to structural misspecification in Bayesian structural equation modeling. The impact of measurement quality, sample size, model size, the magnitude of misspecified path effect, and the choice or prior on the performance of the fit indices was also investigated. The Bayesian fit indices examined in this study included PPP, DIC, BRMSEA, BCFI, BTLI. The results from the simulation study showed that BRMSEA, BCFI, and BTLI failed to detect structural misspecification. The performance of DIC depended majorly on measurement quality, and the sensitivity of PPP depended on sample size, measurement quality, the magnitude of the omitted path effect, and the choice of prior. Informative prior with too narrow precisions resulted in higher FP rates. Empirical implications for applied researchers and future research directions were discussed.

Generalized structured component analysis (GSCA) is a multivariate method for specifying and examining interrelationships between observed variables and components. Despite its data-analytic flexibility honed over the decade, GSCA always defines every component as a linear function of observed variables, which can be less optimal when observed variables for a component are nonlinearly related, often reducing the component's predictive power. To address this issue, we combine deep learning and GSCA into a single framework to allow a component to be a nonlinear function of observed variables without specifying the exact functional form in advance. This new method, termed deep learning generalized structured component analysis (DL-GSCA), aims to maximize the predictive power of components while their directed or undirected network remains interpretable. Our real and simulated data analyses show that DL-GSCA produces components with greater predictive power than those from GSCA in the presence of nonlinear associations between observed variables per component.

Oblique bifactor models, where group factors are allowed to correlate with one another, are commonly used. However, the lack of research on the statistical properties of oblique bifactor models renders the statistical validity of empirical findings questionable. Therefore, the present study took the first step to examine the statistical properties of oblique bifactor models through Monte Carlo simulations. Study 1 showed that the classic oblique bifactor measurement models had severe convergence issues in many conditions. Even for converged replications, both factor loading and group factor correlation estimates were severely biased. Study 2 further showed that the classic oblique bifactor predictive models still had serious convergence problems, and structural parameters suffered from issues of severe estimation bias and low power. Fortunately, the augmentation approach, where one or multiple indicators are specified to load onto only the general factor but not any of the group factors, was useful in ameliorating these issues in both oblique bifactor measurement and predictive models. Tentative recommendations regarding the selection between oblique and orthogonal bifactor models and the approaches to finding augmenting indicators were provided.

A recent review found that 11% of published factor models are hierarchical with second-order factors. However, dedicated recommendations for evaluating hierarchical model fit have yet to emerge. Traditional benchmarks like RMSEA<0.06 or CFI>0.95 are often consulted, but they were never intended to generalize to hierarchical models. Through simulation, we show that traditional benchmarks perform poorly at identifying misspecification in hierarchical models. This corroborates previous studies showing that traditional benchmarks do not maintain optimal sensitivity to misspecification as model characteristics deviate from those used to derive the benchmarks. Instead, we propose a hierarchical extension to the dynamic fit index (DFI) framework, which automates custom simulations to derive cutoffs with optimal sensitivity for specific model characteristics. In simulations to evaluate performance, results showed that the hierarchical DFI extension routinely exceeded 95% classification accuracy and 90% sensitivity to misspecification whereas traditional benchmarks rarely exceeded 50% classification accuracy and 20% sensitivity.

The recently proposed continuous-time latent curve model with structured residuals (CT-LCM-SR) addresses several challenges associated with longitudinal data analysis in the behavioral sciences. First, it provides information about process trends and dynamics. Second, using the continuous-time framework, the CT-LCM-SR can handle unequally spaced measurement occasions and describes processes independently of the length of the time intervals used in a given study. Third, it is a hierarchical model. Thus, multiple subjects can be analyzed simultaneously. However, subjects might also differ in dynamics and trends. Therefore, in the present paper, we extend the CT-LCM-SR to capture these differences as well. This newly proposed random coefficients continuous-time latent curve model with structured residuals (RC-CT-LCM-SR) is introduced theoretically and technically. Additionally, we provide an illustrative example with data from the Health and Retirement Study (HRS), and we show how the RC-CT-LCM-SR can be used to study multiple sources of between-subject differences over time.

Parallel process latent growth curve mediation models (PP-LGCMMs) are frequently used to longitudinally investigate the mediation effects of treatment on the level and change of outcome through the level and change of mediator. An important but often violated assumption in empirical PP-LGCMM analysis is the absence of omitted confounders of the relationships among treatment, mediator, and outcome. In this study, we analytically examined how omitting pretreatment confounders impacts the inference of mediation from the PP-LGCMM. Using the analytical results, we developed three sensitivity analysis approaches for the PP-LGCMM, including the frequentist, Bayesian, and Monte Carlo approaches. The three approaches help investigate different questions regarding the robustness of mediation results from the PP-LGCMM, and handle the uncertainty in the sensitivity parameters differently. Applications of the three sensitivity analyses are illustrated using a real-data example. A user-friendly Shiny web application is developed to conduct the sensitivity analyses.

Accelerated longitudinal designs (ALD) allow studying developmental processes usually spanning multiple years in a much shorter time framework by including participants from different age cohorts, which are assumed to share the same population parameters. However, different cohorts may have been exposed to dissimilar contextual factors, resulting in different developmental trajectories. If such differences are not accounted for, the generating process will not be adequately characterized. In this paper, we propose a continuous-time latent change score model as an approach to capture cohort differences affecting the speed of maturation of psychological processes in ALDs. This approach fills an important gap in the literature because, until now, no method existed for this goal. Using a Monte-Carlo simulation study, we show that the proposed model detects cohort differences adequately, regardless of their size in the population. Our proposed model can help developmental researchers control for cohort effects in the context of ALDs.

Several interesting variations on the Common Fate Model (CFM) for dyadic analysis have emerged over the past decade. For instance, the multilevel-CFM is characterized by directional paths between Person level observed X and Y variables in addition to the Dyad level direct path between the corresponding latent variables. Although this model appears to provide a decomposition of the Y on X regression into Dyad and Person level components, close examination reveals that this specification yields a Dyad level coefficient that captures the contextual effect, or the discrepancy in the between- and within-cluster coefficients. The present work elucidates key features of the multilevel-CFM that produce the contextual effect parametrization, and introduces an alternative specification, the between-within-CFM that yields a decomposition of the Y on X regression consistent with the general multilevel SEM literature. These variations on the CFM are illustrated using empirical examples, supported by syntax for commonly used SEM software.

The Bivariate Latent Change Score (BLCS) model is a popular framework for the study of dynamics in longitudinal research. Despite its popularity, there is little evidence of the ability of this model to recover latent dynamics when the latent trajectories are affected by stochastic innovations (i.e., dynamic error). The deterministic specification of the BLCS model does not account for the effect of these innovations in the system. In contrast, the stochastic specification of the BLCS model includes parameters that capture the effect of such innovations at the latent level. Through Monte Carlo simulation, we generated two developmental processes and examined the recovery of the parameters in the deterministic and stochastic BLCS models under a broad range of empirically relevant conditions. Based on our findings, we provide specific guidelines and recommendations for the application of BLCS models in developmental research.

In their article, Yuan and Deng argue that a structural parameter under partial least squares structural equation modeling (PLS-SEM) is zero if and only if the same structural parameter is zero under covariance-based structural equation modeling (CB-SEM). Yuan and Deng then conclude that statistical tests on individual structural parameters assessing the null hypothesis of no effect can achieve the same purpose in CB-SEM and PLS-SEM. Our response to their article highlights that the relationship they find between PLS-SEM and CB-SEM structural parameters is not universally valid, and that consequently, tests on individual parameters in CB-SEM and PLS-SEM generally do not fulfill the same purpose.