Causal Effects in Mediation Modeling: An Introduction With Applications to Latent Variables

Causal inference in mediation analysis offers counterfactually based causal definitions of direct and indirect effects, drawing on research by Robins, Greenland, Pearl, VanderWeele, Vansteelandt, Imai, and others. This type of mediation effect estimation is little known and seldom used among analysts using structural equation modeling (SEM). The aim of this article is to describe the new analysis opportunities in a way that is accessible to SEM analysts and show examples of how to perform the analyses. An application is presented with an extension to a latent mediator measured with multiple indicators.     

Monte Carlo Confidence Intervals for Complex Functions of Indirect Effects

One challenge in mediation analysis is to generate a confidence interval (CI) with high coverage and power that maintains a nominal significance level for any well-defined function of indirect and direct effects in the general context of structural equation modeling (SEM). This study discusses a proposed Monte Carlo extension that finds the CIs for any well-defined function of the coefficients of SEM such as the product of k coefficients and the ratio of the contrasts of indirect effects, using the Monte Carlo method. Finally, we conduct a small-scale simulation study to compare CIs produced by the Monte Carlo, nonparametric bootstrap, and asymptotic-delta methods. Based on our simulation study, we recommend researchers use the Monte Carlo method to test a complex function of indirect effects.     

Moderated Mediation Analysis Using Bayesian Methods

Conventionally, moderated mediation analysis is conducted through adding relevant interaction terms into a mediation model of interest. In this study, we illustrate how to conduct moderated mediation analysis by directly modeling the relation between the indirect effect components including a and b and the moderators, to permit easier specification and interpretation of moderated mediation. With this idea, we introduce a general moderated mediation model that can be used to model many different moderated mediation scenarios including the scenarios described in Preacher, Rucker, and Hayes (2007). Then we discuss how to estimate and test the conditional indirect effects and to test whether a mediation effect is moderated using Bayesian approaches. How to implement the estimation in both BUGS and Mplus is also discussed. Performance of Bayesian methods is evaluated and compared to that of frequentist methods including maximum likelihood (ML) with 1st-order and 2nd-order delta method standard errors and mL with bootstrap (percentile or bias-corrected confidence intervals) via a simulation study. The results show that Bayesian methods with diffuse (vague) priors implemented in both BUGS and Mplus yielded unbiased estimates, higher power than the ML methods with delta method standard errors, and the ML method with bootstrap percentile confidence intervals, and comparable power to the ML method with bootstrap bias-corrected confidence intervals. We also illustrate the application of these methods with the real data example used in Preacher et al. (2007). Advantages and limitations of applying Bayesian methods to moderated mediation analysis are also discussed.     

Mediation Analysis in a Latent Growth Curve Modeling Framework

This article presents several longitudinal mediation models in the framework of latent growth curve modeling and provides a detailed account of how such models can be constructed. Logical and statistical challenges that might arise when such analyses are conducted are also discussed. Specifically, we discuss how the initial status (intercept) and change (slope) of the putative mediator variable can be appropriately included in the causal chain between the independent and dependent variables in longitudinal mediation models. We further address whether the slope of the dependent variable should be controlled for the dependent variable's intercept to improve the conceptual relevance of the mediation models. The models proposed are illustrated by analyzing a longitudinal data set. We conclude that for certain research questions in developmental science, a multiple mediation model where the dependent variable's slope is controlled for its intercept can be considered an adequate analytical model. However, such models also show several limitations.     

Maximum Likelihood Dynamic Factor Modeling for Arbitrary N and T Using SEM

This article has 3 objectives that build on each other. First, we demonstrate how to obtain maximum likelihood estimates for dynamic factor models (the direct autoregressive factor score model) with arbitrary T and N by means of structural equation modeling (SEM) and compare the approach to existing methods. Second, we go beyond standard time series analysis (T large and N = 1) and conventional SEM (N large and T = 1 or small) by integrating both approaches. The resulting combined model offers a variety of new modeling options including a direct test of the ergodicity hypothesis, according to which the factorial structure of an individual observed at many time points is identical to the factorial structure of a group of individuals observed at a single point in time. Third, we illustrate the flexibility of SEM time series modeling by extending the approach to account for complex error structures. We end with a discussion of current limitations and future applications of SEM-based time series modeling for arbitrary T and N.     

Multimediation Method With Balanced Repeated Replications For Analysis Of Complex Surveys

We propose a method suitable for analysis of cross-sectional studies with complex sampling and continuous variables. The method consists of R + 4 steps, where R denotes the number of replications. In the first R + 1 step, the main and R replicate weights are used (one at a time) to estimate the product of coefficients for all mediation effects using a structural equation model. In step R + 2, the standard errors of these estimates are computed via balanced repeated replications. In step R + 3, the raw p values corresponding to mediation effects are computed based on the generalized Sobel’s tests. In the final step, R + 4, the p values are adjusted for multiplicity and statistical inferences regarding mediation effects are drawn. To illustrate the approach we examined significance of attitudes toward smoking bans as mediators in the association between smoking restrictions at work and nicotine dependence among male daily smokers.     

Digital ITEMS Module 2: Scale Reliability in Structural Equation Modeling

In this ITEMS module, we frame the topic of scale reliability within a confirmatory factor analysis and structural equation modeling (SEM) context and address some of the limitations of Cronbach's α. This modeling approach has two major advantages: (1) it allows researchers to make explicit the relation between their items and the latent variables representing the constructs those items intend to measure, and (2) it facilitates a more principled and formal practice of scale reliability evaluation. Specifically, we begin the module by discussing key conceptual and statistical foundations of the classical test theory model and then framing it within an SEM context; we do so first with a single item and then expand this approach to a multi‐item scale. This allows us to set the stage for presenting different measurement structures that might underlie a scale and, more importantly, for assessing and comparing those structures formally within the SEM context. We then make explicit the connection between measurement model parameters and different measures of reliability, emphasizing the challenges and benefits of key measures while ultimately endorsing the flexible McDonald's ω over Cronbach's α. We then demonstrate how to estimate key measures in both a commercial software program (Mplus) and three packages within an open‐source environment (R). In closing, we make recommendations for practitioners about best practices in reliability estimation based on the ideas presented in the module.     

Code-switching: L1-coded mediation in a kindergarten foreign language classroom

This paper is based on a qualitative inquiry that investigated the role of teachers' mediation in three different modes of coding in a kindergarten foreign language classroom in China (i.e. L2-coded intralinguistic mediation, L1-coded cross-lingual mediation, and L2-and-L1-mixed mediation). Through an exploratory examination of the varying effects of the three kinds of mediation on activating children's noticing and prior knowledge in learning a designated language game, the study shows that the L2-and-L1-mixed mediation optimises not only the children's exposure to the target L2, but also capitalises on the efficiency and effectiveness of mediation. The study also calls attention to the dynamic, interactive and mutually reflexive relationships among the factors involved, that is, although L2-coded mediation is a preferred choice in the setting, its effects are dependent on the realisations of five factors: the activation of learners' noticing, that of prior knowledge, the teacher's proficiency in the L2, the delivery of mediation, and the engagement of learners' interest.     

A Tutorial in Bayesian Potential Outcomes Mediation Analysis

Statistical mediation analysis is used to investigate intermediate variables in the relation between independent and dependent variables. Causal interpretation of mediation analyses is challenging because randomization of subjects to levels of the independent variable does not rule out the possibility of unmeasured confounders of the mediator to outcome relation. Furthermore, commonly used frequentist methods for mediation analysis compute the probability of the data given the null hypothesis, which is not the probability of a hypothesis given the data as in Bayesian analysis. Under certain assumptions, applying the potential outcomes framework to mediation analysis allows for the computation of causal effects, and statistical mediation in the Bayesian framework gives indirect effects probabilistic interpretations. This tutorial combines causal inference and Bayesian methods for mediation analysis so the indirect and direct effects have both causal and probabilistic interpretations. Steps in Bayesian causal mediation analysis are shown in the application to an empirical example.     

Alternative Methods for Assessing Mediation in Multilevel Data: The Advantages of Multilevel SEM

Multilevel modeling (MLM) is a popular way of assessing mediation effects with clustered data. Two important limitations of this approach have been identified in prior research and a theoretical rationale has been provided for why multilevel structural equation modeling (MSEM) should be preferred. However, to date, no empirical evidence of MSEM's advantages relative to MLM approaches for multilevel mediation analysis has been provided. Nor has it been demonstrated that MSEM performs adequately for mediation analysis in an absolute sense. This study addresses these gaps and finds that the MSEM method outperforms 2 MLM-based techniques in 2-level models in terms of bias and confidence interval coverage while displaying adequate efficiency, convergence rates, and power under a variety of conditions. Simulation results support prior theoretical work regarding the advantages of MSEM over MLM for mediation in clustered data.     

Parental experiences of dealing with disputes in Additional Support Needs in Scotland: why are parents not engaging with mediation?

Partnerships with parents, particularly in the field of education, have featured prominently in policy rhetoric for many years, but routes of redress have not had much attention until relatively recently. The development of Alternative Dispute Resolution in the UK reflects the situation in several jurisdictions (e.g. Norway, Germany, the Netherlands) where citizens can choose not to go to court to resolve administrative disputes. Under the Education (Additional Support for Learning [ASL]) (Scotland) Act 2004 local authorities must establish and publicise procedures for identifying and meeting the needs of children requiring additional support for learning. The Act and Code of Practice advocate early intervention to prevent disagreements about the provision for additional support from escalating into more serious disputes, with local authorities required to provide information about, and access to, independent mediation for parents. In Scotland the ASL Act 2004 has resulted in four routes available for redress in the area of Additional Support Needs which include both mediation and litigation type processes: ? Informal mediation

? Formal mediation

? Adjudication

? Tribunal

This article uses case study information from parents in three local authorities in Scotland to explore why independent mediation is being under-used by schools and parents, and what factors influence this. Questions are raised regarding the large numbers of parents who are unaware of mediation, the attitudes towards and use of independent mediation by local authorities and the suitability of independent mediation, particularly when the dispute is over resources.     

Popular Structural Equation Modeling Programs for Behavior Genetics

Behavior genetic modeling is a prominent application of multi-group structural equation modeling (SEM). It decomposes phenotypic variance into genetic and environmental sources by leveraging the covariation within and between kin pairs. Although any SEM program with multi-group capabilities can be employed, the software program, Mx, has dominated behavior genetics research. Indeed, even though Mx has not been maintained since 2011, it remains the most popular SEM program in Behavior Genetics articles published in 2016 and 2017. Given the persistence of Mx, the aim of this article is to understand Mx’s performance relative to other popular behavior genetic programs. Through this process, programs employed in behavior genetics research are identified, and their relevant technical features and accessibility are compared. Finally, the relative strengths and limitations of the programs are discussed, and recommendations are provided for behavior genetics researchers.     

Cumulative childhood trauma and psychological maladjustment of sexually abused children in Korea: Mediating effects of emotion regulation

The purpose of the present study was to identify the mediating effects of emotion regulation on the association between cumulative childhood trauma and behavior problems in sexually abused children in Korea, using structural equation modeling (SEM). Data were collected on 171 children (ages 6–13 years) referred to a public counseling center for sexual abuse in Seoul, Korea. Cumulative childhood traumas were defined on the basis of number of traumas (physical abuse, witnessing domestic violence, neglect, traumatic separation from parent, and sexual abuse) and the severity and duration of traumas. Children were evaluated by their parents on emotion regulation using the Emotion Regulation Checklist and internalizing and externalizing behavior problems using the Korean-Child Behavior Checklist. SEM analyses confirmed the complete mediation model, in which emotion dysregulation fully mediates the relationship between cumulative childhood traumas and internalizing/externalizing behavior problems. These findings indicate that emotion regulation is an important mechanism that can explain the negative effects of cumulative childhood traumas and that there is a need to focus on emotion regulation in sexually abused children exposed to cumulative trauma.     

Revisiting the Model Size Effect in Structural Equation Modeling

Fitting a large structural equation modeling (SEM) model with moderate to small sample sizes results in an inflated Type I error rate for the likelihood ratio test statistic under the chi-square reference distribution, known as the model size effect. In this article, we show that the number of observed variables (p) and the number of free parameters (q) have unique effects on the Type I error rate of the likelihood ratio test statistic. In addition, the effects of p and q cannot be fully explained using degrees of freedom (df). We also evaluated the performance of 4 correctional methods for the model size effect, including Bartlett’s (1950), Swain’s (1975), and Yuan’s (2005) corrected statistics, and Yuan, Tian, and Yanagihara’s (2015) empirically corrected statistic. We found that Yuan et al.’s (2015) empirically corrected statistic generally yields the best performance in controlling the Type I error rate when fitting large SEM models.     

Standardized Effect Size Measures for Mediation Analysis in Cluster-Randomized Trials

This article presents 3 standardized effect size measures to use when sharing results of an analysis of mediation of treatment effects for cluster-randomized trials. The authors discuss 3 examples of mediation analysis (upper-level mediation, cross-level mediation, and cross-level mediation with a contextual effect) with demonstration of the calculation and interpretation of the effect size measures using a simulated dataset and an empirical dataset from a cluster-randomized trial of peer tutoring. SAS syntax is provided for parametric percentile bootstrapped confidence intervals of the effect sizes. The use of any of the 3 standardized effect size measures depends on the nature of the inference the researcher wishes to make within a single site, across the broad population, or at the site level.     

Mediation Effects In 2-1-1 Multilevel Model: Evaluation Of Alternative Estimation Methods

We compared six common methods in estimating the 2-1-1 (level-2 independent, level-1 mediator, level-1 dependent) multilevel mediation model with a random slope. They were the Bayesian with informative priors, the Bayesian with non-informative priors, the Monte-Carlo, the distribution of the product, the bias-corrected, and the bias-uncorrected parametric percentile residual bootstrap. The Bayesian method with informative priors was superior in relative mean square error (RMSE), power, interval width, and interval imbalance. The prior variance and prior mean were also varied and examined. Decreasing the prior variance increased the power, reduced RMSE and interval width when the prior mean was the true value, but decreasing the prior variance reduced the power when the prior mean was set incorrectly. The influence of misspecification of prior information of the b coefficient on multilevel mediation analysis was greater than that on coefficient a. An illustrate example with the Bayesian multilevel mediation was provided.     

Labeling,Affect, and Teachers' Hypothetical Approaches to Conflict Resolution: An Exploratory Study

The current study investigated how teachers would intervene in hypothetical conflicts experienced by students in the classroom and how informal labeling of students and affect relate to teachers' hypothetical interventions. Thirty-one teachers from various early childhood learning centers were recruited for participation. Teachers were presented with 3 hypothetical situations depicting children involved in peer conflicts. They were asked to rate the child who had initiated the conflict according to lists of positive and negative characteristics, as well as to rate how much positive and negative affect was elicited from the situation. Next, teachers recorded how they would intervene in each conflict, with responses coded as either mediation or cessation. Results suggested that teachers tended to use more cessation than mediation in dealing with classroom conflict and that interventions varied depending on the described behavioral background of the child presented. Labeling and affect also varied among the 3 different child characterizations of easy, difficult, and ambiguous. Findings lend support to a relationship between both labeling and affect with teachers' negotiation interventions. Understanding the implications of this study in the context of its limitations is highlighted.     

Power Analysis for Complex Mediational Designs Using Monte Carlo Methods

Applied researchers often include mediation effects in applications of advanced methods such as latent variable models and linear growth curve models. Guidance on how to estimate statistical power to detect mediation for these models has not yet been addressed in the literature. We describe a general framework for power analyses for complex mediational models. The approach is based on the well-known technique of generating a large number of samples in a Monte Carlo study, and estimating power as the percentage of cases in which an estimate of interest is significantly different from zero. Examples of power calculation for commonly used mediational models are provided. Power analyses for the single mediator, multiple mediators, 3-path mediation, mediation with latent variables, moderated mediation, and mediation in longitudinal designs are described. Annotated sample syntax for Mplus is appended and tabled values of required sample sizes are shown for some models.     

The Construction and Interpretation of Differential Ability Patterns

Increasing the correlation between the independent variable and the mediator (a coefficient) increases the effect size (ab) for mediation analysis; however, increasing a by definition increases collinearity in mediation models. As a result, the standard error of product tests increase. The variance inflation caused by increases in a at some point outweighs the increase of the effect size (ab) and results in a loss of statistical power. This phenomenon also occurs with nonparametric bootstrapping approaches because the variance of the bootstrap distribution of ab approximates the variance expected from normal theory. Both variances increase dramatically when a exceeds the b coefficient, thus explaining the power decline with increases in a. Implications for statistical analysis and applied researchers are discussed.     

Accuracy of Estimates and Statistical Power for Testing Meditation in Latent Growth Curve Modeling

The latent growth curve modeling (LGCM) approach has been increasingly utilized to investigate longitudinal mediation. However, little is known about the accuracy of the estimates and statistical power when mediation is evaluated in the LGCM framework. A simulation study was conducted to address these issues under various conditions including sample size, effect size of mediated effect, number of measurement occasions, and R ₂ of measured variables. In general, the results showed that relatively large samples were needed to accurately estimate the mediated effects and to have adequate statistical power, when testing mediation in the LGCM framework. Guidelines for designing studies to examine longitudinal mediation and ways to improve the accuracy of the estimates and statistical power were discussed.