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.     

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.     

Longitudinal Measurement Non-Invariance with Ordered-Categorical Indicators: How are the Parameters in Second-Order Latent Linear Growth Models Affected?

Models of change typically assume longitudinal measurement invariance. Key constructs are often measured by ordered-categorical indicators (e.g., Likert scale items). If tests based on such indicators do not support longitudinal measurement invariance, it would be useful to gauge the practical significance of the detected non-invariance. The authors focus on the commonly used second-order latent growth curve model, proposing a sensitivity analysis that compares the growth parameter estimates from a model assuming the highest achieved level of measurement invariance to those from a model assuming a higher, incorrect level of measurement invariance as a measure of practical significance. A simulation study investigated the practical significance of non-invariance in different locations (loadings, thresholds, uniquenesses) in second-order latent linear growth models. The mean linear slope was affected by non-invariance in the loadings and thresholds, the intercept variance was affected by non-invariance in the uniquenesses, and the linear slope variance and intercept–slope covariance were affected by non-invariance in all three locations.     

Bias-Adjusted Three-Step Latent Markov Modeling With Covariates

Latent Markov models with covariates can be estimated via 1-step maximum likelihood. However, this 1-step approach has various disadvantages, such as that the inclusion of covariates in the model might alter the formation of the latent states and that parameter estimation could become infeasible with large numbers of time points, responses, and covariates. This is why researchers typically prefer performing the analysis in a stepwise manner; that is, they first construct the measurement model, then obtain the latent state classifications, and subsequently study the relationship between covariates and latent state memberships. However, such a stepwise approach yields downward-biased estimates of the covariate effects on initial state and transition probabilities. This article, shows how to overcome this problem using a generalization of the bias-corrected 3-step estimation method proposed for latent class analysis (Asparouhov & Muthén, 2014; Bolck, Croon, & Hagenaars, 2004; Vermunt, 2010). We give a formal derivation of the generalization to latent Markov models and discuss how it can be used with many time points by incorporating it into a Baum–Welch type of expectation-maximization algorithm. We evaluate the method through a simulation study and illustrate it using an application on household financial portfolio change. Our study shows that the proposed correction method yields unbiased parameter estimates and accurate standard errors, except for situations with very poorly separated classes and a small sample.     

Three-Level Models for Indirect Effects in School- and Class-Randomized Experiments in Education

Due to the clustered nature of field data, multi-level modeling has become commonly used to analyze data arising from educational field experiments. While recent methodological literature has focused on multi-level mediation analysis, relatively little attention has been devoted to mediation analysis when three levels (e.g., student, class, school) are present in a study setting. This article presents analysis models that can be used to test indirect effects in experimental designs having three levels where random assignment is at the third (school) or second (class) level and where the indirect effect may be random. In the presentation, simulated datasets are used to illustrate model specification and results interpretation for hypothetical three-level educational experiments involving mediation and moderation of treatment effects.     

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.     

Comparison of Approaches to Constructing Confidence Intervals for Mediating Effects Using Structural Equation Models

Regression mixture models, which have only recently begun to be used in applied research, are a new approach for finding differential effects. This approach comes at the cost of the assumption that error terms are normally distributed within classes. This study uses Monte Carlo simulations to explore the effects of relatively minor violations of this assumption. The use of an ordered polytomous outcome is then examined as an alternative that makes somewhat weaker assumptions, and finally both approaches are demonstrated with an applied example looking at differences in the effects of family management on the highly skewed outcome of drug use. Results show that violating the assumption of normal errors results in systematic bias in both latent class enumeration and parameter estimates. Additional classes that reflect violations of distributional assumptions are found. Under some conditions it is possible to come to conclusions that are consistent with the effects in the population, but when errors are skewed in both classes the results typically no longer reflect even the pattern of effects in the population. The polytomous regression model performs better under all scenarios examined and comes to reasonable results with the highly skewed outcome in the applied example. We recommend that careful evaluation of model sensitivity to distributional assumptions be the norm when conducting regression mixture models.     

Assessing Mediation in Dyadic Data Using the Actor-Partner Interdependence Model

The assessment of mediation in dyadic data is an important issue if researchers are to test process models. Using an extended version of the actor–partner interdependence model the estimation and testing of mediation is complex, especially when dyad members are distinguishable (e.g., heterosexual couples). We show how the complexity of the model can be reduced by assuming specific dyadic patterns. Using structural equation modeling, we demonstrate how specific mediating effects and contrasts among effects can be tested by phantom models that permit point and bootstrap interval estimates. We illustrate the assessment of mediation and the strategies to simplify the model using data from heterosexual couples.     

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.     

Nonlongitudinal Estimation of Growth Curve Parameters: The Cohort Growth Model

The cohort growth model (CGM) is a method for estimating the parameters of a latent growth model (LGM) based on cross-sectional data. The CGM models the interindividual differences in the growth rate, and it models how subjects’ growth rate is related to their initial status. We derive model identification for the CGM and illustrate, in a simulation study, that the CGM provides unbiased parameter estimates in most simulation conditions. Based on empirical data we compare the estimates of the CGM with the estimates of the LGM. The results were comparable for both models. Although the estimates of the (co)-variances were different, the estimates of both models led to similar conclusions on the developmental change. Finally, we discuss the advantages and limitations of the CGM, and we provide recommendations for its use in empirical research.     

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.     

Assessing Sensitivity to Unmeasured Confounding Using a Simulated Potential Confounder

ABSTRACT

A major obstacle to developing evidenced-based policy is the difficulty of implementing randomized experiments to answer all causal questions of interest. When using a nonexperimental study, it is critical to assess how much the results could be affected by unmeasured confounding. We present a set of graphical and numeric tools to explore the sensitivity of causal estimates to the presence of an unmeasured confounder. We characterize the confounder through two parameters that describe the relationships between (a) the confounder and the treatment assignment and (b) the confounder and the outcome variable. Our approach has two primary advantages over similar approaches that are currently implemented in standard software. First, it can be applied to both continuous and binary treatment variables. Second, our method for binary treatment variables allows the researcher to specify three possible estimands (average treatment effect, effect of the treatment on the treated, effect of the treatment on the controls). These options are all implemented in an R package called treatSens. We demonstrate the efficacy of the method through simulations. We illustrate its potential usefulness in practice in the context of two policy applications.     

A General 3-Step Maximum Likelihood Approach to Estimate the Effects of Multiple Latent Categorical Variables on a Distal Outcome

The 3-step approach has been recently advocated over the simultaneous 1-step approach to model a distal outcome predicted by a latent categorical variable. We generalize the 3-step approach to situations where the distal outcome is predicted by multiple and possibly associated latent categorical variables. Although the simultaneous 1-step approach has been criticized, simulation studies have found that the performance of the two approaches is similar in most situations (Bakk & Vermunt, 2016). This is consistent with our findings for a 2-LV extension when all model assumptions are satisfied. Results also indicate that under various degrees of violation of the normality and conditional independence assumption for the distal outcome and indicators, both approaches are subject to bias but the 3-step approach is less sensitive. The differences in estimates using the two approaches are illustrated in an analysis of the effects of various childhood socioeconomic circumstances on body mass index at age 50.     

On the Estimation of Nonlinear Mixed-Effects Models and Latent Curve Models for Longitudinal Data

Nonlinear models are effective tools for the analysis of longitudinal data. These models provide a flexible means for describing data that follow complex forms of change. Exponential and logistic functions that include a parameter to represent an asymptote, for instance, are useful for describing responses that tend to level off with time. There are forms of nonlinear latent curve models and nonlinear mixed-effects model that are equivalent, and so given the same set of data, growth function, distributional assumptions, and method of estimation, the 2 models yield equivalent results. There are also forms that are strikingly different and can yield different interpretations for a given set of data. This article discusses cases in which nonlinear mixed-effects models and nonlinear latent curve models are equivalent and those in which they are different and clarifies the estimation needs of the different models. Examples based on empirical data help to illustrate these points.     

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.     

Not quite normal: Consequences of violating the assumption of normality in regression mixture models

Regression mixture models are a new approach for finding differential effects which have only recently begun to be used in applied research. This approach comes at the cost of the assumption that error terms are normally distributed within classes. The current study uses Monte Carlo simulations to explore the effects of relatively minor violations of this assumption, the use of an ordered polytomous outcome is then examined as an alternative which makes somewhat weaker assumptions, and finally both approaches are demonstrated with an applied example looking at differences in the effects of family management on the highly skewed outcome of drug use. Results show that violating the assumption of normal errors results in systematic bias in both latent class enumeration and parameter estimates. Additional classes which reflect violations of distributional assumptions are found. Under some conditions it is possible to come to conclusions that are consistent with the effects in the population, but when errors are skewed in both classes the results typically no longer reflect even the pattern of effects in the population. The polytomous regression model performs better under all scenarios examined and comes to reasonable results with the highly skewed outcome in the applied example. We recommend that careful evaluation of model sensitivity to distributional assumptions be the norm when conducting regression mixture models.     

Modeling the processes of development via latent variable growth curve methodology

Recent advances in statistical methodology, in particular, latent growth modeling, allow for the testing of complex models regarding developmental trends from both an inter‐ and intraindividual perspective. An example application of latent growth curve methodology, analyzing the effects of gender and parental monitoring on developmental change in adolescent alcohol consumption, is presented. Furthermore, the analyses are conducted within a cohort‐sequential design, incorporating an approach to the analysis of missing data due to attrition. Findings are discussed with particular reference to the utility of latent growth curve models for assessing developmental processes at both the inter‐and intraindividual level across a variety of behavioral domains.     

Measurement Invariance Across Groups in Latent Growth Modeling

This Monte Carlo study investigated the impacts of measurement noninvariance across groups on major parameter estimates in latent growth modeling when researchers test group differences in initial status and latent growth. The average initial status and latent growth and the group effects on initial status and latent growth were investigated in terms of Type I error and bias. The location and magnitude of noninvariance across groups was related to the location and magnitude of bias and Type I error in the parameter estimates. That is, noninvariance in factor loadings and intercepts was associated with the Type I error inflation and bias in the parameter estimates of the slope factor (or latent growth) and the intercept factor (or initial status), respectively. As noninvariance became large, the degree of Type I error and bias also increased. On the other hand, a correctly specified second-order latent growth model yielded unbiased parameter estimates and correct statistical inferences. Other findings and implications on future studies were discussed.     

An Evaluation of Latent Growth Models for Propensity Score Matched Groups

Because random assignment is not possible in observational studies, estimates of treatment effects might be biased due to selection on observable and unobservable variables. To strengthen causal inference in longitudinal observational studies of multiple treatments, we present 4 latent growth models for propensity score matched groups, and evaluate their performance with a Monte Carlo simulation study. We found that the 4 models performed similarly with respect to model fit, bias of parameter estimates, Type I error, and power to test the treatment effect. To demonstrate a multigroup latent growth model with dummy treatment indicators, we estimated the effect of students changing schools during elementary school years on their reading and mathematics achievement, using data from the Early Childhood Longitudinal Study Kindergarten Cohort.     

Investigation of Mediational Processes Using Parallel Process Latent Growth Curve Modeling

This study investigated a method to evaluate mediational processes using latent growth curve modeling. The mediator and the outcome measured across multiple time points were viewed as 2 separate parallel processes. The mediational process was defined as the independent variable influencing the growth of the mediator, which, in turn, affected the growth of the outcome. To illustrate modeling procedures, empirical data from a longitudinal drug prevention program, Adolescents Training and Learning to Avoid Steroids, were used. The program effects on the growth of the mediator and the growth of the outcome were examined first in a 2-group structural equation model. The mediational process was then modeled and tested in a parallel process latent growth curve model by relating the prevention program condition, the growth rate factor of the mediator, and the growth rate factor of the outcome.