Bayesian Causal Mediation Analysis for Group Randomized Designs with Homogeneous and Heterogeneous Effects: Simulation and Case Study

A fully Bayesian approach to causal mediation analysis for group-randomized designs is presented. A unique contribution of this article is the combination of Bayesian inferential methods with G-computation to address the problem of heterogeneous treatment or mediator effects. A detailed simulation study shows that this approach has excellent frequentist properties, particularly in the case of small sample sizes with accurate informative priors. The simulation study also demonstrates that the proposed approach can take into account heterogeneous treatment or mediator effects without bias. A case study using data from a school-based randomized intervention designed to increase parent social capital leading to improved behavioral and academic outcomes in children is offered to illustrate the Bayesian approach to causal mediation in group-randomized designs.     

Analysis of structural equation model with ignorable missing continuous and polytomous data

The main purpose of this article is to develop a Bayesian approach for structural equation models with ignorable missing continuous and polytomous data. Joint Bayesian estimates of thresholds, structural parameters and latent factor scores are obtained simultaneously. The idea of data augmentation is used to solve the computational difficulties involved. In the posterior analysis, in addition to the real missing data, latent variables and latent continuous measurements underlying the polytomous data are treated as hypothetical missing data. An algorithm that embeds the Metropolis-Hastings algorithm within the Gibbs sampler is implemented to produce the Bayesian estimates. A goodness-of-fit statistic for testing the posited model is presented. It is shown that the proposed approach is not sensitive to prior distributions and can handle situations with a large number of missing patterns whose underlying sample sizes may be small. Computational efficiency of the proposed procedure is illustrated by simulation studies and a real example.The work described in this paper was fully supported by a grant from the Research Grants Council of the HKSAR (Project No. CUHK 4088/99H). The authors are greatly indebted to the Editor and anonymous reviewers for valuable comments in improving the paper; and also to D. E. Morisky and J.A. Stein for the use of their AIDS data set.     

Methods for Mediation Analysis with Missing Data

Despite wide applications of both mediation models and missing data techniques, formal discussion of mediation analysis with missing data is still rare. We introduce and compare four approaches to dealing with missing data in mediation analysis including listwise deletion, pairwise deletion, multiple imputation (MI), and a two-stage maximum likelihood (TS-ML) method. An R package bmem is developed to implement the four methods for mediation analysis with missing data in the structural equation modeling framework, and two real examples are used to illustrate the application of the four methods. The four methods are evaluated and compared under MCAR, MAR, and MNAR missing data mechanisms through simulation studies. Both MI and TS-ML perform well for MCAR and MAR data regardless of the inclusion of auxiliary variables and for AV-MNAR data with auxiliary variables. Although listwise deletion and pairwise deletion have low power and large parameter estimation bias in many studied conditions, they may provide useful information for exploring missing mechanisms.     

A Class of Distribution-Free Models for Longitudinal Mediation Analysis

Mediation analysis constitutes an important part of treatment study to identify the mechanisms by which an intervention achieves its effect. Structural equation model (SEM) is a popular framework for modeling such causal relationship. However, current methods impose various restrictions on the study designs and data distributions, limiting the utility of the information they provide in real study applications. In particular, in longitudinal studies missing data is commonly addressed under the assumption of missing at random (MAR), where current methods are unable to handle such missing data if parametric assumptions are violated. In this paper, we propose a new, robust approach to address the limitations of current SEM within the context of longitudinal mediation analysis by utilizing a class of functional response models (FRM). Being distribution-free, the FRM-based approach does not impose any parametric assumption on data distributions. In addition, by extending the inverse probability weighted (IPW) estimates to the current context, the FRM-based SEM provides valid inference for longitudinal mediation analysis under the two most popular missing data mechanisms; missing completely at random (MCAR) and missing at random (MAR). We illustrate the approach with both real and simulated data.     

Bayesian analysis of mixtures in structural equation models with non‐ignorable missing data

Structural equation models (SEMs) have become widely used to determine the interrelationships between latent and observed variables in social, psychological, and behavioural sciences. As heterogeneous data are very common in practical research in these fields, the analysis of mixture models has received a lot of attention in the literature. An important issue in the analysis of mixture SEMs is the presence of missing data, in particular of data missing with a non‐ignorable mechanism. However, only a limited amount of work has been done in analysing mixture SEMs with non‐ignorable missing data. The main objective of this paper is to develop a Bayesian approach for analysing mixture SEMs with an unknown number of components and non‐ignorable missing data. A simulation study shows that Bayesian estimates obtained by the proposed Markov chain Monte Carlo methods are accurate and the Bayes factor computed via a path sampling procedure is useful for identifying the correct number of components, selecting an appropriate missingness mechanism, and investigating various effects of latent variables in the mixture SEMs. A real data set on a study of job satisfaction is used to demonstrate the methodology.     

Bayesian Inference for Growth Mixture Models with Latent Class Dependent Missing Data

Growth mixture models (GMMs) with nonignorable missing data have drawn increasing attention in research communities but have not been fully studied. The goal of this article is to propose and to evaluate a Bayesian method to estimate the GMMs with latent class dependent missing data. An extended GMM is first presented in which class probabilities depend on some observed explanatory variables and data missingness depends on both the explanatory variables and a latent class variable. A full Bayesian method is then proposed to estimate the model. Through the data augmentation method, conditional posterior distributions for all model parameters and missing data are obtained. A Gibbs sampling procedure is then used to generate Markov chains of model parameters for statistical inference. The application of the model and the method is first demonstrated through the analysis of mathematical ability growth data from the National Longitudinal Survey of Youth 1997 (Bureau of Labor Statistics, U.S. Department of Labor, 1997). A simulation study considering 3 main factors (the sample size, the class probability, and the missing data mechanism) is then conducted and the results show that the proposed Bayesian estimation approach performs very well under the studied conditions. Finally, some implications of this study, including the misspecified missingness mechanism, the sample size, the sensitivity of the model, the number of latent classes, the model comparison, and the future directions of the approach, are discussed.     

Latent Class Dynamic Mediation Model with Application to Smoking Cessation Data

Traditional mediation analysis assumes that a study population is homogeneous and the mediation effect is constant over time, which may not hold in some applications. Motivated by smoking cessation data, we propose a latent class dynamic mediation model that explicitly accounts for the fact that the study population may consist of different subgroups and the mediation effect may vary over time. We use a proportional odds model to accommodate the subject heterogeneities and identify latent subgroups. Conditional on the subgroups, we employ a Bayesian hierarchical nonparametric time-varying coefficient model to capture the time-varying mediation process, while allowing each subgroup to have its individual dynamic mediation process. A simulation study shows that the proposed method has good performance in estimating the mediation effect. We illustrate the proposed methodology by applying it to analyze smoking cessation data.

     

Bayesian model comparison of nonlinear structural equation models with missing continuous and ordinal categorical data

Missing data are very common in behavioural and psychological research. In this paper, we develop a Bayesian approach in the context of a general nonlinear structural equation model with missing continuous and ordinal categorical data. In the development, the missing data are treated as latent quantities, and provision for the incompleteness of the data is made by a hybrid algorithm that combines the Gibbs sampler and the Metropolis‐Hastings algorithm. We show by means of a simulation study that the Bayesian estimates are accurate. A Bayesian model comparison procedure based on the Bayes factor and path sampling is proposed. The required observations from the posterior distribution for computing the Bayes factor are simulated by the hybrid algorithm in Bayesian estimation. Our simulation results indicate that the correct model is selected more frequently when the incomplete records are used in the analysis than when they are ignored. The methodology is further illustrated with a real data set from a study concerned with an AIDS preventative intervention for Filipina sex workers.     

Bayesian Analysis of Nonlinear Structural Equation Models with Nonignorable Missing Data

A Bayesian approach is developed for analyzing nonlinear structural equation models with nonignorable missing data. The nonignorable missingness mechanism is specified by a logistic regression model. A hybrid algorithm that combines the Gibbs sampler and the Metropolis–Hastings algorithm is used to produce the joint Bayesian estimates of structural parameters, latent variables, parameters in the nonignorable missing model, as well as their standard errors estimates. A goodness-of-fit statistic for assessing the plausibility of the posited nonlinear structural equation model is introduced, and a procedure for computing the Bayes factor for model comparison is developed via path sampling. Results obtained with respect to different missing data models, and different prior inputs are compared via simulation studies. In particular, it is shown that in the presence of nonignorable missing data, results obtained by the proposed method with a nonignorable missing data model are significantly better than those that are obtained under the missing at random assumption. A real example is presented to illustrate the newly developed Bayesian methodologies. This research is fully supported by a grant (CUHK 4243/03H) from the Research Grant Council of the Hong Kong Special Administration Region. The authors are thankful to the editor and reviewers for valuable comments for improving the paper, and also to ICPSR and the relevant funding agency for allowing the use of the data. Requests for reprints should be sent to Professor S.Y. Lee, Department of Statistics, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong.     

BANOVA: Bayesian Analysis of Experiments in Consumer Psychology

This article introduces a Bayesian extension of ANOVA for the analysis of experimental data in consumer psychology. The approach, called BANOVA (Bayesian ANOVA), addresses some common challenges that consumer psychologists encounter in their experimental work, and is specifically suited for the analysis of repeated measures designs. There appears to be a recent surge in interest in those designs based on the recognition that they are sensitive to individual differences in response to experimental treatments and that they offer advantages for assessing causal mediating mechanisms, even at the individual level. BANOVA enables the analysis of repeated measures data derived from mixed within–between‐subjects experiments with Normal and nonNormal‐dependent variables and accommodates unobserved individual differences. It allows for the calculation of effect sizes, planned comparisons, simple effects, spotlight and floodlight analyses, and includes a wide range of mediation, moderation, and moderated mediation analyses. An R software package implements these analyses, and aims to provide a one‐stop shop for the analysis of experiments in consumer psychology. The package is illustrated through applications to a number of data sets from previously published studies.     

基于不对称区间估计的有调节的中介模型检验

有调节的中介模型是中介过程受到调节变量影响的模型。评介了基于Bootstrap不对称置信区间和贝叶斯不对称可靠区间进行有调节的中介模型检验的3种方法, 包括亚组分析法、差异分析法和系数乘积法。模拟研究发现, 偏差校正的百分位Bootstrap置信区间和无先验信息的贝叶斯可靠区间在有调节的中介模型检验中表现相当, 都优于百分位Bootstrap置信区间的表现。建议使用系数乘积法进行第一阶段或第二阶段被调节的中介模型检验, 使用差异分析法进行两阶段被调节的中介模型检验, 并用一个实际例子演示如何用不对称区间估计检验有调节的中介模型。随后评述了3种有调节的中介模型检验方法在国内心理学的应用现状, 并展望了检验的拓展方向。     

Bayesian Factor Analysis as a Variable-Selection Problem: Alternative Priors and Consequences

Factor analysis is a popular statistical technique for multivariate data analysis. Developments in the structural equation modeling framework have enabled the use of hybrid confirmatory/exploratory approaches in which factor-loading structures can be explored relatively flexibly within a confirmatory factor analysis (CFA) framework. Recently, Muthén & Asparouhov proposed a Bayesian structural equation modeling (BSEM) approach to explore the presence of cross loadings in CFA models. We show that the issue of determining factor-loading patterns may be formulated as a Bayesian variable selection problem in which Muthén and Asparouhov's approach can be regarded as a BSEM approach with ridge regression prior (BSEM-RP). We propose another Bayesian approach, denoted herein as the Bayesian structural equation modeling with spike-and-slab prior (BSEM-SSP), which serves as a one-stage alternative to the BSEM-RP. We review the theoretical advantages and disadvantages of both approaches and compare their empirical performance relative to two modification indices-based approaches and exploratory factor analysis with target rotation. A teacher stress scale data set is used to demonstrate our approach.     

A Regression Framework for Causal Mediation Analysis with Applications to Behavioral Science

We introduce and extend the classical regression framework for conducting mediation analysis from the fit of only one model. Using the essential mediation components (EMCs) allows us to estimate causal mediation effects and their analytical variance. This single-equation approach reduces computation time and permits the use of a rich suite of regression tools that are not easily implemented on a system of three equations. Additionally, we extend this framework to non-nested mediation systems, provide a joint measure of mediation for complex mediation hypotheses, propose new visualizations for mediation effects, and explain why estimates of the total effect may differ depending on the approach used. Using data from social science studies, we also provide extensive illustrations of the usefulness of this framework and its advantages over traditional approaches to mediation analysis. The example data are freely available for download online and we include the R code necessary to reproduce our results.     

Testing indirect effect with a complete or incomplete dichotomous mediator

Past methodological research on mediation analysis mainly focused on situations where all variables were complete and continuous. When issues of categorical data occur combined with missing data, more methodological considerations are involved. Specifically, appropriate decisions need to be made on estimation methods of the indirect effects and on confidence intervals for testing the indirect effects with accommodations of missing data. We compare strategies that address these issues based on a model with a dichotomous mediator, aiming to provide guidelines for researchers facing such challenges in practice.     

Nonlinear Bayesian analysis for single case designs

Several authors have suggested the use of multilevel models for the analysis of data from single case designs. Multilevel models are a logical approach to analyzing such data, and deal well with the possible different time points and treatment phases for different subjects. However, they are limited in several ways that are addressed by Bayesian methods. For small samples Bayesian methods fully take into account uncertainty in random effects when estimating fixed effects; the computational methods now in use can fit complex models that represent accurately the behavior being modeled; groups of parameters can be more accurately estimated with shrinkage methods; prior information can be included; and interpretation is more straightforward. The computer programs for Bayesian analysis allow many (nonstandard) nonlinear models to be fit; an example using floor and ceiling effects is discussed here.     

基于两水平被试内设计的中介效应分析方法

在心理学和其他社科研究领域,通常的中介效应分析都基于被试间设计,研究者对于如何分析基于被试内设计的中介效应往往并不清楚。本文阐述了两水平被试内设计的中介效应分析方法(依次检验法和路径分析法),综合各方法优缺点给出一个分析流程,并用应用研究实例演示如何分析两水平被试内中介效应,最后展望了基于被试内设计的中介效应分析研究的拓展方向。     

Missing data: our view of the state of the art

Statistical procedures for missing data have vastly improved, yet misconception and unsound practice still abound. The authors frame the missing-data problem, review methods, offer advice, and raise issues that remain unresolved. They clear up common misunderstandings regarding the missing at random (MAR) concept. They summarize the evidence against older procedures and, with few exceptions, discourage their use. They present, in both technical and practical language, 2 general approaches that come highly recommended: maximum likelihood (ML) and Bayesian multiple imputation (MI). Newer developments are discussed, including some for dealing with missing data that are not MAR. Although not yet in the mainstream, these procedures may eventually extend the ML and MI methods that currently represent the state of the art.     

Mediation analysis with structural equation models: Combining theory,design, and statistics

Statistical tests of indirect effects can hardly distinguish between genuine and spurious mediation effects. The present research demonstrates, however, that mediation analysis can be improved by combining a significance test of the indirect effect with assessing the fit of causal models. Testing only the indirect effect can be misleading, because significant results may also be obtained when the underlying causal model is different from the mediation model. We use simulated data to demonstrate that additionally assessing the fit of causal models with structural equation models can be used to exclude subsets of models that are incompatible with the observed data. The results suggest that combining structural equation modeling with appropriate research design and theoretically stringent mediation analysis can improve scientific insights. Finally, we discuss limitations of the structural equation modeling approach, and we emphasize the importance of non‐statistical methods for scientific discovery.     

Modeling Incomplete Longitudinal Substance Use Data Using Latent Variable Growth Curve Methodology

Longitudinal data sets typically suffer from attrition and other forms of missing data. When this common problem occurs, several researchers have demonstrated that correct maximum likelihood estimation with missing data can be obtained under mild assumptions concerning the missing data mechanism. With reasonable substantive theory, a mixture of cross-sectional and longitudinal methods developed within multiple-group structural equation modeling can provide a strong basis for inference about developmental change. Using an approach to the analysis of missing data, the present study investigated developmental trends in adolescent (N = 759) alcohol, marijuana, and cigarette use across a 5-year period using multiple-group latent growth modeling. An associative model revealed that common developmental trends existed for all three substances. Age and gender were included in the model as predictors of initial status and developmental change. Findings discuss the utility of latent variable structural equation modeling techniques and missing data approaches in the study of developmental change.     

A Robust Bayesian Approach for Structural Equation Models with Missing Data

In this paper, normal/independent distributions, including but not limited to the multivariate t distribution, the multivariate contaminated distribution, and the multivariate slash distribution, are used to develop a robust Bayesian approach for analyzing structural equation models with complete or missing data. In the context of a nonlinear structural equation model with fixed covariates, robust Bayesian methods are developed for estimation and model comparison. Results from simulation studies are reported to reveal the characteristics of estimation. The methods are illustrated by using a real data set obtained from diabetes patients.