Factorial invariance in multilevel confirmatory factor analysis

This paper presents a procedure to test factorial invariance in multilevel confirmatory factor analysis. When the group membership is at level 2, multilevel factorial invariance can be tested by a simple extension of the standard procedure. However level‐1 group membership raises problems which cannot be appropriately handled by the standard procedure, because the dependency between members of different level‐1 groups is not appropriately taken into account. The procedure presented in this article provides a solution to this problem. This paper also shows Muthén's maximum likelihood (MUML) estimation for testing multilevel factorial invariance across level‐1 groups as a viable alternative to maximum likelihood estimation. Testing multilevel factorial invariance across level‐2 groups and testing multilevel factorial invariance across level‐1 groups are illustrated using empirical examples. SAS macro and Mplus syntax are provided.     

Testing Group Mean Differences of Latent Variables in Multilevel Data Using Multiple-Group Multilevel CFA and Multilevel MIMIC Modeling

Considering that group comparisons are common in social science, we examined two latent group mean testing methods when groups of interest were either at the between or within level of multilevel data: multiple-group multilevel confirmatory factor analysis (MG ML CFA) and multilevel multiple-indicators multiple-causes modeling (ML MIMIC). The performance of these methods were investigated through three Monte Carlo studies. In Studies 1 and 2, either factor variances or residual variances were manipulated to be heterogeneous between groups. In Study 3, which focused on within-level multiple-group analysis, six different model specifications were considered depending on how to model the intra-class group correlation (i.e., correlation between random effect factors for groups within cluster). The results of simulations generally supported the adequacy of MG ML CFA and ML MIMIC for multiple-group analysis with multilevel data. The two methods did not show any notable difference in the latent group mean testing across three studies. Finally, a demonstration with real data and guidelines in selecting an appropriate approach to multilevel multiple-group analysis are provided.     

A Computationally Efficient State Space Approach to Estimating Multilevel Regression Models and Multilevel Confirmatory Factor Models

Although the state space approach for estimating multilevel regression models has been well established for decades in the time series literature, it does not receive much attention from educational and psychological researchers. In this article, we (a) introduce the state space approach for estimating multilevel regression models and (b) extend the state space approach for estimating multilevel factor models. A brief outline of the state space formulation is provided and then state space forms for univariate and multivariate multilevel regression models, and a multilevel confirmatory factor model, are illustrated. The utility of the state space approach is demonstrated with either a simulated or real example for each multilevel model. It is concluded that the results from the state space approach are essentially identical to those from specialized multilevel regression modeling and structural equation modeling software. More importantly, the state space approach offers researchers a computationally more efficient alternative to fit multilevel regression models with a large number of Level 1 units within each Level 2 unit or a large number of observations on each subject in a longitudinal study.     

有调节的多层中介效应分析

多层中介和多层调节效应分析在社科领域已常有应用,但如果将多层中介和调节整合在一起,可以产生2（多层中介类型）×2（调节变量的层次）×3（调节的中介路径）共12种有调节的多层中介模型。面对有调节的多层中介效应分析,研究者往往束手无策。详述了基于多层线性模型的12种有调节的多层中介的分析方法和基于多层结构方程模型的4类有调节的多层中介分析方法,包括正交分割法,随机系数预测法,潜调节结构方程法和贝叶斯合理值法。这四类方法的核心议题在于如何处理潜调节项。当样本量足够大时,建议选择潜调节结构方程法;当样本量不足时,建议选择贝叶斯合理值法。随后用一个实际例子演示如何进行有调节的多层中介效应分析并有相应的Mplus程序。最后展望了有调节的多层中介效应分析研究的拓展方向。     

Using mixture models with known class membership to address incomplete covariance structures in multiple‐group growth models

Multi‐group latent growth modelling in the structural equation modelling framework has been widely utilized for examining differences in growth trajectories across multiple manifest groups. Despite its usefulness, the traditional maximum likelihood estimation for multi‐group latent growth modelling is not feasible when one of the groups has no response at any given data collection point, or when all participants within a group have the same response at one of the time points. In other words, multi‐group latent growth modelling requires a complete covariance structure for each observed group. The primary purpose of the present study is to show how to circumvent these data problems by developing a simple but creative approach using an existing estimation procedure for growth mixture modelling. A Monte Carlo simulation study was carried out to see whether the modified estimation approach provided tangible results and to see how these results were comparable to the standard multi‐group results. The proposed approach produced results that were valid and reliable under the mentioned problematic data conditions. We also present a real data example and demonstrate that the proposed estimation approach can be used for the chi‐square difference test to check various types of measurement invariance as conducted in a standard multi‐group analysis.     

Multilevel component analysis

A general framework for the exploratory component analysis of multilevel data (MLCA) is proposed. In this framework, a separate component model is specified for each group of objects at a certain level. The similarities between the groups of objects at a given level can be expressed by imposing constraints on component models of the groups using the approach adopted in simultaneous component analysis. The constraints used are based on the loading matrices and on the covariances of the component scores of each group. MLCA is related to three‐way component analysis and to currently available multilevel structural equation models. It is shown that the latter are less flexible than MLCA. The use of MLCA is illustrated by means of an empirical example.     

Incorporating Mobility in Growth Modeling for Multilevel and Longitudinal Item Response Data

Multilevel data often cannot be represented by the strict form of hierarchy typically assumed in multilevel modeling. A common example is the case in which subjects change their group membership in longitudinal studies (e.g., students transfer schools; employees transition between different departments). In this study, cross-classified and multiple membership models for multilevel and longitudinal item response data (CCMM-MLIRD) are developed to incorporate such mobility, focusing on students' school change in large-scale longitudinal studies. Furthermore, we investigate the effect of incorrectly modeling school membership in the analysis of multilevel and longitudinal item response data. Two types of school mobility are described, and corresponding models are specified. Results of the simulation studies suggested that appropriate modeling of the two types of school mobility using the CCMM-MLIRD yielded good recovery of the parameters and improvement over models that did not incorporate mobility properly. In addition, the consequences of incorrectly modeling the school effects on the variance estimates of the random effects and the standard errors of the fixed effects depended upon mobility patterns and model specifications. Two sets of large-scale longitudinal data are analyzed to illustrate applications of the CCMM-MLIRD for each type of school mobility.     

Multilevel Dynamic Generalized Structured Component Analysis for Brain Connectivity Analysis in Functional Neuroimaging Data

We extend dynamic generalized structured component analysis (GSCA) to enhance its data-analytic capability in structural equation modeling of multi-subject time series data. Time series data of multiple subjects are typically hierarchically structured, where time points are nested within subjects who are in turn nested within a group. The proposed approach, named multilevel dynamic GSCA, accommodates the nested structure in time series data. Explicitly taking the nested structure into account, the proposed method allows investigating subject-wise variability of the loadings and path coefficients by looking at the variance estimates of the corresponding random effects, as well as fixed loadings between observed and latent variables and fixed path coefficients between latent variables. We demonstrate the effectiveness of the proposed approach by applying the method to the multi-subject functional neuroimaging data for brain connectivity analysis, where time series data-level measurements are nested within subjects.     

What does it mean to belong? Interpersonal bonds and intragroup similarities as predictors of felt belonging in different types of groups

Belonging is a fundamental human need, deemed essential for optimal psychological functioning. There is, however, little consensus about how people gain feelings of belonging from social groups, with theories suggesting different antecedents depending upon how groups are conceptualised. The social identity perspective conceptualises groups as social categories and proposes that feelings of group belonging arise from perceived intragroup similarity. However, if groups are construed as interpersonal networks, feelings of belonging would be expected to arise from the quality of relationships and interactions among members. We tested these predictions using multilevel structural equation modelling of longitudinal data from 113 participants. We found that perceived intragroup similarity prospectively predicted feelings of belonging within groups perceived as social categories but not within those perceived as networks, whereas the quality of interpersonal bonds predicted feelings of belonging to both kinds of groups. We discuss the importance of distinguishing types of groups and suggest implications for research into group membership and well‐being. Copyright © 2013 John Wiley & Sons, Ltd.     

Leveling up the analysis of the reminiscence bump in autobiographical memory: A new approach based on multilevel multinomial models

In many studies of autobiographical memory, participants are asked to generate more than one autobiographical memory. The resulting data then have a hierarchical or multilevel structure, in the sense that the autobiographical memories (Level 1) generated by the same person (Level 2) tend to be more similar. Transferred to an analysis of the reminiscence bump in autobiographical memory, at Level 1 the prediction of whether an autobiographical memory will fall within the reminiscence bump is based on the characteristics of that memory. At Level 2, the prediction of whether an individual will report more autobiographical memories that fall in the reminiscence bump is based on the characteristics of the individual. We suggest a multilevel multinomial model that allows for analyzing whether an autobiographical memory falls in the reminiscence bump at both levels of analysis simultaneously. The data come from 100 older participants who reported up to 33 autobiographical memories. Our results showed that about 12% of the total variance was between persons (Level 2). Moreover, at Level 1, memories of first-time experiences were more likely to fall in the reminiscence bump than were emotionally more positive memories. At Level 2, persons who reported more emotionally positive memories tended to report fewer memories from the life period after the reminiscence bump. In addition, cross-level interactions showed that the effects at Level 1 partly depended on the Level 2 effects. We discuss possible extensions of the model we present and the meaning of our findings for two prominent explanatory approaches to the reminiscence bump, as well as future directions.     

基于结构方程模型的多层中介效应分析

近年社科领域常见使用多层线性模型进行多层中介研究。尽管多层线性模型区分了多层中介的组间和组内效应, 仍然存在抽样误差和测量误差。比较好的方法是, 将多层线性模型整合到结构方程模型中, 在多层结构方程模型框架下设置潜变量和多指标, 可有效校正抽样误差和测量误差、得到比较准确的中介效应值, 还能适用于更多种类的多层中介分析并提供模型的拟合指数。在介绍新方法后, 总结出一套多层中介的分析流程, 通过一个例子来演示如何用MPLUS软件进行多层中介分析。最后展望了多层结构方程和多层中介研究的拓展方向。     

Latent growth curve analysis with dichotomous items: Comparing four approaches

A Monte Carlo study was used to compare four approaches to growth curve analysis of subjects assessed repeatedly with the same set of dichotomous items: A two‐step procedure first estimating latent trait measures using MULTILOG and then using a hierarchical linear model to examine the changing trajectories with the estimated abilities as the outcome variable; a structural equation model using modified weighted least squares (WLSMV) estimation; and two approaches in the framework of multilevel item response models, including a hierarchical generalized linear model using Laplace estimation, and Bayesian analysis using Markov chain Monte Carlo (MCMC). These four methods have similar power in detecting the average linear slope across time. MCMC and Laplace estimates perform relatively better on the bias of the average linear slope and corresponding standard error, as well as the item location parameters. For the variance of the random intercept, and the covariance between the random intercept and slope, all estimates are biased in most conditions. For the random slope variance, only Laplace estimates are unbiased when there are eight time points.     

The developmental ecology of urban males' youth violence

Data from a longitudinal study of 294 African American and Latino adolescent boys and their caregivers living in poor urban communities were used to test a developmental-ecological model of violence. Six annual waves of data were applied to evaluate the relations between microsystem influences of parenting and peer deviance (peer violence and gang membership), macrosystem influences of community structural characteristics and neighborhood social organization, and individual involvement in violence (level and growth). Structural equation modeling analyses showed that community structural characteristics significantly predicted neighborhood social processes. Parenting practices partially mediated the relation between neighborhood social processes and gang membership. Parenting practices was fully mediated in its relation to peer violence by gang membership. Gang membership was partially mediated by peer violence level in its relation to individual violence level. Although the overall set of relations does not satisfy mediation requirements fully in all instances, the model was validated for the most part, supporting a focus on a multilevel ecological model of influences on risk development.     

Bayesian analysis of two‐level nonlinear structural equation models with continuous and polytomous data

Two‐level structural equation models with mixed continuous and polytomous data and nonlinear structural equations at both the between‐groups and within‐groups levels are important but difficult to deal with. A Bayesian approach is developed for analysing this kind of model. A Markov chain Monte Carlo procedure based on the Gibbs sampler and the Metropolis‐Hasting algorithm is proposed for producing joint Bayesian estimates of the thresholds, structural parameters and latent variables at both levels. Standard errors and highest posterior density intervals are also computed. A procedure for computing Bayes factor, based on the key idea of path sampling, is established for model comparison.     

新世纪20年国内心理统计方法研究回顾

新世纪头20年, 国内心理学11本专业期刊一共发表了213篇统计方法研究论文。研究范围主要包括以下10类(按论文篇数排序)：结构方程模型、测验信度、中介效应、效应量与检验力、纵向研究、调节效应、探索性因子分析、潜在类别模型、共同方法偏差和多层线性模型。对各类做了简单的回顾与梳理。结果发现, 国内心理统计方法研究的广度和深度都不断增加, 研究热点在相互融合中共同发展; 但综述类论文比例较大, 原创性研究论文比例有待提高, 研究力量也有待加强。     

Hierarchical modeling of paired comparison data

The method of paired comparisons belongs to a small group of techniques that provide explicit information about the consistency of individual and aggregated choices. This article investigates the link between the individual- and group-level judgments by extending R. D. Luce's (1959) model, which was originally developed for individual choice behavior, to a mixed-effects paired comparison model. It is shown that standard multilevel software for binary data can be used to estimate the model. The interpretation of the paired comparison parameters and statistical model tests are discussed in detail. An extensive analysis of an experimental study illustrates the usefulness of a hierarchical approach in modeling multiple pairwise judgments.     

Assessing the Validity of Data Synthesis Methods to Estimate Religious Populations

The present study tests the validity of a data synthesis approach to population estimates of religiously defined groups. This is particularly important in places like the United States, where there is no definitive source of official data on its population's religious composition, and researchers must rely on costly, large‐scale surveys, or congregational membership studies. Each approach has limitations, especially for estimation of small religious groups and for estimation within small geographic areas. Without official statistics, the degree of bias in estimates is unknown. Data synthesis, specifically Bayesian multilevel estimation with poststratification, offers a useful alternative that maximizes the utility of data across all sources to estimate multiple groups from the same sources of data. This method also facilitates comparison of groups. This study provides evidence of the validity of the approach by synthesizing data from Canada, a country that includes questions about religious identification in its national census.     

Friendship Dissolution Within Social Networks Modeled Through Multilevel Event History Analysis

A social network perspective can bring important insight into the processes that shape human behavior. Longitudinal social network data, measuring relations between individuals over time, has become increasingly common—as have the methods available to analyze such data. A friendship duration model utilizing discrete-time multilevel survival analysis with a multiple membership random effect structure is developed and applied here to study the processes leading to undirected friendship dissolution within a larger social network. While the modeling framework is introduced in terms of understanding friendship dissolution, it can be used to understand microlevel dynamics of a social network more generally. These models can be fit with standard generalized linear mixed-model software, after transforming the data to a pair-period data set. An empirical example highlights how the model can be applied to understand the processes leading to friendship dissolution between high school students, and a simulation study is used to test the use of the modeling framework under representative conditions that would be found in social network data. Advantages of the modeling framework are highlighted, and potential limitations and future directions are discussed.     

Three-level meta-analysis of dependent effect sizes

Although dependence in effect sizes is ubiquitous, commonly used meta-analytic methods assume independent effect sizes. We describe and illustrate three-level extensions of a mixed effects meta-analytic model that accounts for various sources of dependence within and across studies, because multilevel extensions of meta-analytic models still are not well known. We also present a three-level model for the common case where, within studies, multiple effect sizes are calculated using the same sample. Whereas this approach is relatively simple and does not require imputing values for the unknown sampling covariances, it has hardly been used, and its performance has not been empirically investigated. Therefore, we set up a simulation study, showing that also in this situation, a three-level approach yields valid results: Estimates of the treatment effects and the corresponding standard errors are unbiased.     

The multilevel latent covariate model: a new, more reliable approach to group-level effects in contextual studies

In multilevel modeling (MLM), group-level (L2) characteristics are often measured by aggregating individual-level (L1) characteristics within each group so as to assess contextual effects (e.g., group-average effects of socioeconomic status, achievement, climate). Most previous applications have used a multilevel manifest covariate (MMC) approach, in which the observed (manifest) group mean is assumed to be perfectly reliable. This article demonstrates mathematically and with simulation results that this MMC approach can result in substantially biased estimates of contextual effects and can substantially underestimate the associated standard errors, depending on the number of L1 individuals per group, the number of groups, the intraclass correlation, the sampling ratio (the percentage of cases within each group sampled), and the nature of the data. To address this pervasive problem, the authors introduce a new multilevel latent covariate (MLC) approach that corrects for unreliability at L2 and results in unbiased estimates of L2 constructs under appropriate conditions. However, under some circumstances when the sampling ratio approaches 100%, the MMC approach provides more accurate estimates. Based on 3 simulations and 2 real-data applications, the authors evaluate the MMC and MLC approaches and suggest when researchers should most appropriately use one, the other, or a combination of both approaches.