On the Computation of the RMSEA and CFI from the Mean-And-Variance Corrected Test Statistic with Nonnormal Data in SEM

A new type of nonnormality correction to the RMSEA has recently been developed, which has several advantages over existing corrections. In particular, the new correction adjusts the sample estimate of the RMSEA for the inflation due to nonnormality, while leaving its population value unchanged, so that established cutoff criteria can still be used to judge the degree of approximate fit. A confidence interval (CI) for the new robust RMSEA based on the mean-corrected (“Satorra-Bentler”) test statistic has also been proposed. Follow up work has provided the same type of nonnormality correction for the CFI (Brosseau-Liard & Savalei, 2014). These developments have recently been implemented in lavaan. This note has three goals: a) to show how to compute the new robust RMSEA and CFI from the mean-and-variance corrected test statistic; b) to offer a new CI for the robust RMSEA based on the mean-and-variance corrected test statistic; and c) to caution that the logic of the new nonnormality corrections to RMSEA and CFI is most appropriate for the maximum likelihood (ML) estimator, and cannot easily be generalized to the most commonly used categorical data estimators.     

Asymptotic biases in exploratory factor analysis and structural equation modeling

Formulas for the asymptotic biases of the parameter estimates in structural equation models are provided in the case of the Wishart maximum likelihood estimation for normally and nonnormally distributed variables. When multivariate normality is satisfied, considerable simplification is obtained for the models of unstandardized variables. Formulas for the models of standardized variables are also provided. Numerical examples with Monte Carlo simulations in factor analysis show the accuracy of the formulas and suggest the asymptotic robustness of the asymptotic biases with normality assumption against nonnormal data. Some relationships between the asymptotic biases and other asymptotic values are discussed.The author is indebted to the editor and anonymous reviewers for their comments, corrections, and suggestions on this paper, and to Yutaka Kano for discussion on biases.     

A Limited Information Estimator for Dynamic Factor Models

Structural equation modeling (SEM) is an increasingly popular method for examining multivariate time series data. As in cross-sectional data analysis, structural misspecification of time series models is inevitable, and further complicated by the fact that errors occur in both the time series and measurement components of the model. In this article, we introduce a new limited information estimator and local fit diagnostic for dynamic factor models within the SEM framework. We demonstrate the implementation of this estimator and examine its performance under both correct and incorrect model specifications via a small simulation study. The estimates from this estimator are compared to those from the most common system-wide estimators and are found to be more robust to the structural misspecifications considered.     

On normal theory based inference for multilevel models with distributional violations

Data in social and behavioral sciences are often hierarchically organized though seldom normal, yet normal theory based inference procedures are routinely used for analyzing multilevel models. Based on this observation, simple adjustments to normal theory based results are proposed to minimize the consequences of violating normality assumptions. For characterizing the distribution of parameter estimates, sandwich-type covariance matrices are derived. Standard errors based on these covariance matrices remain consistent under distributional violations. Implications of various covariance estimators are also discussed. For evaluating the quality of a multilevel model, a rescaled statistic is given for both the hierarchical linear model and the hierarchical structural equation model. The rescaled statistic, improving the likelihood ratio statistic by estimating one extra parameter, approaches the same mean as its reference distribution. A simulation study with a 2-level factor model implies that the rescaled statistic is preferable.This research was supported by grants DA01070 and DA00017 from the National Institute on Drug Abuse and a University of North Texas faculty research grant. We would like to thank the Associate Editor and two reviewers for suggestions that helped to improve the paper.     

Evaluating Structural Equation Models for Categorical Outcomes: A New Test Statistic and a Practical Challenge of Interpretation

This research is concerned with two topics in assessing model fit for categorical data analysis. The first topic involves the application of a limited-information overall test, introduced in the item response theory literature, to structural equation modeling (SEM) of categorical outcome variables. Most popular SEM test statistics assess how well the model reproduces estimated polychoric correlations. In contrast, limited-information test statistics assess how well the underlying categorical data are reproduced. Here, the recently introduced C₂ statistic of Cai and Monroe (2014) is applied. The second topic concerns how the root mean square error of approximation (RMSEA) fit index can be affected by the number of categories in the outcome variable. This relationship creates challenges for interpreting RMSEA. While the two topics initially appear unrelated, they may conveniently be studied in tandem since RMSEA is based on an overall test statistic, such as C₂. The results are illustrated with an empirical application to data from a large-scale educational survey.     

A Comparison of ML,WLSMV, and Bayesian Methods for Multilevel Structural Equation Models in Small Samples: A Simulation Study

Multilevel structural equation models are increasingly applied in psychological research. With increasing model complexity, estimation becomes computationally demanding, and small sample sizes pose further challenges on estimation methods relying on asymptotic theory. Recent developments of Bayesian estimation techniques may help to overcome the shortcomings of classical estimation techniques. The use of potentially inaccurate prior information may, however, have detrimental effects, especially in small samples. The present Monte Carlo simulation study compares the statistical performance of classical estimation techniques with Bayesian estimation using different prior specifications for a two-level SEM with either continuous or ordinal indicators. Using two software programs (Mplus and Stan), differential effects of between- and within-level sample sizes on estimation accuracy were investigated. Moreover, it was tested to which extent inaccurate priors may have detrimental effects on parameter estimates in categorical indicator models. For continuous indicators, Bayesian estimation did not show performance advantages over ML. For categorical indicators, Bayesian estimation outperformed WLSMV solely in case of strongly informative accurate priors. Weakly informative inaccurate priors did not deteriorate performance of the Bayesian approach, while strong informative inaccurate priors led to severely biased estimates even with large sample sizes. With diffuse priors, Stan yielded better results than Mplus in terms of parameter estimates.     

At the Frontiers of Modeling Intensive Longitudinal Data: Dynamic Structural Equation Models for the Affective Measurements from the COGITO Study

With the growing popularity of intensive longitudinal research, the modeling techniques and software options for such data are also expanding rapidly. Here we use dynamic multilevel modeling, as it is incorporated in the new dynamic structural equation modeling (DSEM) toolbox in Mplus, to analyze the affective data from the COGITO study. These data consist of two samples of over 100 individuals each who were measured for about 100 days. We use composite scores of positive and negative affect and apply a multilevel vector autoregressive model to allow for individual differences in means, autoregressions, and cross-lagged effects. Then we extend the model to include random residual variances and covariance, and finally we investigate whether prior depression affects later depression scores through the random effects of the daily diary measures. We end with discussing several urgent—but mostly unresolved—issues in the area of dynamic multilevel modeling.     

Dyadic measurement invariance and its importance for replicability in romantic relationship science

Comparisons of group means, variances, correlations, and/or regression slopes involving psychological variables rely on an assumption of measurement invariance—that the latent variables under investigation have equivalent meaning and measurement across group. When measures are noninvariant, replicability suffers, as comparisons are either conceptually meaningless, or hindered by inflated Type I error rates. We propose that the failure to account for interdependence among dyad members when testing measurement invariance may be a potential source of unreplicable findings in relationship research. We developed fully dyadic versions of invariance models, created an R package (dySEM) to make specifying dyadic invariance models easier and reporting more reproducible, and executed a Registered Report for gauging the extent of dyadic (non)invariance in romantic relationship research across measures of relationship well‐being, personality, and sexuality in a sample of 282 heterosexual couples. We found that although a number of popular measures display good evidence of dyadic invariance, a few display concerning levels and interesting patterns of noninvariance, while others appeared either noninvariant or poorly fitting for both men and women. We discuss our findings in terms of their meaning for the replicability dyadic close relationship research. We close by arguing that increased theorizing and research on dyadic invariance, and inclusive methods for analyzing invariance with indistinguishable dyads, are needed to capitalize on the opportunity to advance our field's understanding of dyadic constructions of relational concepts.     

A Computationally More Efficient and More Accurate Stepwise Approach for Correcting for Sampling Error and Measurement Error

Over the last decade or two, multilevel structural equation modeling (ML-SEM) has become a prominent modeling approach in the social sciences because it allows researchers to correct for sampling and measurement errors and thus to estimate the effects of Level 2 (L2) constructs without bias. Because the latent variable modeling software Mplus uses maximum likelihood (ML) by default, many researchers in the social sciences have applied ML to obtain estimates of L2 regression coefficients. However, one drawback of ML is that covariance matrices of the predictor variables at L2 tend to be degenerate, and thus, estimates of L2 regression coefficients tend to be rather inaccurate when sample sizes are small. In this article, I show how an approach for stabilizing covariance matrices at L2 can be used to obtain more accurate estimates of L2 regression coefficients. A simulation study is conducted to compare the proposed approach with ML, and I illustrate its application with an example from organizational research.     

A unified approach to exploratory factor analysis with missing data,nonnormal data,and in the presence of outliers

Factor analysis is regularly used for analyzing survey data. Missing data, data with outliers and consequently nonnormal data are very common for data obtained through questionnaires. Based on covariance matrix estimates for such nonstandard samples, a unified approach for factor analysis is developed. By generalizing the approach of maximum likelihood under constraints, statistical properties of the estimates for factor loadings and error variances are obtained. A rescaled Bartlett-corrected statistic is proposed for evaluating the number of factors. Equivariance and invariance of parameter estimates and their standard errors for canonical, varimax, and normalized varimax rotations are discussed. Numerical results illustrate the sensitivity of classical methods and advantages of the proposed procedures.This project was supported by a University of North Texas Faculty Research Grant, Grant #R49/CCR610528 for Disease Control and Prevention from the National Center for Injury Prevention and Control, and Grant DA01070 from the National Institute on Drug Abuse. The results do not necessarily represent the official view of the funding agencies. The authors are grateful to three reviewers for suggestions that improved the presentation of this paper.     

A Comprehensive Examination of the Model Underlying Acceptance and Commitment Therapy for Chronic Pain

The therapeutic model underlying Acceptance and Commitment Therapy (ACT) is reasonably well-established as it applies to chronic pain. Several studies have examined measures of single ACT processes, or subsets of processes, and have almost uniformly indicated reliable relations with patient functioning. To date, however, no study has performed a comprehensive examination of the entire ACT model, including all of its component processes, as it relates to functioning. The present study performed this examination in 274 individuals with chronic pain presenting for an assessment appointment. Participants completed a battery of self-report questionnaires, assessing multiple aspects of the ACT model, as well as pain intensity, disability, and emotional distress. Initial exploratory factor analyses examined measures of the ACT model and measures of patient functioning separately with each analysis identifying three factors. Next, the fit of a model including ACT processes on the one hand and patient functioning on the other was examined using Structural Equation Modeling. Overall model fit was acceptable and indicated moderate correlations among the ACT processes themselves, as well as significant relations with pain intensity, emotional distress, and disability. These analyses build on the existing literature by providing, to our knowledge, the most comprehensive evaluation of the ACT theoretical model in chronic pain to date.     

Antecedents and Consequences of Victimization of Lesbian, Gay, and Bisexual Young People: A Structural Model Comparing Rural University and Urban Samples

Research indicates that antigay victimization is widespread and that lesbian, gay, and bisexual young people may be very vulnerable to such victimization. The current study builds upon previous work by Hershberger and D'Augelli (1995), who studied the consequences of sexual orientation-based victimization in 194 urban lesbian, gay, and bisexual youths. Using structural equation modeling, the present study models both antecedents and consequences (including psychological distress, self-esteem, and suicidality) of victimization via a secondary analysis of their data set. In addition, a second sample of 54 lesbian, gay, and bisexual youths from a rural university setting was examined to cross-validate and generalize the relationships found in urban settings. Results indicated that a revised model of victimization exhibited sufficient fit to the urban sample data and provided preliminary support for the generalizability of the model beyond the initial sample. Additional similarities were found between the urban and rural university community samples, including a high prevalence of reported suicide attempts: 42% of the urban sample and 32% of the rural university sample had attempted suicide at least once. Results indicated that victimization based on sexual orientation has similar correlates for young people in different community settings.     

The Performance of Multivariate Methods for Two-Group Comparisons with Small Samples and Incomplete Data

Abstract

In intervention studies having multiple outcomes, researchers often use a series of univariate tests (e.g., ANOVAs) to assess group mean differences. Previous research found that this approach properly controls Type I error and generally provides greater power compared to MANOVA, especially under realistic effect size and correlation combinations. However, when group differences are assessed for a specific outcome, these procedures are strictly univariate and do not consider the outcome correlations, which may be problematic with missing outcome data. Linear mixed or multivariate multilevel models (MVMMs), implemented with maximum likelihood estimation, present an alternative analysis option where outcome correlations are taken into account when specific group mean differences are estimated. In this study, we use simulation methods to compare the performance of separate independent samples t tests estimated with ordinary least squares and analogous t tests from MVMMs to assess two-group mean differences with multiple outcomes under small sample and missingness conditions. Study results indicated that a MVMM implemented with restricted maximum likelihood estimation combined with the Kenward–Roger correction had the best performance. Therefore, for intervention studies with small N and normally distributed multivariate outcomes, the Kenward–Roger procedure is recommended over traditional methods and conventional MVMM analyses, particularly with incomplete data.     

The two roads of passionate goal pursuit: links with appraisal,coping, and academic achievement

Background and Objectives: In this research, we tested the role of cognitive appraisals in explaining why harmonious and obsessive passion dimensions are related to distinct forms of coping and explored if performance was impacted by these appraisal and coping processes. Design: Undergraduate students (N = 489) participated in a longitudinal study and completed three surveys throughout the course of an academic year. Methods: Participants completed assessments of both passion dimensions (Time 1), reported how they were appraising and coping with the mid-year examination period (Time 2), and provided consent to obtain their final grade in Introductory Psychology (Time 3). The hypothesized model was tested using structural equation modeling. Results: Harmonious and obsessive passion dimensions were linked with approach and avoidant coping responses, respectively. Cognitive appraisals, particularly appraisals of challenge and uncontrollability, played an indirect role in these relationships. In addition, both appraisals and coping responses had an indirect effect in the relationship between passion dimensions and final grade. Conclusions: These results identify cognitive appraisal as a reason why passion dimensions are linked with distinct coping tendencies and demonstrate the role of appraisal and coping processes in the journey to passionate goal attainment.     

Effect size, power, and sample size determination for structured means modeling and mimic approaches to between-groups hypothesis testing of means on a single latent construct

While effect size estimates, post hoc power estimates, and a priori sample size determination are becoming a routine part of univariate analyses involving measured variables (e.g., ANOVA), such measures and methods have not been articulated for analyses involving latent means. The current article presents standardized effect size measures for latent mean differences inferred from both structured means modeling and MIMIC approaches to hypothesis testing about differences among means on a single latent construct. These measures are then related to post hoc power analysis, a priori sample size determination, and a relevant measure of construct reliability.I wish to convey my appreciation to the reviewers and Associate Editor, whose suggestions extended and strengthened the article's content immensely, and to Ralph Mueller of The George Washington University for enhancing the clarity of its presentation.     

Classical test theory based and factor analytic structural models for analyzing change: A note on differences and similarities

This note is concerned with differences and similarities between structural models for analyzing change, which are conceptualized within two different modelling traditions: the one based on the classical test theory, and that within the factor-analytic approach. It is shown that these two possibilities lead to models for studying change, which are indistinguishable when using for data analytic purposes structural modeling programs, such as LISREL, EQS, COSAN, LISCOMP, RAMONA, EzPATH, SAS PROC CALIS. The reason for this data-analytic equivalence of the two conceptually different types of models is the confounding of their differences in the corresponding implied covariance matrix structures.     

Modeling Interactions Between Latent Variables in Research on Type D Personality: A Monte Carlo Simulation and Clinical Study of Depression and Anxiety

Several approaches exist to model interactions between latent variables. However, it is unclear how these perform when item scores are skewed and ordinal. Research on Type D personality serves as a good case study for that matter. In Study 1, we fitted a multivariate interaction model to predict depression and anxiety with Type D personality, operationalized as an interaction between its two subcomponents negative affectivity (NA) and social inhibition (SI). We constructed this interaction according to four approaches: (1) sum score product; (2) single product indicator; (3) matched product indicators; and (4) latent moderated structural equations (LMS). In Study 2, we compared these interaction models in a simulation study by assessing for each method the bias and precision of the estimated interaction effect under varying conditions. In Study 1, all methods showed a significant Type D effect on both depression and anxiety, although this effect diminished after including the NA and SI quadratic effects. Study 2 showed that the LMS approach performed best with respect to minimizing bias and maximizing power, even when item scores were ordinal and skewed. However, when latent traits were skewed LMS resulted in more false-positive conclusions, while the Matched PI approach adequately controlled the false-positive rate.     

The Influence of Number of Categories and Threshold Values on Fit Indices in Structural Equation Modeling with Ordered Categorical Data

This study examines the unscaled and scaled root mean square error of approximation (RMSEA), comparative fit index (CFI), and Tucker–Lewis index (TLI) of diagonally weighted least squares (DWLS) and unweighted least squares (ULS) estimators in structural equation modeling with ordered categorical data. We show that the number of categories and threshold values for categorization can unappealingly impact the DWLS unscaled and scaled fit indices, as well as the ULS scaled fit indices in the population, given that analysis models are misspecified and that the threshold structure is saturated. Consequently, a severely misspecified model may be considered acceptable, depending on how the underlying continuous variables are categorized. The corresponding CFI and TLI are less dependent on the categorization than RMSEA but are less sensitive to model misspecification in general. In contrast, the number of categories and threshold values do not impact the ULS unscaled fit indices in the population.     

互联网使用动机、行为与其社会-心理健康的模型构建

以581名大学生为被试,采用结构方程模型技术构建了互联网使用动机、病理性互联网使用行为与其相关社会-心理健康的关系模型。其中,互联网使用动机包括信息获取性动机和人际情感性动机两种模式;病理性互联网使用行为包括上网冲动性、分离/逃避和网上优越感三个初级因素;互联网相关社会-心理健康包括孤独感、社会参与度、一般抑郁、生活幸福感和生活满意度五个初级因素,并进一步概括为社会健康和心理健康两个维度。研究结果表明,基于信息获取性动机而使用互联网有助于相关社会-心理健康水平的提高;基于人际情感性动机而使用互联网更容易导致病理性互联网使用行为,并由此对使用者的社会-心理健康产生负面影响;大学生使用互联网的积极效应大于消极效应,且信息获取性动机对社会健康具有更大的积极效应,而人际情感性动机对对心理健康具有更大的消极效应