The Poor Fit of Model Fit for Selecting Number of Factors in Exploratory Factor Analysis for Scale Evaluation

Model fit indices are being increasingly recommended and used to select the number of factors in an exploratory factor analysis. Growing evidence suggests that the recommended cutoff values for common model fit indices are not appropriate for use in an exploratory factor analysis context. A particularly prominent problem in scale evaluation is the ubiquity of correlated residuals and imperfect model specification. Our research focuses on a scale evaluation context and the performance of four standard model fit indices: root mean square error of approximate (RMSEA), standardized root mean square residual (SRMR), comparative fit index (CFI), and Tucker–Lewis index (TLI), and two equivalence test-based model fit indices: RMSEAt and CFIt. We use Monte Carlo simulation to generate and analyze data based on a substantive example using the positive and negative affective schedule (N = 1,000). We systematically vary the number and magnitude of correlated residuals as well as nonspecific misspecification, to evaluate the impact on model fit indices in fitting a two-factor exploratory factor analysis. Our results show that all fit indices, except SRMR, are overly sensitive to correlated residuals and nonspecific error, resulting in solutions that are overfactored. SRMR performed well, consistently selecting the correct number of factors; however, previous research suggests it does not perform well with categorical data. In general, we do not recommend using model fit indices to select number of factors in a scale evaluation framework.     

On the Performance of Likelihood-Based Difference Tests in Nonlinear Structural Equation Models

This article investigates likelihood-based difference statistics for testing nonlinear effects in structural equation modeling using the latent moderated structural equations (LMS) approach. In addition to the standard difference statistic T_D, 2 robust statistics have been developed in the literature to ensure valid results under the conditions of nonnormality or small sample sizes: the robust T_DR and the “strictly positive” T_DRP. These robust statistics have not been examined in combination with LMS yet. In 2 Monte Carlo studies we investigate the performance of these methods for testing quadratic or interaction effects subject to different sources of nonnormality, nonnormality due to the nonlinear terms, and nonnormality due to the distribution of the predictor variables. The results indicate that T_D is preferable to both T_DR and T_DRP. Under the condition of strong nonlinear effects and nonnormal predictors, T_DR often produced negative differences and T_DRP showed no desirable power.     

Using Fit Indexes to Select a Covariance Model for Longitudinal Data

This study investigated the performance of fit indexes in selecting a covariance structure for longitudinal data. Data were simulated to follow a compound symmetry, first-order autoregressive, first-order moving average, or random-coefficients covariance structure. We examined the ability of the likelihood ratio test (LRT), root mean square error of approximation (RMSEA), comparative fit index (CFI), and Tucker–Lewis Index (TLI) to reject misspecified models with varying degrees of misspecification. With a sample size of 20, RMSEA, CFI, and TLI are high in both Type I and Type II error rates, whereas LRT has a high Type II error rate. With a sample size of 100, these indexes generally have satisfactory performance, but CFI and TLI are affected by a confounding effect of their baseline model. Akaike's Information Criterion (AIC) and Bayesian Information Criterion (BIC) have high success rates in identifying the true model when sample size is 100. A comparison with the mixed model approach indicates that separately modeling the means and covariance structures in structural equation modeling dramatically improves the success rate of AIC and BIC.     

New Ways to Evaluate Goodness of Fit: A Note on Using Equivalence Testing to Assess Structural Equation Models

Structural equation models are typically evaluated on the basis of goodness-of-fit indexes. Despite their popularity, agreeing what value these indexes should attain to confidently decide between the acceptance and rejection of a model has been greatly debated. A recently proposed approach by means of equivalence testing has been recommended as a superior way to evaluate the goodness of fit of models. The approach has also been proposed as providing a necessary vehicle that can be used to advance the inferential nature of structural equation modeling as a confirmatory tool. The purpose of this article is to introduce readers to key ideas in equivalence testing and illustrate its use for conducting model–data fit assessments. Two confirmatory factor analysis models in which a priori specified latent variable models with known structure and tested against data are used as examples. It is advocated that whenever the goodness of fit of a model is to be assessed researchers should always examine the resulting values obtained via the equivalence testing approach.     

Modeling Nonlinear Structural Equation Models: A Comparison of the Two-Stage Generalized Additive Models and the Finite Mixture Structural Equation Model

Researchers have devoted some time and effort to developing methods for fitting nonlinear relationships among latent variables. In particular, most of these have focused on correctly modeling interactions between 2 exogenous latent variables, and quadratic relationships between exogenous and endogenous variables. All of these approaches require prespecification of the nonlinearity by the researcher, and are limited to fairly simple nonlinear relationships. Other work has been done using mixture structural equation models (SEMM) in an attempt to fit more complex nonlinear relationships. This study expands on this earlier work by introducing the 2-stage generalized additive model (2SGAM) approach for fitting regression splines in the context of structural equation models. The model is first described and then investigated through the use of simulated data, in which it was compared with the SEMM approach. Results demonstrate that the 2SGAM is an effective tool for fitting a variety of nonlinear relationships between latent variables, and can be easily and accurately extended to models including multiple latent variables. Implications of these results are discussed.     

Using the Standardized Root Mean Squared Residual (SRMR) to Assess Exact Fit in Structural Equation Models

We examine the accuracy of p values obtained using the asymptotic mean and variance (MV) correction to the distribution of the sample standardized root mean squared residual (SRMR) proposed by Maydeu-Olivares to assess the exact fit of SEM models. In a simulation study, we found that under normality, the MV-corrected SRMR statistic provides reasonably accurate Type I errors even in small samples and for large models, clearly outperforming the current standard, that is, the likelihood ratio (LR) test. When data shows excess kurtosis, MV-corrected SRMR p values are only accurate in small models (p = 10), or in medium-sized models (p = 30) if no skewness is present and sample sizes are at least 500. Overall, when data are not normal, the MV-corrected LR test seems to outperform the MV-corrected SRMR. We elaborate on these findings by showing that the asymptotic approximation to the mean of the SRMR sampling distribution is quite accurate, while the asymptotic approximation to the standard deviation is not.     

正态广义非线性模型参数置信域的曲率表示

本文从几何观点研究正态广义非线性模型参数和子集参数的置信域的曲率表示,由于本文研究的非线性模型中随机误差的协方差阵可以是正定和非负定的,因此,有关结论推广与发展了现有文献中的许多结论。     

Effects of Missing Data Methods in SEM Under Conditions of Incomplete and Nonnormal Data

Using Monte Carlo simulations, this research examined the performance of four missing data methods in SEM under different multivariate distributional conditions. The effects of four independent variables (sample size, missing proportion, distribution shape, and factor loading magnitude) were investigated on six outcome variables: convergence rate, parameter estimate bias, MSE of parameter estimates, standard error coverage, model rejection rate, and model goodness of fit—RMSEA. A three-factor CFA model was used. Findings indicated that FIML outperformed the other methods in MCAR, and MI should be used to increase the plausibility of MAR. SRPI was not comparable to the other three methods in either MCAR or MAR.     

Going Beyond Convergence in Bayesian Estimation: Why Precision Matters Too and How to Assess It

Most of the software that is available to implement Bayesian approaches uses Markov chain Monte Carlo (MCMC) methods. It is our impression that many researchers are primarily concerned with convergence as assessed by the Potential Scale Reduction (PSR) and that other aspects of MCMC are largely ignored. In this article, we argue that the precision with which the Bayesian estimates are approximated by summary statistics for the MCMC chain is essential to ensure good statistical properties. We discuss the Effective Sample Size (ESS), which indicates how well an estimate is approximated, and present evidence from two simulation studies and an example from organizational research to support our claim that researchers should be concerned not only with convergence but also with precision, particularly when a multilevel model is estimated. In addition, we demonstrate how Mplus can be modified so that users can control the ESS, and we conclude with recommendations.     

Investigating the Sensitivity of Goodness-of-Fit Indices to Detect Measurement Invariance in a Bifactor Model

A Monte Carlo simulation study was conducted to evaluate the sensitivities of the likelihood ratio test and five commonly used delta goodness-of-fit (ΔGOF) indices (i.e., ΔGamma, ΔMcDonald’s, ΔCFI, ΔRMSEA, and ΔSRMR) to detect a lack of metric invariance in a bifactor model. Experimental conditions included factor loading differences, location and number of noninvariant items, and sample size. The results indicated all ΔGOF indices held Type I error to a minimum and overall had adequate power for the study. For detecting the violation of metric invariance, only ΔGamma and ΔCFI, in addition to Δχ², are recommended to use in the bifactor model with values of ?.016 to ?.023 and ?.003 to ?.004, respectively. Moreover, in the variance component analysis, the magnitude of the factor loading differences contributed the most variation to all ΔGOF indices, whereas sample size affected Δχ² the most.     

Validating Student Score Inferences With Person‐Fit Statistic and Verbal Reports: A Person‐Fit Study for Cognitive Diagnostic Assessment

The goal of this study was to investigate the usefulness of person‐fit analysis in validating student score inferences in a cognitive diagnostic assessment. In this study, a two‐stage procedure was used to evaluate person fit for a diagnostic test in the domain of statistical hypothesis testing. In the first stage, the person‐fit statistic, the hierarchy consistency index (HCI; Cui, 2007 ; Cui & Leighton, 2009 ), was used to identify the misfitting student item‐score vectors. In the second stage, students’ verbal reports were collected to provide additional information about students’ response processes so as to reveal the actual causes of misfits. This two‐stage procedure helped to identify the misfits of item‐score vectors to the cognitive model used in the design and analysis of the diagnostic test, and to discover the reasons of misfits so that students’ problem‐solving strategies were better understood and their performances were interpreted in a more meaningful way.     

Combining IRT and SEM: A Hybrid Model for Fitting Responses and Response Certainties

This article proposes a model-based procedure, intended for personality measures, for exploiting the auxiliary information provided by the certainty with which individuals answer every item (response certainty). This information is used to (a) obtain more accurate estimates of individual trait levels, and (b) provide a more detailed assessment of the consistency with which the individual responds to the test. The basis model consists of 2 submodels: an item response theory submodel for the responses, and a linear-in-the-coefficients submodel that describes the response certainties. The latter is based on the distance-difficulty hypothesis, and is parameterized as a factor-analytic model. Procedures for (a) estimating the structural parameters, (b) assessing model–data fit, (c) estimating the individual parameters, and (d) assessing individual fit are discussed. The proposal was used in an empirical study. Model–data fit was acceptable and estimates were meaningful. Furthermore, the precision of the individual trait estimates and the assessment of the individual consistency improved noticeably.     

Testing Time-Invariance of Variable Specificity in Repeated Measure Designs Using Structural Equation Modeling

A structural equation modeling method for examining time-invariance of variable specificity in longitudinal studies with multiple measures is outlined, which is developed within a confirmatory factor-analytic framework. The approach represents a likelihood ratio test for the hypothesis of stability in the specificity part of the residual term associated with repeated administration of each measure. The procedure can be used in the search for parsimonious versions of multiwave multiple-indicator models, to test for variable specificity in them, and to examine assumptions underlying particular parameter estimation procedures in repeated measure designs. The outlined method is illustrated with empirical data.     

Assessing the Sensitivity of Weighted Least Squares Model Fit Indexes to Local Dependence in Item Response Theory Models

Given the relationships of item response theory (IRT) models to confirmatory factor analysis (CFA) models, IRT model misspecifications might be detectable through model fit indexes commonly used in categorical CFA. The purpose of this study is to investigate the sensitivity of weighted least squares with adjusted means and variance (WLSMV)-based root mean square error of approximation, comparative fit index, and Tucker–Lewis Index model fit indexes to IRT models that are misspecified due to local dependence (LD). It was found that WLSMV-based fit indexes have some functional relationships to parameter estimate bias in 2-parameter logistic models caused by violations of LD. Continued exploration into these functional relationships and development of LD-detection methods based on such relationships could hold much promise for providing IRT practitioners with global information on violations of local independence.     

A Survey of the Social Development of the 10th, 11th,and 12th Grade Pupils in a Small High School

A potential weakness of quantitative methods is the intrusive nature of testing in general, and pretesting specifically. The Solomon four-group design ameliorates this difficulty, but because of statistical issues there are few published examples. W. Braver and Braver (1988) suggested the use of meta-analytic Stouffer's Z to combine the data from all four groups. Sawilowsky and Markman (1990a, 1990b, 1990c, 1990d) and Braver and W. Braver (1990a, 1990b) exchanged differing opinions on this approach. The present study is a Monte Carlo demonstration that the experiment-wise error rate inflates nearly triple nominal alpha. However, when not conducted as conditional tests, traditional procedures are more powerful than this meta-analytic approach, despite the ability of Stouffer's Z to combine all available data into a single statistic.     

The Model Size Effect in SEM: Inflated Goodness-of-Fit Statistics Are Due to the Size of the Covariance Matrix

The size of a model has been shown to critically affect the goodness of approximation of the model fit statistic T to the asymptotic chi-square distribution in finite samples. It is not clear, however, whether this “model size effect” is a function of the number of manifest variables, the number of free parameters, or both. It is demonstrated by means of 2 Monte Carlo computer simulation studies that neither the number of free parameters to be estimated nor the model degrees of freedom systematically affect the T statistic when the number of manifest variables is held constant. Increasing the number of manifest variables, however, is associated with a severe bias. These results imply that model fit drastically depends on the size of the covariance matrix and that future studies involving goodness-of-fit statistics should always consider the number of manifest variables, but can safely neglect the influence of particular model specifications.     

Bayesian Meta-Analytic SEM: A One-Stage Approach to Modeling Between-Studies Heterogeneity in Structural Parameters

Meta-analytic structural equation modeling (MASEM) refers to a set of meta-analysis techniques for combining and comparing structural equation modeling (SEM) results from multiple studies. Existing approaches to MASEM cannot appropriately model between-studies heterogeneity in structural parameters because of missing correlations, lack model fit assessment, and suffer from several theoretical limitations. In this study, we address the major shortcomings of existing approaches by proposing a novel Bayesian multilevel SEM approach. Simulation results showed that the proposed approach performed satisfactorily in terms of parameter estimation and model fit evaluation when the number of studies and the within-study sample size were sufficiently large and when correlations were missing completely at random. An empirical example about the structure of personality based on a subset of data was provided. Results favored the third factor structure over the hierarchical structure. We end the article with discussions and future directions.     

The Misspecification of the Covariance Structures in Multilevel Models for Single-Case Data: A Monte Carlo Simulation Study

The impact of misspecifying covariance matrices at the second and third levels of the three-level model is evaluated. Results indicate that ignoring existing covariance has no effect on the treatment effect estimate. In addition, the between-case variance estimates are unbiased when covariance is either modeled or ignored. If the research interest lies in the between-study variance estimate, including at least 30 studies is warranted. Modeling covariance does not result in less biased between-study variance estimates as the between-study covariance estimate is biased. When the research interest lies in the between-case covariance, the model including covariance results in unbiased between-case variance estimates. The three-level model appears to be less appropriate for estimating between-study variance if fewer than 30 studies are included.     

A Specification Error Test That Uses Instrumental Variables to Detect Latent Quadratic and Latent Interaction Effects

The relations between the latent variables in structural equation models are typically assumed to be linear in form. This article aims to explain how a specification error test using instrumental variables (IVs) can be employed to detect unmodeled interactions between latent variables or quadratic effects of latent variables. An empirical example is presented, and the results of a simulation study are reported to evaluate the sensitivity and specificity of the test and compare it with the commonly employed chi-square model test. The results show that the proposed test can identify most unmodeled latent interactions or latent quadratic effects in moderate to large samples. Furthermore, its power is higher when the number of indicators used to define the latent variables is large. Altogether, this article shows how the IV-based test can be applied to structural equation models and that it is a valuable tool for researchers using structural equation models.     

Evaluating Model Fit and Structural Coefficient Bias: A Bayesian Approach to Multilevel Bifactor Model Misspecification

Appropriate model specification is fundamental to unbiased parameter estimates and accurate model interpretations in structural equation modeling. Thus detecting potential model misspecification has drawn the attention of many researchers. This simulation study evaluates the efficacy of the Bayesian approach (the posterior predictive checking, or PPC procedure) under multilevel bifactor model misspecification (i.e., ignoring a specific factor at the within level). The impact of model misspecification on structural coefficients was also examined in terms of bias and power. Results showed that the PPC procedure performed better in detecting multilevel bifactor model misspecification, when the misspecification became more severe and sample size was larger. Structural coefficients were increasingly negatively biased at the within level, as model misspecification became more severe. Model misspecification at the within level affected the between-level structural coefficient estimates more when data dependency was lower and the number of clusters was smaller. Implications for researchers are discussed.