Effect Sizes of Model Misfit in Structural Equation Models

Methodology ◽  
2017 ◽  
Vol 13 (Supplement 1) ◽  
pp. 23-30 ◽  
Author(s):  
Alberto Maydeu-Olivares ◽  
Dexin Shi
Keyword(s):  
Structural Equation ◽  
Structural Equation Models ◽  
Model Misspecification ◽  
Covariance Structure ◽  
Effect Sizes ◽  
Small Samples ◽  
Type I ◽  
Model Misfit ◽  
Covariance Structure Models ◽  
Residual Correlations

Abstract. Residual correlations and covariances provide effect sizes of the misfit of covariance structure models. In a simulation study, we found that accurate confidence intervals (CIs) for standardized residual covariances are obtained even in small samples (N = 100), regardless of model size, degree of model misspecification, and data distribution. Standardized residual covariances also provide information about the source of misfit in poorly fitting models. From this viewpoint, they may be considered an alternative to modification indices. We compared the empirical Type I errors and power rates of standardized residual covariances and modification indices and found that both procedures provide nearly identical rates across the simulation conditions investigated. Residual correlations and standardized residual covariances provide very similar results.

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

2020 ◽  
pp. 001316442092623 ◽  
Author(s):  
Goran Pavlov ◽  
Alberto Maydeu-Olivares ◽  
Dexin Shi
Keyword(s):  
Asymptotic Approximation ◽  
Structural Equation ◽  
Structural Equation Models ◽  
Small Samples ◽  
Type I ◽  
Current Standard ◽  
Type I Errors ◽  
P Values ◽  
Lr Test ◽  
Small 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.

The Robustness of the Likelihood Ratio Chi-Square Test for Structural Equation Models: A Meta-Analysis

2001 ◽  
Vol 26 (1) ◽  
pp. 105-132 ◽  
Author(s):  
Douglas A. Powell ◽  
William D. Schafer
Keyword(s):  
Structural Equation ◽  
Structural Equation Models ◽  
Type I Error ◽  
Meta Analysis ◽  
Generalized Least Squares ◽  
Error Rates ◽  
Type I ◽  
Chi Square ◽  
Distribution Free ◽  
Projection Techniques

The robustness literature for the structural equation model was synthesized following the method of Harwell which employs meta-analysis as developed by Hedges and Vevea. The study focused on the explanation of empirical Type I error rates for six principal classes of estimators: two that assume multivariate normality (maximum likelihood and generalized least squares), elliptical estimators, two distribution-free estimators (asymptotic and others), and latent projection. Generally, the chi-square tests for overall model fit were found to be sensitive to non-normality and the size of the model for all estimators (with the possible exception of the elliptical estimators with respect to model size and the latent projection techniques with respect to non-normality). The asymptotic distribution-free (ADF) and latent projection techniques were also found to be sensitive to sample sizes. Distribution-free methods other than ADF showed, in general, much less sensitivity to all factors considered.

3. Multilevel Covariance Structure Analysis by Fitting Multiple Single-Level Models

2007 ◽  
Vol 37 (1) ◽  
pp. 53-82 ◽  
Author(s):  
Ke-Hai Yuan ◽  
Peter M. Bentler
Keyword(s):  
Structure Analysis ◽  
Model Misspecification ◽  
Covariance Structure ◽  
Covariance Structure Analysis ◽  
Test Statistics ◽  
Single Level ◽  
Misspecified Model ◽  
Simultaneous Procedure ◽  
Error Formulas ◽  
Covariance Structure Models

Data in social and behavioral sciences are often hierarchically organized. Multilevel statistical procedures have been developed to analyze such data while taking into account the dependence of observations. When simultaneously evaluating models at all levels, a significant statistic provides no information on the level at which the model is misspecified. Model misspecification can exist at one or several levels simultaneously. When one level is misspecified, the other levels may be affected even when they are correctly specified. Motivated by these observations, we propose to separate a multilevel covariance structure into multiple single-level covariance structure models and to fit these single-level models as in conventional covariance structure analysis. A procedure for segregating the multilevel model into single-level models is developed. Five test statistics for evaluating a model at each level are provided. Standard error formulas for the separate estimators are also provided, and their efficiency is compared to simultaneous estimators. Empirical and Monte Carlo results demonstrate the advantages of the segregated procedure over the simultaneous procedure. Computer programs that will allow the developed procedure to be used in practice are also presented.

scholarly journals Correcting measurement error bias in interaction models with small samples

2006 ◽  
Vol 3 (2) ◽  
Author(s):  
Josep Bisbe ◽  
Germà Coenders ◽  
Willem Saris ◽  
Joan Batista-Foguet
Keyword(s):  
Measurement Error ◽  
Least Squares ◽  
Structural Equation ◽  
Structural Equation Models ◽  
Ordinary Least Squares ◽  
Alternative Methods ◽  
Small Samples ◽  
Least Squares Regression ◽  
Interaction Terms ◽  
Error Bias

Several methods have been suggested to estimate non-linear models with interaction terms in the presence of measurement error. Structural equation models eliminate measurement error bias, but require large samples. Ordinary least squares regression on summated scales, regression on factor scores and partial least squares are appropriate for small samples but do not correct measurement error bias. Two stage least squares regression does correct measurement error bias but the results strongly depend on the instrumental variable choice. This article discusses the old disattenuated regression method as an alternative for correcting measurement error in small samples. The method is extended to the case of interaction terms and is illustrated on a model that examines the interaction effect of innovation and style of use of budgets on business performance. Alternative reliability estimates that can be used to disattenuate the estimates are discussed. A comparison is made with the alternative methods. Methods that do not correct for measurement error bias perform very similarly and considerably worse than disattenuated regression.

scholarly journals The Use of Mixed Models for the Analysis of Mediated Data with Time-Dependent Predictors

2011 ◽  
Vol 2011 ◽  
pp. 1-12 ◽  
Author(s):  
Emily A. Blood ◽  
Debbie M. Cheng
Keyword(s):  
Longitudinal Data ◽  
Mixed Models ◽  
Structural Equation ◽  
Structural Equation Models ◽  
Type I Error ◽  
Time Dependent ◽  
Type I ◽  
Delayed Effects ◽  
Causal Pathways ◽  
Observational Cohort

Linear mixed models (LMMs) are frequently used to analyze longitudinal data. Although these models can be used to evaluate mediation, they do not directly model causal pathways. Structural equation models (SEMs) are an alternative technique that allows explicit modeling of mediation. The goal of this paper is to evaluate the performance of LMMs relative to SEMs in the analysis of mediated longitudinal data with time-dependent predictors and mediators. We simulated mediated longitudinal data from an SEM and specified delayed effects of the predictor. A variety of model specifications were assessed, and the LMMs and SEMs were evaluated with respect to bias, coverage probability, power, and Type I error. Models evaluated in the simulation were also applied to data from an observational cohort of HIV-infected individuals. We found that when carefully constructed, the LMM adequately models mediated exposure effects that change over time in the presence of mediation, even when the data arise from an SEM.

The Service–Profit Chain: A Meta-Analytic Test of a Comprehensive Theoretical Framework

2017 ◽  
Vol 81 (3) ◽  
pp. 41-61 ◽  
Author(s):  
Jens Hogreve ◽  
Anja Iseke ◽  
Klaus Derfuss ◽  
Tönnjes Eller
Keyword(s):  
Service Quality ◽  
Employee Satisfaction ◽  
Structural Equation ◽  
Structural Equation Models ◽  
Meta Analysis ◽  
Customer Relationship ◽  
Effect Sizes ◽  
Comprehensive Test ◽  
Service Managers ◽  
Service Profit Chain

The service–profit chain (SPC) has served as a prominent guidepost for service managers and researchers alike. This meta-analysis provides the first comprehensive test of the SPC, showing that all the proposed links are statistically significant and substantial. However, the effect sizes vary considerably, partly according to the type of service provided. Meta-analytic structural equation models show that internal service quality translates into service performance through various mechanisms beyond employee satisfaction, and they highlight the importance of the service encounter and customer relationship characteristics for customer responses. The findings not only indicate the need to integrate complementary paths in the SPC framework but also challenge the implicit SPC rationale that firms should always maximize employee satisfaction and external service quality to optimize firm performance.

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