Influence of sample size, estimation method, and model specification on goodness-of-fit assessments in structural equation models.

[Correction Notice: An erratum for this article was reported in Vol 75(1) of Journal of Applied Psychology (see record 2008-10492-001). An error exists in Figure 2 and the accompanying text of the article. The corrected information is included in the erratum.] The problem of assessing fit of structural equation models is reviewed, and two sampling studies are reported that examine the effects of sample size, estimation method, and model misspecification on fit indices. In the first study, the behavior of indices in a known-population confirmatory factor analysis model is considered. In the second study, the same problem in an empirical data set is examined by looking at antecedents and consequences of work motivation. The findings across the two studies suggest that (a) as might be expected, sample size is an important determinant in assessing model fit; (b) estimator-specific, as opposed to estimator-general, fit indices provide more accurate indications of model fit; and (c) the studied fit indices are differentially sensitive to model misspecification. Some recommendations for the use of structural equation model fit indices are given. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

On specifying the null model for incremental fit indices in structural equation modeling.

In structural equation modeling, incremental fit indices are based on the comparison of the fit of a substantive model to that of a null model. The standard null model yields unconstrained estimates of the variance (and mean, if included) of each manifest variable. For many models, however, the standard null model is an improper comparison model. In these cases, incremental fit index values reported automatically by structural modeling software have no interpretation and should be disregarded. The authors explain how to formulate an acceptable, modified null model, predict changes in fit index values accompanying its use, provide examples illustrating effects on fit index values when using such a model, and discuss implications for theory and practice of structural equation modeling. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

On measures of explained variance in nonrecursive structural equation models.

Whereas measures of explained variance in a regression and an equation of a recursive structural equation model can be simply summarized by a standard R2 measure, this is not possible in nonrecursive models in which there are reciprocal interdependencies among variables. This article provides a general approach to defining variance explained in latent dependent variables of nonrecursive linear structural equation models. A new method of its estimation, easily implemented in EQS or LISREL and available in EQS 6, is described and illustrated. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

An introduction to using structural equation models in rehabilitation psychology.

Objective: To provide an overview of structural equation modeling (SEM) using an example drawn from the rehabilitation psychology literature. Design: To illustrate the 5 steps in SEM (model specification, identification, estimation methods, interpretation of results, and model modification), an example is presented, with details on determining whether alternative models result in a significant improvement to fit to the observed data. Data are from a sample of 274 people with spinal cord injury. Issues commonly encountered in preparing data for SEM analyses (e.g., missing data, nonnormality) are reviewed, as is the debate surrounding some aspects of SEM (e.g., acceptable sample size). Conclusion: SEM can be a powerful procedure for empirically representing complex and sophisticated theoretical models of interest to rehabilitation psychologists. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

A cautionary note on measurement error corrections in structural equation models.

It is commonly thought that structural equation modeling corrects estimated relationships among latent variables for the biasing effects of measurement error. The purpose of this article is to review the manner in which structural equation models control for measurement error and to demonstrate the conditions in which structural equation models do and do not correct for unreliability. Generalizability theory is used to demonstrate that there are multiple sources of error in most measurement systems and that applications of structural equation modeling rarely account for more than a single source of error. As a result, the parameter estimates in a structural equation model may be severely biased by unassessed sources of measurement error. Recommendations for modeling multiple sources of error in structural equation models are provided. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Personal space: The conceptual and measurement implications of structural equation models.

Investigated the stop-distance procedure for measuring personal space, using maximum likelihood estimation of structural equation models. Three data sets previously described by the author (see PA, Vols 67:5708 and 68:1091) were examined. Results challenge the factor model underlying recent estimates of the reliability of the stop-distance procedure and convincingly demonstrate the reliability of stop-distance measurements. About 97% of the variance in observed distances arose from sources other than structural measurement error. About 8% of the variance in measured distances was due to sources that reflected unidentified causal determinants of spacing. The repeated failures of the factor model and the repeated successes of the simplex model require a fundamental shift in the conceptualization of personal space. The failure of the factor model implies that momentary spatial preferences should not be considered as merely reflecting a stable underlying preference or as a repeated and momentary construction based on stable situational features. Personal space should be considered as a dynamic process that is continually open to modification but that shows considerable stability due to the persistence of previously maintained distances. (French abstract) (12 ref) (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Formulating clinical research hypotheses as structural equation models: A conceptual overview.

Structural equation modeling is a comprehensive, flexible approach to research design and data analysis. Although in recent years there has been phenomenal growth in the literature on technical aspects of structural equation modeling, relatively little attention has been devoted to conceiving research hypotheses as structural equation models. The aim of this article is to provide a conceptual overview of clinical research hypotheses that invite evaluation as structural equation models. Particular attention is devoted to hypotheses that are not adequately evaluated using traditional statistical models. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Using projection for testing goodness-of-fit in regression models for repeated measures

Repeated measures data often occur in practice. This has led to considerable progress in the development of methods for inference in models for such data. In this paper, projection methods are proposed for examining goodness-of-fit in regression models for repeated measures. Rao's (1959, Biometrika 46, 49-58) F-test for testing a postulated mean structure using an independently identically normally distributed random sample is extended to a broad class of models including both fixed and random effects. The paper also shows how projection methods may be utilized for checking multivariate normality. In addition, application of projection to test the adequacy of extremely unbalanced models is considered. Two examples are given to demonstrate the underlying techniques.     

The impact of nonnormality on full information maximum-likelihood estimation for structural equation models with missing data.

A Monte Carlo simulation examined full information maximum-likelihood estimation (FIML) in structural equation models with nonnormal indicator variables. The impacts of 4 independent variables were examined (missing data algorithm, missing data rate, sample size, and distribution shape) on 4 outcome measures (parameter estimate bias, parameter estimate efficiency, standard error coverage, and model rejection rates). Across missing completely at random and missing at random patterns, FIML parameter estimates involved less bias and were generally more efficient than those of ad hoc missing data techniques. However, similar to complete-data maximum-likelihood estimation in structural equation modeling, standard errors were negatively biased and model rejection rates were inflated. Simulation results suggest that recently developed correctives for missing data (e.g., rescaled statistics and the bootstrap) can mitigate problems that stem from nonnormal data. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

A double-structure structural equation model for three-mode data.

Structural equation models are commonly used to analyze 2-mode data sets, in which a set of objects is measured on a set of variables. The underlying structure within the object mode is evaluated using latent variables, which are measured by indicators coming from the variable mode. Additionally, when the objects are measured under different conditions, 3-mode data arise, and with this, the simultaneous study of the correlational structure of 2 modes may be of interest. In this article the authors present a model with a simultaneous latent structure for 2 of the 3 modes of such a data set. They present an empirical illustration of the method using a 3-mode data set (person by situation by response) exploring the structure of anger and irritation across different interpersonal situations as well as across persons. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Quantified choice of root-mean-square errors of approximation for evaluation and power analysis of small differences between structural equation models.

MacCallum, Browne, and Cai (2006) proposed a new framework for evaluation and power analysis of small differences between nested structural equation models (SEMs). In their framework, the null and alternative hypotheses for testing a small difference in fit and its related power analyses were defined by some chosen root-mean-square error of approximation (RMSEA) pairs. In this article, we develop a new method that quantifies those chosen RMSEA pairs and allows a quantitative comparison of them. Our method proposes the use of single RMSEA values to replace the choice of RMSEA pairs for model comparison and power analysis, thus avoiding the differential meaning of the chosen RMSEA pairs inherent in the approach of MacCallum et al. (2006). With this choice, the conventional cutoff values in model overall evaluation can directly be transferred and applied to the evaluation and power analysis of model differences. (PsycINFO Database Record (c) 2011 APA, all rights reserved)     

Measurement and structural models for children's problem behaviors.

This article considers an analytic strategy for measuring and modeling child and adolescent problem behaviors. The strategy embeds an item response model within a hierarchical model to define an interval scale for the outcomes, to assess dimensionality, and to study how individual and contextual factors relate to multiple dimensions of problem behaviors. To illustrate, the authors analyze data from the primary caregiver ratings of 2,177 children aged 9-15 in 79 urban neighborhoods on externalizing behavior problems using the Child Behavior Checklist 4-18 (T. M. Achenbach, 1991a). Two subscales, Aggression and Delinquency, are highly correlated, and yet unidimensionality must be rejected because these subscales have different associations with key theoretically related covariates. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Analysis of interaction terms in structural equation models: A non-technical demonstration using the deviation score approach.

One of the major shortcomings in using structural equation modeling (SEM) data analytic techniques has been the difficulty in handling interaction terms in the modeling process. The issue is that interaction terms that are created by cross-multiplying raw scores result in the matrix of covariances or correlations being singular (there is at least 1 linear dependency in the matrix). The data analyses will not proceed as the matrix is not positive definite. This paper shows a valid and easy way to cope with the problem of interaction terms by using deviation scores or "centred" variables as the interaction terms. The authors expand on the work of L. S. Aiken and S. G. West (1991), which discussed interaction effects within a multiple regression framework, by taking it into the SEM domain for the benefit of users of SEM programs. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Estimating, testing, and comparing specific effects in structural equation models: The phantom model approach.

The phantom model approach for estimating, testing, and comparing specific effects within structural equation models (SEMs) is presented. The rationale underlying this novel method consists in representing the specific effect to be assessed as a total effect within a separate latent variable model, the phantom model that is added to the main model. The following favorable features characterize the method: (a) It enables the estimation, testing, and comparison of arbitrary specific effects for recursive and nonrecursive models with latent and manifest variables; (b) it enables the bootstrapping of confidence intervals; and (c) it can be applied with all standard SEM programs permitting latent variables, the specification of equality constraints, and the bootstrapping of total effects. These features along with the fact that no manipulation of matrices and formulas is required make the approach particularly suitable for applied researchers. The method is illustrated by means of 3 examples with real data sets. (PsycINFO Database Record (c) 2011 APA, all rights reserved)     

A structural equation test of the value-attitude-behavior hierarchy.

The role of values has received limited empirical attention relative to its potential significance, especially within a causal modeling approach. A series of multivariate and structural equation analyses supported the hypotheses that values have internal and external dimensions that influence attitudes. In turn, attitudes were found to influence behaviors, as the final phase in the value-attitude-behavior hierarchy. These analyses were performed on data derived from a survey about natural food shopping. As hypothesized, we found that people who have more internally oriented and less externally oriented value structures like natural foods more than other people, and these attitudes then lead to behaviors appropriate to the structure. Theoretical implications are discussed. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Comparative fit indexes in structural models.

Normed and nonnormed fit indexes are frequently used as adjuncts to chi-square statistics for evaluating the fit of a structural model. A drawback of existing indexes is that they estimate no known population parameters. A new coefficient is proposed to summarize the relative reduction in the noncentrality parameters of 2 nested models. Two estimators of the coefficient yield new normed (CFIN) and nonnormed (FIN) fit indexes. CFIN avoids the underestimation of fit often noted in small samples for P. M. Bentler and D. G. Bonett's (see record 1981-06898-001) normed fit index (NFIN). FIN is a linear function of Bentler and Bonett's nonnormed fit index (NNFIN) that avoids the extreme underestimation and overestimation often found in NNFIN. Asymptotically, CFIN, FIN, NFIN, and a new index developed by K. A. Bollen (1989) are equivalent measures of comparative fit, whereas NNFIN measures relative fit by comparing noncentrality per degree of freedom. All of the indexes are generalized to permit use of Wald and Lagrange multiplier statistics. An example illustrates the behavior of these indexes under conditions of correct specification and misspecification. The new fit indexes perform very well at all sample sizes. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Constrained versus unconstrained estimation in structural equation modeling.

Recently, R. D. Stoel, F. G. Garre, C. Dolan, and G. van den Wittenboer (2006) reviewed approaches for obtaining reference mixture distributions for difference tests when a parameter is on the boundary. The authors of the present study argue that this methodology is incomplete without a discussion of when the mixtures are needed and show that they only become relevant when constrained difference tests are conducted. Because constrained difference tests can hide important model misspecification, a reliable way to assess global model fit under constrained estimation would be needed. Examination of the options for assessing model fit under constrained estimation reveals that no perfect solutions exist, although the conditional approach of releasing a degree of freedom for each active constraint appears to be the most methodologically sound one. The authors discuss pros and cons of constrained and unconstrained estimation and their implementation in 5 popular structural equation modeling packages and argue that unconstrained estimation is a simpler method that is also more informative about sources of misfit. In practice, researchers will have trouble conducting constrained difference tests appropriately, as this requires a commitment to ignore Heywood cases. Consequently, mixture distributions for difference tests are rarely appropriate. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Principles and practice in reporting structural equation analyses.

Principles for reporting analyses using structural equation modeling are reviewed, with the goal of supplying readers with complete and accurate information. It is recommended that every report give a detailed justification of the model used, along with plausible alternatives and an account of identifiability. Nonnormality and missing data problems should also be addressed. A complete set of parameters and their standard errors is desirable, and it will often be convenient to supply the correlation matrix and discrepancies, as well as goodness-of-fit indices, so that readers can exercise independent critical judgment. A survey of fairly representative studies compares recent practice with the principles of reporting recommended here. (PsycINFO Database Record (c) 2011 APA, all rights reserved)