Evaluating model fit for growth curve models: Integration of fit indices from SEM and MLM frameworks.

Evaluating overall model fit for growth curve models involves 3 challenging issues. (a) Three types of longitudinal data with different implications for model fit may be distinguished: balanced on time with complete data, balanced on time with data missing at random, and unbalanced on time. (b) Traditional work on fit from the structural equation modeling (SEM) perspective has focused only on the covariance structure, but growth curve models have four potential sources of misspecification: within-individual covariance matrix, between-individuals covariance matrix, marginal mean structure, and conditional mean structure. (c) Growth curve models can be estimated in both the SEM and multilevel modeling (MLM) frameworks; these have different emphases for the evaluation of model fit. In this article, the authors discuss the challenges presented by these 3 issues in the calculation and interpretation of SEM- and MLM-based fit indices for growth curve models and conclude by identifying some lines for future research. (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)     

A theoretical note on models of interlimb coordination.

This article clarifies some points about models of interlimb coordination. The authors discuss in detail the predictions and extensions of the H. Haken et al (1985) model and compare these to a truncated version used by R. C. Schmidt and colleagues. It is demonstrated that procedures commonly used to fit experimental data to model parameters require the fulfillment of certain assumptions. Finally, the authors show how recent experimental findings can be modeled straightforwardly using a system of coupled nonlinear oscillators. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Masking misfit in confirmatory factor analysis by increasing unique variances: A cautionary note on the usefulness of cutoff values of fit indices.

Fit indices are widely used in order to test the model fit for structural equation models. In a highly influential study, Hu and Bentler (1999) showed that certain cutoff values for these indices could be derived, which, over time, has led to the reification of these suggested thresholds as “golden rules” for establishing the fit or other aspects of structural equation models. The current study shows how differences in unique variances influence the value of the global chi-square model test and the most commonly used fit indices: Root-mean-square error of approximation, standardized root-mean-square residual, and the comparative fit index. Using data simulation, the authors illustrate how the value of the chi-square test, the root-mean-square error of approximation, and the standardized root-mean-square residual are decreased when unique variances are increased although model misspecification is present. For a broader understanding of the phenomenon, the authors used different sample sizes, number of observed variables per factor, and types of misspecification. A theoretical explanation is provided, and implications for the application of structural equation modeling are discussed. (PsycINFO Database Record (c) 2011 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)     

Structural Equation Modeling of Paired-Comparison and Ranking Data.

L. L. Thurstone's (1927) model provides a powerful framework for modeling individual differences in choice behavior. An overview of Thurstonian models for comparative data is provided, including the classical Case V and Case III models as well as more general choice models with unrestricted and factor-analytic covariance structures. A flow chart summarizes the model selection process. The authors show how to embed these models within a more familiar structural equation modeling (SEM) framework. The different special cases of Thurstone's model can be estimated with a popular SEM statistical package, including factor analysis models for paired comparisons and rankings. Only minor modifications are needed to accommodate both types of data. As a result, complex models for comparative judgments can be both estimated and tested efficiently. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Testing parameters in structural equation modeling: Every "one" matters.

A problem with standard errors estimated by many structural equation modeling programs is described. In such programs, a parameter's standard error is sensitive to how the model is identified (i.e., how scale is set). Alternative but equivalent ways to identify a model may yield different standard errors, and hence different Z tests for a parameter, even though the identifications produce the same overall model fit. This lack of invariance due to model identification creates the possibility that different analysts may reach different conclusions about a parameter's significance level even though they test equivalent models on the same data. The authors suggest that parameters be tested for statistical significance through the likelihood ratio test, which is invariant to the identification choice. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

"Psychometric approaches for developing commensurate measures across independent studies: Traditional and new models": Clarification to Bauer and Hussong (2009).

Reports a clarification to "Psychometric approaches for developing commensurate measures across independent studies: Traditional and new models" by Daniel J. Bauer and Andrea M. Hussong (Psychological Methods, 2009[Jun], Vol 14[2], 101-125). In this article, the authors wrote, "To our knowledge, the multisample framework is the only available option within these [latent variable] programs that allows for the moderation of all types of parameters, and this approach requires a single categorical moderator variable to define the samples.” Bengt Muthén has clarified for the authors that some programs, including Mplus and Mx, can allow for continuous moderation through the implementation of nonlinear constraints involving observed variables, further enlarging the class of MNLFA models that can be fit with these programs. (The following abstract of the original article appeared in record 2009-08072-001.) When conducting an integrative analysis of data obtained from multiple independent studies, a fundamental problem is to establish commensurate measures for the constructs of interest. Fortunately, procedures for evaluating and establishing measurement equivalence across samples are well developed for the linear factor model and commonly used item response theory models. A newly proposed moderated nonlinear factor analysis model generalizes these models and procedures, allowing for items of different scale types (continuous or discrete) and differential item functioning across levels of categorical and/or continuous variables. The potential of this new model to resolve the problem of measurement in integrative data analysis is shown via an empirical example examining changes in alcohol involvement from ages 10 to 22 years across 2 longitudinal studies. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

On the likelihood ratio test in structural equation modeling when parameters are subject to boundary constraints.

The authors show how the use of inequality constraints on parameters in structural equation models may affect the distribution of the likelihood ratio test. Inequality constraints are implicitly used in the testing of commonly applied structural equation models, such as the common factor model, the autoregressive model, and the latent growth curve model, although this is not commonly acknowledged. Such constraints are the result of the null hypothesis in which the parameter value or values are placed on the boundary of the parameter space. For instance, this occurs in testing whether the variance of a growth parameter is significantly different from 0. It is shown that in these cases, the asymptotic distribution of the chi-square difference cannot be treated as that of a central chi-square-distributed random variable with degrees of freedom equal to the number of constraints. The correct distribution for testing 1 or a few parameters at a time is inferred for the 3 structural equation models mentioned above. Subsequently, the authors describe and illustrate the steps that one should take to obtain this distribution. An important message is that using the correct distribution may lead to appreciably greater statistical power. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Distinguishing between moderator and quadratic effects in multiple regression.

Moderated regression analysis is commonly used to test for multiplicative influences of independent variables in regression models. D. Lubinski and L. G. Humphreys (1990) have shown that significant moderator effects can exist even when stronger quadratic effects are present. They recommend comparing effect sizes associated with both effect types and selecting the model that yields the strongest effect. The authors show that this procedure of comparing effect sizes is biased in favor of the moderated model when multicollinearity is high because of the differential reliability of the quadratic and multiplicative terms in the regression models. Fortunately, levels of multicollinearity under which this bias is most problematic may be outside the range encountered in many empirical studies. The authors discuss causes and implications of this phenomenon as well as alternative procedures for evaluating structural relationships among variables. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Structural equation modeling with interchangeable dyads.

Structural equation modeling (SEM) can be adapted in a relatively straightforward fashion to analyze data from interchangeable dyads (i.e., dyads in which the 2 members cannot be differentiated). The authors describe a general strategy for SEM model estimation, comparison, and fit assessment that can be used with either dyad-level or pairwise (double-entered) dyadic data. They present applications illustrating this approach with the actor-partner interdependence model, confirmatory factor analysis, and latent growth curve analysis. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

"The influence of sample size, estimation methods, and model specification on goodness-of-fit assessments in structural equation models": Correction to la Du and Tanaka (1989).

Reports an error in "Influence of sample size, estimation method, and model specification on goodness-of-fit assessments in structural equation models" by Terence J. la Du and J. S. Tanaka (Journal of Applied Psychology, 1989[Aug], Vol 74[4], 625-635). Figure 2 (p. 631) summarizes Katzell's work motivation model and indicates where the trivial misspecification (dashed line) and nontrivial misspecification (starred line) occurred in our model specification condition. The error is in the latter. The starred line should be from Operations and Resources to Extrinsic Rewards and not from Rewards for Performance to Fruity. Our findings are not changed by this error, because we were using Katzell's model and accompanying data base to conduct a sampling study on goodness-of-fit indices and not testing his model. Hence, any of the paths were candidates for the nontrivial misspecification condition. (The following abstract of the original article appeared in record 1989-38703-001.) 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)     

Choosing a multivariate model: Noncentrality and goodness of fit.

Anumber of goodness-of-fit indices for the evaluation of multivariate structural models are expressed as functions of the noncentrality parameter in order to elucidate their mathematical properties and, in particular, to explain previous numerical findings. Most of the indices considered are shown to vary systematically with sample size. It is suggested that H. Akaike's (1974; see record 1989-17660-001) information criterion cannot be used for model selection in real applications and that there are problems attending the definition of parsimonious fit indices. A normed function of the noncentrality parameter is recommended as an unbiased absolute goodness-of-fit index, and the Tucker–Lewis (see record 1973-30255-001) index and a new unbiased counterpart of the Bentler–Bonett (see record 1981-06898-001) index are recommended for those investigators who might wish to evaluate fit relative to a null model. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Modeling caregiver adaptation over time: The longitudinal impact of behavior problems.

Although cross-sectional research has established the link between care demands and various indicators of caregiver adaptation, few studies have examined the impact of care recipients' problematic behavior over time. The present analysis determines the importance of behavior problems when predicting rates of change in subjective stressors (role overload and role captivity) and depression. Using 4-wave longitudinal data (N?=?137) on dementia caregivers, the authors fit individual growth curve models for care demands (i.e., behavior problems, activities of daily living dependencies, and cognitive impairment), subjective stressors, and depression. Subsequent structural equation models found that increases in behavior problems were most likely to predict increases in role overload. The findings emphasize the deleterious long-term impact of behavior problems on individuals' emotional adaptation to caregiving. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Construct Validity of the Posttraumatic Stress Disorder Checklist in Cancer Survivors: Analyses Based on Two Samples.

The measurement of posttraumatic stress disorder (PTSD) is critically important for the identification and treatment of this disorder. The PTSD Checklist (PCL; F. W. Weathers & J. Ford, 1996) is a self-report measure that is increasingly used. In this study, the authors investigated the factorial validity of the PCL with data from 236 cancer survivors who received a bone marrow or stem cell transplantation. The authors examined the fit of these data with the clinical model of 3 symptom clusters for PTSD, as proposed in the Diagnostic and Statistical Manual of Mental Disorders (4th ed.; American Psychiatric Association, 1994), and alternative models tested in prior research. By using confirmatory factor analysis the authors found that a 4-first-order-factor model of PTSD provided the best fit. The relations of PTSD symptoms with sociodemographic and medical variables were also explored. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

The four-factor model of depressive symptoms in dementia caregivers: A structural equation model of ethnic differences.

Previous studies have suggested that 4 latent constructs (depressed affect, well-being, interpersonal problems, somatic symptoms) underlie the item responses on the Center for Epidemiological Studies Depression (CES-D) Scale. This instrument has been widely used in dementia caregiving research, but the fit of this multifactor model and the explanatory contributions of multifactor models have not been sufficiently examined for caregiving samples. The authors subjected CES-D data (N = 1,183) from the initial Resources for Enhancing Alzheimer's Caregiver Health Study to confirmatory factor analysis methods and found that the 4-factor model provided excellent fit to the observed data. Invariance analyses suggested only minimal item-loading differences across race subgroups and supported the validity of race comparisons on the latent factors. Significant race differences were found on 3 of the 4 latent factors both before and after controlling for demographic covariates. African Americans reported less depressed affect and better well-being than White caregivers, who reported better well-being and fewer interpersonal problems than Hispanic caregivers. These findings clarify and extend previous studies of race differences in depression among diverse samples of dementia caregivers. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Locating eating pathology within an empirical diagnostic taxonomy: Evidence from a community-based sample.

Existing structural models of psychopathology need to be expanded to include additional diagnostic constructs beyond mood, anxiety, substance use, and antisocial behavior disorders. The goal of this study was to locate eating disorders within a hierarchical structural model of psychopathology that is anchored by broad Internalizing and Externalizing factors. Participants were female adolescent twins (N = 1,434) from the Minnesota Twin Family Study. The authors compared the fit of 4 models in which eating disorders (a) defined their own diagnostic class, (b) represented a subclass within Internalizing, (c) formed a subclass within Externalizing, and (d) were allowed to cross-load on both Internalizing and Externalizing. In the best fitting model, eating disorders formed a subfactor within Internalizing. These findings underscore the value of developing more comprehensive empirically based models of psychopathology to increase researchers' understanding of diverse mental disorders. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

A meta-analytical structural equations analysis of a model of employee turnover.

Combined meta-analysis with structural equations modeling (SEM) to validate the W. H. Mobley et al (1978) turnover theory as well as alternative structural networks proposed by A. Dalessio et al (1986), P. W. Hom et al (1984), and B. D. Bannister and R. W. Griffeth (1986). The authors aggregated correlations from 17 studies (N?=?5,013 employees), correcting for unreliability and sampling error. Then they used SEM to assess the models, comparing their relative fits to data. SEM analyses corroborated the Mobley et al model better than did past research, but these analyses also showed that the Dalessio et al and the Hom et al theories explained sample data more plausibly. Additional SEM tests found that turnover base rates, time lags between turnover and model assessments, unemployment rates, and occupational differences moderated the models' pathways. The present findings suggest various implications for these theories and for turnover research. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Structure of interests in employed male and female members of US racial-ethnic minority and nonminority groups.

This study explored the structure of interests across large samples of employed U.S. racial-ethnic minority and nonminority adults drawn from over 38,000 individuals who were part of the 1994 revision of the Strong Interest Inventory (SII; L. W. Harmon, J. C. Hansen, F. H. Borgen, & A. C. Hammer, 1994; N?=?805 African Americans, 795 Asian Americans, 36,632 Caucasians, and 686 Latinos-Hispanics). Correlation matrices from the general occupational themes of the SII were subjected to 2 analyses commonly used in structural analyses of Holland's themes: randomization test of hypothesized order relations and multidimensional scaling. Analyses tested whether the data fit the circular and equidistant hexagonal structure models. All of the data fit the circular model that corresponds to Holland's calculus assumption, but the data for women and for some of the male racial-ethnic groups did not fit the more stringent equidistant hexagonal structure. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Psychometric approaches for developing commensurate measures across independent studies: Traditional and new models.

[Correction Notice: An erratum for this article was reported in Vol 14(4) of Psychological Methods (see record 2009-22665-007). In this article, the authors wrote, "To our knowledge, the multisample framework is the only available option within these [latent variable] programs that allows for the moderation of all types of parameters, and this approach requires a single categorical moderator variable to define the samples.” Bengt Muthén has clarified for the authors that some programs, including Mplus and Mx, can allow for continuous moderation through the implementation of nonlinear constraints involving observed variables, further enlarging the class of MNLFA models that can be fit with these programs.] When conducting an integrative analysis of data obtained from multiple independent studies, a fundamental problem is to establish commensurate measures for the constructs of interest. Fortunately, procedures for evaluating and establishing measurement equivalence across samples are well developed for the linear factor model and commonly used item response theory models. A newly proposed moderated nonlinear factor analysis model generalizes these models and procedures, allowing for items of different scale types (continuous or discrete) and differential item functioning across levels of categorical and/or continuous variables. The potential of this new model to resolve the problem of measurement in integrative data analysis is shown via an empirical example examining changes in alcohol involvement from ages 10 to 22 years across 2 longitudinal studies. (PsycINFO Database Record (c) 2010 APA, all rights reserved)