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)     

The Use of Latent Trajectory Models in Psychopathology Research.

Despite the recent surge in the development of powerful modeling strategies to test questions about individual differences in stability and change over time, these methods are not currently widely used in psychopathology research. In an attempt to further the dissemination of these new methods, the authors present a pedagogical introduction to the structural equation modeling based latent trajectory model, or LTM. They review several different types of LTMs, discuss matching an optimal LTM to a given question of interest, and highlight several issues that might be particularly salient for research in psychopathology. The authors augment each section with a review of published applications of these methods in psychopathology-related research to demonstrate the implementation and interpretation of LTMs in practice. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Introduction to the Special Section on Structural Equation Modeling.

Structural equation modeling (SEM) is a frequently used data-analytic technique in psychopathology research. This popularity is due to the unique capabilities and broad applicability of SEM and to recent advances in model and software development. Unfortunately, the popularity and accessibility of SEM is matched by its complexities and ambiguities. Thus, users are often faced with difficult decisions regarding a variety of issues. This special section is designed to increase the effective use of SEM by reviewing recently developed modeling capabilities, identifying common problems in application, and recommending appropriate strategies for analysis and evaluation. (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)     

The Integration of Continuous and Discrete Latent Variable Models: Potential Problems and Promising Opportunities.

Structural equation mixture modeling (SEMM) integrates continuous and discrete latent variable models. Drawing on prior research on the relationships between continuous and discrete latent variable models, the authors identify 3 conditions that may lead to the estimation of spurious latent classes in SEMM: misspecification of the structural model, nonnormal continuous measures, and nonlinear relationships among observed and/or latent variables. When the objective of a SEMM analysis is the identification of latent classes, these conditions should be considered as alternative hypotheses and results should be interpreted cautiously. However, armed with greater knowledge about the estimation of SEMMs in practice, researchers can exploit the flexibility of the model to gain a fuller understanding of the phenomenon under study. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Empirical and Conceptual Problems With Longitudinal Trait-State Models: Introducing a Trait-State-Occasion Model.

The latent trait-state-error model (TSE) and the latent state-trait model with autoregression (LST-AR) represent creative structural equation methods for examining the longitudinal structure of psychological constructs. Application of these models has been somewhat limited by empirical or conceptual problems. In the present study, Monte Carlo analysis revealed that TSE models tend to generate improper solutions when N is too small, when waves are too few, and when occasion factor stability is either too large or too small. Mathematical analysis of the LST-AR model revealed its limitation to constructs that become more highly auto-correlated over time. The trait-state-occasion model has fewer empirical problems than does the TSE model and is more broadly applicable than is the LST-AR model. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

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)     

"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)     

How things fall apart: Understanding the nature of internalizing through its relationship with impairment.

The literature suggests that internalizing psychopathology relates to impairment incrementally and gradually. However, the form of this relationship has not been characterized. This form is critical to understanding internalizing psychopathology, as it is possible that internalizing may accelerate in effect at some level of severity, defining a natural boundary of abnormality. Here, a novel method—semiparametric structural equation modeling—was used to model the relationship between internalizing and impairment in a sample of 8,580 individuals from the 2000 British Office for National Statistics Survey of Psychiatric Morbidity, a large, population-representative study of psychopathology. This method allows one to model relationships between latent internalizing and impairment without assuming any particular form a priori and to compare models in which the relationship is constant and linear. Results suggest that the relationship between internalizing and impairment is in fact linear and constant across the entire range of internalizing variation and that it is impossible to nonarbitrarily define a specific level of internalizing beyond which consequences suddenly become catastrophic in nature. Results demonstrate the phenomenological continuity of internalizing psychopathology, highlight the importance of impairment as well as symptoms, and have clear implications for defining mental disorder. (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)     

Introduction to the Special Section on Applied Longitudinal Methods in Aging Research.

This article introduces articles that appear in the special section on applied longitudinal methods in aging research. These articles apply quantitative statistical techniques such as multilevel modeling and structural equation models to the analysis of longitudinal data. They exemplify how applications of these techniques can advance scientific research on the effects of aging on psychological constructs. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Structural Equation Models for Interchangeable Dyads: Being the Same Makes a Difference.

Structural equation modeling (SEM) offers a flexible method for studying the patterns of interdependence in partners' behavior, which lie at the heart of interactions and relationships. Although SEM has been applied to the study of distinguishable dyads, in which partners are distinguishable by type, such as male and female, it has rarely been applied to the study of interchangeable dyads, such as male-male or female-female pairs. The authors integrate a wide range of dyadic interdependence models--including actor-partner interdependence models, mutual-influence models, and common-fate or dyadic personality models--into an SEM framework for use with interchangeable dyads. The authors also address the use of latent variables at both the dyadic and individual levels, whereby substantive relationships in these models can be corrected for errors of measurement. Furthermore, the authors discuss the conceptual underpinnings of dyadic models and give examples of their application. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Testing differences between nested covariance structure models: Power analysis and null hypotheses.

For comparing nested covariance structure models, the standard procedure is the likelihood ratio test of the difference in fit, where the null hypothesis is that the models fit identically in the population. A procedure for determining statistical power of this test is presented where effect size is based on a specified difference in overall fit of the models. A modification of the standard null hypothesis of zero difference in fit is proposed allowing for testing an interval hypothesis that the difference in fit between models is small, rather than zero. These developments are combined yielding a procedure for estimating power of a test of a null hypothesis of small difference in fit versus an alternative hypothesis of larger difference. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Regulatory Fit and Persuasion: Transfer From "Feeling Right."

The authors propose that when a message recipient "feels right" from regulatory fit (E. T. Higgins, 2000), this subjective experience transfers to the persuasion context and serves as information for relevant evaluations, including perceived message persuasiveness and opinions of the topic. Fit was induced either by strategic framing of message arguments in a way that fit/did not fit with the recipient's regulatory state or by a source unrelated to the message itself. Across 4 studies, regulatory fit enhanced perceived persuasiveness and opinion ratings. These effects were eliminated when the correct source of feeling right was made salient before message exposure, supporting the misattribution account. These effects reversed when message-related thoughts were negative, supporting the claim that fit provides information about the "rightness" of one's (positive or negative) evaluations. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Testing Mediational Models With Longitudinal Data: Questions and Tips in the Use of Structural Equation Modeling.

R. M. Baron and D. A. Kenny (1986; see record 1987-13085-001) provided clarion conceptual and methodological guidelines for testing mediational models with cross-sectional data. Graduating from cross-sectional to longitudinal designs enables researchers to make more rigorous inferences about the causal relations implied by such models. In this transition, misconceptions and erroneous assumptions are the norm. First, we describe some of the questions that arise (and misconceptions that sometimes emerge) in longitudinal tests of mediational models. We also provide a collection of tips for structural equation modeling (SEM) of mediational processes. Finally, we suggest a series of 5 steps when using SEM to test mediational processes in longitudinal designs: testing the measurement model, testing for added components, testing for omitted paths, testing the stationarity assumption, and estimating the mediational effects. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Missing Data Techniques for Structural Equation Modeling.

As with other statistical methods, missing data often create major problems for the estimation of structural equation models (SEMs). Conventional methods such as listwise or pairwise deletion generally do a poor job of using all the available information. However, structural equation modelers are fortunate that many programs for estimating SEMs now have maximum likelihood methods for handling missing data in an optimal fashion. In addition to maximum likelihood, this article also discusses multiple imputation. This method has statistical properties that are almost as good as those for maximum likelihood and can be applied to a much wider array of models and estimation methods. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Commentaire sur le colloque "Alternatives to classical statistical procedures."

A commentary is presented on the colloquium "Alternatives to Classical Statistical Procedures" organized by Zumbo. Each presentation of the colloquium is summarized and briefly critically reviewed. Together, the presentations yield an array of diverse methodological solutions to the fact that psychological data do not conform to the ideal of the normal distribution nor to the assumptions of parametric analysis (e.g., independent observations). These considerations suggest that psychologists should question how their variables of interest are measured. The present commentary adds to this suggestion the following one: psychologists should also critically review the effect size of the phenomena they study. The consideration of both measurement problems and effect size should bring psychologists towards a coherent use of inferential statistics. (PsycINFO Database Record (c) 2010 APA, all rights reserved)