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)     

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)     

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)     

Using covariance structure analysis to detect correlates and predictors of individual change over time.

Recently, methodologists have shown how 2 disparate conceptual arenas (individual growth modeling and covariance structure analysis) can be integrated. The integration brings the flexibility of covariance analysis to bear on the investigation of systematic interindividual differences in change and provides another powerful data-analytic tool for answering questions about the relationship between individual true change and potential predictors of that change. The individual growth modeling framework uses a pair of hierarchical statistical models to represent (1) within-person true status as a function of time and (2) between-person differences in true change as a function of predictors. This article explains how these models can be reformatted to correspond, respectively, to the measurement and structural components of the general LISREL model with mean structures and illustrates, by means of worked example, how the new method can be applied to a sample of longitudinal panel data. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Taxometrics, polytomous constructs, and the comparison curve fit index: A Monte Carlo analysis.

The taxometric method effectively distinguishes between dimensional (1-class) and taxonic (2-class) latent structure, but there is virtually no information on how it responds to polytomous (3-class) latent structure. A Monte Carlo analysis showed that the mean comparison curve fit index (CCFI; Ruscio, Haslam, & Ruscio, 2006) obtained with 3 taxometric procedures—mean above minus below a cut (MAMBAC), maximum covariance (MAXCOV), and latent mode factor analysis (L-Mode)—accurately identified 1-class (dimensional) and 2-class (taxonic) samples and produced taxonic results when applied to 3-class (polytomous) samples. From these results it is concluded that using the simulated data curve approach and averaging across procedures is an effective way of distinguishing between dimensional (1-class) and categorical (2 or more classes) latent structure. (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)     

Reporting results of latent growth modeling and multilevel modeling analyses: Some recommendations for rehabilitation psychology.

Objective: There has been a general increase in interest and use of modeling techniques that treat data as nested, whether it is people nested within larger units, such as families or treatment centers, or observations nested under people. The popularity can be witnessed by noting the number of new textbooks and articles related to latent growth curve modeling and multilevel modeling. This paper discusses both of these techniques in the context of longitudinal research designs, with the main purposes of highlighting some benefits and issues related to the use of these models and outlining guidelines for reporting results from studies using multilevel modeling or latent growth modeling. Implications: These longitudinal analytic techniques can be greatly beneficial to researchers conducting rehabilitation studies, but there are several issues related to their use and reporting that need to be taken into consideration. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Linear models for the analysis and construction of instruments in a facet design.

Describes linear models for the situation wherein a measurement instrument is constructed for all elements of the Cartesian product of several facets when the elements of each facet are not ordered. The structure of the covariance matrix of the instrument is derived from the models. By using covariance structure analysis, the models can be tested, and estimates of the parameters can be obtained. Models for 20 tests were formulated and tested and were constructed from a design with a behavioral and a situational facet measuring social anxiety in children; for 15 tests a model proved to fit the data. It is concluded that covariance structure analysis is useful for the analysis and construction of measurement instruments. (34 ref) (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)     

Overall fit in covariance structure models: Two types of sample size effects.

Acontroversial area in covariance structure models is the assessment of overall model fit. Researchers have expressed concern over the influence of sample size on measures of fit. Many contradictory claims have been made regarding which fit statistics are affected by N. Part of the confusion is due to there being two types of sample size effects that are confounded. The first is whether N directly enters the calculation of a fit measure. The second is whether the means of the sampling distributions of a fit index are associated with sample size. These types of sample size effects are explained and illustrated with the major structural equation fit indices. In addition, the current debate on sample size influences is examined in light of this distinction. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Model modifications in covariance structure analysis: The problem of capitalization on chance.

In applications of covariance structure modeling in which an initial model does not fit sample data well, it has become common practice to modify that model to improve its fit. Because this process is data driven, it is inherently susceptible to capitalization on chance characteristics of the data, thus raising the question of whether model modifications generalize to other samples or to the population. This issue is discussed in detail and is explored empirically through sampling studies using 2 large sets of data. Results demonstrate that over repeated samples, model modifications may be very inconsistent and cross-validation results may behave erratically. These findings lead to skepticism about generalizability of models resulting from data-driven modifications of an initial model. The use of alternative a priori models is recommended as a preferred strategy. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Comparing the relative fit of categorical and dimensional latent variable models using consistency tests.

A number of recent studies have used Meehl’s (1995) taxometric method to determine empirically whether one should model assessment-related constructs as categories or dimensions. The taxometric method includes multiple data-analytic procedures designed to check the consistency of results. The goal is to differentiate between strong evidence of categorical structure, strong evidence of dimensional structure, and ambiguous evidence that suggests withholding judgment. Many taxometric consistency tests have been proposed, but their use has not been operationalized and studied rigorously. What tests should be performed, how should results be combined, and what thresholds should be applied? We present an approach to consistency testing that builds on prior work demonstrating that parallel analyses of categorical and dimensional comparison data provide an accurate index of the relative fit of competing structural models. Using a large simulation study spanning a wide range of data conditions, we examine many critical elements of this approach. The results provide empirical support for what marks the first rigorous operationalization of consistency testing. We discuss and empirically illustrate guidelines for implementing this approach and suggest avenues for future research to extend the practice of consistency testing to other techniques for modeling latent variables in the realm of psychological assessment. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Applications of covariance structure modeling in psychology: Cause for concern?

Methods of covariance structure modeling are frequently applied in psychological research. These methods merge the logic of confirmatory factor analysis, multiple regression, and path analysis within a single data analytic framework. Among the many applications are estimation of disattenuated correlation and regression coefficients, evaluation of multitrait–multimethod matrices, and assessment of hypothesized causal structures. Shortcomings of these methods are commonly acknowledged in the mathematical literature and in textbooks. Nevertheless, serious flaws remain in many published applications. For example, it is rarely noted that the fit of a favored model is identical for a potentially large number of equivalent models. A review of the personality and social psychology literature illustrates the nature of this and other problems in reported applications of covariance structure models. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Commentary on Steinley and Brusco (2011): Recommendations and cautions.

I discuss the recommendations and cautions in Steinley and Brusco's (2011) article on the use of finite models to cluster a data set. In their article, much use is made of comparison with the K-means procedure. As noted by researchers for over 30 years, the K-means procedure can be viewed as a special case of finite mixture modeling in which the components are in equal (fixed) proportions and are taken to be normal with a common spherical covariance matrix. In this commentary, I pay particular attention to this link and to the use of normal mixture models with arbitrary component-covariance matrices. (PsycINFO Database Record (c) 2011 APA, all rights reserved)     

The use of covariance functions and random regressions for genetic evaluation of milk production based on test day records

In the analysis of test day records for dairy cattle, covariance functions allow a continuous change of variances and covariances of test day yields on different lactation days. The equivalence between covariance functions as an infinite dimensional extension of multivariate models and random regression models is shown in this paper. A canonical transformation procedure is proposed for random regression models in large-scale genetic evaluations. Two methods were used to estimate covariance function coefficients for first parity test day yields of Holsteins: 1) a two-step procedure fitting covariance functions to matrices with estimated genetic and residual covariances between predetermined periods of lactation and 2) REML directly from data with a random regression model. The first method gave more reliable estimates, particularly for the periphery of the trajectory. The goodness of fit of a random regression model based on covariables describing the shape of the lactation curve was nearly the same as random regression on Legendre polynomials. In the latter model, two and three regression coefficients were sufficient to fit the covariance structure for additive genetic and permanent environment, respectively. The eigenfunction pattern revealed the possibility of selection for persistency. Covariance functions can be usefully implemented in large-scale test day models by means of random regressions.     

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)