Showing papers in "Structural Equation Modeling in 2004"

In Search of Golden Rules: Comment on Hypothesis-Testing Approaches to Setting Cutoff Values for Fit Indexes and Dangers in Overgeneralizing Hu and Bentler's (1999) Findings

Abstract: Goodness-of-fit (GOF) indexes provide "rules of thumb"—recommended cutoff values for assessing fit in structural equation modeling. Hu and Bentler (1999) proposed a more rigorous approach to evaluating decision rules based on GOF indexes and, on this basis, proposed new and more stringent cutoff values for many indexes. This article discusses potential problems underlying the hypothesis-testing rationale of their research, which is more appropriate to testing statistical significance than evaluating GOF. Many of their misspecified models resulted in a fit that should have been deemed acceptable according to even their new, more demanding criteria. Hence, rejection of these acceptable-misspecified models should have constituted a Type 1 error (incorrect rejection of an "acceptable" model), leading to the seemingly paradoxical results whereby the probability of correctly rejecting misspecified models decreased substantially with increasing N. In contrast to the application of cutoff values to evaluate each ...

Testing for Multigroup Invariance Using AMOS Graphics: A Road Less Traveled

Abstract: The purpose of this article is to illustrate the steps involved in testing for multigroup invariance using Amos Graphics. Based on analysis of covariance (ANCOV) structures, 2 applications are demonstrated, each of which represents a different set of circumstances. Application 1 focuses on the equivalence of a measuring instrument and tests for its invariance across 3 teacher panels, given baseline models that are identical across groups. Application 2 centers on the equivalence of a postulated theoretical structure across adolescent boys and girls in light of baseline models that are differentially specified across groups. Taken together, these illustrated examples should be of substantial assistance to researchers interested in testing for multigroup invariance using the Amos program.

Applying Multigroup Confirmatory Factor Models for Continuous Outcomes to Likert Scale Data Complicates Meaningful Group Comparisons

Abstract: Treating Likert rating scale data as continuous outcomes in confirmatory factor analysis violates the assumption of multivariate normality. Given certain requirements pertaining to the number of categories, skewness, size of the factor loadings, and so forth, it seems nevertheless possible to recover true parameter values if the data stem from a single homogeneous population. It is shown that, in a multigroup context, an analysis of Likert data under the assumption of multivariate normality may distort the factor structure differently across groups. In that case, investigations of measurement invariance (MI), which are necessary for meaningful group comparisons, are problematic. Analyzing subscale scores computed from Likert items does not seem to solve the problem.

Using the Delta Method for Approximate Interval Estimation of Parameter Functions in SEM

Abstract: In applications of structural equation modeling, it is often desirable to obtain measures of uncertainty for special functions of model parameters. This article provides a didactic discussion of how a method widely used in applied statistics can be employed for approximate standard error and confidence interval evaluation of such functions. The described approach is illustrated with data from a cognitive intervention study, in which it is used to estimate time-invariant reliability in multiwave, multiple indicator models.

A Monte-Carlo Study of Confirmatory Factor Analytic Tests of Measurement Equivalence/Invariance.

Abstract: In recent years, confirmatory factor analytic (CFA) techniques have become the most common method of testing for measurement equivalence/invariance (ME/I). However, no study has simulated data with known differences to determine how well these CFA techniques perform. This study utilizes data with a variety of known simulated differences in factor loadings to determine how well traditional tests of ME/I can detect these specific simulated differences. Results show that traditional CFA tests of ME/I perform well under ideal situations but that large sample sizes, a sufficient number of manifest indicators, and at least moderate communalities are crucial for assurance that ME/I conditions exist.

Factor Models for Ordinal Variables With Covariate Effects on the Manifest and Latent Variables: A Comparison of LISREL and IRT Approaches

Abstract: We consider a general type of model for analyzing ordinal variables with covariate effects and 2 approaches for analyzing data for such models, the item response theory (IRT) approach and the PRELIS-LISREL (PLA) approach. We compare these 2 approaches on the basis of 2 examples, 1 involving only covariate effects directly on the ordinal variables and 1 involving covariate effects on the latent variables in addition.

Using an EM Covariance Matrix to Estimate Structural Equation Models with Missing Data: Choosing an Adjusted Sample Size to Improve the Accuracy of Inferences.

Abstract: Two methods, direct maximum likelihood (ML) and the expectation maximization (EM) algorithm, can be used to obtain ML parameter estimates for structural equation models with missing data (MD). Although the 2 methods frequently produce identical parameter estimates, it may be easier to satisfy missing at random assumptions using EM. However, no single value of N is applicable to the EM covariance matrix, and this may compromise inferences gained from the model fit statistic and parameter standard errors. The purpose of this study was to identify a value of N that provides accurate inferences when using EM. If all confirmatory factor analysis model indicators have MD, results suggest that the minimum N per covariance term yields honest Type 1 error rates. If MD are restricted to a subset of indicators, the minimum N per variance works well. With respect to standard errors, the harmonic mean N per variance term produces honest confidence interval coverage rates.

Survey of Attitudes Toward Statistics: Factor Structure Invariance by Gender and by Administration Time

Abstract: In addition to student learning, positive student attitudes have become an important course outcome for many introductory statistics instructors. To adequately assess changes in mean attitudes across introductory statistics courses, the attitude instruments used should be invariant by administration time. Attitudes toward statistics from 4,910 students enrolled in an introductory statistics course were measured using the Survey of Attitudes Toward Statistics (SATS) both at the beginning and at the end of the semester. Confirmatory factor analysis on the covariance structure was used to determine the gender and time invariance properties of the SATS. Results indicate that the SATS is gender, time, and Gender x Time invariant with respect to factor loadings and factor correlations. Gender was invariant with respect to 3 of the 4 factor variances; variances from these same 3 factors were larger at the end than at the beginning of the course. Having established that the SATS is factorially invariant with resp...

To Bayes or Not to Bayes, From Whether to When: Applications of Bayesian Methodology to Modeling

Abstract: This article presents relevant research on Bayesian methods and their major applications to modeling in an effort to lay out differences between the frequentist and Bayesian paradigms and to look at the practical implications of these differences. Before research is reviewed, basic tenets and methods of the Bayesian approach to modeling are presented and contrasted with basic estimation results from a frequentist perspective. It is argued that Bayesian methods have become a viable alternative to traditional maximum likelihood-based estimation techniques and may be the only solution for more complex psychometric data structures. Hence, neither the applied nor the theoretical measurement community can afford to neglect the exciting new possibilities that have opened up on the psychometric horizon.

How Well Do Item Parcels Represent Conceptually Defined Latent Constructs? A Two-Facet Approach

Abstract: This article presents a methodology for examining the content and nature of item parcels as indicators of a conceptually defined latent construct. An essential component of this methodology is the 2-facet measurement model, which includes items and parcels as facets of construct indicators. The 2-facet model tests assumptions required for accepting parcels as aggregates of item covariation in representing the latent construct. According to this methodology, parcels are acceptable indicators of the latent construct if the 2-facet model meets parametric assumptions for unidimensionality and if items and parcels have content validity as measures of the latent construct. The proposed methodology is illustrated using a 1-factor model of the Worry construct in the test anxiety measurement tradition

Modeling Latent Growth Curves With Incomplete Data Using Different Types of Structural Equation Modeling and Multilevel Software

Abstract: This article offers different examples of how to fit latent growth curve (LGC) models to longitudinal data using a variety of different software programs (i.e., LISREL, Mx, Mplus, AMOS, SAS). The article shows how the same model can be fitted using both structural equation modeling and multilevel software, with nearly identical results, even in the case of models of latent growth fitted to incomplete data. The general purpose of this article is to provide a demonstration that integrates programming features from different software. The most immediate goal is to help researchers implement these LGC models as a useful way to test hypotheses of growth.

Divorce and Child Behavior Problems: Applying Latent Change Score Models to Life Event Data.

Abstract: Effects of parents' divorce on children's adjustment have been studied extensively. This article applies new advances in trajectory modeling to the problem of disentangling the effects of divorce on children's adjustment from related factors such as the child's age at the time of divorce and the child's gender. Latent change score models were used to examine trajectories of externalizing behavior problems in relation to children's experience of their parents' divorce. Participants included 356 boys and girls whose biological parents were married at kindergarten entry. The children were assessed annually through Grade 9. Mothers reported whether they had divorced or separated in each 12-month period, and teachers reported children's externalizing behavior problems each year. Girls' externalizing behavior problem trajectories were not affected by experiencing their parents' divorce, regardless of the timing of the divorce. In contrast, boys who were in elementary school when their parents divorced showed an increase in externalizing behavior problems in the year of the divorce. This increase persisted in the years following the divorce. Boys who were in middle school when their parents divorced showed an increase in externalizing behavior problems in the year of the divorce followed by a decrease to below baseline levels in the year after the divorce. This decrease persisted in the following years.

A cautionary note on the use of information fit indexes in covariance structure modeling with means

Abstract: Information fit indexes such as Akaike Information Criterion, Consistent Akaike Information Criterion, Bayesian Information Criterion, and the expected cross validation index can be valuable in assessing the relative fit of structural equation models that differ regarding restrictiveness. In cases in which models without mean restrictions (i.e., saturated mean structure) are compared to models with restricted (i.e., modeled) means, one should take account of the presence of means, even if the model is saturated with respect to the means. The failure to do this can result in an incorrect rank order of models in terms of the information fit indexes. We demonstrate this point by an analysis of measurement invariance in a multigroup confirmatory factor model.

Bias in the correlated uniqueness model for MTMM data

Abstract: This simulation investigates bias in trait factor loadings and intercorrelations when analyzing multitrait-multimethod (MTMM) data using the correlated uniqueness (CU) confirmatory factor analysis (CFA) model. A theoretical weakness of the CU model is the assumption of uncorrelated methods. However, previous simulation studies have shown little bias in trait estimates even when true method correlations are large. We hypothesized that there would be substantial bias when both method factor correlations and method factor loadings were large. We generated simulated sample data using population parameters based on our review of actual MTMM results. Results confirmed the prediction; substantial bias occurred in trait factor loadings and correlations when both method loadings and method correlations were large.

Point and Interval Estimation of Reliability for Multiple-Component Measuring Instruments via Linear Constraint Covariance Structure Modeling

Abstract: A widely and readily applicable covariance structure modeling approach is outlined that allows point and interval estimation of scale reliability with fixed components. The procedure employs only linear constraints introduced in a congeneric model, which after reparameterization permit expression of composite reliability as a function of appropriate parameters. Unlike the popular Cronbach's coefficient alpha that already at the population level is generally a misestimator of scale reliability, this method is based on the formal definition of the reliability coefficient. The proposed approach is illustrated by means of a numerical example.

Dynamic Structure of Emotions Among Individuals with Parkinson's Disease

Abstract: With few exceptions, the dynamics underlying the mood structures of individuals with Parkinson's Disease have consistently been overlooked Based on 12 participants' daily self-reports over 72 days, we identified 10 participants whose covariance matrices for positive and negative affect were similar enough to warrant pooling Dynamic factor models that included factor autoregression and cross-regressions were fitted to the pooled, lagged covariance matrix representing approximately 700 occasions of measurement Although results from the pooled data indicated that both positive and negative affect had a strong lag-1 autoregressive impact on current positive and negative affect, most individuals showed stronger autoregressive effects for positive than negative affect when examined individually There was also a weak cross-regression effect of positive affect on negative affect, but the reverse was not true Through model fitting, we demonstrated that failure to incorporate lagged relations among factors cou

Oral Language and Reading Success: A Structural Equation Modeling Approach

Abstract: Oral language skills and habits may serve as important resources for success or failure in school-related tasks such as learning to read. This article tests this hypothesis utilizing a unique data set, the original Woodcock-Johnson Psycho-Educational Battery-Revised norming sample. This article assesses the importance of oral language by focusing on auditory processing, a variable strongly affected by the oral language of the family and peer group within which the youth is raised. It estimates a structural equation model in which this variable, along with other measures of basic cognitive skills, serve as mediators between race and mother's schooling background and basic and advanced reading skill. The model fits very well, and the youth's basic skill at auditory processing is both a major determinant of basic reading success, and by far the most important of the mediating variables. In particular, for children ages 5 to 10, this measure accounts for much of the race effect, and for more than one half of ...

A note on extending Steiger's (1998) multiple sample RMSEA adjustment to other noncentrality parameter-based statistics

Abstract: This article considers the implications for other noncentrality parameter-based statistics from Steiger's (1998) multiple sample adjustment to the root mean square error of approximation (RMSEA) measure. When a structural equation model is fitted simultaneously in more than 1 sample, it is shown that the calculation of the noncentrality parameter used in tests of approximate fit and in point and interval estimators of other noncentral fit statistics (except the expected cross-validation index) also requires a likeminded adjustment. Furthermore, it is shown that an adjustment is needed in multiple sample models for correctly calculating MacCallum, Browne, and Sugawara's (1996) approach to power analysis. The accuracy of these proposals is investigated and demonstrated in a small Monte Carlo study in which particular attention is paid to using appropriately constructed covariance matrices that give specified nonzero population discrepancy values under maximum likelihood estimation.

Including Time-Invariant Covariates in the Latent Growth Curve Model

Abstract: Within the latent growth curve model, time-invariant covariates are generally modeled on the subject level, thereby estimating the effect of the covariate on the latent growth parameters. Incorporating the time-invariant covariate in this manner may have some advantages regarding the interpretation of the effect but may also be incorrect in certain instances. In this article we discuss a more general approach for modeling time-invariant covariates in latent growth curve models in which the covariate is directly regressed on the observed indicators. The approach can be used on its own to get estimates of the growth curves corrected for the influence of a 3rd variable, or it can be used to test the appropriateness of the standard way of modeling the time-invariant covariates. It thus provides a test of the assumption of full mediation, which states that the relation between the covariate and the observed indicators is fully mediated by the latent growth parameters.

A Direct Estimation Method on Analyzing Ipsative Data With Chan and Bentler's (1993) Method.

Abstract: Ipsative data (individual scores subject to a constant-sum constraint), suggested to minimize response bias, are sometimes observed in behavioral sciences. Chan and Bentler (1993, 1996) proposed a method to analyze ipsative data in a single-group case. Cheung and Chan (2002) extended the method to multiple-group analysis. However, these methods require tedious procedures on formulating within- and between-group constraints and recovering the parameter estimates and their standard errors. A direct estimation method, which is equivalent to Chan and Bentler's method with an alternative model specification, is proposed in this article. The 1st-order factor-analytic ipsative model in Chan and Bentler's method is reparameterized as a restricted 2nd-order factor-analytic model with fixed factor loading matrix reflecting the ipsative properties in the direct estimation method. The direct estimation method can be easily extended to test measurement invariance properties in multiple-group analysis. Issues related t...

A SAS Macro for Estimating and Visualizing Individual Growth Curves

Abstract: Longitudinal data analyses can be usefully supplemented by the plotting of individual growth curves. Unfortunately, such graphics can be challenging and tedious to produce. This article presents and demonstrates a SAS macro designed to automate this task. The OLStraj macro graphically depicts ordinary least squares (OLS)-estimated individual trajectories, describes interindividual variability in OLS-estimated growth parameters, and identifies possible outlier observations. Analytical developments are briefly outlined, and the use of the macro is demonstrated, with particular attention paid to the potential utility of the macro as both a data screening and post hoc diagnostic device. Potential limitations of the macro and suggestions for future developments are discussed. It is hoped that the program will be of use to applied researchers who seek to maximize the effectiveness of growth curve models in answering questions about stability and change.

Review of SAS for Monte Carlo Studies: A Guide for Quantitative Researchers, by Xitao Fan, Akos Felsovalyi, Stephen A. Sivo, and Sean C. Keenan

Abstract: Simulation studies are increasingly useful to quantitative methodologists in the social sciences. As the methods being used by researchers increase in complexity and sophistication, there are often not analytically tractable solutions to questions of model performance or estimation procedure. While simulation studies should not take the place of careful analytic work, they provide a valuable tool for quantitative researchers. With that in mind, the intended audience of SAS for Monte Carlo Studies is mainly the quantitative researcher, but it could extend to the methodologically sophisticated substantive researcher. This book is not an introduction to SAS or the SAS Macro language. Rather, it is an intermediate-to-advanced guide for the researcher who is comfortable programming in the SAS environment. The goals of the book are to introduce the SAS user to Monte Carlo simulation studies, to show that SAS environments meet theoretical concerns (such as random number and distributional generation) for Monte Carlo Simulation (MCS), and feature examples that explicate MCS in SAS to serve as a template for researchers. This book can be roughly divided into three sections. It begins with an introduction to the idea and reasons for simulation studies. This continues with chapters dedicated to programming issues common to all simulation work, such as random number generators and automating Monte Carlo studies using macros. The final third of the book uses illustrative examples of Monte Carlo simulation to clarify and expound the application of this research technique. These examples address methodological issues germane to quantitative researchers in the social sciences as well as to the fields of finance and economics. STRUCTURAL EQUATION MODELING, 11(2), 301–304 Copyright © 2004, Lawrence Erlbaum Associates, Inc.

Respecifying Improper Structures

Abstract: Improper structures arising from the estimation of parameters in structural equation models (SEMs) are commonly an indication that the model is incorrectly specified. The use of boundary solutions cannot in general be recommended. Partly on the basis of theory given by Van Driel, and partly by example, suggestions are made for using the data as evidence of the type of misspecification and as a guide to the appropriate model. Three types of improper structure suggested by Van Driel are reviewed, and a 4th type is introduced.

A Bistable View of Single Constructs Measured Using Balanced Questionnaires: Application to Trait Anxiety.

Abstract: Single constructs measured using positively and negatively worded items are often incompatible with a congeneric model, but require 2 correlated factors. Imperfect correlation entails that 2 independent dimensions are required for representing the true variance. If 2 dimensions are sought, how can they be interpreted? This study shows how to extract a group factor orthogonal to the common factor, from either the positive or the negative variables. Applied to trait anxiety measured using the State–Trait Anxiety Inventory (STAI), the approach generates a bistable view of the construct, stressing basic definitional ambiguity.

Randomized Response Analysis in Mplus

Abstract: This article describes a technique to analyze randomized response data using available structural equation modeling (SEM) software. The randomized response technique was developed to obtain estimates that are more valid when studying sensitive topics. The basic feature of all randomized response methods is that the data are deliberately contaminated with error. This makes it difficult to relate randomized responses to explanatory variables. In this tutorial, we present an approach to this problem, in which the analysis of randomized response data is viewed as a latent class problem, with different latent classes for the random and the truthful responses. To illustrate this technique, an example is presented using the program Mplus.

Book review of The Elements of Statistical Learning: Data Mining, Inference and Prediction, by Trevor Hastie, Robert Tibshirani, and Jerome Friedman; and Generalizability Theory, by Robert L. Brennan

Abstract: (2004). Book review of The Elements of Statistical Learning: Data Mining, Inference and Prediction, by Trevor Hastie, Robert Tibshirani, and Jerome Friedman; and Generalizability Theory, by Robert L. Brennan. Structural Equation Modeling: A Multidisciplinary Journal: Vol. 11, No. 1, pp. 150-152.

Temporal Stability in the Effects of College Attendance on Locus of Control, 1972–1992

Abstract: Locus of control is fairly stable over time but does change as a result of natural events, such as the acquisition of college education. Previous research found this to be so in a study of the high school class of 1972. This investigation asked whether a model of stability and change in locus of control had changed since the benchmark study of the 1972 high school class. Using a carefully designed structural equation model comparing high school classes of 1972 and 1992, we concluded that the influence of college on locus of control, although small to begin with, has not changed in 20 years.

Analysis of Paired Comparison Data Using Mx.

Abstract: By postulating that the random utilities associated with the choice options follow a multivariate normal distribution, Thurstonian models (Thurstone, 1927) provide a straightforward representation of paired comparison data. The use of Monte Carlo Expectation-Maximization (MCEM) algorithms and limited information approaches have been proposed to overcome the estimation intractability in analyzing data with a large number of choice items. However, these approaches have not yet been implemented into standard statistical software. For paired comparison data with a medium number of items (�6), it is possible to use the free software program Mx to obtain parameter estimates. This article shows how Mx can be used to obtain parameter estimates for Thurstonian paired comparison models. A number of simulations are conducted to assess its validity in obtaining the estimates in comparison to MCEM. In addition, 2 datasets are analyzed to demonstrate the use of Mx in real applications.

A Forward Search Procedure for Identifying Influential Observations in the Estimation of a Covariance Matrix

Abstract: This study uses a Cook's distance type diagnostic statistic to identify unusual observations in a data set that unduly influence the estimation of a covariance matrix. Similar to many other deletion-type diagnostic statistics, this proposed measure is susceptible to masking or swamping effect in the presence of several unusual observations. In view of this, a forward search procedure based on the proposed measure to detect extreme observations that respectively influence the covariance matrix estimates computed from different subsets of the data set was developed. These identified observations are summarized in a stalactite plot, giving a comprehensive picture about the suspicious data points. The stalactite plot is further examined with an objective to identifying multiple influential observations in the data set. Several data sets taken from the literature are used to illustrate the practicability and applicability of the proposed procedure.

The Application of SEM to Behavioral Research in Oncology: Past Accomplishments and Future Opportunities

Abstract: The past decade has seen a tremendous growth in the use of structural equation modeling (SEM) to address research questions in 2 subfields of behavioral science: cancer prevention and control (e.g., determinants of cancer screening adherence) and behavioral oncology (e.g., determinants of psychosocial adjustment among cancer patients or survivors). The application of SEM in these areas can have far-reaching implications for lowering cancer morbidity and mortality and for improving the psychological well-being and quality of life of those afflicted with, and recovering from, cancer. This article reviews the progress made in the application of SEM to cancer prevention and control and behavioral oncology research, highlighting 2 central uses: theory development and evaluation and scale construction. Directions for the future application of SEM to address research questions concerning cancer prevention and control and behavioral oncology are also discussed.