"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)     

Evaluation of goodness-of-fit indices for structural equation models.

Discusses how current goodness-of-fit indices fail to assess parsimony and hence disconfirmability of a model and are insensitive to misspecifications of causal relations (a) among latent variables when measurement model with many indicators is correct and (b) when causal relations corresponding to free parameters expected to be nonzero turn out to be zero or near zero. A discussion of philosophy of parsimony elucidates relations of parsimony to parameter estimation, disconfirmability, and goodness of fit. AGFI in {lisrel} is rejected. A method of adjusting goodness-of-fit indices by a parsimony ratio is described. Also discusses less biased estimates of goodness of fit and a relative normed-fit index for testing fit of structural model exclusive of the measurement model. (PsycINFO Database Record (c) 2011 APA, all rights reserved)     

A goodness-of-fit approach to inference procedures for the kappa statistic: confidence interval construction, significance-testing and sample size estimation

We propose a new procedure for constructing a confidence interval about the kappa statistic in the case of two raters and a dichotomous outcome. The procedure is based on a chi-square goodness-of-fit test as applied to a model frequently used for clustered binary data. The procedure provides coverage levels that are accurate in samples of smaller size than those required for other procedures. The procedure also has use for significance-testing and the planning of corresponding sample size requirements.     

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)     

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)     

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)     

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)     

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)     

Accuracy in parameter estimation for targeted effects in structural equation modeling: Sample size planning for narrow confidence intervals.

In addition to evaluating a structural equation model (SEM) as a whole, often the model parameters are of interest and confidence intervals for those parameters are formed. Given a model with a good overall fit, it is entirely possible for the targeted effects of interest to have very wide confidence intervals, thus giving little information about the magnitude of the population targeted effects. With the goal of obtaining sufficiently narrow confidence intervals for the model parameters of interest, sample size planning methods for SEM are developed from the accuracy in parameter estimation approach. One method plans for the sample size so that the expected confidence interval width is sufficiently narrow. An extended procedure ensures that the obtained confidence interval will be no wider than desired, with some specified degree of assurance. A Monte Carlo simulation study was conducted that verified the effectiveness of the procedures in realistic situations. The methods developed have been implemented in the MBESS package in R so that they can be easily applied by researchers. (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)     

A structural equation model of expertise in college physics.

A model of expertise in physics was tested on a sample of 374 college students in 2 different level physics courses. Structural equation modeling was used to test hypothesized relationships among variables linked to expert performance in physics including strategy use, pictorial representation, categorization skills, and motivation, and these variables were examined for their influence on physics achievement. Gender was included in the model to examine how it influenced achievement indirectly through its influence on the other variables in the model. Two levels of expertise were examined by testing the model on trigonometry-based physics students and on more advanced, calculus-based physics students. Results were similar across both levels of expertise: For both courses, student motivation had a significant influence on students’ strategy use and categorization skills. Categorization skills, in turn, influenced student achievement directly, and indirectly, through strategy use. Strategy use had a significant influence on achievement. Pictorial representation played little role in the model. Gender contributed primarily through motivation, but for the more advanced level course it also directly predicted strategy use. (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)     

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)     

Influence of a patient decision aid on decisional conflict related to PSA testing: A structural equation model.

Objective: To examine the impact of a decision aid (DA) designed to promote informed decision making for screening with the prostate-specific antigen (PSA) test and to test a theoretical model of factors influencing decisional conflict. Design: Structural equation modeling examined pathways between DA exposure, knowledge, schema, prostate cancer risk perceptions, decisional anxiety, and decisional conflict. Sample participants included 200 men from the general population (exclusive of African Americans) and 200 African American men. Half of the men in each subsample were randomly assigned to receive the DA. Main Outcome Measures: Decisional conflict regarding prostate cancer screening. Results: The DA influences level of decisional conflict by increasing patient knowledge. This effect of knowledge on decisional conflict is indirect, however, through an association with greater perceived risk and lower decisional anxiety. Also, positive PSA schema was associated with lower decisional anxiety and decisional conflict. It is important that exposure to the DA had no impact on PSA schema. Conclusion: Schemas about testing must be considered in developing messages about the risks and benefits of testing. If schemas are counter to message content, mechanisms for modifying schemas must be incorporated into interventions. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

A model for integrating fixed-, random-, and mixed-effects meta-analyses into structural equation modeling.

Meta-analysis and structural equation modeling (SEM) are two important statistical methods in the behavioral, social, and medical sciences. They are generally treated as two unrelated topics in the literature. The present article proposes a model to integrate fixed-, random-, and mixed-effects meta-analyses into the SEM framework. By applying an appropriate transformation on the data, studies in a meta-analysis can be analyzed as subjects in a structural equation model. This article also highlights some practical benefits of using the SEM approach to conduct a meta-analysis. Specifically, the SEM-based meta-analysis can be used to handle missing covariates, to quantify the heterogeneity of effect sizes, and to address the heterogeneity of effect sizes with mixture models. Examples are used to illustrate the equivalence between the conventional meta-analysis and the SEM-based meta-analysis. Future directions on and issues related to the SEM-based meta-analysis are discussed. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Effects of size of sample on eigenvalues, observed communalities, and factor loadings.

Several samples were randomly selected from a finite population of undergraduates over 5 N levels, factor analyzed, and then compared to the factor analysis of the finite population. Results indicate that within the limitations of the sampling design (Ns = 1,600, 400, 100, 25, and 17), samples representing N = 400 of a population of N = 2,322 yield factor structures comparable to that of the population. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

A structural equation model of dyadic relationships within the family system.

Ideas about the association between family interaction patterns and mental health have inspired a great deal of research on family relationships. Conventional statistical models (e.g., ANOVA), however, are not suited for the analysis of the nonindependent observations produced by family systems; yet, these sources of nonindependence (e.g., reciprocity, partner effects) are very interesting to many family researchers. This article demonstrates how structural equation modeling enables the study of family processes in a way that is consistent with a systems perspective. An analysis of perceived coerciveness in 2-parent, 2-child families illustrates the method. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Latent variable interaction and quadratic effect estimation: A two-step technique using structural equation analysis.

The author proposes an alternative estimation technique for latent variable interactions and quadratics. Available techniques for specifying these variables in structural equation models require adding variables or constraint equations that can produce specification tedium and errors or estimation difficulties. The proposed technique avoids these difficulties and may be useful for EQS, LISREL 7, and LISREL 8 users. First, measurement parameters for indicator loadings and errors of linear latent variables are estimated in a measurement model that excludes the interaction and quadratic variables. Next, these estimates are used to calculate values for the indicator loadings and error variances of the interaction and quadratic latent variables. Then, these calculated values are specified as constants in the structural model containing the interaction and quadratic variables. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Laboratory panel estimation of consumer assessments of taste and flavor.

Correlation coefficients of the investigation ranged from .87 to .89 for 29 samples of mineralized domestic water evaluated by scaling procedures. Prediction of consumer assessments using regression equation and panel means as independent variables was evaluated and found to be satisfactory. Implications of findings for research in which consumer assessments of taste quality are systematically estimated by a laboratory taste panel are discussed. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Therapeutic empathy and recovery from depression in cognitive-behavioral therapy: A structural equation model.

Demonstrated that therapeutic empathy has a moderate-to-large causal effect on recovery from depression in a group of 185 patients (aged 18–75 yrs) treated with cognitive-behavioral therapy (CBT). The authors simultaneously estimated the reciprocal effects of depression severity on therapeutic empathy and found that this effect was quite small. In addition, homework compliance had a separate effect on clinical recovery, over and above the effect of therapeutic empathy. The patients of novice therapists improved significantly less than did the patients of more experienced therapists, when controlling for therapeutic empathy and homework compliance. Ss who terminated therapy prematurely were less likely to complete the self-help assignments between sessions, rated their therapists as significantly less empathic, and improved significantly less. Ss with borderline personality disorder improved significantly less, but they rated their therapists as just as empathic and caring as other patients. The significance of these findings for psychotherapy research, treatment, and clinical training is discussed. (PsycINFO Database Record (c) 2010 APA, all rights reserved)