Model Implied Instrumental Variables (MIIVs): An Alternative Orientation to Structural Equation Modeling

Few dispute that our models are approximations to reality. Yet when it comes to structural equation models (SEMs), we use estimators that assume true models (e.g. maximum likelihood) and that can create biased estimates when the model is inexact. This article presents an overview of the Model Implied Instrumental Variable (MIIV) approach to SEMs from Bollen (1996). The MIIV estimator using Two Stage Least Squares (2SLS), MIIV-2SLS, has greater robustness to structural misspecifications than system wide estimators. In addition, the MIIV-2SLS estimator is asymptotically distribution free. Furthermore, MIIV-2SLS has equation-based overidentification tests that can help pinpoint misspecifications. Beyond these features, the MIIV approach has other desirable qualities. MIIV methods apply to higher order factor analyses, categorical measures, growth curve models, dynamic factor analysis, and nonlinear latent variables. Finally, MIIV-2SLS permits researchers to estimate and test only the latent variable model or any other subset of equations. In addition, other MIIV estimators beyond 2SLS are available. Despite these promising features, research is needed to better understand its performance under a variety of conditions that represent empirical applications. Empirical and simulation examples in the article illustrate the MIIV orientation to SEMs and highlight an R package MIIVsem that implements MIIV-2SLS.     

Analyzing multitrait-mulitmethod data with multilevel confirmatory factor analysis: An application to the validation of the State-Trait Cheerfulness Inventory

This article gives a didactic introduction to the analysis of multitrait-multimethod data with models of multilevel confirmatory factor analysis. In particular, the principles of the multilevel CT-C(M-1) model for interchangeable and structural different methods are explained in detail, and the first application of this model to more than two structurally different methods is presented. The model is illustrated by an application to the analysis of the convergent and discriminant validity of the trait subscales of the State-Trait Cheerfulness Inventory (STCI-T, Ruch, Köhler, & van Thriel, 1996). The results show that the STCI is a reliable and valid questionnaire for assessing the temperamental basis of sense of humor.     

Examining the Factor Structure and Discriminant Validity of the 12-Item General Health Questionnaire (GHQ-12) Among Spanish Postpartum Women

In this study, the authors tested alternative factor models of the 12-item General Health Questionnaire (GHQ-12) in a sample of Spanish postpartum women, using confirmatory factor analysis. The authors report the results of modeling three different methods for scoring the GHQ-12 using estimation methods recommended for categorical and binary data. A discriminant function analysis was also performed to test the utility of a multiple factor model. A two-phase cross-sectional study was designed: (a) 1,453 women visiting at 6 weeks postpartum completed the GHQ-12 and the Edinburgh Postnatal Depression Scale questionnaire and (b) based on the Edinburgh Postnatal Depression Scale outcomes, participants were stratified and randomly selected within each stratum for clinical evaluation. Using the Likert-type scoring approach, Hankins's one-factor model with "method effects" obtained the best fit. In addition, Graetz's three-factor model provided little discrimination between diagnostic groups, the factors being highly correlated. These results support the presence of only one latent factor in the GHQ-12.     

Reexamining the factor structure of the 20-item Toronto alexithymia scale: commentary on Gignac, Palmer, and Stough

Gignac, Palmer, and Stough (2007/this issue) test a number of different latent factor models for the TAS-20 using a community sample of 355 participants and conclude that this scale is best represented by a "nested factors model," with five substantive factors and a method factor. Gignac et al. also report that the correlated three-factor model and a comparable higher order model supported by most studies produced poor levels of incremental close fit. In this article, we challenge Gignac et al.'s unheralded and largely unsupported use of nested model fitting and the uncritical acceptance of exceptionally high cutoff levels to assess goodness of fit (GOF). Using more traditional and empirically supported model testing procedures and a more flexible approach to the interpretation of multiple tests of GOF, we interpret Gignac et al.'s results as actually supportive of the traditional three-factor model and one that has been recovered in 17 of the 24 factor analytic studies of the TAS-20.     

Score Metric Equivalence of the Psychopathy Checklist-Revised (PCL-R) across criminal offenders in North America and the United Kingdom: a critique of Cooke, Michie, Hart, and Clark (2005) and new analyses

David Cooke and colleagues have published a series of item response theory (IRT) studies investigating the equivalence of the Psychopathy Checklist-Revised (PCL-R) for European versus North American (NA) male criminal offenders. They have consistently concluded that PCL-R scores are not equivalent, with European offenders receiving scores up to five points lower than those in NA when matched according to the latent trait. In this article, the authors critique the Cooke et al. analyses and demonstrate how their anchor item selection method is responsible for their final conclusions concerning the apparent lack of equivalence. The authors provide a competing IRT analysis using an iterative purification strategy for anchor item selection and show how this more justifiable approach leads to very different conclusions regarding the equivalence of the PCL-R. More generally, it is argued that strong interpretations of IRT analyses in the presence of uncorroborated anchor items can be highly misleading when evaluating score metric equivalence.     

Random intercept item factor analysis

The common factor model assumes that the linear coefficients (intercepts and factor loadings) linking the observed variables to the latent factors are fixed coefficients (i.e., common for all participants). When the observed variables are participants' observed responses to stimuli, such as their responses to the items of a questionnaire, the assumption of common linear coefficients may be too restrictive. For instance, this may occur if participants consistently use the response scale idiosyncratically. To account for this phenomenon, the authors partially relax the fixed coefficients assumption by allowing the intercepts in the factor model to change across participants. The model is attractive when m factors are expected on the basis of substantive theory but m + 1 factors are needed in practice to adequately reproduce the data. Also, this model for single-level data can be fitted with conventional software for structural equation modeling. The authors demonstrate the use of this model with an empirical data set on optimism in which they compare it with competing models such as the bifactor and the correlated trait-correlated method minus 1 models.     

Reexamining the Factor Structure of the 20-Item Toronto Alexithymia Scale: Commentary on Gignac,Palmer, and Stough

Gignac, Palmer, and Stough (2007/this issue) test a number of different latent factor models for the TAS–20 using a community sample of 355 participants and conclude that this scale is best represented by a “nested factors model,” with five substantive factors and a method factor. Gignac et al. also report that the correlated three-factor model and a comparable higher order model supported by most studies produced poor levels of incremental close fit. In this article, we challenge Gignac et al.'s unheralded and largely unsupported use of nested model fitting and the uncritical acceptance of exceptionally high cutoff levels to assess goodness of fit (GOF). Using more traditional and empirically supported model testing procedures and a more flexible approach to the interpretation of multiple tests of GOF, we interpret Gignac et al.'s results as actually supportive of the traditional three-factor model and one that has been recovered in 17 of the 24 factor analytic studies of the TAS–20.     

A confirmatory factor analysis of the California Verbal Learning Test--Second Edition (CVLT-II) in the standardization sample

The purpose of this study is to determine the latent structure of the California Verbal Learning Test-Second Edition (CVLT-II; Delis, Kramer, Kaplan, & Ober, 2000) at three different age levels, using the standardization sample. Maximum likelihood confirmatory factor analyses are performed to test four competing hypothetical models for fit and parsimony. A four-factor model consisting of Attention Span, Learning Efficiency, Delayed Memory, and Inaccurate Memory provides the best fit to the data at two age levels (16-30 years and 31-60 years, respectively) but the factor structure is somewhat less robust in the oldest age group (>60 years). The results clarify the latent structure of the CVLT-II and suggest that a multifactorial interpretation of quantitative and qualitative indexes from this instrument is appropriate for clinical practice with most adults, although findings may need to be interpreted more cautiously among older persons.     

Model selection and psychological theory: a discussion of the differences between the Akaike information criterion (AIC) and the Bayesian information criterion (BIC)

This article reviews the Akaike information criterion (AIC) and the Bayesian information criterion (BIC) in model selection and the appraisal of psychological theory. The focus is on latent variable models, given their growing use in theory testing and construction. Theoretical statistical results in regression are discussed, and more important issues are illustrated with novel simulations involving latent variable models including factor analysis, latent profile analysis, and factor mixture models. Asymptotically, the BIC is consistent, in that it will select the true model if, among other assumptions, the true model is among the candidate models considered. The AIC is not consistent under these circumstances. When the true model is not in the candidate model set the AIC is efficient, in that it will asymptotically choose whichever model minimizes the mean squared error of prediction/estimation. The BIC is not efficient under these circumstances. Unlike the BIC, the AIC also has a minimax property, in that it can minimize the maximum possible risk in finite sample sizes. In sum, the AIC and BIC have quite different properties that require different assumptions, and applied researchers and methodologists alike will benefit from improved understanding of the asymptotic and finite-sample behavior of these criteria. The ultimate decision to use the AIC or BIC depends on many factors, including the loss function employed, the study's methodological design, the substantive research question, and the notion of a true model and its applicability to the study at hand.     

Nonlinear Regime-Switching State-Space (RSSS) Models

Nonlinear dynamic factor analysis models extend standard linear dynamic factor analysis models by allowing time series processes to be nonlinear at the latent level (e.g., involving interaction between two latent processes). In practice, it is often of interest to identify the phases—namely, latent “regimes” or classes—during which a system is characterized by distinctly different dynamics. We propose a new class of models, termed nonlinear regime-switching state-space (RSSS) models, which subsumes regime-switching nonlinear dynamic factor analysis models as a special case. In nonlinear RSSS models, the change processes within regimes, represented using a state-space model, are allowed to be nonlinear. An estimation procedure obtained by combining the extended Kalman filter and the Kim filter is proposed as a way to estimate nonlinear RSSS models. We illustrate the utility of nonlinear RSSS models by fitting a nonlinear dynamic factor analysis model with regime-specific cross-regression parameters to a set of experience sampling affect data. The parallels between nonlinear RSSS models and other well-known discrete change models in the literature are discussed briefly.     

A comparison of four approaches to account for method effects in latent state-trait analyses

Latent state-trait (LST) analysis is frequently applied in psychological research to determine the degree to which observed scores reflect stable person-specific effects, effects of situations and/or person-situation interactions, and random measurement error. Most LST applications use multiple repeatedly measured observed variables as indicators of latent trait and latent state residual factors. In practice, such indicators often show shared indicator-specific (or method) variance over time. In this article, the authors compare 4 approaches to account for such method effects in LST models and discuss the strengths and weaknesses of each approach based on theoretical considerations, simulations, and applications to actual data sets. The simulation study revealed that the LST model with indicator-specific traits (Eid, 1996) and the LST model with M - 1 correlated method factors (Eid, Schneider, & Schwenkmezger, 1999) performed well, whereas the model with M orthogonal method factors used in the early work of Steyer, Ferring, and Schmitt (1992) and the correlated uniqueness approach (Kenny, 1976) showed limitations under conditions of either low or high method-specificity. Recommendations for the choice of an appropriate model are provided.     

On the reliability, consistency, and method-specificity based on the CT-C(M–1) model

In the present study, we discuss reliability, consistency, and method specificity based on the CT-C(M?-?1) model, which provides clear definitions of trait and method factors and can facilitate parameter estimation. Properties of the reliability coefficient, the consistency coefficient, and the method-specificity coefficient of the summated score for a trait factor are addressed. The consistency coefficient and the method-specificity coefficient are both functions of the number of items, the average item consistency, and the average item method specificity. The usefulness of the findings is demonstrated in an alternative approach proposed for scale reduction. The approach, taking into account both traits and methods, helps identify the items leading to the maximum of convergent validity or method effects. The approach, illustrated with a simulated data set, is recommended for scale development based on multitrait-multimethod designs.     

A Polychoric Instrumental Variable (PIV) Estimator for Structural Equation Models with Categorical Variables

This paper presents a new polychoric instrumental variable (PIV) estimator to use in structural equation models (SEMs) with categorical observed variables. The PIV estimator is a generalization of Bollen’s (Psychometrika 61:109–121, 1996) 2SLS/IV estimator for continuous variables to categorical endogenous variables. We derive the PIV estimator and its asymptotic standard errors for the regression coefficients in the latent variable and measurement models. We also provide an estimator of the variance and covariance parameters of the model, asymptotic standard errors for these, and test statistics of overall model fit. We examine this estimator via an empirical study and also via a small simulation study. Our results illustrate the greater robustness of the PIV estimator to structural misspecifications than the system-wide estimators that are commonly applied in SEMs. Kenneth Bollen gratefully acknowledges support from NSF SES 0617276, NIDA 1-RO1-DA13148-01, and DA013148-05A2. Albert Maydeu-Olivares was supported by the Department of Universities, Research and Information Society (DURSI) of the Catalan Government, and by grant BSO2003-08507 from the Spanish Ministry of Science and Technology. We thank Sharon Christ, John Hipp, and Shawn Bauldry for research assistance. The comments of the members of the Carolina Structural Equation Modeling (CSEM) group are greatly appreciated. An earlier version of this paper under a different title was presented by K. Bollen at the Psychometric Society Meetings, June, 2002, Chapel Hill, North Carolina.     

Reconsidering the Use of Autoregressive Latent Trajectory (ALT) Models

The simultaneous estimation of autoregressive (simplex) structures and latent trajectories, so called ALT (autoregressive latent trajectory) models, is becoming an increasingly popular approach to the analysis of change. Although historically autoregressive (AR) and latent growth curve (LGC) models have been developed quite independently from each other, the underlying pattern of change is often highly similar. In this article it is shown that their integration rests on the strong assumption that neither the AR part nor the LGC part contains any misspecification. In practice, however, this assumption is often violated due to nonlinearity in the LGC part. As a consequence, the autoregressive (simplex) process incorrectly accounts for part of this nonlinearity, thus rendering any substantive interpretation of parameter estimates virtually impossible. Accordingly, researchers are advised to exercise extreme caution when using ALT models in practice. All arguments are illustrated by empirical data on skill acquisition, and a simulation study is provided to investigate the conditions and consequences of mistaking nonlinear growth curve patterns as autoregressive processes.     

Agreement among student,parent and teacher ratings of school inclusion: A multitrait-multimethod analysis

One of the central objectives of inclusive education, and education in general, is not only to support every students' academic learning, but also their social and emotional development. It therefore is important to identify difficulties in a child's socio-emotional development at school. The current study investigates students' emotional inclusion and social inclusion, as well as students' academic self-concept from four different perspectives using the Perceptions of Inclusion Questionnaire (PIQ). In particular, we analyzed the degree of agreement between teacher, mother, and father ratings with students' self-reports. Moreover, we tested if students' gender and special educational needs (SEN) are predictors for possible bias in parent and teacher reports. Survey participants included 721 Austrian, Grade 4 students from 48 classes. In addition, data from 46 teachers, 466 mother reports, and 375 father reports were included. We assessed the consistency (i.e., agreement) between the different raters by means of multitrait-multimethod analyses, or more precisely, a correlated trait–correlated method minus one (CT-C[M-1]) model. Results of the CT-C(M-1) analyses indicated a rather strong rater bias (i.e., method effects) for all three dimensions of inclusion. However, the consistency for academic self-concept was higher than for emotional and social inclusion. Furthermore, gender and SEN status affected rater bias, particularly for teacher reports. Results suggested that it matters who reports students' emotional inclusion, social inclusion, and academic self-concept, which has methodological and practical implications.     

Modeling population heterogeneity in appearance- and performance-enhancing drug (APED) use: applications of mixture modeling in 400 regular APED users

Appearance- and performance-enhancing drugs (APEDs) constitute a wide range of substances, including anabolic-androgenic steroids, nonsteroidal anabolics, and licit and illicit ergo/thermogenics. A great deal of heterogeneity exists in APED use patterns among weight-lifting men, and, consequently, little is known about how these patterns are related to side effect profiles or risk potential. In the current study, a sample of 400 adult men who were regular APED users completed an interactive Web-based instrument detailing information about APED use, side effects, and related indicators of risk. To explore the heterogeneity of APED use patterns, the authors subjected data on use patterns to (a) latent class analysis (LCA), (b) latent trait analysis (LTA), and (c) factor mixture analysis to determine the best model of APED use. Results indicated that a 4-class factor mixture model provided a better fit than LCA and LTA models. The authors also found that severity and latent class were uniquely associated with negative outcomes. Each of the 4 classes was associated with unique side effects, motivations, and participant use patterns. Implications for identifying pathological forms of APED use are discussed.     

Spanish version of the Prospective and Retrospective Memory Questionnaire (PRMQ-S)

The concept of prospective memory is relatively recent. Thus, it is necessary to have instruments to assess it, in interaction with retrospective memory. The Prospective and Retrospective Memory Questionnaire (PRMQ; Smith et al., 2000) is an instrument that has been effectively used for assessment in several languages. Nevertheless, there is still no Spanish adaptation. The aim of this study was to translate the PRMQ and obtain information about its psychometric properties. A back translation process was applied to the PRMQ. The resulting PRMQ-S was self-administered to 520 subjects after applying the pilot test. Confirmatory factor analysis (CFA) was used to test competing models of its latent structure. Internal consistency was evaluated using Cronbach's alpha. The results showed an adequate internal consistency of the total scale and the Prospective and Retrospective scales were: .89, .84 and .76, respectively. The factor structure was compared with the original study. Our results suggested a disagreement with the tripartite model defended by the authors of the PRMQ. Since there is a high correlation between two factors in the bi-factor model, and the tripartite model showed non-significant loadings to the two specific factors, we recommend that the PRMQ-S be used as one general memory score. We demonstrated adequate psychometric properties of the PRMQ-S if it is used as unique measurement of memory failures. These results are relevant due to the need for a short and accurate instrument in our language that explores memory failures, which has a major impact on the activities of daily living.     

The Construct Validity of Acceptance: A Multitrait-Multimethod Investigation

Despite increasing clinical and empirical attention, the construct validity of acceptance has not been extensively investigated. The present study utilized a multitrait-multimethod design and a correlated trait-correlated method minus one [CT-C(M-1)] confirmatory factor analytic model to assess acceptance's convergent validity across methods and discriminant validity in comparison to cognitive reappraisal and perceived emotional control in a sample of 210 outpatients with anxiety and mood disorders. In addition, the study evaluated acceptance's concurrent validity by investigating the extent to which it was associated with variables of clinical interest over and above the two rival constructs. Results of confirmatory factor analyses supported acceptance's convergent and discriminant validity in comparison to the two neighboring constructs, and thereby provided partial support for its construct validity. However, contrary to prediction, acceptance was not significantly associated with concurrent validation measures. These results are discussed in the context of acceptance's potential therapeutic utility and functional relationships with associated constructs.     

Longitudinal factor structure of posttraumatic stress symptoms related to intimate partner violence

Confirmatory factor analysis (CFA) studies have suggested that a model of posttraumatic stress disorder (PTSD) that is characterized by 4 factors is preferable to competing models. However, the composition of these 4 factors has varied across studies, with 1 model splitting avoidance and numbing symptoms (e.g., D. W. King, G. A. Leskin, L. A. King, & F. W. Weathers, 1998) and the other including a dysphoria factor that combines numbing and nonspecific hyperarousal symptoms (L. J. Simms, D. Watson, & B. N. Doebbeling, 2002). Using the PTSD Checklist (F. W. Weathers, B. T. Litz, D. S. Herman, J. A. Huska, & T. M. Keane, 1993) and CFA, the authors compared these models with competing models. A model of PTSD with 4 intercorrelated factors of Intrusions, Avoidance, Dysphoria, and Hyperarousal was found superior among 396 medical patients who screened positive for intimate partner violence (IPV) and 405 women seeking services for IPV. Structural invariance testing indicated that this 4-factor model remains stable across service setting and time.     

Construct validity of the California Verbal Learning Test--Children's Version (CVLT-C) after pediatric traumatic brain injury

The purpose of this study was to determine the latent structure of the California Verbal Learning Test--Children's Version (CVLT-C; D. Delis, J. Kramer, E. Kaplan, & B. Ober, 1994) in a sample of 175 children with traumatic brain injury (TBI). Maximum-likelihood confirmatory factor analyses were performed to test 6 competing hypothetical models for fit and parsimony. A 4-factor model consisting of Attention Span, Learning Efficiency, Delayed Recall, and Inaccurate Recall provided the best fit to the data. The results support the construct validity of the CVLT-C in children with TBI and suggest that a multifactorial interpretation of quantitative indexes from this instrument is appropriate for clinical practice.