Bayesian structural equation modeling: A more flexible representation of substantive theory

This article proposes a new approach to factor analysis and structural equation modeling using Bayesian analysis. The new approach replaces parameter specifications of exact zeros with approximate zeros based on informative, small-variance priors. It is argued that this produces an analysis that better reflects substantive theories. The proposed Bayesian approach is particularly beneficial in applications where parameters are added to a conventional model such that a nonidentified model is obtained if maximum-likelihood estimation is applied. This approach is useful for measurement aspects of latent variable modeling, such as with confirmatory factor analysis, and the measurement part of structural equation modeling. Two application areas are studied, cross-loadings and residual correlations in confirmatory factor analysis. An example using a full structural equation model is also presented, showing an efficient way to find model misspecification. The approach encompasses 3 elements: model testing using posterior predictive checking, model estimation, and model modification. Monte Carlo simulations and real data are analyzed using Mplus. The real-data analyses use data from Holzinger and Swineford's (1939) classic mental abilities study, Big Five personality factor data from a British survey, and science achievement data from the National Educational Longitudinal Study of 1988. (PsycINFO Database Record (c) 2012 APA, all rights reserved).     

Fitting the factor analysis model in ℓ1 norm

The well‐known problem of fitting the exploratory factor analysis model is reconsidered where the usual least squares goodness‐of‐fit function is replaced by a more resistant discrepancy measure, based on a smooth approximation of the ?₁ norm. Fitting the factor analysis model to the sample correlation matrix is a complex matrix optimization problem which requires the structure preservation of the unknown parameters (e.g. positive definiteness). The projected gradient approach is a natural way of solving such data matching problems as especially designed to follow the geometry of the model parameters. Two reparameterizations of the factor analysis model are considered. The approach leads to globally convergent procedures for simultaneous estimation of the factor analysis matrix parameters. Numerical examples illustrate the algorithms and factor analysis solutions.     

Bayesian latent variable models for the analysis of experimental psychology data

In this paper, we address the use of Bayesian factor analysis and structural equation models to draw inferences from experimental psychology data. While such application is non-standard, the models are generally useful for the unified analysis of multivariate data that stem from, e.g., subjects’ responses to multiple experimental stimuli. We first review the models and the parameter identification issues inherent in the models. We then provide details on model estimation via JAGS and on Bayes factor estimation. Finally, we use the models to re-analyze experimental data on risky choice, comparing the approach to simpler, alternative methods.     

Examining time‐invariance in reliability in multi‐wave,multi‐indicator models: A covariance structure analysis approach accounting for indicator specificity

A covariance structure analysis method for testing time‐invariance in reliability in multiwave, multiple‐indicator models in outlined. The approach accounts for observed variable specificity and permits, in addition, estimation of reliability in terms of ‘pure’ measurement error variance. The proposed procedure is developed within a confirmatory factor analysis framework and illustrated with data from a cognitive intervention study.     

On the Relation Between the Linear Factor Model and the Latent Profile Model

The relationship between linear factor models and latent profile models is addressed within the context of maximum likelihood estimation based on the joint distribution of the manifest variables. Although the two models are well known to imply equivalent covariance decompositions, in general they do not yield equivalent estimates of the unconditional covariances. In particular, a 2-class latent profile model with Gaussian components underestimates the observed covariances but not the variances, when the data are consistent with a unidimensional Gaussian factor model. In explanation of this phenomenon we provide some results relating the unconditional covariances to the goodness of fit of the latent profile model, and to its excess multivariate kurtosis. The analysis also leads to some useful parameter restrictions related to symmetry.     

Estimation of maximal reliability for multiple‐component instruments in multilevel designs

A procedure for point and interval estimation of maximal reliability of multiple‐component measuring instruments in multi‐level settings is outlined. The approach is applicable to hierarchical designs in which individuals are nested within higher‐order units and exhibit possibly related performance on components of a given homogeneous scale. The method is developed within the framework of multi‐level factor analysis. The proposed procedure is illustrated with an empirical example.     

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.     

An empirical evaluation of alternative methods of estimation for confirmatory factor analysis with ordinal data

Confirmatory factor analysis (CFA) is widely used for examining hypothesized relations among ordinal variables (e.g., Likert-type items). A theoretically appropriate method fits the CFA model to polychoric correlations using either weighted least squares (WLS) or robust WLS. Importantly, this approach assumes that a continuous, normal latent process determines each observed variable. The extent to which violations of this assumption undermine CFA estimation is not well-known. In this article, the authors empirically study this issue using a computer simulation study. The results suggest that estimation of polychoric correlations is robust to modest violations of underlying normality. Further, WLS performed adequately only at the largest sample size but led to substantial estimation difficulties with smaller samples. Finally, robust WLS performed well across all conditions.     

Influence analysis for the factor analysis model with ranking data

Influence analysis is an important component of data analysis, and the local influence approach has been widely applied to many statistical models to identify influential observations and assess minor model perturbations since the pioneering work of Cook (1986) . The approach is often adopted to develop influence analysis procedures for factor analysis models with ranking data. However, as this well‐known approach is based on the observed data likelihood, which involves multidimensional integrals, directly applying it to develop influence analysis procedures for the factor analysis models with ranking data is difficult. To address this difficulty, a Monte Carlo expectation and maximization algorithm (MCEM) is used to obtain the maximum‐likelihood estimate of the model parameters, and measures for influence analysis on the basis of the conditional expectation of the complete data log likelihood at the E‐step of the MCEM algorithm are then obtained. Very little additional computation is needed to compute the influence measures, because it is possible to make use of the by‐products of the estimation procedure. Influence measures that are based on several typical perturbation schemes are discussed in detail, and the proposed method is illustrated with two real examples and an artificial example.     

Selecting polychoric instrumental variables in confirmatory factor analysis: An alternative specification test and effects of instrumental variables

The polychoric instrumental variable (PIV) approach is a recently proposed method to fit a confirmatory factor analysis model with ordinal data. In this paper, we first examine the small-sample properties of the specification tests for testing the validity of instrumental variables (IVs). Second, we investigate the effects of using different numbers of IVs. Our results show that specification tests derived for continuous data are extremely oversized at all sample sizes when applied to ordinal variables. Possible modifications for ordinal data are proposed in the present study. Simulation results show that the modified specification tests with all available IVs are able to detect model misspecification. In terms of estimation accuracy, the PIV approach where the IVs outnumber the endogenous variables by one produces a lower bias but a higher variation than the PIV approach with more IVs for correctly specified factor loadings at small samples.     

Factor analysis with (mixed) observed and latent variables in the exponential family

We develop a general approach to factor analysis that involves observed and latent variables that are assumed to be distributed in the exponential family. This gives rise to a number of factor models not considered previously and enables the study of latent variables in an integrated methodological framework, rather than as a collection of seemingly unrelated special cases. The framework accommodates a great variety of different measurement scales and accommodates cases where different latent variables have different distributions. The models are estimated with the method of simulated likelihood, which allows for higher dimensional factor solutions to be estimated than heretofore. The models are illustrated on synthetic data. We investigate their performance when the distribution of the latent variables is mis-specified and when part of the observations are missing. We study the properties of the simulation estimators relative to maximum likelihood estimation with numerical integration. We provide an empirical application to the analysis of attitudes.     

Full-information item bi-factor analysis

A plausibles-factor solution for many types of psychological and educational tests is one that exhibits a general factor ands − 1 group or method related factors. The bi-factor solution results from the constraint that each item has a nonzero loading on the primary dimension and at most one of thes − 1 group factors. This paper derives a bi-factor item-response model for binary response data. In marginal maximum likelihood estimation of item parameters, the bi-factor restriction leads to a major simplification of likelihood equations and (a) permits analysis of models with large numbers of group factors; (b) permits conditional dependence within identified subsets of items; and (c) provides more parsimonious factor solutions than an unrestricted full-information item factor analysis in some cases. Supported by the Cognitive Science Program, Office of Naval Research, Under grant #N00014-89-J-1104. We would like to thank Darrell Bock for several helpful suggestions.     

The Infinitesimal Jackknife with Exploratory Factor Analysis

The infinitesimal jackknife, a nonparametric method for estimating standard errors, has been used to obtain standard error estimates in covariance structure analysis. In this article, we adapt it for obtaining standard errors for rotated factor loadings and factor correlations in exploratory factor analysis with sample correlation matrices. Both maximum likelihood estimation and ordinary least squares estimation are considered.     

Regression among factor scores

Structural equation models with latent variables are sometimes estimated using an intuitive three-step approach, here denoted factor score regression. Consider a structural equation model composed of an explanatory latent variable and a response latent variable related by a structural parameter of scientific interest. In this simple example estimation of the structural parameter proceeds as follows: First, common factor models areseparately estimated for each latent variable. Second, factor scores areseparately assigned to each latent variable, based on the estimates. Third, ordinary linear regression analysis is performed among the factor scores producing an estimate for the structural parameter. We investigate the asymptotic and finite sample performance of different factor score regression methods for structural equation models with latent variables. It is demonstrated that the conventional approach to factor score regression performs very badly. Revised factor score regression, using Regression factor scores for the explanatory latent variables and Bartlett scores for the response latent variables, produces consistent estimators for all parameters.     

Structural analysis of covariance and correlation matrices

A general approach to the analysis of covariance structures is considered, in which the variances and covariances or correlations of the observed variables are directly expressed in terms of the parameters of interest. The statistical problems of identification, estimation and testing of such covariance or correlation structures are discussed.Several different types of covariance structures are considered as special cases of the general model. These include models for sets of congeneric tests, models for confirmatory and exploratory factor analysis, models for estimation of variance and covariance components, regression models with measurement errors, path analysis models, simplex and circumplex models. Many of the different types of covariance structures are illustrated by means of real data.1978 Psychometric Society Presidential Address.This research has been supported by the Bank of Sweden Tercentenary Foundation under the project entitledStructural Equation Models in the Social Sciences, Karl G. Jöreskog, project director.     

Semi-sparse PCA

It is well known that the classical exploratory factor analysis (EFA) of data with more observations than variables has several types of indeterminacy. We study the factor indeterminacy and show some new aspects of this problem by considering EFA as a specific data matrix decomposition. We adopt a new approach to the EFA estimation and achieve a new characterization of the factor indeterminacy problem. A new alternative model is proposed, which gives determinate factors and can be seen as a semi-sparse principal component analysis (PCA). An alternating algorithm is developed, where in each step a Procrustes problem is solved. It is demonstrated that the new model/algorithm can act as a specific sparse PCA and as a low-rank-plus-sparse matrix decomposition. Numerical examples with several large data sets illustrate the versatility of the new model, and the performance and behaviour of its algorithmic implementation.

     

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.     

Item factor analysis: current approaches and future directions

The rationale underlying factor analysis applies to continuous and categorical variables alike; however, the models and estimation methods for continuous (i.e., interval or ratio scale) data are not appropriate for item-level data that are categorical in nature. The authors provide a targeted review and synthesis of the item factor analysis (IFA) estimation literature for ordered-categorical data (e.g., Likert-type response scales) with specific attention paid to the problems of estimating models with many items and many factors. Popular IFA models and estimation methods found in the structural equation modeling and item response theory literatures are presented. Following this presentation, recent developments in the estimation of IFA parameters (e.g., Markov chain Monte Carlo) are discussed. The authors conclude with considerations for future research on IFA, simulated examples, and advice for applied researchers.     

Multiple Group Analysis in Multilevel Structural Equation Model Across Level 1 Groups

This article introduces and evaluates a procedure for conducting multiple group analysis in multilevel structural equation model across Level 1 groups (MG1-MSEM; Ryu, 2014). When group membership is at Level 1, multiple group analysis raises two issues that cannot be solved by a simple extension of the standard multiple group analysis in single-level structural equation model. First, the Level 2 data are not independent between Level 1 groups. Second, the standard procedure fails to take into account the dependency between members of different Level 1 groups within the same cluster. The MG1-MSEM approach provides solutions to these problems. In MG1-MSEM, the Level 1 mean structure is necessary to represent the differences between Level 1 groups within clusters. The Level 2 model is the same regardless of Level 1 group membership. A simulation study examined the performance of MUML (Muthén's maximum likelihood) estimation in MG1-MSEM. The MG1-MSEM approach is illustrated for both a multilevel path model and a multilevel factor model using empirical data sets.     

Instrumental variables in factor analysis

The factor analysis model is rewritten as a system of linear structural relations with errors in variables. The method of instrumental variables is applied to this revised form of the model to obtain estimates of the factor loading matrix. The relation between this method and interbattery analysis, proportional profile analysis, and canonical factor analysis is pointed out. In addition, an estimation procedure based on replicated sampling different from proportional profile analysis is given.