Model Comparison of Bayesian Semiparametric and Parametric Structural Equation Models

Structural equation models have wide applications. One of the most important issues in analyzing structural equation models is model comparison. This article proposes a Bayesian model comparison statistic, namely the L _ν-measure for both semiparametric and parametric structural equation models. For illustration purposes, we consider a Bayesian semiparametric approach for estimation and model comparison in the context of structural equation models with fixed covariates. A finite dimensional Dirichlet process is used to model the crucial latent variables, and a blocked Gibbs sampler is implemented for estimation. Empirical performance of the L _ν-measure is evaluated through a simulation study. Results obtained indicate that the L _ν-measure, which additionally requires very minor computational effort, gives satisfactory performance. Moreover, the methodologies are demonstrated through an example with a real data set on kidney disease. Finally, the application of the L _ν-measure to Bayesian semiparametric nonlinear structural equation models is outlined.     

On pythagorean triples of the form (i, i + l, k)

A Pythagorean triple is a triad of positive integers (x, y, z) which satisfy the Pythagoras’ equation x ² + y ² = z ². In this article, we shall consider triples of the form (i, i + 1, k), and the recurrence relations governing them. In the process, we also solve completely the equation i ²+ (i + 1)² = k ².     

Two Rules of Identification for Structural Equation Models

Identification of structural equation models remains a challenge to many researchers. Although empirical tests of identification are readily available in structural equation modeling software, these examine local identification and rely on sample estimates of parameters. Rules of identification are available, but do not include all models encountered in practice. In this article we provide 2 rules of identification: the 2+ emitted paths rule and the exogenous X rule. The former is a necessary condition of identification and the latter is a sufficient condition. We explain and prove each of these rules and provide illustrations of their application. These rules extend the coverage of structural equation models that we can check for identification. We also explain how they can be part of a piecewise identification strategy that extends their use even further.     

Structural Equation Models and Mixture Models With Continuous Nonnormal Skewed Distributions

In this article we describe a structural equation modeling (SEM) framework that allows nonnormal skewed distributions for the continuous observed and latent variables. This framework is based on the multivariate restricted skew t distribution. We demonstrate the advantages of skewed SEM over standard SEM modeling and challenge the notion that structural equation models should be based only on sample means and covariances. The skewed continuous distributions are also very useful in finite mixture modeling as they prevent the formation of spurious classes formed purely to compensate for deviations in the distributions from the standard bell curve distribution. This framework is implemented in Mplus Version 7.2.     

Regression through the Origin

This article describes situations in which regression through the origin is appropriate, derives the normal equation for such a regression and explains the controversy regarding its evaluative statistics. Differences between three popular software packages that allow regression through the origin are illustrated using examples from previous issues of Teaching Statistics.     

Maximum Likelihood Dynamic Factor Modeling for Arbitrary N and T Using SEM

This article has 3 objectives that build on each other. First, we demonstrate how to obtain maximum likelihood estimates for dynamic factor models (the direct autoregressive factor score model) with arbitrary T and N by means of structural equation modeling (SEM) and compare the approach to existing methods. Second, we go beyond standard time series analysis (T large and N = 1) and conventional SEM (N large and T = 1 or small) by integrating both approaches. The resulting combined model offers a variety of new modeling options including a direct test of the ergodicity hypothesis, according to which the factorial structure of an individual observed at many time points is identical to the factorial structure of a group of individuals observed at a single point in time. Third, we illustrate the flexibility of SEM time series modeling by extending the approach to account for complex error structures. We end with a discussion of current limitations and future applications of SEM-based time series modeling for arbitrary T and N.     

On the Performance of Likelihood-Based Difference Tests in Nonlinear Structural Equation Models

This article investigates likelihood-based difference statistics for testing nonlinear effects in structural equation modeling using the latent moderated structural equations (LMS) approach. In addition to the standard difference statistic T_D, 2 robust statistics have been developed in the literature to ensure valid results under the conditions of nonnormality or small sample sizes: the robust T_DR and the “strictly positive” T_DRP. These robust statistics have not been examined in combination with LMS yet. In 2 Monte Carlo studies we investigate the performance of these methods for testing quadratic or interaction effects subject to different sources of nonnormality, nonnormality due to the nonlinear terms, and nonnormality due to the distribution of the predictor variables. The results indicate that T_D is preferable to both T_DR and T_DRP. Under the condition of strong nonlinear effects and nonnormal predictors, T_DR often produced negative differences and T_DRP showed no desirable power.     

Molecule matters van der Waals molecules

van der Waals introduced his equation of state in 1867 modifying the ideal gas equation. His equation had two constants a and b, which accounted for the fact that molecules have attractive forces and finite size. The constant a accounted for the attractive forces, the nature of which was not outlined by van der Waals. This series of articles will discuss about the nature of intermolecular forces.     

Revisiting the Model Size Effect in Structural Equation Modeling

Fitting a large structural equation modeling (SEM) model with moderate to small sample sizes results in an inflated Type I error rate for the likelihood ratio test statistic under the chi-square reference distribution, known as the model size effect. In this article, we show that the number of observed variables (p) and the number of free parameters (q) have unique effects on the Type I error rate of the likelihood ratio test statistic. In addition, the effects of p and q cannot be fully explained using degrees of freedom (df). We also evaluated the performance of 4 correctional methods for the model size effect, including Bartlett’s (1950), Swain’s (1975), and Yuan’s (2005) corrected statistics, and Yuan, Tian, and Yanagihara’s (2015) empirically corrected statistic. We found that Yuan et al.’s (2015) empirically corrected statistic generally yields the best performance in controlling the Type I error rate when fitting large SEM models.     

Comparision of DdQm models in image optical flow analysis based on LBM

The optical flow analysis of the image sequence based on the formal lattice Boltzmann equation, with different DdQm models, is discussed in this paper. The algorithm is based on the lattice Boltzmann method (LBM), which is used in computational fluid dynamics theory for the simulation of fluid dynamics. At first, a generalized approximation to the formal lattice Boltzmann equation is discussed. Then the effects of different DdQm models on the results of the optical flow estimation are compared with each other, while calculating the movement vectors of pixels in the image sequence. The experimental results show that the higher dimension DdQm models, e.g., D3Q15 are more effective than those lower dimension ones.     

The effects of college students’ positive thinking,learning motivation and self-regulation through a self-reflection intervention in Taiwan

This quasi-experimental study examined the effects of a self-reflection intervention on college (college in this article refers to university-level education) students’ positive thinking, learning motivation and self-regulation in Taiwan. One hundred and two college students were selected to participate in an 18-week intervention forming the experimental group (EG) which emphasized providing main lecture, role-play, self-reflection activity, group discussion and group work. Another 179 college students from two other courses were selected as a comparison group. This study showed that supportive, resourced discussion with peers and instructor, self-reflection activities and assignment as a facilitative agent improved the EG students’ positive thinking, learning motivation and self-regulation. In addition, three structural equation models revealed that positive thinking had a strong and direct relation to Taiwanese college students’ pretest learning motivation (β?=?.85) and self-regulation (β?=?.77); learning motivation had a strong and direct relation to students’ pretest positive thinking (β?=?.86) and self-regulation (β?=?.81); and self-regulation had a strong and direct relation to students’ pretest positive thinking (β?=?.83) and learning motivation (β?=?.86). Instructional implications and research recommendations are discussed.     

Promotion of Reading and Early Literacy Skills in Schools: a comparison of three European countries

This article gives a short overview based on the EU High Level Group of Experts on Literacy (HLWG) report to address issues and challenges in Europe on improving literacy competencies in schools. Furthermore, an analysis is conducted focusing on the promotion of reading and early literacy skills in schools taking into account psychological and institutional learning conditions in three countries (Denmark, Germany, and France) with different outcomes in PIRLS 2011 (Progress in International Reading Literacy Study). Country comparison is conducted using multiple groups – multilevel structural equation modelling (MG-MSEM). An important finding is that policy action taken in order to evaluate and improve school effectiveness in each country must address particular needs of its educational system. Suggestions for improving reading achievement in schools are derived from the analysis.     

Assessing specific mediational effects in complex theoretical models

Numerous theoretical structures involve the concept of mediation, the mechanism that accounts for the relation between the predictor and the criterion. The, empirical interpretation of mediation has employed multiple independent models to assess components of the overall structure in a piecemeal manner. This article discusses various limitations of using this independent approach and proposes the use of structural equation modeling (SEM) as an alternative. The article furthermore distinguishes current limitations in using SEM software for assessing mediation in complex structures and presents a strategy for supplemental details that more accurately measure the magnitude of mediational effects, in particular, specific indirect effects.     

Reordering and Reflecting Factors for Simulation Studies With Exploratory Factor Analysis

This article describes the REREFACT R package, which provides a postrotation algorithm that reorders or reflects factors for each replication of a simulation study with exploratory factor analysis (EFA). The purpose of REREFACT is to provide a general algorithm written in freely available software, R, dedicated to addressing the possibility that a nonuniform order or sign pattern of the factors could be observed across replications. The algorithm implemented in REREFACT proceeds in 4 steps. Step 1 determines the total number of equivalent forms, I, of the vector of factors, η. Step 2 indexes, i = 1, 2 … I, each equivalent form of η (i.e., ηⁱ) via a unique permutation matrix, P (i.e., Pⁱ). Step 3 determines which ηⁱ each replication follows. Step 4 uses the appropriate Pⁱ to reorder or re-sign parameter estimates within each replication so that all replications uniformly follow the order and sign pattern defined by the population values. Results from two simulation studies provided evidence for the efficacy of the REREFACT to identify and remediate equivalent forms of η in models with EFA only (i.e., Example 1) and in fuller parameterizations of exploratory structural equation modeling (i.e., Example 2). How to use REREFACT is briefly demonstrated prior to the Discussion section by providing annotations for key commands and condensed output using a subset of simulated data from Example 1.     

A Scaled F Distribution as an Approximation to the Distribution of Test Statistics in Covariance Structure Analysis

When the assumption of multivariate normality is violated or when a discrepancy function other than (normal theory) maximum likelihood is used in structural equation models, the null distribution of the test statistic may not be χ² distributed. Most existing methods to approximate this distribution only match up to 2 moments. In this article, we propose 2 additional approximation methods: a scaled F distribution that matches 3 moments simultaneously and a direct Monte Carlo–based weighted sum of i.i.d. χ² variates. We also conduct comprehensive simulation studies to compare the new and existing methods for both maximum likelihood and nonmaximum likelihood discrepancy functions and to separately evaluate the effect of sampling uncertainty in the estimated weights of the weighted sum on the performance of the approximation methods.     

Binomial index of model fit: An elaboration

This article is an elaboration on the use of the binomial test of model fit value, which in this article will be referred to as the binomial index of model fit value, to gauge the degree that the data fit a path analytic or structural equation model. In addition, this article responds to the criticisms and comments made by Hsu (this issue), Drezner and Drezner (this issue), and Raykov and Penev (this issue) regarding the use of this approach to measuring the degree of model fit. We appreciate the comments provided by these authors. Their comments have assisted us in clarifying our reason for developing the binomial index of model fit procedure as well as our perception of its use.     

Mathematics in ancient India

In this series of articles, we intend to have a glimpse of some of the landmarks in ancient Indian mathematics with special emphasis on number theory. This issue features a brief overview of some of the high peaks of mathematics in ancient India. In the next part we shall describe Aryabhata’s general solution in integers of the equationax -by =c. In subsequent instalments we shall discuss in some detail two of the major contributions by Indians in number theory. The climax of the Indian achievements in algebra and number theory was their development of the ingeniouschakravala method for solving, in integers, the equation x² -Dy² = 1, erroneously known as the Pell equation. We shall later describe the partial solution of Brahmagupta and then the complete solution due to Jayadeva and Bhaskaracharya.     

tetrad: A Set of Stata Commands for Confirmatory Tetrad Analysis

This article provides a brief overview of confirmatory tetrad analysis (CTA) and presents a new set of Stata commands for conducting CTA. The tetrad command allows researchers to use model-implied vanishing tetrads to test the overall fit of structural equation models (SEMs) and the relative fit of two SEMs that are tetrad-nested. An extension of the command, tetrad_matrix, allows researchers to conduct CTA using a sample covariance matrix as input rather than relying on raw data. Researchers can also use the tetrad_matrix command to input a polychoric correlation matrix and conduct CTA for SEMs involving dichotomous, ordinal, or censored outcomes. Another extension of the command, tetrad_bootstrap, provides a bootstrapped p value for the chi-square test statistic. With Stata’s recently developed commands for structural equation modeling, researchers can integrate CTA with data preparation, likelihood ratio tests for model fit, and the estimation of model parameters in a single statistical software package.     

Multivariate Meta-Analysis as Structural Equation Models

Multivariate meta-analysis has become increasingly popular in the educational, social, and medical sciences. It is because the outcome measures in a meta-analysis can involve more than one effect size. This article proposes 2 mathematically equivalent models to implement multivariate meta-analysis in structural equation modeling (SEM). Specifically, this article shows how multivariate fixed-, random- and mixed-effects meta-analyses can be formulated as structural equation models. metaSEM (a free R package based on OpenMx) and Mplus are used to implement the proposed procedures. A real data set is used to illustrate the procedures. Formulating multivariate meta-analysis as structural equation models provides many new research opportunities for methodological development in both meta-analysis and SEM. Issues related to and extensions on the SEM-based meta-analysis are discussed.     

How autonomy support mediates the relationship between self-efficacy and approaches to learning

Abstract

The significant interplay between self-efficacy, autonomy support and approaches to learning in adolescent students is widely recognized. However, less is known about whether substantial differences exist between early and middle adolescent students from schools with different environments. To close this research gap, this study used latent mean comparison and multigroup structural equation modeling with questionnaire data from a sample of German adolescent students (N?=?1153; M_AgeT1?=?13.97, SD = 1.37; M_AgeT2?=?14.27, SD = 1.25) in two waves. Multigroup structural equation modeling reveals that autonomy functions as a mediator in the association between self-efficacy and approaches to learning only for students from schools with a student-centered learning environment based on competence-matrices in contrast to a teacher-directed learning environment, particularly for early adolescent students. More specifically, adolescents’ approaches to learning can be enhanced through autonomy-supportive learning independently of their self-efficacy.