Statistical analysis of nonlinear structural equation models with continuous and polytomous data

Longitudinal Modeling of Age-Dependent Latent Traits with Generalized Additive Latent and Mixed Models

We present generalized additive latent and mixed models (GALAMMs) for analysis of clustered data with responses and latent variables depending smoothly on observed variables. A scalable maximum likelihood estimation algorithm is proposed, utilizing the Laplace approximation, sparse matrix computation, and automatic differentiation. Mixed response types, heteroscedasticity, and crossed random effects are naturally incorporated into the framework. The models developed were motivated by applications in cognitive neuroscience, and two case studies are presented. First, we show how GALAMMs can jointly model the complex lifespan trajectories of episodic memory, working memory, and speed/executive function, measured by the California Verbal Learning Test (CVLT), digit span tests, and Stroop tests, respectively. Next, we study the effect of socioeconomic status on brain structure, using data on education and income together with hippocampal volumes estimated by magnetic resonance imaging. By combining semiparametric estimation with latent variable modeling, GALAMMs allow a more realistic representation of how brain and cognition vary across the lifespan, while simultaneously estimating latent traits from measured items. Simulation experiments suggest that model estimates are accurate even with moderate sample sizes.

Statistical estimation of structural equation models with a mixture of continuous and categorical observed variables

In the social and behavioral sciences, observed variables of mixed scale types (i.e., both continuous and categorical observed variables) have long been included in structural equation models. However, little is known about the impact of mixed continuous and categorical observed variables on the performance of existing estimation methods. This study compares two popular estimation methods with robust corrections, robust maximum likelihood (MLR) and diagonally weighted least squares (DWLS), when mixed continuous and categorical observed data are analyzed, evaluating the behavior of DWLS and MLR estimates in both measurement and full structural equation models. Monte Carlo simulation was carried out to examine the performance of DWLS and MLR in estimating model parameters, standard errors, and chi-square statistics. Two population models, a correlated three-factor measurement model and a five-factor structural equation model, were tested in combination with 36 other experimental conditions characterized by the number of observed variables' categories (2, 3, 4, 5, 6, and 7), categorical observed distribution shape (symmetry and slight asymmetry), and sample size (200, 500, and 1000). Data generation and analysis were performed with Mplus 8. Results reveal that (1) DWLS yields more accurate factor loading estimates for categorical observed variables than MLR, whereas DWLS and MLR produce comparable factor loading estimates for continuous observed variables; (2) inter-factor correlations and structural paths are estimated equally well by DWLS and MLR in nearly all conditions; (3) robust standard errors of parameter estimates obtained by MLR are slightly more accurate than those produced by DWLS in almost every condition, but the superiority of MLR over DWLS is not clearly evident once a medium or large sample is used (i.e., n = 500 or 1000); and (4) DWLS is systematically superior to MLR in controlling Type I error rates, but this superiority is attenuated with increasing sample size. The article concludes with a general discussion of the findings and some recommendations for practice and future research.

A two-stage estimation procedure for non-linear structural equation models

Applications of structural equation models (SEMs) are often restricted to linear associations between variables. Maximum likelihood (ML) estimation in non-linear models may be complex and require numerical integration. Furthermore, ML inference is sensitive to distributional assumptions. In this article, we introduce a simple two-stage estimation technique for estimation of non-linear associations between latent variables. Here both steps are based on fitting linear SEMs: first a linear model is fitted to data on the latent predictor and terms describing the non-linear effect are predicted by their conditional means. In the second step, the predictions are included in a linear model for the latent outcome variable. We show that this procedure is consistent and identifies its asymptotic distribution. We also illustrate how this framework easily allows the association between latent variables to be modeled using restricted cubic splines, and we develop a modified estimator which is robust to non-normality of the latent predictor. In a simulation study, we compare the proposed method to MLE and alternative two-stage estimation techniques.

The use of items and item parcels in nonlinear structural equation models

Nonlinear structural equation models within the frequentist framework were developed to work with continuous items. Applied researchers who usually work with Likert-type items choose between two strategies to estimate such models: treat items as continuous variables or create item parcels. Two Monte Carlo studies were conducted to evaluate the effects of each strategy on estimates and Type I errors for models with interaction and quadratic effects estimated using LMS. The first study evaluated the effect of asymmetry type and item quantity. The second assessed the use of item parcels and parcel configuration under equivalent conditions. Results reveal that treating items as continuous variables is not problematic when item categories are symmetrical or have opposite-direction asymmetries; however, meaningful parameter bias and increased Type I errors are produced in the case of same-direction asymmetry. Use of parcels does not overcome these problems. The results are discussed to provide recommendations for applied researchers.

Bayesian structural equation modelling tutorial for novice management researchers

Purpose

This paper aims to popularize the Bayesian methods among novice management researchers. The paper interprets the results of Bayesian method of confirmatory factor analysis (CFA), structural equation modelling (SEM), mediation and moderation analysis, with the intention that the novice researchers will apply this method in their research. The paper made an attempt in discussing various complex mathematical concepts such as Markov Chain Monte Carlo, Bayes factor, Bayesian information criterion and deviance information criterion (DIC), etc. in a lucid manner.

Design/methodology/approach

Data collected from 172 pharmaceutical sales representatives were used. The study will help the management researchers to perform Bayesian CFA, Bayesian SEM, Bayesian moderation analysis and Bayesian mediation analysis using SPSS AMOS software.

Findings

The interpretation of the results of Bayesian CFA, Bayesian SEM and Bayesian mediation analysis were discussed.

Practical implications

The management scholars are non-statisticians and are not much aware of the benefits offered by Bayesian methods. Hitherto, the management scholars use predominantly traditional SEM in validating their models empirically, and this study will give an exposure to “Bayesian statistics” that has practical advantages.

Originality/value

This is one paper, which discusses the following four concepts: Bayesian method of CFA, SEM, mediation and moderation analysis.

Bayesian methods for latent trait modelling of longitudinal data

Latent trait models have long been used in the social science literature for studying variables that can only be measured indirectly through multiple items. However, such models are also very useful in accounting for correlation in multivariate and longitudinal data, particularly when outcomes have mixed measurement scales. Bayesian methods implemented with Markov chain Monte Carlo provide a flexible framework for routine fitting of a broad class of latent variable (LV) models, including very general structural equation models. However, in considering LV models, a number of challenging issues arise, including identifiability, confounding between the mean and variance, uncertainty in different aspects of the model, and difficulty in computation. Motivated by the problem of modelling multidimensional longitudinal data, this article reviews the recent literature, provides some recommendations and highlights areas in need of additional research, focusing on methods for model uncertainty.

The Comparison of Single Item Constructs by Relative Mean and Relative Variance

This article discusses the use of an approach advocated in the psychometrics and statistics literature for analyzing square contingency tables with ordered and comparable categories. Assuming that observed ordinal categorical variables are manifestations of underlying continuous variables, a model is formulated to compare the variables by studying the relative location and the relative dispersion of the underlying continuous variables. The formulation, interpretation, and analysis of the model are discussed, and the implementation of the proposed procedure using easily accessible software is addressed. The proposed approach is then compared with a widely adopted simple method that treats the ordinal measures as if they were interval scales. Analyses of real data and simulation results show that the simple method can be misleading and that the proposed approach is preferable in detecting variable differences.

Estimating and testing interactions when explanatory variables are subject to non-classical measurement error

Assessing interactions in linear regression models when covariates have measurement error (ME) is complex.We previously described regression calibration (RC) methods that yield consistent estimators and standard errors for interaction coefficients of normally distributed covariates having classical ME. Here we extend normal based RC (NBRC) and linear RC (LRC) methods to a non-classical ME model, and describe more efficient versions that combine estimates from the main study and internal sub-study. We apply these methods to data from the Observing Protein and Energy Nutrition (OPEN) study. Using simulations we show that (i) for normally distributed covariates efficient NBRC and LRC were nearly unbiased and performed well with sub-study size ≥200; (ii) efficient NBRC had lower MSE than efficient LRC; (iii) the naïve test for a single interaction had type I error probability close to the nominal significance level, whereas efficient NBRC and LRC were slightly anti-conservative but more powerful; (iv) for markedly non-normal covariates, efficient LRC yielded less biased estimators with smaller variance than efficient NBRC. Our simulations suggest that it is preferable to use: (i) efficient NBRC for estimating and testing interaction effects of normally distributed covariates and (ii) efficient LRC for estimating and testing interactions for markedly non-normal covariates.

Bayesian lasso for semiparametric structural equation models

There has been great interest in developing nonlinear structural equation models and associated statistical inference procedures, including estimation and model selection methods. In this paper a general semiparametric structural equation model (SSEM) is developed in which the structural equation is composed of nonparametric functions of exogenous latent variables and fixed covariates on a set of latent endogenous variables. A basis representation is used to approximate these nonparametric functions in the structural equation and the Bayesian Lasso method coupled with a Markov Chain Monte Carlo (MCMC) algorithm is used for simultaneous estimation and model selection. The proposed method is illustrated using a simulation study and data from the Affective Dynamics and Individual Differences (ADID) study. Results demonstrate that our method can accurately estimate the unknown parameters and correctly identify the true underlying model.

Bayesian estimation of semiparametric nonlinear dynamic factor analysis models using the Dirichlet process prior

Parameters in time series and other dynamic models often show complex range restrictions and their distributions may deviate substantially from multivariate normal or other standard parametric distributions. We use the truncated Dirichlet process (DP) as a non-parametric prior for such dynamic parameters in a novel nonlinear Bayesian dynamic factor analysis model. This is equivalent to specifying the prior distribution to be a mixture distribution composed of an unknown number of discrete point masses (or clusters). The stick-breaking prior and the blocked Gibbs sampler are used to enable efficient simulation of posterior samples. Using a series of empirical and simulation examples, we illustrate the flexibility of the proposed approach in approximating distributions of very diverse shapes.

Bayesian analysis for finite mixture in non-recursive non-linear structural equation models

This paper considers finite mixtures of structural equation models with non-linear effects of exogenous latent variables and non-recursive relations among endogenous latent variables. A Bayesian approach is developed to analyse this kind of model. In order to cope with the label switching problem, the permutation sampler is used to choose an appropriate identification constraint. Furthermore, a hybrid Markov chain Monte Carlo method that combines the Gibbs sampler, Metropolis-Hastings algorithm, and Langevin-Hastings algorithm is implemented to produce the Bayesian outputs. Finally, the proposed approach is illustrated by a simulation study and a real example.

The Rise of Markov Chain Monte Carlo Estimation for Psychometric Modeling

Markov chain Monte Carlo (MCMC) estimation strategies represent a powerful approach to estimation in psychometric models. Popular MCMC samplers and their alignment with Bayesian approaches to modeling are discussed. Key historical and current developments of MCMC are surveyed, emphasizing how MCMC allows the researcher to overcome the limitations of other estimation paradigms, facilitates the estimation of models that might otherwise be intractable, and frees the researcher from certain possible misconceptions about the models.

Estimating and testing interactions in linear regression models when explanatory variables are subject to classical measurement error

Estimating and testing interactions in a linear regression model when normally distributed explanatory variables are subject to classical measurement error is complex, since the interaction term is a product of two variables and involves errors of more complex structure. Our aim is to develop simple methods, based on the method of moments (MM) and regression calibration (RC) that yield consistent estimators of the regression coefficients and their standard errors when the model includes one or more interactions. In contrast to previous work using structural equations models framework, our methods allow errors that are correlated with each other and can deal with measurements of relatively low reliability. Using simulations, we show that, under the normality assumptions, the RC method yields estimators with negligible bias and is superior to MM in both bias and variance. We also show that the RC method also yields the correct type I error rate of the test of the interaction. However, when the true covariates are not normally distributed, we recommend using MM. We provide an example relating homocysteine to serum folate and B12 levels.

Bayesian analysis of structural equation models with mixed exponential family and ordered categorical data

Structural equation models are very popular for studying relationships among observed and latent variables. However, the existing theory and computer packages are developed mainly under the assumption of normality, and hence cannot be satisfactorily applied to non-normal and ordered categorical data that are common in behavioural, social and psychological research. In this paper, we develop a Bayesian approach to the analysis of structural equation models in which the manifest variables are ordered categorical and/or from an exponential family. In this framework, models with a mixture of binomial, ordered categorical and normal variables can be analysed. Bayesian estimates of the unknown parameters are obtained by a computational procedure that combines the Gibbs sampler and the Metropolis-Hastings algorithm. Some goodness-of-fit statistics are proposed to evaluate the fit of the posited model. The methodology is illustrated by results obtained from a simulation study and analysis of a real data set about non-adherence of hypertension patients in a medical treatment scheme.

Multilevel IRT using dichotomous and polytomous response data

A structural multilevel model is presented where some of the variables cannot be observed directly but are measured using tests or questionnaires. Observed dichotomous or ordinal polytomous response data serve to measure the latent variables using an item response theory model. The latent variables can be defined at any level of the multilevel model. A Bayesian procedure Markov chain Monte Carlo (MCMC), to estimate all parameters simultaneously is presented. It is shown that certain model checks and model comparisons can be done using the MCMC output. The techniques are illustrated using a simulation study and an application involving students' achievements on a mathematics test and test results regarding management characteristics of teachers and principles.

Maximum Likelihood Analysis of a General Latent Variable Model with Hierarchically Mixed Data

A general two-level latent variable model is developed to provide a comprehensive framework for model comparison of various submodels. Nonlinear relationships among the latent variables in the structural equations at both levels, as well as the effects of fixed covariates in the measurement and structural equations at both levels, can be analyzed within the framework. Moreover, the methodology can be applied to hierarchically mixed continuous, dichotomous, and polytomous data. A Monte Carlo EM algorithm is implemented to produce the maximum likelihood estimate. The E-step is completed by approximating the conditional expectations through observations that are simulated by Markov chain Monte Carlo methods, while the M-step is completed by conditional maximization. A procedure is proposed for computing the complicated observed-data log likelihood and the BIC for model comparison. The methods are illustrated by using a real data set.

Comparison of Approaches in Estimating Interaction and Quadratic Effects of Latent Variables

Various approaches using the maximum likelihood (ML) option of the LISREL program and products of indicators have been proposed to analyze structural equation models with non-linear latent effects on the basis of Kenny and Judd's formulation. Recently, some methods based on the Bayesian approach and the exact ML approaches have been developed. This article reviews, elaborates and compares several approaches for analyzing nonlinear models with interaction and/or quadratic effects. A total of four approaches are examined, including the product indicator ML approaches proposed by Jaccard and Wan (1995) and Joreskog and Yang (1996), a Bayesian approach and an exact ML approach. The empirical performances of these approaches are assessed using simulation studies in terms of their capabilities in producing reliable parameter and standard error estimates. It is found that whilst the Bayesian and the exact ML approaches produce satisfactory results in all the settings under consideration, and are in general very reliable; the product indicator ML approaches can only produce reasonable results in simple models with large sample sizes.

Bayesian analysis of structural equation models with dichotomous variables

Structural equation modelling has been used extensively in the behavioural and social sciences for studying interrelationships among manifest and latent variables. Recently, its uses have been well recognized in medical research. This paper introduces a Bayesian approach to analysing general structural equation models with dichotomous variables. In the posterior analysis, the observed dichotomous data are augmented with the hypothetical missing values, which involve the latent variables in the model and the unobserved continuous measurements underlying the dichotomous data. An algorithm based on the Gibbs sampler is developed for drawing the parameters values and the hypothetical missing values from the joint posterior distributions. Useful statistics, such as the Bayesian estimates and their standard error estimates, and the highest posterior density intervals, can be obtained from the simulated observations. A posterior predictive p-value is used to test the goodness-of-fit of the posited model. The methodology is applied to a study of hypertensive patient non-adherence to medication.

Full Maximum Likelihood Estimation of Polychoric and Polyserial Correlations With Missing Data

This article develops a full maximum likelihood method for obtaining joint estimates of variances and correlations among continuous and polytomous variables with incomplete data which are missing at random with an ignorable missing mechanism. The approach for obtaining the maximum likelihood estimate of the covariance matrix is via a simple confirmatory analysis model with a fixed identity loading matrix and a fixed diagonal matrix with small of unique variances. A Monte Carlo Expectation-Maximization (MCEM) algorithm is constructed to obtain the solution, in which the E-step is approximated by observations simulated by the Gibbs sampler. Results from a simulation study and a real example are provided to illustrate the methodology.

Bayesian Estimation and Model Selection of Multivariate Linear Model with Polytomous Variables

This article provides a Bayesian analysis of the multivariate linear model with polytomous variables. The computational burden due to the intractable multiple integrals induced by the polytomous variables and the model is solved by augmenting the underlying latent continuous measurements of the observed polytomous data. A Gibbs sampler algorithm is implemented to produce the Bayesian estimate. A Bayes factor approach is proposed for model selection, and it is approximated by the Bayesian information criterion (BIC) via the bridge sampling. The proposed methodology is illustrated by examples using multivariate linear regression and multivariate two-way analysis of variance with real data.