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

Effect of fear appeal mobile phone messaging on health behaviors of caregivers with children under-five in Ghana

Social and Behavior Change Communication is a vital strategy in the control of malaria. However, the effectiveness of fear appeal tactic as a preventive strategy remains uncertain. This study examined the influence of a fear appeal mobile phone-based intervention, guided by Witte's Extended Parallel Process model, on malaria prevention among caregivers with children under-five. We conducted a quasi-experimental study of a 12-month intervention using a sample of 324 caregivers from two rural districts, assigned to either an intervention or control group. The intervention group received fear appeal voice Short Message Service (SMS), once a week for twelve (12) months, while caregivers in the control group received none. The results showed that exposure to the messages was associated with an increased odds of positive attitude [adjusted Odds ratio (aOR) = 2.58; 95% CI 1.61-4.15] and behavioral changes (aOR = 2.03, 95% CI 1.29-3.19). The intervention group exhibited lower odds of defensive avoidance (aOR = 0.44, 95% CI 0.29-0.68) and message minimization (aOR = 0.51, 95% CI 0.33-0.78) compared with the control group. These findings highlight the importance of communicating health messages via mobile phones using fear appeal for improving the health behaviors of caregivers. This strategy, however, may not be useful for influencing the intention of caregivers to engage in positive health practices to protect their children from malaria.

Testing and Interpreting Latent Variable Interactions Using the semTools Package

Examining interactions among predictors is an important part of a developing research program. Estimating interactions using latent variables provides additional power to detect effects over testing interactions in regression. However, when predictors are modeled as latent variables, estimating and testing interactions requires additional steps beyond the models used for regression. We review methods of estimating and testing latent variable interactions with a focus on product indicator methods. Product indicator methods of examining latent interactions provide an accurate method to estimate and test latent interactions and can be implemented in any latent variable modeling software package. Significant latent interactions require additional steps (plotting and probing) to interpret interaction effects. We demonstrate how these methods can be easily implemented using functions in the semTools package with models fit using the lavaan package in R, and we illustrate how these methods work using an applied example concerning teacher stress and testing.

A Quasi-Likelihood Approach to Assess Model Fit in Quadratic and Interaction SEM

For the assessment of model fit in linear structural equation modeling (SEM), several fit measures have been developed that use an unconstrained mean and covariance structure, but cannot be readily applied to SEM with quadratic and interaction effects. In this article, we propose the novel quasi-likelihood ratio test (Q-LRT) to evaluate global fit of nonlinear SEM models. The Q-LRT is based on a simplification of the quasi-maximum likelihood method for the estimation of model parameters. An empirical application of the Q-LRT is demonstrated for data in a study about aging in men. Results from a Monte Carlo study show that the Q-LRT performs reliably when sample size is sufficiently large. Also, simulations suggest robustness of Q-LRT for moderately skewed latent exogenous variables.

Investigating Approaches to Estimating Covariate Effects in Growth Mixture Modeling: A Simulation Study

Researchers continue to be interested in efficient, accurate methods of estimating coefficients of covariates in mixture modeling. Including covariates related to the latent class analysis not only may improve the ability of the mixture model to clearly differentiate between subjects but also makes interpretation of latent group membership more meaningful. Very few studies have been conducted that compare the performance of various approaches to estimating covariate effects in mixture modeling, and fewer yet have considered more complicated models such as growth mixture models where the latent class variable is more difficult to identify. A Monte Carlo simulation was conducted to investigate the performance of four estimation approaches: (1) the conventional three-step approach, (2) the one-step maximum likelihood (ML) approach, (3) the pseudo class (PC) approach, and (4) the three-step ML approach in terms of their ability to recover covariate effects in the logistic regression class membership model within a growth mixture modeling framework. Results showed that when class separation was large, the one-step ML approach and the three-step ML approach displayed much less biased covariate effect estimates than either the conventional three-step approach or the PC approach. When class separation was poor, estimation of the relation between the dichotomous covariate and latent class variable was severely affected when the new three-step ML approach was used.

An overview of structural equation modeling: its beginnings, historical development, usefulness and controversies in the social sciences

This paper is a tribute to researchers who have significantly contributed to improving and advancing structural equation modeling (SEM). It is, therefore, a brief overview of SEM and presents its beginnings, historical development, its usefulness in the social sciences and the statistical and philosophical (theoretical) controversies which have often appeared in the literature pertaining to SEM. Having described the essence of SEM in the context of causal analysis, the author discusses the years of the development of structural modeling as the consequence of many researchers' systematically growing needs (in particular in the social sciences) who strove to effectively understand the structure and interactions of latent phenomena. The early beginnings of SEM models were related to the work of Spearman and Wright, and to that of other prominent researchers who contributed to SEM development. The importance and predominance of theoretical assumptions over technical issues for the successful construction of SEM models are also described. Then, controversies regarding the use of SEM in the social sciences are presented. Finally, the opportunities and threats of this type of analytical strategy as well as selected areas of SEM applications in the social sciences are discussed.

Consistent Partial Least Squares for nonlinear structural equation models

Partial Least Squares as applied to models with latent variables, measured indirectly by indicators, is well-known to be inconsistent. The linear compounds of indicators that PLS substitutes for the latent variables do not obey the equations that the latter satisfy. We propose simple, non-iterative corrections leading to consistent and asymptotically normal (CAN)-estimators for the loadings and for the correlations between the latent variables. Moreover, we show how to obtain CAN-estimators for the parameters of structural recursive systems of equations, containing linear and interaction terms, without the need to specify a particular joint distribution. If quadratic and higher order terms are included, the approach will produce CAN-estimators as well when predictor variables and error terms are jointly normal. We compare the adjusted PLS, denoted by PLSc, with Latent Moderated Structural Equations (LMS), using Monte Carlo studies and an empirical application.

A general non-linear multilevel structural equation mixture model

In the past 2 decades latent variable modeling has become a standard tool in the social sciences. In the same time period, traditional linear structural equation models have been extended to include non-linear interaction and quadratic effects (e.g., Klein and Moosbrugger, 2000), and multilevel modeling (Rabe-Hesketh et al., 2004). We present a general non-linear multilevel structural equation mixture model (GNM-SEMM) that combines recent semiparametric non-linear structural equation models (Kelava and Nagengast, 2012; Kelava et al., 2014) with multilevel structural equation mixture models (Muthén and Asparouhov, 2009) for clustered and non-normally distributed data. The proposed approach allows for semiparametric relationships at the within and at the between levels. We present examples from the educational science to illustrate different submodels from the general framework.

Regression calibration for models with two predictor variables measured with error and their interaction, using instrumental variables and longitudinal data

Regression calibration provides a way to obtain unbiased estimators of fixed effects in regression models when one or more predictors are measured with error. Recent development of measurement error methods has focused on models that include interaction terms between measured-with-error predictors, and separately, methods for estimation in models that account for correlated data. In this work, we derive explicit and novel forms of regression calibration estimators and associated asymptotic variances for longitudinal models that include interaction terms, when data from instrumental and unbiased surrogate variables are available but not the actual predictors of interest. The longitudinal data are fit using linear mixed models that contain random intercepts and account for serial correlation and unequally spaced observations. The motivating application involves a longitudinal study of exposure to two pollutants (predictors) - outdoor fine particulate matter and cigarette smoke - and their association in interactive form with levels of a biomarker of inflammation, leukotriene E4 (LTE 4 , outcome) in asthmatic children. Because the exposure concentrations could not be directly observed, we used measurements from a fixed outdoor monitor and urinary cotinine concentrations as instrumental variables, and we used concentrations of fine ambient particulate matter and cigarette smoke measured with error by personal monitors as unbiased surrogate variables. We applied the derived regression calibration methods to estimate coefficients of the unobserved predictors and their interaction, allowing for direct comparison of toxicity of the different pollutants. We used simulations to verify accuracy of inferential methods based on asymptotic theory.

A Bayesian Model For The Estimation Of Latent Interaction And Quadratic Effects When Latent Variables Are Non-Normally Distributed

Structural equation models with interaction and quadratic effects have become a standard tool for testing nonlinear hypotheses in the social sciences. Most of the current approaches assume normally distributed latent predictor variables. In this article, we present a Bayesian model for the estimation of latent nonlinear effects when the latent predictor variables are nonnormally distributed. The nonnormal predictor distribution is approximated by a finite mixture distribution. We conduct a simulation study that demonstrates the advantages of the proposed Bayesian model over contemporary approaches (Latent Moderated Structural Equations [LMS], Quasi-Maximum-Likelihood [QML], and the extended unconstrained approach) when the latent predictor variables follow a nonnormal distribution. The conventional approaches show biased estimates of the nonlinear effects; the proposed Bayesian model provides unbiased estimates. We present an empirical example from work and stress research and provide syntax for substantive researchers. Advantages and limitations of the new model are discussed.

A comparison of methods for estimating quadratic effects in nonlinear structural equation models

Two Monte Carlo simulations were performed to compare methods for estimating and testing hypotheses of quadratic effects in latent variable regression models. The methods considered in the current study were (a) a 2-stage moderated regression approach using latent variable scores, (b) an unconstrained product indicator approach, (c) a latent moderated structural equation method, (d) a fully Bayesian approach, and (e) marginal maximum likelihood estimation. Of the 5 estimation methods, it was found that overall the methods based on maximum likelihood estimation and the Bayesian approach performed best in terms of bias, root-mean-square error, standard error ratios, power, and Type I error control, although key differences were observed. Similarities as well as disparities among methods are highlight and general recommendations articulated. As a point of comparison, all 5 approaches were fit to a reparameterized version of the latent quadratic model to educational reading data.

Non-linear structural equation models with correlated continuous and discrete data

Structural equation models (SEMs) have been widely applied to examine interrelationships among latent and observed variables in social and psychological research. Motivated by the fact that correlated discrete variables are frequently encountered in practical applications, a non-linear SEM that accommodates covariates, and mixed continuous, ordered, and unordered categorical variables is proposed. Maximum likelihood methods for estimation and model comparison are discussed. One real-life data set about cardiovascular disease is used to illustrate the methodologies.

Bayesian Analysis of Multivariate Latent Curve Models With Nonlinear Longitudinal Latent Effects

In longitudinal studies, investigators often measure multiple variables at multiple time points and are interested in investigating individual differences in patterns of change on those variables. Furthermore, in behavioral, social, psychological, and medical research, investigators often deal with latent variables that cannot be observed directly and should be measured by 2 or more manifest variables. Longitudinal latent variables occur when the corresponding manifest variables are measured at multiple time points. Our primary interests are in studying the dynamic change of longitudinal latent variables and exploring the possible interactive effect among the latent variables.Much of the existing research in longitudinal studies focuses on studying change in a single observed variable at different time points. In this article, we propose a novel latent curve model (LCM) for studying the dynamic change of multivariate manifest and latent variables and their linear and interaction relationships. The proposed LCM has the following useful features: First, it can handle multivariate variables for exploring the dynamic change of their relationships, whereas conventional LCMs usually consider change in a univariate variable. Second, it accommodates both first- and second-order latent variables and their interactions to explore how changes in latent attributes interact to produce a joint effect on the growth of an outcome variable. Third, it accommodates both continuous and ordered categorical data, and missing data.

Generalized latent variable models with non-linear effects

Until recently, item response models such as the factor analysis model for metric responses, the two-parameter logistic model for binary responses and the multinomial model for nominal responses considered only the main effects of latent variables without allowing for interaction or polynomial latent variable effects. However, non-linear relationships among the latent variables might be necessary in real applications. Methods for fitting models with non-linear latent terms have been developed mainly under the structural equation modelling approach. In this paper, we consider a latent variable model framework for mixed responses (metric and categorical) that allows inclusion of both non-linear latent and covariate effects. The model parameters are estimated using full maximum likelihood based on a hybrid integration-maximization algorithm. Finally, a method for obtaining factor scores based on multiple imputation is proposed here for the non-linear model.

Multicollinearity and Missing Constraints

Multicollinearity complicates the simultaneous estimation of interaction and quadratic effects in structural equation modeling (SEM). So far, approaches developed within the Kenny-Judd (1984 ) tradition have failed to specify additional and necessary constraints on the measurement error covariances of the nonlinear indicators. Given that the constraints comprise, in part, latent linear predictor correlations, multicollinearity poses a problem for such approaches. Klein and Moosbrugger’s (2000 ) latent moderated structural equations approach (LMS) approach does not utilize nonlinear indicators and should therefore not be affected by this problem. In the context of a simulation study, we varied predictor correlation and the number of nonlinear effects in order to compare the performance of three approaches developed for the estimation of simultaneous nonlinear effects: Ping’s (1996 ) two-step approach, a correctly extended Jöreskog-Yang (1996 ) approach, and LMS. Results show that in contrast to the Jöreskog-Yang approach and LMS, the two-step approach produces biased parameter estimates and false inferences under heightened multicollinearity. Ping’s approach resulted in overestimated interaction effects and underestimated quadratic effects.

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.

Structural equation modeling: strengths, limitations, and misconceptions

Because structural equation modeling (SEM) has become a very popular data-analytic technique, it is important for clinical scientists to have a balanced perception of its strengths and limitations. We review several strengths of SEM, with a particular focus on recent innovations (e.g., latent growth modeling, multilevel SEM models, and approaches for dealing with missing data and with violations of normality assumptions) that underscore how SEM has become a broad data-analytic framework with flexible and unique capabilities. We also consider several limitations of SEM and some misconceptions that it tends to elicit. Major themes emphasized are the problem of omitted variables, the importance of lower-order model components, potential limitations of models judged to be well fitting, the inaccuracy of some commonly used rules of thumb, and the importance of study design. Throughout, we offer recommendations for the conduct of SEM analyses and the reporting of results.

Bayesian Analysis of Structural Equation Models With Nonlinear Covariates and Latent Variables

In this article, we formulate a nonlinear structural equation model (SEM) that can accommodate covariates in the measurement equation and nonlinear terms of covariates and exogenous latent variables in the structural equation. The covariates can come from continuous or discrete distributions. A Bayesian approach is developed to analyze the proposed model. Markov chain Monte Carlo methods for obtaining Bayesian estimates and their standard error estimates, highest posterior density intervals, and a PP p value are developed. Results obtained from two simulation studies are reported to respectively reveal the empirical performance of the proposed Bayesian estimation in analyzing complex nonlinear SEMs, and in analyzing nonlinear SEMs with the normal assumption of the exogenous latent variables violated. The proposed methodology is further illustrated by a real example. Detailed interpretation about the interaction terms is presented.

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.

Maximum Likelihood Analysis of Nonlinear Structural Equation Models With Dichotomous Variables

In this article, a maximum likelihood approach is developed to analyze structural equation models with dichotomous variables that are common in behavioral, psychological and social research. To assess nonlinear causal effects among the latent variables, the structural equation in the model is defined by a nonlinear function. The basic idea of the development is to augment the observed dichotomous data with the hypothetical missing data that involve the latent underlying continuous measurements and the latent variables in the model. An EM algorithm is implemented. The conditional expectation in the E-step is approximated via observations simulated from the appropriate conditional distributions by a Metropolis-Hastings algorithm within the Gibbs sampler, whilst the M-step is completed by conditional maximization. Convergence is monitored by bridge sampling. Standard errors are also obtained. Results from a simulation study and a real example are presented to illustrate the methodology.

Evaluation of the Bayesian and Maximum Likelihood Approaches in Analyzing Structural Equation Models with Small Sample Sizes

The main objective of this article is to investigate the empirical performances of the Bayesian approach in analyzing structural equation models with small sample sizes. The traditional maximum likelihood (ML) is also included for comparison. In the context of a confirmatory factor analysis model and a structural equation model, simulation studies are conducted with the different magnitudes of parameters and sample sizes n = da, where d = 2, 3, 4 and 5, and a is the number of unknown parameters. The performances are evaluated in terms of the goodness-of-fit statistics, and various measures on the accuracy of the estimates. The conclusion is: for data that are normally distributed, the Bayesian approach can be used with small sample sizes, whilst ML cannot.