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.     

潜变量交互效应分析方法

简要回顾了分析显变量交互效应的常用方法。详细讨论了目前分析潜变量交互效应的主要方法,包括用潜变量的因子得分做回归分析、分组线性结构方程模型分析、加入乘积项的结构方程模型分析和两步最小二乘回归分析,并比较和评价了这些方法的优缺点。最后归纳了潜变量交互效应分析方法的研究趋势,并介绍了新近进展(包括LMS方法和GAPI方法)。     

Full Information Maximum Likelihood Estimation for Latent Variable Interactions With Incomplete Indicators

Researchers have developed missing data handling techniques for estimating interaction effects in multiple regression. Extending to latent variable interactions, we investigated full information maximum likelihood (FIML) estimation to handle incompletely observed indicators for product indicator (PI) and latent moderated structural equations (LMS) methods. Drawing on the analytic work on missing data handling techniques in multiple regression with interaction effects, we compared the performance of FIML for PI and LMS analytically. We performed a simulation study to compare FIML for PI and LMS. We recommend using FIML for LMS when the indicators are missing completely at random (MCAR) or missing at random (MAR) and when they are normally distributed. FIML for LMS produces unbiased parameter estimates with small variances, correct Type I error rates, and high statistical power of interaction effects. We illustrated the use of these methods by analyzing the interaction effect between advanced cancer patients’ depression and change of inner peace well-being on future hopelessness levels.     

潜变量交互效应标准化估计：方法比较与选用策略

标准化估计对模型的解释和效应大小的比较有重要作用。虽然潜变量交互效应的恰当标准化估计公式已经面世超过10年, 国内外都在使用和引用, 但至今未见到关于不同估计方法得到的恰当标准化估计的系统比较。通过模拟实验, 比较了乘积指标法、潜调节结构方程(LMS)、无先验信息和有先验信息的贝叶斯法的潜变量交互效应标准化估计在不同条件下的表现。结果发现, 在正态条件下, LMS和有信息贝叶斯法表现较好; 而在非正态条件下, 乘积指标法比较稳健, 但需要较大的样本(不小于500), 小样本且外生潜变量之间相关很低时可使用无信息贝叶斯法。     

有调节的多层中介效应分析

多层中介和多层调节效应分析在社科领域已常有应用,但如果将多层中介和调节整合在一起,可以产生2（多层中介类型）×2（调节变量的层次）×3（调节的中介路径）共12种有调节的多层中介模型。面对有调节的多层中介效应分析,研究者往往束手无策。详述了基于多层线性模型的12种有调节的多层中介的分析方法和基于多层结构方程模型的4类有调节的多层中介分析方法,包括正交分割法,随机系数预测法,潜调节结构方程法和贝叶斯合理值法。这四类方法的核心议题在于如何处理潜调节项。当样本量足够大时,建议选择潜调节结构方程法;当样本量不足时,建议选择贝叶斯合理值法。随后用一个实际例子演示如何进行有调节的多层中介效应分析并有相应的Mplus程序。最后展望了有调节的多层中介效应分析研究的拓展方向。     

Modeling Interactions Between Latent Variables in Research on Type D Personality: A Monte Carlo Simulation and Clinical Study of Depression and Anxiety

Several approaches exist to model interactions between latent variables. However, it is unclear how these perform when item scores are skewed and ordinal. Research on Type D personality serves as a good case study for that matter. In Study 1, we fitted a multivariate interaction model to predict depression and anxiety with Type D personality, operationalized as an interaction between its two subcomponents negative affectivity (NA) and social inhibition (SI). We constructed this interaction according to four approaches: (1) sum score product; (2) single product indicator; (3) matched product indicators; and (4) latent moderated structural equations (LMS). In Study 2, we compared these interaction models in a simulation study by assessing for each method the bias and precision of the estimated interaction effect under varying conditions. In Study 1, all methods showed a significant Type D effect on both depression and anxiety, although this effect diminished after including the NA and SI quadratic effects. Study 2 showed that the LMS approach performed best with respect to minimizing bias and maximizing power, even when item scores were ordinal and skewed. However, when latent traits were skewed LMS resulted in more false-positive conclusions, while the Matched PI approach adequately controlled the false-positive rate.     

基于结构方程模型的有调节的中介效应分析

有调节的中介模型是中介过程受到调节变量影响的模型。指出了目前有调节的中介效应分析普遍存在的问题：当前有调节的中介效应检验大多使用多元线性回归分析,忽略了测量误差;而基于结构方程模型(SEM)的有调节的中介效应分析需要产生乘积指标,又会面临乘积指标生成和乘积项非正态分布的问题。在简介潜调节结构方程(LMS)方法后,建议使用LMS方法得到偏差校正的bootstrap置信区间来进行基于SEM的有调节的中介效应分析。总结出一个有调节的中介SEM分析流程,并有示例和相应的Mplus程序。文末展望了LMS和有调节的中介模型的发展方向。     

Examining Nonnormal Latent Variable Distributions for Non-Ignorable Missing Data

Missing not at random (MNAR) modeling for non-ignorable missing responses usually assumes that the latent variable distribution is a bivariate normal distribution. Such an assumption is rarely verified and often employed as a standard in practice. Recent studies for “complete” item responses (i.e., no missing data) have shown that ignoring the nonnormal distribution of a unidimensional latent variable, especially skewed or bimodal, can yield biased estimates and misleading conclusion. However, dealing with the bivariate nonnormal latent variable distribution with present MNAR data has not been looked into. This article proposes to extend unidimensional empirical histogram and Davidian curve methods to simultaneously deal with nonnormal latent variable distribution and MNAR data. A simulation study is carried out to demonstrate the consequence of ignoring bivariate nonnormal distribution on parameter estimates, followed by an empirical analysis of “don’t know” item responses. The results presented in this article show that examining the assumption of bivariate nonnormal latent variable distribution should be considered as a routine for MNAR data to minimize the impact of nonnormality on parameter estimates.     

Ramsay-curve item response theory (RC-IRT) to detect and correct for nonnormal latent variables

Popular methods for fitting unidimensional item response theory (IRT) models to data assume that the latent variable is normally distributed in the population of respondents, but this can be unreasonable for some variables. Ramsay-curve IRT (RC-IRT) was developed to detect and correct for this nonnormality. The primary aims of this article are to introduce RC-IRT less technically than it has been described elsewhere; to evaluate RC-IRT for ordinal data via simulation, including new approaches for model selection; and to illustrate RC-IRT with empirical examples. The empirical examples demonstrate the utility of RC-IRT for real data, and the simulation study indicates that when the latent distribution is skewed, RC-IRT results can be more accurate than those based on the normal model. Along with a plot of candidate curves, the Hannan-Quinn criterion is recommended for model selection.     

潜变量交互效应结构方程：分布分析方法

对于潜变量交互效应结构方程分析,目前应用较多的是乘积指标方法.分布分析方法国内还罕有应用,包括潜调节结构方程（LMS）方法和准极大似然（QML）方法.该研究以乘积指标方法的模型假设为参照,介绍了分布分析方法的模型假设.并简要叙述了LMS方法及其Mplus程序,QML方法及其QML程序.综合现有研究结果,总结出LMS和QML方法、无约束和约束方法的特点,从中可以看出各方法的优缺点,推荐了不同条件下合适的分析方法.     

Reexamining Nonlinear Structural Equation Modeling Procedures: The Effect of Parallel and Congeneric Measures

The current study examines the performance of the extended unconstrained approach (EXUC) and the latent moderated structural equation modeling procedure (LMS) in situations where quadratic and interaction terms are tested simultaneously and investigates their limitations with regard to the employment of parallel and congeneric measures, relatively low indicator reliabilities, and relatively large numbers of indicators. By means of a Monte Carlo study, we found LMS to be the best option for testing multiple nonlinear effects given sufficient sample size (n ≥ 500) and normally distributed exogenous variables. Its advantages became more prominent when indicator reliabilities were heterogeneous and small. The EXUC was a viable option for estimating the model when indicators were parallel and exhibited large indicator reliabilities. An empirical example of the results is provided, and the relevance of measurement model characteristics to assess nonlinear relationships is discussed.     

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.     

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.     

A maximum likelihood method for latent class regression involving a censored dependent variable

The standard tobit or censored regression model is typically utilized for regression analysis when the dependent variable is censored. This model is generalized by developing a conditional mixture, maximum likelihood method for latent class censored regression. The proposed method simultaneously estimates separate regression functions and subject membership in K latent classes or groups given a censored dependent variable for a cross-section of subjects. Maximum likelihood estimates are obtained using an EM algorithm. The proposed method is illustrated via a consumer psychology application.     

A Bayesian Approach to More Stable Estimates of Group-Level Effects in Contextual Studies

Multilevel analyses are often used to estimate the effects of group-level constructs. However, when using aggregated individual data (e.g., student ratings) to assess a group-level construct (e.g., classroom climate), the observed group mean might not provide a reliable measure of the unobserved latent group mean. In the present article, we propose a Bayesian approach that can be used to estimate a multilevel latent covariate model, which corrects for the unreliable assessment of the latent group mean when estimating the group-level effect. A simulation study was conducted to evaluate the choice of different priors for the group-level variance of the predictor variable and to compare the Bayesian approach with the maximum likelihood approach implemented in the software Mplus. Results showed that, under problematic conditions (i.e., small number of groups, predictor variable with a small ICC), the Bayesian approach produced more accurate estimates of the group-level effect than the maximum likelihood approach did.     

Abstract: Comparing Semiparametric and Parametric Methods for Modeling Interactions Among Latent Variables

Many approaches have been proposed to estimate interactions among latent variables. These methods often assume a specific functional form for the interaction, such as a bilinear interaction. Theory is seldom specific enough to provide a functional form for an interaction, however, so a more exploratory, diagnostic approach may often be required. Bauer (2005) proposed a semiparametric approach that allows for the estimation of interaction effects of unknown functional form among latent variables. A structural equation mixture model (SEMM) is first fit to the data. Then an approximation of the interaction is obtained by aggregating over the mixing components. A simulation study is used to examine the performance of this semiparametric approach to two parametric approaches: the latent moderated structures approach (Klein & Moosbrugger, 2000) and the unconstrained product-indicator approach (Marsh, Wen, & Hau, 2004). Data were generated from four functional forms: main effects only, quadratic trend, bilinear interaction, and exponential interaction. Estimates of bias and root mean squared error of approximation were calculated by comparing the surface used to generate the data and the model-implied surface constructed from each approach. As expected, the parametric approaches were more efficient than the SEMM. For the main effects model, bias was similar for both the SEMM and parametric approaches. For the bilinear interaction, the parametric approaches provided nearly identical results, although the SEMM approach was slightly more biased. When the parametric approaches assumed a bilinear interaction and the data were generated from a quadratic trend or an exponential interaction, the parametric approaches generated biased estimates of the true surface. The SEMM approach approximated the true data generation surface with a similarly low level of bias for all the nonlinear surfaces. For example, Figure 1 shows the true surface for the bilinear interaction along with the SEMM estimated average surface. The results suggest that the SEMM approach can provide a relatively unbiased approximation to variety of nonlinear relationships among latent variables.     

Relations Between Instructional Practices and On-Task Behavior in Classrooms Serving American Indian Students

Achievement disparities between American Indian students and non-American Indian peers are persistent and well documented. Student engagement is a promising target for intervention given its relation to academic achievement. This study investigated the relation between specific teacher practices (opportunities to respond [OTRs], praise, and reprimands) and classroom on-task behavior in an urban, public K–8 school that serves primarily American Indian students. OTRs and praise were positively associated with student on-task behavior, whereas reprimands were negatively associated with on-task behavior. Results from multilevel logistic regression indicated that OTRs significantly increased the likelihood that a classroom was highly on-task, whereas the reprimands significantly decreased the likelihood. Praise did not have a significant effect after controlling for the other variables in the model. Results are interpreted in a context of evidence-based instructional practices for increasing OTRs and praise, decreasing reprimands, and ultimately enhancing on-task behavior in an urban classrooms serving primarily American Indian youth.     

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.     

A Composite Likelihood Inference in Latent Variable Models for Ordinal Longitudinal Responses

The paper proposes a composite likelihood estimation approach that uses bivariate instead of multivariate marginal probabilities for ordinal longitudinal responses using a latent variable model. The model considers time-dependent latent variables and item-specific random effects to be accountable for the interdependencies of the multivariate ordinal items. Time-dependent latent variables are linked with an autoregressive model. Simulation results have shown composite likelihood estimators to have a small amount of bias and mean square error and as such they are feasible alternatives to full maximum likelihood. Model selection criteria developed for composite likelihood estimation are used in the applications. Furthermore, lower-order residuals are used as measures-of-fit for the selected models.     

Contributions to Estimation of Polychoric Correlations

This research concerns the estimation of polychoric correlations in the context of fitting structural equation models to observed ordinal variables by multistage estimation. The first main contribution of this research is to propose and evaluate a Monte Carlo estimator for the asymptotic covariance matrix (ACM) of the polychoric correlation estimates. In multistage estimation, the ACM plays a prominent role, as overall test statistics, derived fit indices, and parameter standard errors all depend on this quantity. The ACM, however, must itself be estimated. Established approaches to estimating the ACM use a sample-based version, which can yield poor estimates with small samples. A simulation study demonstrates that the proposed Monte Carlo estimator can be more efficient than its sample-based counterpart. This leads to better calibration for established test statistics, in particular with small samples. The second main contribution of this research is a further exploration of the consequences of violating the normality assumption for the underlying response variables. We show the consequences depend on the type of nonnormality, and the number and location of thresholds. The simulation study also demonstrates that overall test statistics have little power to detect the studied forms of nonnormality, regardless of the ACM estimator.