Diagnosing Nonlinearity With Confidence Envelopes for a Semiparametric Approach to Modeling Bivariate Nonlinear Relations Among Latent Variables

During the early phases of research, semiparametric models (SPMs) have the advantage of recovering latent nonlinearity over parametric counterparts. Structural equation mixture models (Bauer, 2005) can be applied as SPMs to flexibly recover and describe the form of the unknown latent relationship with minimal distributional assumptions. This short report extends the work on this SPM (Bauer, 2005; Pek, Losardo & Bauer, 2011) by developing approximate simultaneous confidence bands or confidence envelopes (CEs) to evaluate potential nonlinearity of the unknown latent function. A line-finding algorithm to be used in conjunction with these CEs is also developed as an implementation of an informal test to diagnose nonlinearity. Coverage of the CEs and performance of the algorithm in terms of rates of detecting latent nonlinearity are evaluated by Monte Carlo. Recommendations for the use of these CEs and the algorithm for detecting nonlinearity are suggested.     

A Nonlinear Structural Equation Mixture Modeling Approach for Nonnormally Distributed Latent Predictor Variables

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 describe a nonlinear structural equation mixture approach that integrates the strength of parametric approaches (specification of the nonlinear functional relationship) and the flexibility of semiparametric structural equation mixture approaches for approximating the nonnormality of latent predictor variables. In a comparative simulation study, the advantages of the proposed mixture procedure over contemporary approaches [Latent Moderated Structural Equations approach (LMS) and the extended unconstrained approach] are shown for varying degrees of skewness of the latent predictor variables. Whereas the conventional approaches show either biased parameter estimates or standard errors of the nonlinear effects, the proposed mixture approach provides unbiased estimates and standard errors. We present an empirical example from educational research. Guidelines for applications of the approaches and limitations are discussed.     

Bootstrapping Confidence Intervals for Fit Indexes in Structural Equation Modeling

Bootstrapping approximate fit indexes in structural equation modeling (SEM) is of great importance because most fit indexes do not have tractable analytic distributions. Model-based bootstrap, which has been proposed to obtain the distribution of the model chi-square statistic under the null hypothesis (Bollen & Stine, 1992), is not theoretically appropriate for obtaining confidence intervals (CIs) for fit indexes because it assumes the null is exactly true. On the other hand, naive bootstrap is not expected to work well for those fit indexes that are based on the chi-square statistic, such as the root mean square error of approximation (RMSEA) and the comparative fit index (CFI), because sample noncentrality is a biased estimate of the population noncentrality. In this article we argue that a recently proposed bootstrap approach due to Yuan, Hayashi, and Yanagihara (YHY; 2007) is ideal for bootstrapping fit indexes that are based on the chi-square. This method transforms the data so that the “parent” population has the population noncentrality parameter equal to the estimated noncentrality in the original sample. We conducted a simulation study to evaluate the performance of the YHY bootstrap and the naive bootstrap for 4 indexes: RMSEA, CFI, goodness-of-fit index (GFI), and standardized root mean square residual (SRMR). We found that for RMSEA and CFI, the CIs under the YHY bootstrap had relatively good coverage rates for all conditions, whereas the CIs under the naive bootstrap had very low coverage rates when the fitted model had large degrees of freedom. However, for GFI and SRMR, the CIs under both bootstrap methods had poor coverage rates in most conditions.     

Confidence Intervals of Fit Indexes by Inverting a Bootstrap Test

Fit indexes are an important tool in the evaluation of model fit in structural equation modeling (SEM). Currently, the newest confidence interval (CI) for fit indexes proposed by Zhang and Savalei (2016) is based on the quantiles of a bootstrap sampling distribution at a single level of misspecification. This method, despite a great improvement over naive and model-based bootstrap methods, still suffers from unsatisfactory coverage. In this work, we propose a new method of constructing bootstrap CIs for various fit indexes. This method directly inverts a bootstrap test and produces a CI that involves levels of misspecification that would not be rejected in a bootstrap test. Similar in rationale to a parametric CI of root mean square error of approximation (RMSEA) based on a noncentral χ2 distribution and a profile-likelihood CI of model parameters, this approach is shown to have better performance than the approach of Zhang and Savalei (2016), with more accurate coverage and more efficient widths.     

Bootstrap方法在置信区间和标准差估计中的应用

Bootstrap方法是上世纪80年代出现和发展起来的一种新型再抽样统计方法，在统计各领域已有广泛应用．把Bootstrap方法应用于置信区间和标准差估计，在某些情况下，使得置信区间和标准差估计得到改进和改善。     

A Specification Error Test That Uses Instrumental Variables to Detect Latent Quadratic and Latent Interaction Effects

The relations between the latent variables in structural equation models are typically assumed to be linear in form. This article aims to explain how a specification error test using instrumental variables (IVs) can be employed to detect unmodeled interactions between latent variables or quadratic effects of latent variables. An empirical example is presented, and the results of a simulation study are reported to evaluate the sensitivity and specificity of the test and compare it with the commonly employed chi-square model test. The results show that the proposed test can identify most unmodeled latent interactions or latent quadratic effects in moderate to large samples. Furthermore, its power is higher when the number of indicators used to define the latent variables is large. Altogether, this article shows how the IV-based test can be applied to structural equation models and that it is a valuable tool for researchers using structural equation models.     

Causal Effects in Mediation Modeling: An Introduction With Applications to Latent Variables

Causal inference in mediation analysis offers counterfactually based causal definitions of direct and indirect effects, drawing on research by Robins, Greenland, Pearl, VanderWeele, Vansteelandt, Imai, and others. This type of mediation effect estimation is little known and seldom used among analysts using structural equation modeling (SEM). The aim of this article is to describe the new analysis opportunities in a way that is accessible to SEM analysts and show examples of how to perform the analyses. An application is presented with an extension to a latent mediator measured with multiple indicators.     

Effect Analysis Using Nonlinear Structural Equation Mixture Modeling

In this article, we present an approach for comprehensive analysis of the effectiveness of interventions based on nonlinear structural equation mixture models (NSEMM). We provide definitions of average and conditional effects and show how they can be computed. We extend the traditional moderated regression approach to include latent continous and discrete (mixture) variables as well as their higher order interactions, quadratic or more general nonlinear relationships. This new approach can be considered a combination of the recently proposed EffectLiteR approach and the NSEMM approach. A key advantage of this synthesis is that it gives applied researchers the opportunity to gain greater insight into the effectiveness of the intervention. For example, it makes it possible to consider structural equation models for situations where the treatment is noneffective for extreme values of a latent covariate but is effective for medium values, as we illustrate using an example from the educational sciences.     

Advanced Nonlinear Latent Variable Modeling: Distribution Analytic LMS and QML Estimators of Interaction and Quadratic Effects

Interaction and quadratic effects in latent variable models have to date only rarely been tested in practice. Traditional product indicator approaches need to create product indicators (e.g., x ₁ ², x ₁ x ₄) to serve as indicators of each nonlinear latent construct. These approaches require the use of complex nonlinear constraints and additional model specifications and do not directly address the nonnormal distribution of the product terms. In contrast, recently developed, easy-to-use distribution analytic approaches do not use product indicators, but rather directly model the nonlinear multivariate distribution of the measured indicators. This article outlines the theoretical properties of the distribution analytic Latent Moderated Structural Equations (LMS; Klein & Moosbrugger, 2000) and Quasi-Maximum Likelihood (QML; Klein & Muthén, 2007) estimators. It compares the properties of LMS and QML to those of the product indicator approaches. A small simulation study compares the two approaches and illustrates the advantages of the distribution analytic approaches as multicollinearity increases, particularly in complex models with multiple nonlinear terms. An empirical example from the field of work stress applies LMS and QML to a model with an interaction and 2 quadratic effects. Example syntax for the analyses with both approaches is provided.     

Comparing Exploratory Structural Equation Modeling and Existing Approaches for Multiple Regression with Latent Variables

Exploratory structural equation modeling (ESEM) is an approach for analysis of latent variables using exploratory factor analysis to evaluate the measurement model. This study compared ESEM with two dominant approaches for multiple regression with latent variables, structural equation modeling (SEM) and manifest regression analysis (MRA). Main findings included: (1) ESEM in general provided the least biased estimation of the regression coefficients; SEM was more biased than MRA given large cross-factor loadings. (2) MRA produced the most precise estimation, followed by ESEM and then SEM. (3) SEM was the least powerful in the significance tests; statistical power was lower for ESEM than MRA with relatively small target-factor loadings, but higher for ESEM than MRA with relatively large target-factor loadings. (4) ESEM showed difficulties in convergence and occasionally created an inflated type I error rate under some conditions. ESEM is recommended when non-ignorable cross-factor loadings exist.     

解非线性方程的免导数牛顿算法

通过函数值的运算近似牛顿法中的导数项,构造了一个免导数的牛顿法.该算法与牛顿法一样,具有二阶收敛速度,但不需要用到函数的导数.通过与二分法结合,实现该算法的全局收敛性.数值结果表明该算法是有效的.     

A Bayesian Approach for Estimating Multilevel Latent Contextual Models

In many applications of multilevel modeling, group-level (L2) variables for assessing group-level effects are generated by aggregating variables from a lower level (L1). However, the observed group mean might not be a reliable measure of the unobserved true group mean. In this article, we propose a Bayesian approach for estimating a multilevel latent contextual model that corrects for measurement error and sampling error (i.e., sampling only a small number of L1 units from a L2 unit) when estimating group-level effects of aggregated L1 variables. Two simulation studies were conducted to compare the Bayesian approach with the maximum likelihood approach implemented in Mplus. The Bayesian approach showed fewer estimation problems (e.g., inadmissible solutions) and more accurate estimates of the group-level effect than the maximum likelihood approach under problematic conditions (i.e., small number of groups, predictor variable with a small intraclass correlation). An application from educational psychology is used to illustrate the different estimation approaches.     

一类非线性偏微分方程的摄动解法

首先用行波变换将非线性偏微分方程转化为非线性常微分方程,然后采用摄动方法直接求解该非线性常微分方程,最后求得了非线性Klein-Gordon方程的二级近似解.这种方法也可进一步推广用于求其它非线性偏微分方程的近似解析解.     

Estimating Latent Variable Interactions with the Unconstrained Approach: A Comparison of Methods to Form Product Indicators for Large,Unequal Numbers of Items

This Monte Carlo simulation study investigated methods of forming product indicators for the unconstrained approach for latent variable interaction estimation when the exogenous factors are measured by large and unequal numbers of indicators. Product indicators were created based on multiplying parcels of the larger scale by indicators of the smaller scale, multiplying the three most reliable indicators of each scale matched by reliability, and matching items by reliability to create as many product indicators as the number of indicators of the smallest scale. The unconstrained approach was compared with the latent moderated structural equations (LMS) approach. All methods considered provided unbiased parameter estimates. Unbiased standard errors were obtained in all conditions with the LMS approach and when the sample size was large with the unconstrained approach. Power levels to test the latent interaction and Type I error rates were similar for all methods but slightly better for the LMS approach.     

从Newton定律到广义Hamiltonian系统(Ⅲ)——关于Korteweg-de Vries(KdV)类型的非线性发展方程的一个估计

Newton定律是描述物体运动的基本定律,Hamiltonian方程则为运动的基本规律提供了另外一种表达。由Hamiltonian方程发展而来的Hamiltonian可积系统是现代孤立子理论的重要组成部分。文中证明了一个关于Korteweg-de Vries(KdV)类型的非线性发展方程的在加权Sobolev空间中的估计式。这一估计式对证明一类一般的非线性扩散型发展方程的不变性质是非常有用的。     

Fit for a Bayesian: An Evaluation of PPP and DIC for Structural Equation Modeling

Despite its importance to structural equation modeling, model evaluation remains underdeveloped in the Bayesian SEM framework. Posterior predictive p-values (PPP) and deviance information criteria (DIC) are now available in popular software for Bayesian model evaluation, but they remain underutilized. This is largely due to the lack of recommendations for their use. To address this problem, PPP and DIC were evaluated in a series of Monte Carlo simulation studies. The results show that both PPP and DIC are influenced by severity of model misspecification, sample size, model size, and choice of prior. The cutoffs PPP < 0.10 and ?DIC > 7 work best in the conditions and models tested here to maintain low false detection rates and misspecified model selection rates, respectively. The recommendations provided in this study will help researchers evaluate their models in a Bayesian SEM analysis and set the stage for future development and evaluation of Bayesian SEM fit indices.     

木材干燥过程中一个非线性模型的二阶收敛差分格式

针对描述木材干燥过程中的一个非线性微分方程模型,用降阶法对其建立了一个差分格式.此模型是由一个非线性常微分方程和一个非线性抛物方程组成的耦合微分方程组.首先引进一个新变量把原问题转化为一阶微分方程组问题,然后对此一阶微分方程组建立了一个线性化差分格式,应用能量方法证明了差分格式的可解性、稳定性和收敛性,并给出了误差估计式.差分格式关于时间步长和空间步长均为二阶.在实际计算时,将引入的新变量分离开,得到仅含原变量的差分格式,降低了计算量.数值计算结果验证了理论结果的可靠性.     

用摄动法求解一类小参数方程的近似周期解

主要讨论了强非线性拟保守自治系统x g(x)=εf(x,x)的近似周期解及其稳定性,并应用以能量函数为基础的摄动法,分析了两个强非线性方程的近似周期解.实例表明,该摄动法不仅有效而且结果精度较高.