Comparing Multilevel and Classical Confirmatory Factor Analysis Parameterizations of Multirater Data: A Monte Carlo Simulation Study

This simulation study assesses the statistical performance of two mathematically equivalent parameterizations for multitrait–multimethod data with interchangeable raters—a multilevel confirmatory factor analysis (CFA) and a classical CFA parameterization. The sample sizes of targets and raters, the factorial structure of the trait factors, and rater missingness are varied. The classical CFA approach yields a high proportion of improper solutions under conditions with small sample sizes and indicator-specific trait factors. In general, trait factor related parameters are more sensitive to bias than other types of parameters. For multilevel CFAs, there is a drastic bias in fit statistics under conditions with unidimensional trait factors on the between level, where root mean square error of approximation (RMSEA) and χ2 distributions reveal a downward bias, whereas the between standardized root mean square residual is biased upwards. In contrast, RMSEA and χ2 for classical CFA models are severely upwardly biased in conditions with a high number of raters and a small number of targets.     

Approaches for Specifying the Level-1 Error Structure When Synthesizing Single-Case Data

Multilevel modeling has been utilized for combining single-case experimental design (SCED) data assuming simple level-1 error structures. The purpose of this study is to compare various multilevel analysis approaches for handling potential complexity in the level-1 error structure within SCED data, including approaches assuming simple and complex error structures (heterogeneous, autocorrelation, and both) and those using fit indices to select between alternative error structures. A Monte Carlo study was conducted to empirically validate the suggested multilevel modeling approaches. Results indicate that each approach leads to fixed effect estimates with little to no bias and that inferences for fixed effects were frequently accurate, particularly when a simple homogeneous level-1 error structure or a first-order autoregressive structure was assumed and the inferences were based on the Kenward-Roger method. Practical implications and recommendations are discussed.     

Testing Factorial Invariance in Multilevel Data: A Monte Carlo Study

Testing factorial invariance has recently gained more attention in different social science disciplines. Nevertheless, when examining factorial invariance, it is generally assumed that the observations are independent of each other, which might not be always true. In this study, we examined the impact of testing factorial invariance in multilevel data, especially when the dependency issue is not taken into account. We considered a set of design factors, including number of clusters, cluster size, and intraclass correlation (ICC) at different levels. The simulation results showed that the test of factorial invariance became more liberal (or had inflated Type I error rate) in terms of rejecting the null hypothesis of invariance held between groups when the dependency was not considered in the analysis. Additionally, the magnitude of the inflation in the Type I error rate was a function of both ICC and cluster size. Implications of the findings and limitations are discussed.     

Examining and Controlling for Wording Effect in a Self-Report Measure: A Monte Carlo Simulation Study

Wording effect refers to the systematic method variance caused by positive and negative item wordings on a self-report measure. This Monte Carlo simulation study investigated the impact of ignoring wording effect on the reliability and validity estimates of a self-report measure. Four factors were considered in the simulation design: (a) the number of positively and negatively worded items, (b) the loadings on the trait and the wording effect factors, (c) sample size, and (d) the magnitude of population validity coefficient. The findings suggest that the unidimensional model that ignores the negative wording effect would underestimate the composite reliability and criterion-related validity, but overestimate the homogeneity coefficient. The magnitude of relative bias of the composite reliability was generally small and acceptable, whereas the relative bias for the homogeneity coefficient and criterion-related validity coefficient was negatively correlated with the strength of the general trait factor.     

The Asymptotic Covariance Matrix and its Use in Simulation Studies

The asymptotic performance of structural equation modeling tests and standard errors are influenced by two factors: the model and the asymptotic covariance matrix Γ of the sample covariances. Although most simulation studies clearly specify model conditions, specification of Γ is usually limited to values of univariate skewness and kurtosis. We illustrate that marginal skewness and kurtosis are not sufficient to adequately specify a nonnormal simulation condition by showing that asymptotic standard errors and test statistics vary substantially among distributions with skewness and kurtosis that are identical. We argue therefore that Γ should be reported when presenting the design of simulation studies. We show how Γ can be exactly calculated under the widely used Vale–Maurelli transform. We suggest plotting the elements of Γ and reporting the eigenvalues associated with the test statistic. R code is provided.     

Alternative Methods for Assessing Mediation in Multilevel Data: The Advantages of Multilevel SEM

Multilevel modeling (MLM) is a popular way of assessing mediation effects with clustered data. Two important limitations of this approach have been identified in prior research and a theoretical rationale has been provided for why multilevel structural equation modeling (MSEM) should be preferred. However, to date, no empirical evidence of MSEM's advantages relative to MLM approaches for multilevel mediation analysis has been provided. Nor has it been demonstrated that MSEM performs adequately for mediation analysis in an absolute sense. This study addresses these gaps and finds that the MSEM method outperforms 2 MLM-based techniques in 2-level models in terms of bias and confidence interval coverage while displaying adequate efficiency, convergence rates, and power under a variety of conditions. Simulation results support prior theoretical work regarding the advantages of MSEM over MLM for mediation in clustered data.     

蒙特卡罗方法在数学建模中的应用

蒙特卡罗方法是一个以概率模型为基础,利用计算机通过多次反复模拟实验完成问题求解的一种数值计算方法。它特别适用于用传统的解析法难以解决甚至是无法解决的问题。文章主要介绍蒙特卡罗方法及基本原理,并通过实例说明蒙特卡罗方法在数学建模中的应用。     

理想气体扩散的Monte Carlo模拟

用Monte Carlo方法模拟了理想气体系统通过一小孔的扩散。模拟直观展现了气体分子通过小孔的扩散过程。模拟结果表明,一旦经过扩散达到平衡,系统将不会再回到最初的分布,这说明扩散过程是一个不可逆过程;在扩散过程中,系统的混乱程度不断增大。气体扩散快慢与初始分子数以及小孔的尺寸有关。初始分子数越多小尺寸越大,扩散过程进行的就越快。模拟结果与菲克定律一致。     

Testing the Intervention Effect in Single-Case Experiments: A Monte Carlo Simulation Study

This article reports on a Monte Carlo simulation study, evaluating two approaches for testing the intervention effect in replicated randomized AB designs: two-level hierarchical linear modeling (HLM) and using the additive method to combine randomization test p values (RTcombiP). Four factors were manipulated: mean intervention effect, number of cases included in a study, number of measurement occasions for each case, and between-case variance. Under the simulated conditions, Type I error rate was under control at the nominal 5% level for both HLM and RTcombiP. Furthermore, for both procedures, a larger number of combined cases resulted in higher statistical power, with many realistic conditions reaching statistical power of 80% or higher. Smaller values for the between-case variance resulted in higher power for HLM. A larger number of data points resulted in higher power for RTcombiP.     

晶粒生长过程的蒙特卡罗模拟方法

计算机模拟晶粒生长所用的模型及模拟方法大体可分3种:蒙特卡罗(Monte Carlo)方法或改进的蒙特卡罗方法(简称MC法)、使用连续扩散界面场模型以及将细胞状的晶粒结构看作Laguere棋盘形布局来处理的Laguerre模型.特别是应用于模拟焊接热影响区(HAZ)晶粒长大的二维和三维过程取得了较好结果,但在生长模型的边界处理等方面有待继续完善,特别是晶粒生长与晶化温度、时间、气氛等参数密切相关,这种复杂工艺条件下的晶粒生长过程的模拟,是当前该领域正待解决的难题.     

Modeling Unobserved Heterogeneity Using Latent Profile Analysis: A Monte Carlo Simulation

Latent profile analysis (LPA) has become a popular statistical method for modeling unobserved population heterogeneity in cross-sectionally sampled data, but very few empirical studies have examined the question of how well enumeration indexes accurately identify the correct number of latent profiles present. This Monte Carlo simulation study examined the ability of several classes of enumeration indexes to correctly identify the number of latent population profiles present under 3 different research design conditions: sample size, the number of observed variables used for LPA, and the separation distance among the latent profiles measured in Mahalanobis D units. Results showed that, for the homogeneous population (i.e., the population has k = 1 latent profile) conditions, many of the enumeration indexes used in LPA were able to correctly identify the single latent profile if variances and covariances were freely estimated. However, for a heterogeneous population (i.e., the population has k = 3 distinct latent profiles), the correct identification rate for the enumeration indexes in the k = 3 latent profile conditions was typically very low. These results are compared with the previous cross-sectional mixture modeling studies, and the limitations of this study, as well as future cross-sectional mixture modeling and enumeration index research possibilities, are discussed.     

蒙特卡罗法预测含油气区石油资源量

石油勘探阶段,较准确地估算勘探地区的石油资源量是十分重要的。根据在勘探前所采集到的有限资料,并考虑到蒙特卡罗模拟在分析和求解复杂系统或样本数据少的情况下,更是一种不可缺少的分析技术。本文在随机变量的概率分布,计算局部含油地质单元的石油资源量,计算全局含油地质的石油资源量方面进行了较深入的研究,并通过利用各种数学理论和概率统计方面的知识和数学模型解决问题,按照模型的步骤采用Matlab7．1实现。     

一维长程相互作用吸附模型的临界点与作用强度之间的Monte Carlo模拟研究

利用动力学Monte Carlo方法对一维长程相互作用吸附模型进行计算机模拟研究, 得出了其临界点λc与作用力程r-1-α中的α在α>1．0时具有指数关系:λc=λc0 Be-α/γ．     

The accuracy of performance task scores after resolution of rater disagreement: A Monte Carlo study

When multiple raters score a writing sample, on occasion they will award discrepant scores. To report a single score to the examinee, some method of resolving those differences must be applied to the ratings before an operational score can be reported. Several forms of resolving score discrepancies have been described in the literature. Initial studies of the various methods, however, have demonstrated that decisions about student performance may differ depending on the resolution method applied. Thus, studies are needed to investigate the quality of the scores associated with each model. To study score quality associated with each model, we conducted a Monte Carlo study and varied the factors associated with scoring and resolution to determine the conditions under which a particular resolution method might be superior.     

The Impact of Over-Simplifying the Between-Subject Covariance Structure on Inferences of Fixed Effects in Modeling Nested Data

This study discusses the effects of oversimplifying the between-subject covariance structure on inferences for fixed effects in modeling nested data. Linear and quadratic growth curve models (GCMs) with both full and simplified between-subject covariance structures were fit to real longitudinal data. The results were contradictory to the statement that using oversimplified between-subject covariance structures (e.g., uni-level analysis) leads to underestimated standard errors of fixed effect estimates and thus inflated Type I error rates. We analytically derived simple mathematical forms to systematically examine the oversimplification effects for the linear GCMs. The derivation results were aligned with the real data analysis results and further revealed the conditions under which the standard errors of the fixed-effect intercept and slope estimates could be underestimated or overestimated for over-simplified linear GCMs. Therefore, our results showed that the underestimation statement is a myth and can be misleading. Implications are discussed and recommendations are provided.     

蒙特卡罗方法及其在辐射剂量计算中的应用

概述了蒙特卡罗方法的产生与发展，阐述了蒙特卡罗方法的基本特点，最后就蒙特卡罗方法在辐射剂量计算上的应用进行了讨论。     

Sample Size Requirements in Single- and Multiphase Growth Mixture Models: A Monte Carlo Simulation Study

Just as growth mixture models are useful with single-phase longitudinal data, multiphase growth mixture models can be used with multiple-phase longitudinal data. One of the practically important issues in single- and multiphase growth mixture models is the sample size requirements for accurate estimation. In a Monte Carlo simulation study, the sample sizes required for using these models are investigated under various theoretical and realistic conditions. In particular, the relationship between the sample size requirement and the number of indicator variables is examined, because the number of indicators can be relatively easily controlled by researchers in many multiphase data collection settings such as ecological momentary assessment. The findings not only provide tangible information about required sample sizes under various conditions to help researchers, but they also increase understanding of sample size requirements in single- and multiphase growth mixture models.     

Digital Module 13: Monte Carlo Simulation Studies in Item Response Theory

In this digital ITEMS module, Dr. Brian Leventhal and Dr. Allison Ames provide an overview of Monte Carlo simulation studies (MCSS) in item response theory (IRT). MCSS are utilized for a variety of reasons, one of the most compelling being that they can be used when analytic solutions are impractical or nonexistent because they allow researchers to specify and manipulate an array of parameter values and experimental conditions (e.g., sample size, test length, and test characteristics). Dr. Leventhal and Dr. Ames review the conceptual foundation of MCSS in IRT and walk through the processes of simulating total scores as well as item responses using the two-parameter logistic, graded response, and bifactor models. They provide guidance for how to implement MCSS using other item response models and best practices for efficient syntax and executing an MCSS. The digital module contains sample SAS code, diagnostic quiz questions, activities, curated resources, and a glossary.     

A Test for Cluster Bias: Detecting Violations of Measurement Invariance Across Clusters in Multilevel Data

We present a test for cluster bias, which can be used to detect violations of measurement invariance across clusters in 2-level data. We show how measurement invariance assumptions across clusters imply measurement invariance across levels in a 2-level factor model. Cluster bias is investigated by testing whether the within-level factor loadings are equal to the between-level factor loadings, and whether the between-level residual variances are zero. The test is illustrated with an example from school research. In a simulation study, we show that the cluster bias test has sufficient power, and the proportions of false positives are close to the chosen levels of significance.     

A δ-function-like peak in the specific heat of two-dimensional vortex lattice: Monte Carlo study

A repulsive vortex-vortex interaction model was used to numerically study the melting transition of the two-dimensional vortex system with Monte Carlo method. Then a δ-function-like peak in the specific heat was observed and the internal energy showed a sharp drop at the melting temperature, which indicated that there exists a first-order melting transition at finite temperatures. The Lindemann criterion was also investigated and valid, but different from previous simulation results. Project supported by the Science and Technology Ministry of China (No. NKBRST-G19990646) and Zhejiang Provincial Natural Science Foundation (No. 199031)