Meta-analytic structural equation modeling: a two-stage approach

To synthesize studies that use structural equation modeling (SEM), researchers usually use Pearson correlations (univariate r), Fisher z scores (univariate z), or generalized least squares (GLS) to combine the correlation matrices. The pooled correlation matrix is then analyzed by the use of SEM. Questionable inferences may occur for these ad hoc procedures. A 2-stage structural equation modeling (TSSEM) method is proposed to incorporate meta-analytic techniques and SEM into a unified framework. Simulation results reveal that the univariate-r, univariate-z, and TSSEM methods perform well in testing the homogeneity of correlation matrices and estimating the pooled correlation matrix. When fitting SEM, only TSSEM works well. The GLS method performed poorly in small to medium samples.     

Accounting for Missing Correlation Coefficients in Fixed-Effects MASEM

Meta-analytic structural equation modeling (MASEM) is increasingly applied to advance theories by synthesizing existing findings. MASEM essentially consists of two stages. In Stage 1, a pooled correlation matrix is estimated based on the reported correlation coefficients in the individual studies. In Stage 2, a structural model (such as a path model) is fitted to explain the pooled correlations. Frequently, the individual studies do not provide all the correlation coefficients between the research variables. In this study, we modify the currently optimal MASEM-method to deal with missing correlation coefficients, and compare its performance with existing methods. This study is the first to evaluate the performance of fixed-effects MASEM methods under different levels of missing correlation coefficients. We found that the often used univariate methods performed very poorly, while the multivariate methods performed well overall.     

The Performance of Multivariate Methods for Two-Group Comparisons with Small Samples and Incomplete Data

Abstract

In intervention studies having multiple outcomes, researchers often use a series of univariate tests (e.g., ANOVAs) to assess group mean differences. Previous research found that this approach properly controls Type I error and generally provides greater power compared to MANOVA, especially under realistic effect size and correlation combinations. However, when group differences are assessed for a specific outcome, these procedures are strictly univariate and do not consider the outcome correlations, which may be problematic with missing outcome data. Linear mixed or multivariate multilevel models (MVMMs), implemented with maximum likelihood estimation, present an alternative analysis option where outcome correlations are taken into account when specific group mean differences are estimated. In this study, we use simulation methods to compare the performance of separate independent samples t tests estimated with ordinary least squares and analogous t tests from MVMMs to assess two-group mean differences with multiple outcomes under small sample and missingness conditions. Study results indicated that a MVMM implemented with restricted maximum likelihood estimation combined with the Kenward–Roger correction had the best performance. Therefore, for intervention studies with small N and normally distributed multivariate outcomes, the Kenward–Roger procedure is recommended over traditional methods and conventional MVMM analyses, particularly with incomplete data.     

Revisiting the Gain–Loss Separability Assumption in Prospect Theory

A fundamental assumption of prospect theory is gain–loss separability (GLS)—the assertion that the overall utility of a prospect can be expressed as a function of the utilities of its positive and negative components. Violations of GLS may potentially limit the generalization of results from studies of single‐domain prospects to mixed prospects and systematically distort the predictions of the theory. Violations also have implications for how choices with positive and negative components should be presented to decision makers. Previous studies, using different elicitation methods, have documented different rates, and types, of systematic violations of GLS. We discuss the differences between two specific elicitation methods—binary choice and certainty equivalents—and report results of a new study of GLS using both methods and randomly generated prospects. We compare the extent and nature of GLS violations under the two elicitation methods using between‐subject and within‐subject analyses. We find (i) systematic violations of GLS under both methods, (ii) higher rates of violations under choice, (iii) higher sensitivity to the outcomes for the certainty equivalents, which is consistent with the predictions of the scale‐compatibility hypothesis, and (iv) different patterns of violations under the two methods, which are explained by method‐specific preferences. We discuss the psychological mechanisms underlying the findings and the implications for presenting information with gain and loss components. Copyright © 2012 John Wiley & Sons, Ltd.     

无均值结构的潜变量交互效应模型的标准化估计

潜变量交互效应建模研究近年来有两项重要进展, 一是提出了潜变量交互效应模型的标准化估计及其计算公式; 二是发现无均值结构模型可以取代传统的有均值结构模型, 建模大为简化。但标准化估计是在传统的有均值结构模型中建立的, 在简化的模型中同样适用吗？本文在无均值结构模型的框架内, 给出了潜变量交互效应模型的标准化形式、计算公式和建模步骤, 并通过模拟研究比较了极大似然和广义最小二乘两种估计方法、配对乘积指标和全部乘积指标两种指标类型, 结果表明, 在计算交互效应的标准化估计时, 应当使用配对乘积指标建模, 并且首选极大似然估计。     

Structural Equation Modeling in the Communication Sciences, 1995–2000

Structural equation modeling (SEM) is a viable multivariate tool used by communication researchers for the past quarter century. Building off Cappella (1975) as well as McPhee and Babrow (1987), this study summarizes the use of this technique from 1995–2000 in 37 communication‐based academic journals. We identify and critically assess 3 unique methods for testing structural relationships via SEM in terms of the specification, estimation, and evaluation of their respective structural equation models. We provide general guidelines for the use of SEM and make recommendations concerning latent variable models, sample size, reporting parameter estimates, model fit statistics, cross‐sectional data, univariate normality, cross‐validation, nonrecursive modeling, and the decomposition of effects (direct, indirect, and total).     

Fitting Rasch model using appropriateness measure statistics

In this paper, the distributional properties and power rates of the Lz, Eci2z, and Eci4z statistics when they are used as item fit statistics were explored. The results were compared to t-transformation of Outfit and Infit mean square. Four sample sizes were selected: 100, 250, 500, and 1000 examinees. The abilities were uniform and normal with mean 0 and standard deviation 1, and uniform and normal with mean -1 and standard deviation 1. The pseudo-guessing parameter was fixed at .25. Two ranges of difficulty parameters were selected: +/- 1 logits and +/- 2 logits. Two test lengths were selected: 15 and 30 items. The results showed important differences between the T-infit, T-outfit, Lz, Eci2z, and Eci4z statistics. The T-oufit, T-infit, and Lz statistics showed poor standardization with estimated parameters because their distributional properties were not close to the expected values. However, the Eci2z and Eci4z statistics showed satisfactory standardization on all conditions. Further, the power rates of Eci2z and Eci4z were 5% to 10% higher than the power rates of Lz, T-outfit, and T-infit to detect items that do not fit Rasch model.     

More efficient parameter estimates for factor analysis of ordinal variables by ridge generalized least squares

Data in psychology are often collected using Likert‐type scales, and it has been shown that factor analysis of Likert‐type data is better performed on the polychoric correlation matrix than on the product‐moment covariance matrix, especially when the distributions of the observed variables are skewed. In theory, factor analysis of the polychoric correlation matrix is best conducted using generalized least squares with an asymptotically correct weight matrix (AGLS). However, simulation studies showed that both least squares (LS) and diagonally weighted least squares (DWLS) perform better than AGLS, and thus LS or DWLS is routinely used in practice. In either LS or DWLS, the associations among the polychoric correlation coefficients are completely ignored. To mend such a gap between statistical theory and empirical work, this paper proposes new methods, called ridge GLS, for factor analysis of ordinal data. Monte Carlo results show that, for a wide range of sample sizes, ridge GLS methods yield uniformly more accurate parameter estimates than existing methods (LS, DWLS, AGLS). A real‐data example indicates that estimates by ridge GLS are 9–20% more efficient than those by existing methods. Rescaled and adjusted test statistics as well as sandwich‐type standard errors following the ridge GLS methods also perform reasonably well.     

Validation and error in multiplicative utility functions

An approach to the concept of error in utility assessment is proposed. Four kinds of errors are considered and each kind is related to four separate elicitation methods - all in the context of a general multiplicative multiattribute utility model. The methods are a Keeney-Raiffa (1976) procedure; SMART, for Simple Multi-Attribute Rating Technique (Edwards 1977); SJT, for a Social Judgment Theory based regression model (Hammond et al. 1975); and HOPE, for Holistic Orthogonal Parameter Estimation (Barron and Person 1979).The individual judgments elicited are either holistic - in which the entity to be evaluated is considered as a whole - or decomposed - in which attention is directed to one or two aspects at a time.If a general multiplicative model can be assumed to be an appropriate representation of the decision maker's basic preference structure, error can occur in the direct estimation of the scaling constants and univariate utility functions for decomposition methods (Keeney-Raiffa and SMART), or in the holistic assessments for holistic methods (SJT and HOPE). Individual estimates may be merely subject to random noise or may be substantially incorrect. The utility model may be incorrectly specified; finally, all four methods may be subject to systematic error. The four assessment methods are considered in conjunction with errors of each kind.     

MIRT模型中多维能力及其相关矩阵估计的影响因素

多维项目反应理论因其模型本身的天然优势及其兼具因素分析与项目反应理论于一身的优点,而被广大研究者及应用者所重视.本研究在前人研究基础上,重点讨论MIRT多维能力及能力间相关矩阵的参数估计问题.研究采用Monte Carlo模拟方法进行,在三因素完全随机设计（4 ×3×3）下,使用MCMC算法,探讨测验维度数、维度间的相关大小和测验项目数三个因素对MIRT能力及其相关矩阵估计的影响.     

Accuracy in parameter estimation for targeted effects in structural equation modeling: sample size planning for narrow confidence intervals

In addition to evaluating a structural equation model (SEM) as a whole, often the model parameters are of interest and confidence intervals for those parameters are formed. Given a model with a good overall fit, it is entirely possible for the targeted effects of interest to have very wide confidence intervals, thus giving little information about the magnitude of the population targeted effects. With the goal of obtaining sufficiently narrow confidence intervals for the model parameters of interest, sample size planning methods for SEM are developed from the accuracy in parameter estimation approach. One method plans for the sample size so that the expected confidence interval width is sufficiently narrow. An extended procedure ensures that the obtained confidence interval will be no wider than desired, with some specified degree of assurance. A Monte Carlo simulation study was conducted that verified the effectiveness of the procedures in realistic situations. The methods developed have been implemented in the MBESS package in R so that they can be easily applied by researchers.     

The Mental Changes Due to Frontal Lobe Damage

The literature suggests association between arousal, general activation, and anxiety on the one hand, and time judgments on the other hand, implying that reported differences in time judgment between nosological groups may be confounded by group differences in arousal-anxiety.

Self-report measures of anxiety, as well as magnitude estimates and magnitude productions of standards ranging from 500 to 2000 msec in 250 msec steps and presented in 10 randomized blocks, were obtained from 16 male normals and from 16 male hospitalized patients with a tentative diagnosis of chronic undifferentiated schizophrenia. Only 10 of the 16 patients were later found to have the same confirmed diagnosis. Data from nine normals and from seven chronic undifferentiated schizophrenics met a criterion of linearity of response functions for both time judgment methods and were further analyzed.

Magnitude estimates and magnitude productions showed underestimation of elapsed time, both types of judgment exhibited satisfactory reliability, estimates showed “shortening,” and productions showed “lengthening” over blocks of trials.

Intercepts of the response × standards functions were not generally equal to zero, were more negative in estimation than in production, had marginal reliability or were unreliable, did not correlate significantly between methods, and did not show significant trends over blocks of trials.

Following a model by Carlson and Feinberg, slopes of response × standard functions were used as estimates of the rate of the “internal clock.” (In estimation, the rate is equal to the slope; in production, it is equal to the reciprocal of the slope.) Average rates of the internal clock did not differ between methods for normals, but were higher in production than in estimation for the seven patients. Clock rates did not differ significantly between groups, were reliable, and exhibited positive correlation between methods. Clock rates exhibited trends over blocks of trials: arctan equivalents of clock rates were linearly related to ordinal numbers of blocks of trials and showed decreases, or “slowing” of the internal clock, in both methods.

Differences in mean anxiety between groups were not significant. In each group, anxiety scores showed positive average correlations with magnitude estimates and negative average correlations with magnitude productions, failed to correlate significantly with intercepts, but showed positive correlations with clock rates.

The data also suggest that anxiety and intrasubject variability may be interrelated.

To conclude: Reported differences in time judgments between nosological groups may not solely be due to nosological differences per se, but instead may be due to group differences in anxiety.     

Measurement bias and error correction in a two-stage estimation for multilevel IRT models

Among current state-of-the-art estimation methods for multilevel IRT models, the two-stage divide-and-conquer strategy has practical advantages, such as clearer definition of factors, convenience for secondary data analysis, convenience for model calibration and fit evaluation, and avoidance of improper solutions. However, various studies have shown that, under the two-stage framework, ignoring measurement error in the dependent variable in stage II leads to incorrect statistical inferences. To this end, we proposed a novel method to correct both measurement bias and measurement error of latent trait estimates from stage I in the stage II estimation. In this paper, the HO-IRT model is considered as the measurement model, and a linear mixed effects model on overall (i.e., higher-order) abilities is considered as the structural model. The performance of the proposed correction method is illustrated and compared via a simulation study and a real data example using the National Educational Longitudinal Survey data (NELS 88). Results indicate that structural parameters can be recovered better after correcting measurement biases and errors.     

多维项目反应理论补偿性模型参数估计:基于广义回归神经网络集合

运用广义回归神经网络(GRNN)方法对小样本多维项目反应理论(MIRT)补偿性模型的项目参数进行估计,尝试解决传统参数估计方法样本数量要求较大的问题。MIRT双参数Logistic补偿模型被设置为二级计分的二维模型。首先,模拟二维能力参数、项目参数值与考生作答矩阵。其次,把通过主成分分析得到的前两个因子在每个题目上的载荷作为区分度的初始值以及题目通过率作为难度的初始值,这两个指标的初始值作为神经网络的输入。集成100个神经网络,其输出值的均值作为MIRT的项目参数估计值。最后,设置2×2种(能力相关水平:0.3和0.7; 两种估计方法:GRNN和MCMC方法)实验处理,对GRNN和MCMC估计方法的返真性进行比较。结果表明,小样本的情况下,基于GRNN集成方法的参数估计结果优于MCMC方法。     

Hopes and cautions in implementing Bayesian structural equation modeling

Muthén and Asparouhov (2012) have proposed and demonstrated an approach to model specification and estimation in structural equation modeling (SEM) using Bayesian methods. Their contribution builds on previous work in this area by (a) focusing on the translation of conventional SEM models into a Bayesian framework wherein parameters fixed at zero in a conventional model can be respecified using small-variance priors and (b) implementing their approach in software that is widely accessible. We recognize potential benefits for applied researchers as discussed by Muthén and Asparouhov, and we also see a tradeoff in that effective use of the proposed approach introduces increased demands in terms of expertise of users to navigate new complexities in model specification, parameter estimation, and evaluation of results. We also raise cautions regarding the issues of model modification and model fit. Although we see significant potential value in the use of Bayesian SEM, we also believe that effective use will require an awareness of these complexities. (PsycINFO Database Record (c) 2012 APA, all rights reserved).     

当结构假设和分布假设不满足时的验证性因子分析：稳健极大似然法估计和贝叶斯估计的比较研究

结构方程模型已被广泛应用于心理学、教育学、以及社会科学领域的统计分析中。结构方程模型分析中最常用的估计方法是基于正 态分布的估计量,比如极大似然估计法。这些方法需要满足两个假设。第一, 理论模型必须正确地反映变量与变量之间的关系,称为结构假 设。第二,数据必须符合多元正态分布,称为分布假设。如果这些假设不满足,基于正态分布的估计量就有可能导致不正确的卡方指数、不 正确的拟合度、以及有偏差的参数估计和参数估计的标准误。在实际应用中,几乎所有的理论模型都不能准确地解释变量与变量之间的关系, 数据也常常呈非多元正态分布。为此,一些新的估计方法得以发展。这些方法要么在理论上不要求数据呈多元正态分布,要么对因数据呈非 正态分布而导致的不正确结果进行纠正。当前较为流行的两种方法是稳健极大似然估计和贝叶斯估计。稳健极大似然估计是应用 Satorra and Bentler (1994) 的方法对不正确的卡方指数和参数估计的标准误进行调整,而参数估计和用极大似然方法得出的完全等同。贝叶斯估计方法则是 基于贝叶斯定理,其要点是：参数的后验分布是由参数的先验分布和数据似然值相乘而得来。后验分布常用马尔科夫蒙特卡洛算法来进行模拟。 对于稳健极大似然估计和贝叶斯估计这两种方法之间的优劣比较,先前的研究只局限于理论模型是正确的情境。而本研究则着重于理论模型 是错误的情境,同时也考虑到数据呈非正态分布的情境。本研究所采用的模型是验证性因子模型,数据全部由计算机模拟而来。数据的生成 取决于三个因素：8 类因子结构,3 种变量分布,和3 组样本量。这三个因素产生72 个模拟条件（72=8x3x3）。每个模拟条件下生成2000 个 数据组,每个数据组都拟合两个模型,一个是正确模型、一个是错误模型。每个模型都用两种估计方法来拟合：稳健极大似然估计法和贝叶 斯估计方法。贝叶斯估计方法中所使用的先验分布是无信息先验分布。结果分析主要着重于模型拒绝率、拟合度、参数估计、和参数估计的 标准误。研究的结果表明：在样本量充足的情况下,两种方法得出的参数估计非常相似。当数据呈非正态分布时,贝叶斯估计法比稳健极大 似然估计法更好地拒绝错误模型。但是,当样本量不足且数据呈正态分布时,贝叶斯估计在拒绝错误模型和参数估计上几乎没有优势,甚至 在一些条件下,比稳健极大似然法要差。     

Aggregating and Testing Intra-Individual Correlations: Methods and Comparisons

From a longitudinal study, we have repeatedly measured data from multiple individuals at multiple occasions. For each individual, the relation between 2 variables can be measured by the Pearson’s correlation. The question is how to aggregate the multiple correlations and conduct statistical inference on the aggregated intra-individual correlation. Several methods are proposed to aggregate and test intra-individual correlations: (a) a meta-analysis method based on Fisher’s Z transformed correlations, (b) a meta-analysis method based on the Pearson’s correlations, and (c) a multilevel modeling method using data standardized within each individual. The performance of the methods after bias corrections was compared using simulations with considering factors including numbers of individuals, numbers of time points, population effect sizes, and their distribution forms (homogeneous vs heterogeneous). The results from the simulation studies show that estimation biases were found using the meta-analytic methods and suggestions on when and how to correct biases were provided based on the simulation results. Furthermore, the performance of the 3 methods after necessary bias corrections was found to be comparable and reasonably good, indicating that all 3 methods worked for aggregating and testing intra-individual correlations. An empirical daily diary data set was then used to illustrate the applications of the 3 methods. The assumptions, advantages and disadvantages, and possible extensions of the 3 methods were discussed.     

Correction for range restriction: an expanded typology

A common research problem is the estimation of the population correlation between x and y from an observed correlation rxy obtained from a sample that has been restricted because of some sample selection process. Methods of correcting sample correlations for range restriction in a limited set of conditions are well-known. An expanded classification scheme for range-restriction scenarios is developed that conceptualizes range-restriction scenarios from various combinations of the following facets: (a) the variable(s) on which selection occurs (x, y and/or a 3rd variable z), (b) whether unrestricted variances for the relevant variables are known, and (c) whether a 3rd variable, if involved, is measured or unmeasured. On the basis of these facets, the authors describe potential solutions for 11 different range-restriction scenarios and summarize research to date on these techniques.     

Congenital correlates of distractibility

The present research reports correlations between ratings of distractibility and minor physical anomalies and gross motor incoordination. For an original and two replication samples of preschool children, there were significant average weighted correlations between distractibility and both minor physical anomalies and gross motor incordination for boys. Also, the magnitude of the correlation between distractibility and the congenital characteristics was significantly higher for boys. Although neither parental rejection nor parental restrictiveness was related to boys' distractibility, parental rejection was marginally related to girls' distractibility.