Model Misspecification and Invariance Testing Using Confirmatory Factor Analytic Procedures

Confirmatory factor analytic procedures are routinely implemented to provide evidence of measurement invariance. Current lines of research focus on the accuracy of common analytic steps used in confirmatory factor analysis for invariance testing. However, the few studies that have examined this procedure have done so with perfectly or near perfectly fitting models. In the present study, the authors examined procedures for detecting simulated test structure differences across groups under model misspecification conditions. In particular, they manipulated sample size, number of factors, number of indicators per factor, percentage of a lack of invariance, and model misspecification. Model misspecification was introduced at the factor loading level. They evaluated three criteria for detection of invariance, including the chi-square difference test, the difference in comparative fit index values, and the combination of the two. Results indicate that misspecification was associated with elevated Type I error rates in measurement invariance testing.     

Exploratory and Confirmatory Factor Analysis of the School Level Environment Questionnaire (SLEQ)

The purpose of the present study was to validate an existing school environment instrument, the School Level Environment Questionnaire (SLEQ). The SLEQ consists of 56 items, with seven items in each of eight scales. One thousand, one hundred and six (1106) teachers in 59 elementary schools in a southwestern USA public school district completed the instrument. An exploratory factor analysis was undertaken for a random sample of half of the completed surveys. Using principal axis factoring with oblique rotation, this analysis suggested that 13 items should be dropped and that the remaining 43 items could best be represented by seven rather than eight factors. A confirmatory factor analysis was run with the other half of the original sample using structural equation modeling. Examination of the fit indices indicated that the model came close to fitting the data, with goodness-of-fit (GOF) coefficients just below recommended levels. A second model was then run with two of the seven factors, with their associated items removed. That left five factors with 35 items. Model fit was improved. A third model was tried, using the same five factors with 35 items but with correlated residuals between some of the items within a factor. This model seemed to fit the data well, with GOF coefficients in recommended ranges. These results led to a refined, more parsimonious version of the SLEQ that was then used in a larger study. Future research is needed to see if this model would fit other samples in different elementary schools and in secondary schools both in the USA and in other countries. This revised version was published online in July 2006 with corrections to the Cover Date.     

Examining the Performance Standards for In-Service Teachers: A Confirmatory Factor Analysis of the Assessment of Teachers' NETS-T Expertise

The survey Performance Standards for In-service Teachers is widely used to help describe teacher skills relating to instructional technology for assessment or planning of professional development. It is based on the six constructs of the National Educational Technology Standards for Teachers, and though used broadly in a variety of contexts, it does not appear to have undergone significant independent evaluation of its psychometric properties. Our analysis found that all six subscales possessed strong internal consistency (all α > 0.93). A subsequent confirmatory factor analysis supported both its current six-factor structure as well as a more parsimonious one-factor model.     

Measurement Invariance in Confirmatory Factor Analysis: An Illustration Using IQ Test Performance of Minorities

Measurement invariance with respect to groups is an essential aspect of the fair use of scores of intelligence tests and other psychological measurements. It is widely believed that equal factor loadings are sufficient to establish measurement invariance in confirmatory factor analysis. Here, it is shown why establishing measurement invariance with confirmatory factor analysis requires a statistical test of the equality over groups of measurement intercepts. Without this essential test, measurement bias may be overlooked. A re-analysis of a study by Te Nijenhuis, Tolboom, Resing, and Bleichrodt (2004) on ethnic differences on the RAKIT IQ test illustrates that ignoring intercept differences may lead to the conclusion that bias of IQ tests with respect to minorities is small, while in reality bias is quite severe.     

Cross-National Validation of the What is Happening in this Class? (WIHIC) Questionnaire using Confirmatory Factor Analysis

Learning Environments Research - The What Is Happening In this Class? (WIHIC) questionnaire was validated cross-nationally using a sample of 3980 high school students from Australia, the UK and...     

A Comparison of Method Effects in Two Confirmatory Factor Models for Structurally Different Methods

It is well known that measurement error in observable variables induces bias in estimates in standard regression analysis and that structural equation models are a typical solution to this problem. Often, multiple indicator equations are subsumed as part of the structural equation model, allowing for consistent estimation of the relevant regression parameters. In many instances, however, embedding the measurement model into structural equation models is not possible because the model would not be identified. To correct for measurement error one has no other recourse than to provide the exact values of the variances of the measurement error terms of the model, although in practice such variances cannot be ascertained exactly, but only estimated from an independent study. The usual approach so far has been to treat the estimated values of error variances as if they were known exact population values in the subsequent structural equation modeling (SEM) analysis. In this article we show that fixing measurement error variance estimates as if they were true values can make the reported standard errors of the structural parameters of the model smaller than they should be. Inferences about the parameters of interest will be incorrect if the estimated nature of the variances is not taken into account. For general SEM, we derive an explicit expression that provides the terms to be added to the standard errors provided by the standard SEM software that treats the estimated variances as exact population values. Interestingly, we find there is a differential impact of the corrections to be added to the standard errors depending on which parameter of the model is estimated. The theoretical results are illustrated with simulations and also with empirical data on a typical SEM model.     

Confirmatory factor analysis of achieving the beginning teacher standards inventory

The purpose of this study was to examine the factorial validity and reliability of the Achieving the NASPE Standards Inventory (ANSI) that assesses pre-service physical education teachers' perceptions of achieving the National Association for Sport and Physical Education (NASPE) beginning teacher standards (2003). Four hundred fifty-two pre-service teachers from 15 Physical Education Teacher Education (PETE) programs voluntarily and anonymously completed the inventory. The hypothesized measurement models were analyzed by means of a confirmatory factor analysis. The Goodness-of-fit statistics indicated that the modified 3-factor model with 38 items represented a best fitting model with the data. The scores generated by the total and the 3 subscales of the 38-item ANSI demonstrated a high level of internal consistency.     

Students' Behavioral and Emotional Participation in Academic Activities in the Mathematics Classroom: A Multilevel Confirmatory Factor Analysis

In this research, the authors raised the issue that prior studies had failed to address the nested structure of data in examining the construct validity of an instrument measuring students' behavioral and emotional participation in academic activities in the classroom. To address this question, the authors illustrated the utility of the multilevel confirmatory factor analysis (MCFA) approach to reexamine the construct validity of this instrument. The sample consisted of 2,041 students in 5th grade from 67 classes in Hong Kong. First, the results justified the requirement of MCFA and indicated that the 4-factor model tested with MCFA provided better fit to the data than that tested with a single-level confirmatory factor analysis (CFA). Second, the study also provided adequate support for a multilevel second-order two-factor model that distinguished engagement from disaffection. Third, the factor structure was invariant across the student level and the classroom level for both the 4-factor model and the second-order two-factor model. Fourth, the results highlighted the presence of ambiguity in differentiating between the dimensions at the classroom level and supported the unidimensionality of the classroom-level construct. Fifth, student engagement was significantly and positively correlated with mathematics test scores, teachers' classroom-management practices, teacher support, and student order in the classroom. Finally, the authors discuss the implications of the study and its limitations and offer suggestions for model selection and explorations for future research.     

tetrad: A Set of Stata Commands for Confirmatory Tetrad Analysis

This article provides a brief overview of confirmatory tetrad analysis (CTA) and presents a new set of Stata commands for conducting CTA. The tetrad command allows researchers to use model-implied vanishing tetrads to test the overall fit of structural equation models (SEMs) and the relative fit of two SEMs that are tetrad-nested. An extension of the command, tetrad_matrix, allows researchers to conduct CTA using a sample covariance matrix as input rather than relying on raw data. Researchers can also use the tetrad_matrix command to input a polychoric correlation matrix and conduct CTA for SEMs involving dichotomous, ordinal, or censored outcomes. Another extension of the command, tetrad_bootstrap, provides a bootstrapped p value for the chi-square test statistic. With Stata’s recently developed commands for structural equation modeling, researchers can integrate CTA with data preparation, likelihood ratio tests for model fit, and the estimation of model parameters in a single statistical software package.     

高校教师教学效果的多因素模型的验证性分析

通过贵州省文科、理科和农科专业的二、三、四年级大学生填写的《大学生评价教师教学效果问卷》（学生用）,即SEEQ问卷,运用协方差结构模型验证了贵州地区大学生评价教师教学效果的七、九因素模型的适合性。结果表明：1．Marsh的一阶九因素结构比较稳定;2．七因素完整模型也比较稳定,比九因素结构模型更好;3．教学热情／组织、群体互动、人际和谐、知识宽度、考试／作业与阅读材料这五个维度影响学习／价值感;4．项目T1（教师讲课能在智力上激发学生,富有启发和激励性）可归为教学热情／组织性这个维度。     

品牌延伸评估的要素分析——从品牌延伸的目标角度

品牌延伸评估是近年来营销理论和品牌战略研究的热点之一。品牌延伸评估就是要对品牌延伸目标的实现过程和实现程度进行评估。从品牌延伸目标的角度来看,消费者需求的满足程度、品牌延伸的契合程度、母品牌的特征、目标市场竞争的状况以及企业战略是品牌延伸评估的主要影响因素。     

Differential Item Functioning Effect Size From the Multigroup Confirmatory Factor Analysis for a Meta-Analysis: A Simulation Study

This study presents a new approach to synthesizing differential item functioning (DIF) effect size: First, using correlation matrices from each study, we perform a multigroup confirmatory factor analysis (MGCFA) that examines measurement invariance of a test item between two subgroups (i.e., focal and reference groups). Then we synthesize, across the studies, the differences in the estimated factor loadings between the two subgroups, resulting in a meta-analytic summary of the MGCFA effect sizes (MGCFA-ES). The performance of this new approach was examined using a Monte Carlo simulation, where we created 108 conditions by four factors: (1) three levels of item difficulty, (2) four magnitudes of DIF, (3) three levels of sample size, and (4) three types of correlation matrix (tetrachoric, adjusted Pearson, and Pearson). Results indicate that when MGCFA is fitted to tetrachoric correlation matrices, the meta-analytic summary of the MGCFA-ES performed best in terms of bias and mean square error values, 95% confidence interval coverages, empirical standard errors, Type I error rates, and statistical power; and reasonably well with adjusted Pearson correlation matrices. In addition, when tetrachoric correlation matrices are used, a meta-analytic summary of the MGCFA-ES performed well, particularly, under the condition that a high difficulty item with a large DIF was administered to a large sample size. Our result offers an option for synthesizing the magnitude of DIF on a flagged item across studies in practice.     

The Construct Validity of Student Engagement: A Confirmatory Factor Analysis Approach

As institutions seek to promote student engagement on campus, the National Survey of Student Engagement (NSSE) is increasingly being used to chart progress and compare results using the Five Benchmark Scores. While recent research has begun to decompose the five benchmarks in a variety of ways; few research studies have sought to explore the underlying structure of these five benchmarks, their interdependence, and the extent to which the items do reflect those five dimensions. This study begins to address the instrument’s construct validity by submitting a single, first-time freshman cohort’s NSSE responses to a confirmatory factor analysis, and proposes as an alternative, eight “dimensions” of student engagement that fit this set of data slightly better and in a more useful way. Results have practical implications for institutions utilizing NSSE, but also contain conceptual implications pertaining to the application of these benchmarks.     

On Common Factor and Principal Component Representations of Data: Implications for Theory and for Confirmatory Replications

Common factor analysis (FA) and principal component analysis (PCA) are commonly used to obtain lower-dimensional representations of matrices of correlations among manifest variables. Whereas some experts argue that differences in results from use of FA and PCA are small and relatively unimportant in empirical studies, the fundamental rationales for the two methods are very different. Here, FA and PCA are contrasted on four key issues: the range of possible dimensional loadings, the range of potential correlations among dimensions, the structure of residual covariances and correlations, and the relation between population parameters and the correlational structures with which they are associated. For decades, experts have emphasized indeterminacies of the FA model, particularly indeterminacy of common factor scores. Determinate in most respects, a heretofore unacknowledged, pernicious indeterminacy of PCA is demonstrated: the indeterminacy between PCA structural representations and the correlational structures from which they are derived. Researchers are often advised to use either FA or PCA in exploratory rounds of data analysis to understand and refine the dimensional structure of a domain before moving to Structural Equation Modeling in later theory-testing, confirmatory, replication studies. Results from the current study suggest that PCA is an unreliable method to use for such purposes and may lead to serious misrepresentation of the structure of a domain. Hence, PCA should never be used if the goal is to understand and represent the latent structure of a domain; only FA techniques should be used for this purpose, as only FA provides reliable structural representations as the basis for confirmatory tests in future studies.     

关于方幂和中因子7的指数

设　n=7αC ,　 7 c.本文给出下列方幂和中因子 7的指数公式 :　 ∑n-1k =0(x+ 2k) r,∑n-1k =0(x+ 4k) r     

中国旅行社业知识资本结构解析

知识资本是知识经济时代企业的首要资本。知识资本结构解析是知识资本管理的首要前提。中国旅行社业只有加强知识资本管理,才能提高核心竞争优势,以应对国内外日益紧逼的严峻挑战。     

城市居民消费结构变动的因子分析

在居民消费结构研究中,应用最广泛的是恩格尔系数和线性支出系统,但在进行消费结构变动分析时,因指标较多且各指标的变化趋势不尽相同,很难看出居民消费在某一方面的变动趋势.多元统计分析中的因子分析法正是解决这一问题的有力工具,文中利用因子分析法和DPS数据处理系统,对泰安城市居民消费结构的变动情况进行了分析.     

Factor Retention in Exploratory Factor Analysis With Missing Data

Determining the number of factors in exploratory factor analysis is arguably the most crucial decision a researcher faces when conducting the analysis. While several simulation studies exist that compare various so-called factor retention criteria under different data conditions, little is known about the impact of missing data on this process. Hence, in this study, we evaluated the performance of different factor retention criteria—the Factor Forest, parallel analysis based on a principal component analysis as well as parallel analysis based on the common factor model and the comparison data approach—in combination with different missing data methods, namely an expectation-maximization algorithm called Amelia, predictive mean matching, and random forest imputation within the multiple imputations by chained equations (MICE) framework as well as pairwise deletion with regard to their accuracy in determining the number of factors when data are missing. Data were simulated for different sample sizes, numbers of factors, numbers of manifest variables (indicators), between-factor correlations, missing data mechanisms and proportions of missing values. In the majority of conditions and for all factor retention criteria except the comparison data approach, the missing data mechanism had little impact on the accuracy and pairwise deletion performed comparably well as the more sophisticated imputation methods. In some conditions, especially small-sample cases and when comparison data were used to determine the number of factors, random forest imputation was preferable to other missing data methods, though. Accordingly, depending on data characteristics and the selected factor retention criterion, choosing an appropriate missing data method is crucial to obtain a valid estimate of the number of factors to extract.     

积极人格特质问卷PPTQ在中国大学生中的因素结构研究

探索积极人格特质问卷(PPTQ)在中国大学生中的因素结构。采用积极人格特质问卷(Positive Personality Traits Questionnaire)中文版,对648名大学生施测,对其中一半数据使用PASWStatistics18进行探索性因素分析,另一半数据使用AMOS16.0进行验证性因素分析(CFA)。探索性因素分析得出积极自我意象、外向性和文化认同三因素结构。累计解释率为50.457%,验证性因素分析结果显示:χ2/df=2.230,GFI=0.841。AGFI=0.816,CFI=0.822,RMSEA=0.062,中文版三因素结构在中国大学生人群中较为合理。     

A metric measure for direct comparison of competing models in covariance structure analysis

Comparing the fit of alternative models has become a standard procedure for analyzing covariance structure analysis. Comparison of alternative models is typically accomplished by examining the fit of each model to sample data. It is argued that rather than using this indirect approach, one should do direct comparisons of the similarities and differences among competing models. It is shown that among the existing good‐ness‐of‐fit indexes, the root mean square residual (RMSR) is the only one that can be used for this purpose. However, the RMSR fails to satisfy some important statistical desiderata. Rao's Distance (RD), an alternate measure, is shown to overcome this limitation of RMSR. The preference for RD over RMSR for model comparisons is illustrated through a detailed analysis of a particular sample of multitrait‐multimethod data. A simulation study conducted to empirically investigate the sampling behavior of RD reveals that the true orderings of intermodel proximities are recovered (on average) with a fair degree of accuracy.