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.     

Using Multigroup Confirmatory Factor Analysis to Test Measurement Invariance in Raters: A Clinical Skills Examination Application

Psychometric models based on structural equation modeling framework are commonly used in many multiple-choice test settings to assess measurement invariance of test items across examinee subpopulations. The premise of the current article is that they may also be useful in the context of performance assessment tests to test measurement invariance of raters. The modeling approach and how it can be used for performance tests with less than optimal rater designs are illustrated using a data set from a performance test designed to measure medical students’ patient management skills. The results suggest that group-specific rater statistics can help spot differences in rater performance that might be due to rater bias, identify specific weaknesses and strengths of individual raters, and enhance decisions related to future task development, rater training, and test scoring processes.     

Extension Procedures for Confirmatory Factor Analysis

We present factor extension procedures for confirmatory factor analysis that provide estimates of the relations of common and unique factors with external variables that do not undergo factor analysis. We present identification strategies that build upon restrictions of the pattern of correlations between unique factors and external variables. The first restriction minimizes the sum of squared correlations between unique factors and external variables. This approach is similar to the traditional factor extension procedure. The second restriction minimizes the complexity of the pattern of external correlations of unique factors. This approach has similarities with the simple structure ideal imposed on most factor rotation strategies. The procedures are illustrated with a real data example that demonstrates their applicability to real-world research questions.     

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 Bias Detection Through Factor Analysis

Assessing the correctness of a structural equation model is essential to avoid drawing incorrect conclusions from empirical research. In the past, the chi-square test was recommended for assessing the correctness of the model but this test has been criticized because of its sensitivity to sample size. As a reaction, an abundance of fit indexes have been developed. The result of these developments is that structural equation modeling packages are now producing a large list of fit measures. One would think that this progression has led to a clear understanding of evaluating models with respect to model misspecifications. In this article we question the validity of approaches for model evaluation based on overall goodness-of-fit indexes. The argument against such usage is that they do not provide an adequate indication of the “size” of the model's misspecification. That is, they vary dramatically with the values of incidental parameters that are unrelated with the misspecification in the model. This is illustrated using simple but fundamental models. As an alternative method of model evaluation, we suggest using the expected parameter change in combination with the modification index (MI) and the power of the MI test.     

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.     

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.     

Measurement Invariance Testing Across Between-Level Latent Classes Using Multilevel Factor Mixture Modeling

This simulation study examines the efficacy of multilevel factor mixture modeling (ML FMM) for measurement invariance testing across unobserved groups when the groups are at the between level of multilevel data. To this end, latent classes are generated with class-specific item parameters (i.e., factor loading and intercept) across the between-level classes. The efficacy of ML FMM is evaluated in terms of class enumeration, class assignment, and the detection of noninvariance. Various classification criteria such as Akaike’s information criterion, Bayesian information criterion, and bootstrap likelihood ratio tests are examined for the correct enumeration of between-level latent classes. For the detection of measurement noninvariance, free and constrained baseline approaches are compared with respect to true positive and false positive rates. This study evidences the adequacy of ML FMM. However, its performance heavily depends on the simulation factors such as the classification criteria, sample size, and the magnitude of noninvariance. Practical guidelines for applied researchers are provided.     

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.     

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...     

Using Confirmatory Factor Analysis and the Rasch Model to Assess Measurement Invariance in a High Stakes Reading Assessment

The psychometric properties and multigroup measurement invariance of scores across subgroups, items, and persons on the Reading for Meaning items from the Georgia Criterion Referenced Competency Test (CRCT) were assessed in a sample of 778 seventh-grade students. Specifically, we sought to determine the extent to which score-based inferences on a high stakes state assessment hold across several subgroups within the population of students. To that end, both confirmatory factor analysis (CFA) and Rasch (1980 Rasch, G. 1980. Probabilistic models for some intelligence and attainment tests, Chicago: The University of Chicago Press (Original work published 1960).  [Google Scholar]) models were used to assess measurement invariance. Results revealed a unidimensional construct with factorial-level measurement invariance across disability status (students with and without specific learning disabilities), but not across test accommodations (resource guide, read-aloud, and standard administrations). Item-level analysis using the Rasch Model also revealed minimal differential item functioning across disability status, but not accommodation status.     

The potential of anchoring vignettes to increase intercultural comparability of non-cognitive factors

ABSTRACT

We investigate whether Anchoring Vignettes (AV) improve intercultural comparability of non-cognitive student-directed factors (e.g., procrastination). So far, correlation analyses for anchored and non-anchored scores with a criterion have been used to demonstrate the effectiveness of AV in improving data quality. However, correlation analyses are often used to investigate external validity of a scale. Nonetheless, before testing for validity, the reliability of the measurement of a construct should be examined. In the present study, we tested for measurement invariance across countries and languages and compared anchored and non-anchored student-directed self-reports that are highly relevant for the students’ self and their behaviour and performance. In addition, we apply further criteria for testing reliability. The results indicate that the data quality for some of the constructs can – in fact – be improved slightly by anchoring; whereas, for other self-reports, anchoring is less successful than was hoped. We discuss with regard to possible consequences for research methodology.     

Assessing Measurement Invariance in Multiple-Group Latent Profile Analysis

The study of measurement invariance in latent profile analysis (LPA) indicates whether the latent profiles differ across known subgroups (e.g., gender). The purpose of the present study was to examine the impact of noninvariance on the relative bias of LPA parameter estimates and on the ability of the likelihood ratio test (LRT) and information criteria statistics to reject the hypothesis of invariance. A Monte Carlo simulation study was conducted in which noninvariance was defined as known group differences in the indicator means in each profile. Results indicated that parameter estimates were biased in conditions with medium and large noninvariance. The LRT and AIC detected noninvariance in most conditions with small sample sizes, while the BIC and adjusted BIC needed larger sample sizes to detect noninvariance. Implications of the results are discussed along with recommendations for future research.     

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。     

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.     

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.     

Assessing statistical aspects of test fairness with structural equation modelling

Test fairness and test bias are not synonymous concepts. Test bias refers to statistical evidence that the psychometrics or interpretation of test scores depend on group membership, such as gender or race, when such differences are not expected. A test that is grossly biased may be judged to be unfair, but test fairness concerns the broader, more subjective evaluation of assessment outcomes from perspectives of social justice. Thus, the determination of test fairness is not solely a matter of statistics, but statistical evidence is important when evaluating test fairness. This work introduces the use of the structural equation modelling technique of multiple-group confirmatory factor analysis (MGCFA) to evaluate hypotheses of measurement invariance, or whether a set of observed variables measures the same factors with the same precision over different populations. An example of testing for measurement invariance with MGCFA in an actual, downloadable data set is also demonstrated.     

Testing Measurement Invariance Across Unobserved Groups: The Role of Covariates in Factor Mixture Modeling

Factor mixture modeling (FMM) has been increasingly used to investigate unobserved population heterogeneity. This study examined the issue of covariate effects with FMM in the context of measurement invariance testing. Specifically, the impact of excluding and misspecifying covariate effects on measurement invariance testing and class enumeration was investigated via Monte Carlo simulations. Data were generated based on FMM models with (1) a zero covariate effect, (2) a covariate effect on the latent class variable, and (3) covariate effects on both the latent class variable and the factor. For each population model, different analysis models that excluded or misspecified covariate effects were fitted. Results highlighted the importance of including proper covariates in measurement invariance testing and evidenced the utility of a model comparison approach in searching for the correct specification of covariate effects and the level of measurement invariance. This approach was demonstrated using an empirical data set. Implications for methodological and applied research are discussed.     

An Investigation of the Alignment Method With Polytomous Indicators Under Conditions of Partial Measurement Invariance

The alignment method (Asparouhov & Muthén, 2014) is an alternative to multiple-group factor analysis for estimating measurement models and testing for measurement invariance across groups. Simulation studies evaluating the performance of the alignment for estimating measurement models across groups show promising results for continuous indicators. This simulation study builds on previous research by investigating the performance of the alignment method’s measurement models estimates with polytomous indicators under conditions of systematically increasing, partial measurement invariance. We also present an evaluation of the testing procedure, which has not been the focus of previous simulation studies. Results indicate that the alignment adequately recovers parameter estimates under small and moderate amounts of noninvariance, with issues only arising in extreme conditions. In addition, the statistical tests of invariance were fairly conservative, and had less power for items with more extreme skew. We include recommendations for using the alignment method based on these results.     

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.