A Modified Comparative Fit Index for Factorial Invariance Studies

As a prerequisite for meaningful comparison of latent variables across multiple populations, measurement invariance or specifically factorial invariance has often been evaluated in social science research. Alongside with the changes in the model chi-square values, the comparative fit index (CFI; Bentler, 1990) is a widely used fit index for evaluating different stages of factorial invariance, including metric invariance (equal factor loadings), scalar invariance (equal intercepts), and strict invariance (equal unique factor variances). Although previous literature generally showed that the CFI performed well for single-group structural equation modeling analyses, its applicability to multiple group analyses such as factorial invariance studies has not been examined. In this study we argue that the commonly used default baseline model for the CFI might not be suitable for factorial invariance studies because (a) it is not nested within the scalar invariance model, and thus (b) the resulting CFI values might not be sensitive to the group differences in the measurement model. We therefore proposed a modified version of the CFI with an alternative (and less restrictive) baseline model that allows observed variables to be correlated. Monte Carlo simulation studies were conducted to evaluate the utility of this modified CFI across various conditions including varying degree of noninvariance and different factorial invariance models. Results showed that the modified CFI outperformed both the conventional CFI and the ΔCFI (Cheung & Rensvold, 2002) in terms of sensitivity to small and medium noninvariance.     

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.     

CFI versus RMSEA: A comparison of two fit indexes for structural equation modeling

This article compares two structural equation modeling fit indexes—Bentler's ( 1990; Bentler & Bonett, 1980) Confirmatory Fit Index (CFI) and Steiger and Lind's (1980; Browne & Cudeck, 1993) Root Mean Square Error of Approximation (RMSEA). These two fit indexes are both conceptually linked to the noncentral chi‐square distribution, but CFI has seen much wider use in applied research, whereas RMSEA has only recently been gaining attention. The article suggests that use of CFI is problematic because of its baseline model. CFI seems to be appropriate in more exploratory contexts, whereas RMSEA is appropriate in more confirmatory contexts. On the other hand, CFI does have an established parsimony adjustment, although the adjustment included in RMSEA may be inadequate.     

Assessing Structural Equation Models by Equivalence Testing With Adjusted Fit Indexes

Conventional null hypothesis testing (NHT) is a very important tool if the ultimate goal is to find a difference or to reject a model. However, the purpose of structural equation modeling (SEM) is to identify a model and use it to account for the relationship among substantive variables. With the setup of NHT, a nonsignificant test statistic does not necessarily imply that the model is correctly specified or the size of misspecification is properly controlled. To overcome this problem, this article proposes to replace NHT by equivalence testing, the goal of which is to endorse a model under a null hypothesis rather than to reject it. Differences and similarities between equivalence testing and NHT are discussed, and new “T-size” terminology is introduced to convey the goodness of the current model under equivalence testing. Adjusted cutoff values of root mean square error of approximation (RMSEA) and comparative fit index (CFI) corresponding to those conventionally used in the literature are obtained to facilitate the understanding of T-size RMSEA and CFI. The single most notable property of equivalence testing is that it allows a researcher to confidently claim that the size of misspecification in the current model is below the T-size RMSEA or CFI, which gives SEM a desirable property to be a scientific methodology. R code for conducting equivalence testing is provided in an appendix.     

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.     

Validity evidence for the Chinese version Classroom Appraisal of Resources and Demands (CARD)

The Classroom Appraisal of Resources and Demands (CARD) was designed to evaluate teacher stress based on subjective evaluations of classroom demands and resources. However, the CARD has been mostly utilized in western countries. The aim of the current study was to provide aspects of the validity of responses to a Chinese version of the CARD that considers Chinese teachers’ unique vocational conditions in the classroom. A sample of 580 Chinese elementary school teachers (510 female teachers and 70 male teachers) were asked to respond to the Chinese version of the CARD. Confirmatory factor analyses showed that the data fit the theoretical model very well (e.g., CFI: .982; NFI: .977; GFI: .968; SRMR: .028; RMSEA: .075; where CFI is comparative fit index, NFI is normed fit index, GFI is goodness of fit, SRMR is standardized root mean square residual, RMSEA is root mean square error of approximation), thus providing evidence of construct validity. Latent constructs of the Chinese version of the CARD were also found to be significantly associated with other measures that are related to teacher stress such as self‐efficacy, job satisfaction, personal habits to deal with stress, and intention to leave their current job.     

Using Fit Indexes to Select a Covariance Model for Longitudinal Data

This study investigated the performance of fit indexes in selecting a covariance structure for longitudinal data. Data were simulated to follow a compound symmetry, first-order autoregressive, first-order moving average, or random-coefficients covariance structure. We examined the ability of the likelihood ratio test (LRT), root mean square error of approximation (RMSEA), comparative fit index (CFI), and Tucker–Lewis Index (TLI) to reject misspecified models with varying degrees of misspecification. With a sample size of 20, RMSEA, CFI, and TLI are high in both Type I and Type II error rates, whereas LRT has a high Type II error rate. With a sample size of 100, these indexes generally have satisfactory performance, but CFI and TLI are affected by a confounding effect of their baseline model. Akaike's Information Criterion (AIC) and Bayesian Information Criterion (BIC) have high success rates in identifying the true model when sample size is 100. A comparison with the mixed model approach indicates that separately modeling the means and covariance structures in structural equation modeling dramatically improves the success rate of AIC and BIC.     

Levels of aggregation in higher level confirmatory factor analysis: Application for academic self‐concept

The effects of levels of aggregation on measures of goodness of fit and higher order parameter estimates obtained from confirmatory factor analysis (CFA) were investigated. For a higher order model of academic self‐concept, 3 levels of aggregation were considered—disaggregated, partially disaggregated, and partially aggregated. In the disaggregated model, measured variables represented individual items. In the partially disaggregated model, testlets (groups of 4 items) represented measured variables. In the partially aggregated model, subscale scores represented measured variables. Three indexes of fit were employed: the Tucker‐Lewis Index (TLI), the Comparative Fit Index (CFI), and chi‐square. Solutions for the disaggregated models consistently evidenced poor fit. TLI and CFI values for partially disaggregated and partially aggregated solutions were satisfactory. Standardized parameter estimates were similar across all solutions. Implications of these findings are discussed with consideration of other research on model complexity in CFA.     

Psychometric properties of the Chinese version of the Childhood Trauma Questionnaire-Short Form (CTQ-SF) among undergraduates and depressive patients

ObjectiveThe Childhood Trauma Questionnaire-Short Form (CTQ-SF) is a self-report questionnaire that retrospectively provides screening for a history of childhood abuse and neglect, and which is widely used throughout the world. The current study aimed to examine the psychometric properties of the Chinese version of the CTQ-SF.MethodsParticipants included 3431 undergraduates from Hunan provinces and 234 depressive patients from psychological clinics. Confirmatory factor analysis was performed to examine how well the original five-factor model fit the data and the measurement equivalence of CTQ-SF across gender. Internal consistency was also evaluated.ResultsThe five-factor model achieved satisfactory fit (Undergraduate sample TLI = 0.925, CFI = 0.936, RMSEA = 0.034, SRMR = 0.046; depressive sample TLI = 0.912, CFI = 0.923, RMSEA = 0.044, SRMR = 0.062). Measurement invariance of the five-factor model across gender was supported fully assuming different degrees of invariance. The CTQ-SF also showed acceptable internal consistency and good stability.ConclusionThe current study provides that the Chinese version of the Childhood Trauma questionnaire-short form has good reliability and validity among Chinese undergraduates and depressive samples, which also indicates that the CTQ-SF is a good tool for child trauma assessment.     

Four New Corrected Statistics for SEM With Small Samples and Nonnormally Distributed Data

When the assumption of multivariate normality is violated and the sample sizes are relatively small, existing test statistics such as the likelihood ratio statistic and Satorra–Bentler’s rescaled and adjusted statistics often fail to provide reliable assessment of overall model fit. This article proposes four new corrected statistics, aiming for better model evaluation with nonnormally distributed data at small sample sizes. A Monte Carlo study is conducted to compare the performances of the four corrected statistics against those of existing statistics regarding Type I error rate. Results show that the performances of the four new statistics are relatively stable compared with those of existing statistics. In particular, Type I error rates of a new statistic are close to the nominal level across all sample sizes under a condition of asymptotic robustness. Other new statistics also exhibit improved Type I error control, especially with nonnormally distributed data at small sample sizes.     

Parenting quality: Confirmation of a higher-order latent construct with mothers of Head Start children

We proposed a higher order latent construct of parenting young children, parenting quality. This higher-order latent construct comprises five component constructs: demographic protection, psychological distress, psychosocial maturity, moral and cognitive reflectivity, and parenting attitudes and beliefs. We evaluated this model with data provided by 199 mothers of 4-year-old children enrolled in Head Start. The model was confirmed with only one adjustment suggested by modification indices. Final RMSEA was .05, CFI .96, and NNFI .94, indicating good model fit. Results were interpreted as emphasizing the interdependence of psychological and environmental demands on parenting. Implications of the model for teachers, early interventionists, and public policy are discussed.     

Standardized Parameters in Misspecified Structural Equation Models: Empirical Performance in Point Estimates,Standard Errors,and Confidence Intervals

In practice, models always have misfit, and it is not well known in what situations methods that provide point estimates, standard errors (SEs), or confidence intervals (CIs) of standardized structural equation modeling (SEM) parameters are trustworthy. In this article we carried out simulations to evaluate the empirical performance of currently available methods. We studied maximum likelihood point estimates, as well as SE estimators based on the delta method, nonparametric bootstrap (NP-B), and semiparametric bootstrap (SP-B). For CIs we studied Wald CI based on delta, and percentile and BC_a intervals based on NP-B and SP-B. We conducted simulation studies using both confirmatory factor analysis and SEM models. Depending on (a) whether point estimate, SE, or CI is of interest; (b) amount of model misfit; (c) sample size; and (d) model complexity, different methods can be the one that renders best performance. Based on the simulation results, we discuss how to choose proper methods in practice.     

Performance of Bootstrapping Approaches to Model Test Statistics and Parameter Standard Error Estimation in Structural Equation Modeling

Though the common default maximum likelihood estimator used in structural equation modeling is predicated on the assumption of multivariate normality, applied researchers often find themselves with data clearly violating this assumption and without sufficient sample size to utilize distribution-free estimation methods. Fortunately, promising alternatives are being integrated into popular software packages. Bootstrap resampling, which is offered in AMOS (Arbuckle, 1997), is one potential solution for estimating model test statistic p values and parameter standard errors under nonnormal data conditions. This study is an evaluation of the bootstrap method under varied conditions of nonnormality, sample size, model specification, and number of bootstrap samples drawn from the resampling space. Accuracy of the test statistic p values is evaluated in terms of model rejection rates, whereas accuracy of bootstrap standard error estimates takes the form of bias and variability of the standard error estimates themselves.     

One of Us Cannot Be Wrong: The paradox of achievement gaps

There are two general groups of methods of calculating achievement gaps (between groups of students in education) in common current usage, similar to those used to calculate social segregation in space and social mobility over time. Each type of method clearly seems valid to its proponents, yet their results in practice are radically different, and often contradictory. This brief paper considers both of these methods and some related problems in the calculation of achievement gaps, in an attempt to resolve the contradiction. The issue is a relatively simple one, but one with significant implications for social researchers as well as commentators in many areas of public policy using similar indicators of performance.     

Sensitivity of Fit Indexes to Misspecified Structural or Measurement Model Components: Rationale of Two-Index Strategy Revisited

In previous research (Hu & Bentler, 1998, 1999), 2 conclusions were drawn: standardized root mean squared residual (SRMR) was the most sensitive to misspecified factor covariances, and a group of other fit indexes were most sensitive to misspecified factor loadings. Based on these findings, a 2-index strategy-that is, SRMR coupled with another index-was proposed in model fit assessment to detect potential misspecification in both the structural and measurement model parameters. Based on our reasoning and empirical work presented in this article, we conclude that SRMR is not necessarily most sensitive to misspecified factor covariances (structural model misspecification), the group of indexes (TLI, BL89, RNI, CFI, Gamma hat, Mc, or RMSEA) are not necessarily more sensitive to misspecified factor loadings (measurement model misspecification), and the rationale for the 2-index presentation strategy appears to have questionable validity.     

The Relation Among Fit Indexes,Power, and Sample Size in Structural Equation Modeling

The relation among fit indexes, power, and sample size in structural equation modeling is examined. The noncentrality parameter is required to compute power. The 2 existing methods of computing power have estimated the noncentrality parameter by specifying an alternative hypothesis or alternative fit. These methods cannot be implemented easily and reliably. In this study, 4 fit indexes (RMSEA, CFI, McDonald's Fit Index, and Steiger's gamma) were used to compute the noncentrality parameter and sample size to achieve certain level of power. The resulting power and sample size varied as a function of (a) choice of fit index, (b) number of variables/degrees of freedom, (c) relation among the variables, and (d) value of the fit index. However, if the level of misspecification were held constant, then the resulting power and sample size would be identical.     

数学专业本科师范生实践教学改革的研究

南京晓庄学院数学系的培养目标主要为中学培养数学师资,故中学数学教学论课程及实践课程的教学大纲必须以培养适应我国当前的数学新课程标准,能运用新理念、新方法进行教学的合格师资服务。针对数学实践课具体特点,经过较全面的调查分析及广泛征求意见,形成了初等数学研究与实践课程的整合方案,经过四年多的实践,逐步得到完善。     

正态性检验方法在教学研究中的应用

针对目前很多研究者在进行正态性检验时仅会依据自己的习惯或喜好来选择方法这一状况,文章从常用方法中选取Jarque-Bera检验、Shapiro-Wilk检验、D'Agostino检验、KolmogorovSmirnov检验以及Lilliefors检验这五种正态性检验方法进行简要论述,利用Monte Carlo法分析比较五种检验在不同样本量的不同分布下的检验功效或Ⅰ型错误率,再结合SAS、SPSS和R这三种常用的教学统计软件,讨论正态性检验方法的选取问题,以期为科研工作者选择正态性检验方法时提供参考。     

The Impact of Moderate Priors For Bayesian Estimation and Testing of Item Factor Analysis Models When Maximum Likelihood is Unsuitable

In psychological research, available data are often insufficient to estimate item factor analysis (IFA) models using traditional estimation methods, such as maximum likelihood (ML) or limited information estimators. Bayesian estimation with common-sense, moderately informative priors can greatly improve efficiency of parameter estimates and stabilize estimation. There are a variety of methods available to evaluate model fit in a Bayesian framework; however, past work investigating Bayesian model fit assessment for IFA models has assumed flat priors, which have no advantage over ML in limited data settings. In this paper, we evaluated the impact of moderately informative priors on ability to detect model misfit for several candidate indices: posterior predictive checks based on the observed score distribution, leave-one-out cross-validation, and widely available information criterion (WAIC). We found that although Bayesian estimation with moderately informative priors is an excellent aid for estimating challenging IFA models, methods for testing model fit in these circumstances are inadequate.