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1.
We compare the accuracy of confidence intervals (CIs) and tests of close fit based on the root mean square error of approximation (RMSEA) with those based on the standardized root mean square residual (SRMR). Investigations used normal and nonnormal data with models ranging from p = 10 to 60 observed variables. CIs and tests of close fit based on the SRMR are generally accurate across all conditions (even at p = 60 with nonnormal data). In contrast, CIs and tests of close fit based on the RMSEA are only accurate in small models. In larger models (p ≥ 30), they incorrectly suggest that models do not fit closely, particularly if sample size is less than 500.  相似文献   

2.
Recently a new mean scaled and skewness adjusted test statistic was developed for evaluating structural equation models in small samples and with potentially nonnormal data, but this statistic has received only limited evaluation. The performance of this statistic is compared to normal theory maximum likelihood and 2 well-known robust test statistics. A modification to the Satorra–Bentler scaled statistic is developed for the condition that sample size is smaller than degrees of freedom. The behavior of the 4 test statistics is evaluated with a Monte Carlo confirmatory factor analysis study that varies 7 sample sizes and 3 distributional conditions obtained using Headrick's fifth-order transformation to nonnormality. The new statistic performs badly in most conditions except under the normal distribution. The goodness-of-fit χ2 test based on maximum-likelihood estimation performed well under normal distributions as well as under a condition of asymptotic robustness. The Satorra–Bentler scaled test statistic performed best overall, whereas the mean scaled and variance adjusted test statistic outperformed the others at small and moderate sample sizes under certain distributional conditions.  相似文献   

3.
Statistical theories of goodness-of-fit tests in structural equation modeling are based on asymptotic distributions of test statistics. When the model includes a large number of variables or the population is not from a multivariate normal distribution, the asymptotic distributions do not approximate the distribution of the test statistics very well at small sample sizes. A variety of methods have been developed to improve the accuracy of hypothesis testing at small sample sizes. However, all these methods have their limitations, specially for nonnormal distributed data. We propose a Monte Carlo test that is able to control Type I error with more accuracy compared to existing approaches in both normal and nonnormally distributed data at small sample sizes. Extensive simulation studies show that the suggested Monte Carlo test has a more accurate observed significance level as compared to other tests with a reasonable power to reject misspecified models.  相似文献   

4.
Classical accounts of maximum likelihood (ML) estimation of structural equation models for continuous outcomes involve normality assumptions: standard errors (SEs) are obtained using the expected information matrix and the goodness of fit of the model is tested using the likelihood ratio (LR) statistic. Satorra and Bentler (1994) introduced SEs and mean adjustments or mean and variance adjustments to the LR statistic (involving also the expected information matrix) that are robust to nonnormality. However, in recent years, SEs obtained using the observed information matrix and alternative test statistics have become available. We investigate what choice of SE and test statistic yields better results using an extensive simulation study. We found that robust SEs computed using the expected information matrix coupled with a mean- and variance-adjusted LR test statistic (i.e., MLMV) is the optimal choice, even with normally distributed data, as it yielded the best combination of accurate SEs and Type I errors.  相似文献   

5.
This study examined the effect of model size on the chi-square test statistics obtained from ordinal factor analysis models. The performance of six robust chi-square test statistics were compared across various conditions, including number of observed variables (p), number of factors, sample size, model (mis)specification, number of categories, and threshold distribution. Results showed that the unweighted least squares (ULS) robust chi-square statistics generally outperform the diagonally weighted least squares (DWLS) robust chi-square statistics. The ULSM estimator performed the best overall. However, when fitting ordinal factor analysis models with a large number of observed variables and small sample size, the ULSM-based chi-square tests may yield empirical variances that are noticeably larger than the theoretical values and inflated Type I error rates. On the other hand, when the number of observed variables is very large, the mean- and variance-corrected chi-square test statistics (e.g., based on ULSMV and WLSMV) could produce empirical variances conspicuously smaller than the theoretical values and Type I error rates lower than the nominal level, and demonstrate lower power rates to reject misspecified models. Recommendations for applied researchers and future empirical studies involving large models are provided.  相似文献   

6.
Model fit indices are being increasingly recommended and used to select the number of factors in an exploratory factor analysis. Growing evidence suggests that the recommended cutoff values for common model fit indices are not appropriate for use in an exploratory factor analysis context. A particularly prominent problem in scale evaluation is the ubiquity of correlated residuals and imperfect model specification. Our research focuses on a scale evaluation context and the performance of four standard model fit indices: root mean square error of approximate (RMSEA), standardized root mean square residual (SRMR), comparative fit index (CFI), and Tucker–Lewis index (TLI), and two equivalence test-based model fit indices: RMSEAt and CFIt. We use Monte Carlo simulation to generate and analyze data based on a substantive example using the positive and negative affective schedule (N = 1,000). We systematically vary the number and magnitude of correlated residuals as well as nonspecific misspecification, to evaluate the impact on model fit indices in fitting a two-factor exploratory factor analysis. Our results show that all fit indices, except SRMR, are overly sensitive to correlated residuals and nonspecific error, resulting in solutions that are overfactored. SRMR performed well, consistently selecting the correct number of factors; however, previous research suggests it does not perform well with categorical data. In general, we do not recommend using model fit indices to select number of factors in a scale evaluation framework.  相似文献   

7.
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.  相似文献   

8.
Mean and mean-and-variance corrections are the 2 major principles to develop test statistics with violation of conditions. In structural equation modeling (SEM), mean-rescaled and mean-and-variance-adjusted test statistics have been recommended under different contexts. However, recent studies indicated that their Type I error rates vary from 0% to 100% as the number of variables p increases. Can we still trust the 2 principles and what alternative rules can be used to develop test statistics for SEM with “big data”? This article addresses the issues by a large-scale Monte Carlo study. Results indicate that empirical means and standard deviations of each statistic can differ from their expected values many times in standardized units when p is large. Thus, the problems in Type I error control with the 2 statistics are because they do not possess the properties to which they are entitled, not because of the wrongdoing of the mean and mean-and-variance corrections. However, the 2 principles need to be implemented using small sample methodology instead of asymptotics. Results also indicate that distributions other than chi-square might better describe the behavior of test statistics in SEM with big data.  相似文献   

9.
Structured means analysis is a very useful approach for testing hypotheses about population means on latent constructs. In such models, a z test is most commonly used for testing the statistical significance of the relevant parameter estimates or of the differences between parameter estimates, where a z value is computed based on the asymptotic standard error estimate associated with the parameter of interest. In the current article, a series of population analyses demonstrate that the z tests for latent mean structure parameters or, more directly, the standard error estimates upon which those z tests are based are, not invariant to how factors are scaled. As such, circumstances exist in which latent mean inference is compromised solely as a result of scaling decisions. This problem is illustrated in the context of between-subjects (i.e., multisample) latent means models and within-subjects latent means models. Recommendations for practice are also offered.  相似文献   

10.
Cindy L. James   《Assessing Writing》2006,11(3):167-178
How do scores from writing samples generated by computerized essay scorers compare to those generated by “untrained” human scorers and what combination of scores, if any, is more accurate at placing students in composition courses? This study endeavored to answer this two-part question by evaluating the correspondence between writing sample scores generated by the IntelliMetric™ automated scoring system and scores generated by University Preparation English faculty, as well as examining the predictive validity of both the automated and human scores. The results revealed significant correlations between the faculty scores and the IntelliMetric™ scores of the ACCUPLACER™ OnLine WritePlacer Plus test. Moreover, logistic regression models that utilized the IntelliMetric™ scores and average faculty scores were more accurate at placing students (77% overall correct placement rate) than were models incorporating only the average faculty score or the IntelliMetric™ scores.  相似文献   

11.
The absolute normal scores test (K) is described as a test for the symmetry of a distribution of scores about a location parameter designated as 0. The test is compared to the sign test (S) and the Wilcoxon test (W) as an alternative to the t-test. Power comparisons are made among the K, S, W, and t tests. An example is presented where the sample size is less than 20. The large sample approximation to the normal distribution is also illustrated.  相似文献   

12.
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.  相似文献   

13.
Fitting a large structural equation modeling (SEM) model with moderate to small sample sizes results in an inflated Type I error rate for the likelihood ratio test statistic under the chi-square reference distribution, known as the model size effect. In this article, we show that the number of observed variables (p) and the number of free parameters (q) have unique effects on the Type I error rate of the likelihood ratio test statistic. In addition, the effects of p and q cannot be fully explained using degrees of freedom (df). We also evaluated the performance of 4 correctional methods for the model size effect, including Bartlett’s (1950), Swain’s (1975), and Yuan’s (2005) corrected statistics, and Yuan, Tian, and Yanagihara’s (2015) empirically corrected statistic. We found that Yuan et al.’s (2015) empirically corrected statistic generally yields the best performance in controlling the Type I error rate when fitting large SEM models.  相似文献   

14.
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.  相似文献   

15.
This study used Monte Carlo methods to investigate the accuracy and utility of estimators of overall error and error due to approximation in structural equation models. The effects of sample size, indicator reliabilities, and degree of misspecification were examined. The rescaled noncentrality parameter (McDonald & Marsh, 1990) was examined as a measure of approximation error, whereas the one‐ and two‐sample cross‐validation indices and a sample estimator of overall error (EFo) proposed by Browne and Cudeck (1989, 1993) were presented as measures of overall error. The rescaled noncentrality parameter and EFo provided extremely accurate estimates of the amounts of approximation and overall error, respectively. However, although models with errors of omission produced larger estimates of approximation and overall error, the presence of errors of inclusion had little or no effect on estimates of either type of error. The cross‐validation indices and sample estimator of overall error reached minimum values for the same model as an empirically derived measure of overall error only for models with large amounts of specification error. Implications for the use of these estimators in choosing among competing models were discussed.  相似文献   

16.
The asymptotically distribution free (ADF) method is often used to estimate parameters or test models without a normal distribution assumption on variables, both in covariance structure analysis and in correlation structure analysis. However, little has been done to study the differences in behaviors of the ADF method in covariance versus correlation structure analysis. The behaviors of 3 test statistics frequently used to evaluate structural equation models with nonnormally distributed variables, χ2 test TAGLS and its small-sample variants TYB and TF(AGLS) were compared. Results showed that the ADF method in correlation structure analysis with test statistic TAGLS performs much better at small sample sizes than the corresponding test for covariance structures. In contrast, test statistics TYB and TF(AGLS) under the same conditions generally perform better with covariance structures than with correlation structures. It is proposed that excessively large and variable condition numbers of weight matrices are a cause of poor behavior of ADF test statistics in small samples, and results showed that these condition numbers are systematically increased with substantial increase in variance as sample size decreases. Implications for research and practice are discussed.  相似文献   

17.
How should researchers choose between competing scales in predicting a criterion variable? This article proposes the use of nonnested tests for the 2SLS estimator of latent variable models to discriminate between scales. The finite sample performance of these tests is compared to structural equation modeling information-based criteria such as root mean squared error of approximation (RMSEA) and Akaike's Information Criterion (AIC). The Cox and encompassing tests and augmented versions of these tests are compared to the inconsistent ordinary least squares (OLS) J test. An augmented version of the encompassing test performs best for sample sizes of 100 or more and can be recommended for use on scales with high reliability (0.9) and sample sizes of 200 or more, under varying regressor and error distributions. The OLS J test performs best for small samples of N = 50 and can be recommended for use in small samples when scales have high reliability (0.9). Relative to the nonnested tests, the information-based criteria perform poorly.  相似文献   

18.
A Monte Carlo approach was used to examine bias in the estimation of indirect effects and their associated standard errors. In the simulation design, (a) sample size, (b) the level of nonnormality characterizing the data, (c) the population values of the model parameters, and (d) the type of estimator were systematically varied. Estimates of model parameters were generally unaffected by either nonnormality or small sample size. Under severely nonnormal conditions, normal theory maximum likelihood estimates of the standard error of the mediated effect exhibited less bias (approximately 10% to 20% too small) compared to the standard errors of the structural regression coefficients (20% to 45% too small). Asymptotically distribution free standard errors of both the mediated effect and the structural parameters were substantially affected by sample size, but not nonnormality. Robust standard errors consistently yielded the most accurate estimates of sampling variability.  相似文献   

19.
Score equating based on small samples of examinees is often inaccurate for the examinee populations. We conducted a series of resampling studies to investigate the accuracy of five methods of equating in a common-item design. The methods were chained equipercentile equating of smoothed distributions, chained linear equating, chained mean equating, the symmetric circle-arc method, and the simplified circle-arc method. Four operational test forms, each containing at least 110 items, were used for the equating, with new-form samples of 100, 50, 25, and 10 examinees and reference-form samples three times as large. Accuracy was described in terms of the root-mean-squared difference (over 1,000 replications) of the sample equatings from the criterion equating. Overall, chained mean equating produced the most accurate results for low scores, but the two circle-arc methods produced the most accurate results, particularly in the upper half of the score distribution. The difference in equating accuracy between the two circle-arc methods was negligible.  相似文献   

20.
This study investigated the self-reported instructional assessment practices of a selected sample of secondary school science teachers in Barbados. The study sought to determine if there were statistically significant differences in the instructional assessment practices of teachers based on their sex and teacher quality (teaching experience, professional qualification and teacher academic qualification). It also sought to determine the extent to which each of these four selected variables individually and jointly affected the teachers’ report of their instructional assessment practices. A sample of 55 science teachers from nine secondary schools in Barbados was randomly selected to participate in this study. Data was collected by means of a survey and was analyzed using the means and standard deviations of the instructional assessment practices scores and linear, multiple and binary logistic regression. The results of the study were such that the majority of the sample reported good overall instructional assessment practices while only a few participants reported moderate assessment practices. The instructional assessment practices in the area of student knowledge were mostly moderate as indicated by the sample. There were no statistically significant differences between or among the mean scores of the teachers’ reported instructional assessment practices based on sex ( t?=?0.10; df?=?53; p?=?0.992), teaching experience ( F[4,50]?=?1.766; p?=?0.150), the level of professional qualification (F[3,45]?=?0.2117; p?=?0.111) or the level of academic qualification (F[2,52]?=?0.504; p?=?0.607). The independent variables (teacher sex, teaching experience, teacher professional qualification or teacher academic qualification) were not significant predictors of the instructional assessment practices scores. However, teacher sex was a significant predictor of the teachers’ report of good instructional assessment practices. The study also found that the joint effect of the variables teacher sex, teaching experience, teacher professional qualification and teacher academic qualification was not significant in predicting the instructional assessment practices scores of the science teachers. However, the joint effect of these variables was statistically significant (X 2?=?18.482; df?=?10; p?=?0.047) in predicting the teachers’ reported use of good instructional assessment practices. The best predictor of teachers’ report of good instructional assessment practices, though not statistically significant, was the diploma in education professional qualification.  相似文献   

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