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1.
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   

2.
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.  相似文献   

3.
Using a complex simulation study we investigated parameter recovery, classification accuracy, and performance of two item‐fit statistics for correct and misspecified diagnostic classification models within a log‐linear modeling framework. The basic manipulated test design factors included the number of respondents (1,000 vs. 10,000), attributes (3 vs. 5), and items (25 vs. 50) as well as different attribute correlations (.50 vs. .80) and marginal attribute difficulties (equal vs. different). We investigated misspecifications of interaction effect parameters under correct Q‐matrix specification and two types of Q‐matrix misspecification. While the misspecification of interaction effects had little impact on classification accuracy, invalid Q‐matrix specifications led to notably decreased classification accuracy. Two proposed item‐fit indexes were more strongly sensitive to overspecification of Q‐matrix entries for items than to underspecification. Information‐based fit indexes AIC and BIC were sensitive to both over‐ and underspecification.  相似文献   

4.
This study examined the performance of 4 correlation-based fit indexes (marginal and conditional pseudo R 2s; average and conditional concordance correlations) in detecting misspecification in mean structures in growth curve models. Their performance was also compared to that of 4 traditional SEM fit indexes. We found that the marginal pseudo R 2 and average concordance correlation were able to detect misspecification in the marginal mean structure (average change trajectory). The conditional pseudo R 2 and concordance correlation could detect misspecification when it occurred in the conditional mean structure (individual change trajectory) or in both mean structures. Compared to the SEM fit indexes, the correlation-based fit indexes were more robust to sample size but were less robust to data properties such as magnitude of population mean and measurement error. Theoretical and practical implications of the results and directions for future research are discussed.  相似文献   

5.
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.  相似文献   

6.
Proper model specification is an issue for researchers, regardless of the estimation framework being utilized. Typically, indexes are used to compare the fit of one model to the fit of an alternate model. These indexes only provide an indication of relative fit and do not necessarily point toward proper model specification. There is a procedure in the Bayesian framework called posterior predictive checking that is designed theoretically to detect model misspecification for observed data. However, the performance of the posterior predictive check procedure has thus far not been directly examined under different conditions of mixture model misspecification. This article addresses this task and aims to provide additional insight into whether or not posterior predictive checks can detect model misspecification within the context of Bayesian growth mixture modeling. Results indicate that this procedure can only identify mixture model misspecification under very extreme cases of misspecification.  相似文献   

7.
A Monte Carlo simulation study was conducted to investigate the effects on structural equation modeling (SEM) fit indexes of sample size, estimation method, and model specification. Based on a balanced experimental design, samples were generated from a prespecified population covariance matrix and fitted to structural equation models with different degrees of model misspecification. Ten SEM fit indexes were studied. Two primary conclusions were suggested: (a) some fit indexes appear to be noncomparable in terms of the information they provide about model fit for misspecified models and (b) estimation method strongly influenced almost all the fit indexes examined, especially for misspecified models. These 2 issues do not seem to have drawn enough attention from SEM practitioners. Future research should study not only different models vis‐à‐vis model complexity, but a wider range of model specification conditions, including correctly specified models and models specified incorrectly to varying degrees.  相似文献   

8.
Two Lagrange multiplier (LM) methods may be used in specification searches for adding parameters to models: one based on univariate LM tests and respecification of the model (LM‐respecified method) and the other based on a partitioning of multivariate LM tests (LM‐incremental method). These methods may result in extraneous parameters being included in models due to either sampling error or the model being misspecified. A 2‐stage specification search may be used to reduce errors due to misspecification. In the 1st stage, parameters are added to models based on LM tests to maximize fit. Second, parameters added in the 1st stage are deleted if they are no longer necessary to maintain model fit. Illustrations are presented to demonstrate that errors due to misspecification occur with the LM‐respecified method and are even more likely with the LM‐incremental approach. These illustrations also show how the deletion stage can help eliminate some of these errors.  相似文献   

9.
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.  相似文献   

10.
Goodness-of-fit (GOF) indexes provide "rules of thumb"—recommended cutoff values for assessing fit in structural equation modeling. Hu and Bentler (1999) proposed a more rigorous approach to evaluating decision rules based on GOF indexes and, on this basis, proposed new and more stringent cutoff values for many indexes. This article discusses potential problems underlying the hypothesis-testing rationale of their research, which is more appropriate to testing statistical significance than evaluating GOF. Many of their misspecified models resulted in a fit that should have been deemed acceptable according to even their new, more demanding criteria. Hence, rejection of these acceptable-misspecified models should have constituted a Type 1 error (incorrect rejection of an "acceptable" model), leading to the seemingly paradoxical results whereby the probability of correctly rejecting misspecified models decreased substantially with increasing N. In contrast to the application of cutoff values to evaluate each solution in isolation, all the GOF indexes were more effective at identifying differences in misspecification based on nested models. Whereas Hu and Bentler (1999) offered cautions about the use of GOF indexes, current practice seems to have incorporated their new guidelines without sufficient attention to the limitations noted by Hu and Bentler (1999).  相似文献   

11.
McDonald goodness‐of‐fit indices based on maximum likelihood, asymptotic distribution free, and the Satorra‐Bentler scale correction estimation methods are investigated. Sampling experiments are conducted to assess the magnitude of error for each index under variations in distributional misspecification, structural misspecification, and sample size. The Satorra‐Bentler correction‐based index is shown to have the least error under each distributional misspecification level when the model has correct structural specification. The scaled index also performs adequately when there is minor structural misspecification and distributional misspecification. However, when a model has major structural misspecification with distributional misspecification, none of the estimation methods perform adequately.  相似文献   

12.
Posterior predictive model checking (PPMC) is a Bayesian model checking method that compares the observed data to (plausible) future observations from the posterior predictive distribution. We propose an alternative to PPMC in the context of structural equation modeling, which we term the poor person’s PPMC (PP-PPMC), for the situation wherein one cannot afford (or is unwilling) to draw samples from the full posterior. Using only by-products of likelihood-based estimation (maximum likelihood estimate and information matrix), the PP-PPMC offers a natural method to handle parameter uncertainty in model fit assessment. In particular, a coupling relationship between the classical p values from the model fit chi-square test and the predictive p values from the PP-PPMC method is carefully examined, suggesting that PP-PPMC might offer an alternative, principled approach for model fit assessment. We also illustrate the flexibility of the PP-PPMC approach by applying it to case-influence diagnostics.  相似文献   

13.
This simulation study demonstrates how the choice of estimation method affects indexes of fit and parameter bias for different sample sizes when nested models vary in terms of specification error and the data demonstrate different levels of kurtosis. Using a fully crossed design, data were generated for 11 conditions of peakedness, 3 conditions of misspecification, and 5 different sample sizes. Three estimation methods (maximum likelihood [ML], generalized least squares [GLS], and weighted least squares [WLS]) were compared in terms of overall fit and the discrepancy between estimated parameter values and the true parameter values used to generate the data. Consistent with earlier findings, the results show that ML compared to GLS under conditions of misspecification provides more realistic indexes of overall fit and less biased parameter values for paths that overlap with the true model. However, despite recommendations found in the literature that WLS should be used when data are not normally distributed, we find that WLS under no conditions was preferable to the 2 other estimation procedures in terms of parameter bias and fit. In fact, only for large sample sizes (N = 1,000 and 2,000) and mildly misspecified models did WLS provide estimates and fit indexes close to the ones obtained for ML and GLS. For wrongly specified models WLS tended to give unreliable estimates and over-optimistic values of fit.  相似文献   

14.
One challenge in mediation analysis is to generate a confidence interval (CI) with high coverage and power that maintains a nominal significance level for any well-defined function of indirect and direct effects in the general context of structural equation modeling (SEM). This study discusses a proposed Monte Carlo extension that finds the CIs for any well-defined function of the coefficients of SEM such as the product of k coefficients and the ratio of the contrasts of indirect effects, using the Monte Carlo method. Finally, we conduct a small-scale simulation study to compare CIs produced by the Monte Carlo, nonparametric bootstrap, and asymptotic-delta methods. Based on our simulation study, we recommend researchers use the Monte Carlo method to test a complex function of indirect effects.  相似文献   

15.
The comparative fit index (CFI) is one of the most widely-used fit indices in structural equation modeling (SEM). When applying the CFI to model evaluation, although it is universally recognized that the focus should be the population fit, in practice one often considers only the CFI value within a sample and neglects the uncertainty in point estimation. Confidence interval (CI) methods for CFI appeared only recently, but these methods assume multivariate normality, which often fails to hold in practice. In addition, the current methods are applications of the bootstrap and are thus computationally intensive. To better handle nonnormal data and simplify CI construction, in this paper we propose an analytic CI method for CFI without assuming normality. We then carry out simulation studies to compare the new and current methods at various levels of model misfit and nonnormality. Simulation results verify the effectiveness and advantages of the new method.  相似文献   

16.
We compared six common methods in estimating the 2-1-1 (level-2 independent, level-1 mediator, level-1 dependent) multilevel mediation model with a random slope. They were the Bayesian with informative priors, the Bayesian with non-informative priors, the Monte-Carlo, the distribution of the product, the bias-corrected, and the bias-uncorrected parametric percentile residual bootstrap. The Bayesian method with informative priors was superior in relative mean square error (RMSE), power, interval width, and interval imbalance. The prior variance and prior mean were also varied and examined. Decreasing the prior variance increased the power, reduced RMSE and interval width when the prior mean was the true value, but decreasing the prior variance reduced the power when the prior mean was set incorrectly. The influence of misspecification of prior information of the b coefficient on multilevel mediation analysis was greater than that on coefficient a. An illustrate example with the Bayesian multilevel mediation was provided.  相似文献   

17.
When the multivariate normality assumption is violated in structural equation modeling, a leading remedy involves estimation via normal theory maximum likelihood with robust corrections to standard errors. We propose that this approach might not be best for forming confidence intervals for quantities with sampling distributions that are slow to approach normality, or for functions of model parameters. We implement and study a robust analog to likelihood-based confidence intervals based on inverting the robust chi-square difference test of Satorra (2000). We compare robust standard errors and the robust likelihood-based approach versus resampling methods in confirmatory factor analysis (Studies 1 & 2) and mediation analysis models (Study 3) for both single parameters and functions of model parameters, and under a variety of nonnormal data generation conditions. The percentile bootstrap emerged as the method with the best calibrated coverage rates and should be preferred if resampling is possible, followed by the robust likelihood-based approach.  相似文献   

18.
The information matrix can equivalently be determined via the expectation of the Hessian matrix or the expectation of the outer product of the score vector. The identity of these two matrices, however, is only valid in case of a correctly specified model. Therefore, differences between the two versions of the observed information matrix indicate model misfit. The equality of both matrices can be tested with the so‐called information matrix test as a general test of misspecification. This test can be adapted to item response models in order to evaluate the fit of single items and the fit of the whole scale. The performance of different versions of the test is compared in a simulation study with existing tests of model fit, among them the test of Orlando and Thissen, the score test of local independence due to Glas and Suarez‐Falcon, and the limited information approach of Maydeu‐Olivares and Joe. In general, the different versions of the information matrix test adhere to the nominal Type I error rate and have high power for detecting misspecified item characteristic curves. Additionally, some versions of the test can be used in order to detect violations of the local independence assumption.  相似文献   

19.
This study examined the performance of the weighted root mean square residual (WRMR) through a simulation study using confirmatory factor analysis with ordinal data. Values and cut scores for the WRMR were examined, along with a comparison of its performance relative to commonly cited fit indexes. The findings showed that WRMR illustrated worse fit when sample size increased or model misspecification increased. Lower (i.e., better) values of WRMR were observed when nonnormal data were present, there were lower loadings, and when few categories were analyzed. WRMR generally illustrated expected patterns of relations to other well-known fit indexes. In general, a cutoff value of 1.0 appeared to work adequately under the tested conditions and the WRMR values of “good fit” were generally in agreement with other indexes. Users are cautioned that when the fitted model is misspeficifed, the index might provide misleading results under situations where extremely large sample sizes are used.  相似文献   

20.
Approximations to the distributions of goodness-of-fit indexes in structural equation modeling are derived with the assumption of multivariate normality and slight misspecification of models. The fit indexes considered in this article are Joreskog and Sorbom's goodness-of-fit index (GFI) and the adjusted GFI, McDonald's absolute GFI, Steiger and Lind's root mean squared error of approximation, Steiger's Γ1 and Γ2, Bentler and Bonett's normed fit index, Bollen's incremental fit index and ρ1, Tucker and Lewis's index ρ2, and Bentler's fit index (McDonald and Marsh's relative noncentrality index). An approximation to the asymptotic covariance matrix for the fit indexes is derived by using the delta method. Furthermore, approximations to the densities of the fit indexes are obtained from the transformations of the asymptotically noncentral chi-square distributed variable. A simulation is carried out to confirm the accuracy of the approximations.  相似文献   

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