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1.
Ordinal variables are common in many empirical investigations in the social and behavioral sciences. Researchers often apply the maximum likelihood method to fit structural equation models to ordinal data. This assumes that the observed measures have normal distributions, which is not the case when the variables are ordinal. A better approach is to use polychoric correlations and fit the models using methods such as unweighted least squares (ULS), maximum likelihood (ML), weighted least squares (WLS), or diagonally weighted least squares (DWLS). In this simulation evaluation we study the behavior of these methods in combination with polychoric correlations when the models are misspecified. We also study the effect of model size and number of categories on the parameter estimates, their standard errors, and the common chi-square measures of fit when the models are both correct and misspecified. When used routinely, these methods give consistent parameter estimates but ULS, ML, and DWLS give incorrect standard errors. Correct standard errors can be obtained for these methods by robustification using an estimate of the asymptotic covariance matrix W of the polychoric correlations. When used in this way the methods are here called RULS, RML, and RDWLS.  相似文献   

2.
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   

3.
Factor analysis models with ordinal indicators are often estimated using a 3-stage procedure where the last stage involves obtaining parameter estimates by least squares from the sample polychoric correlations. A simulation study involving 324 conditions (1,000 replications per condition) was performed to compare the performance of diagonally weighted least squares (DWLS) and unweighted least squares (ULS) in the procedure's third stage. Overall, both methods provided accurate and similar results. However, ULS was found to provide more accurate and less variable parameter estimates, as well as more precise standard errors and better coverage rates. Nevertheless, convergence rates for DWLS are higher. Our recommendation is therefore to use ULS, and, in the case of nonconvergence, to use DWLS, as this method might converge when ULS does not.  相似文献   

4.
Data collected from questionnaires are often in ordinal scale. Unweighted least squares (ULS), diagonally weighted least squares (DWLS) and normal-theory maximum likelihood (ML) are commonly used methods to fit structural equation models. Consistency of these estimators demands no structural misspecification. In this article, we conduct a simulation study to compare the equation-by-equation polychoric instrumental variable (PIV) estimation with ULS, DWLS, and ML. Accuracy of PIV for the correctly specified model and robustness of PIV for misspecified models are investigated through a confirmatory factor analysis (CFA) model and a structural equation model with ordinal indicators. The effects of sample size and nonnormality of the underlying continuous variables are also examined. The simulation results show that PIV produces robust factor loading estimates in the CFA model and in structural equation models. PIV also produces robust path coefficient estimates in the model where valid instruments are used. However, robustness highly depends on the validity of instruments.  相似文献   

5.
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   

6.
This study examined the efficacy of 4 different parceling methods for modeling categorical data with 2, 3, and 4 categories and with normal, moderately nonnormal, and severely nonnormal distributions. The parceling methods investigated were isolated parceling in which items were parceled with other items sharing the same source of variance, and distributed parceling in which items were parceled with items influenced by different factors. These parceling strategies were crossed with strategies in which items were either parceled with similarly distributed or differently distributed items, to create 4 different parceling methods. Overall, parceling together items influenced by different factors and with different distributions resulted in better model fit, but high levels of parameter estimate bias. Across all parceling methods, parameter estimate bias ranged from 20% to over 130%. Parceling strategies were contrasted with use of the WLSMV estimator for categorical, unparceled data. Results based on this estimator are encouraging, although some bias was found when high levels of nonnormality were present. Values of the chi-square and root mean squared error of approximation based on WLSMV also resulted in Type II error rates for misspecified models when data were severely nonnormally distributed.  相似文献   

7.
A paucity of research has compared estimation methods within a measurement invariance (MI) framework and determined if research conclusions using normal-theory maximum likelihood (ML) generalizes to the robust ML (MLR) and weighted least squares means and variance adjusted (WLSMV) estimators. Using ordered categorical data, this simulation study aimed to address these queries by investigating 342 conditions. When testing for metric and scalar invariance, Δχ2 results revealed that Type I error rates varied across estimators (ML, MLR, and WLSMV) with symmetric and asymmetric data. The Δχ2 power varied substantially based on the estimator selected, type of noninvariant indicator, number of noninvariant indicators, and sample size. Although some the changes in approximate fit indexes (ΔAFI) are relatively sample size independent, researchers who use the ΔAFI with WLSMV should use caution, as these statistics do not perform well with misspecified models. As a supplemental analysis, our results evaluate and suggest cutoff values based on previous research.  相似文献   

8.
Research in covariance structure analysis suggests that nonnormal data will invalidate chi‐square tests and produce erroneous standard errors. However, much remains unknown about the extent to and the conditions under which highly skewed and kurtotic data can affect the parameter estimates, standard errors, and fit indices. Using actual kurtotic and skewed data and varying sample sizes and estimation methods, we found that (a) normal theory maximum likelihood (ML) and generalized least squares estimators were fairly consistent and almost identical, (b) standard errors tended to underestimate the true variation of the estimators, but the problem was not very serious for large samples (n = 1,000) and conservative (99%) confidence intervals, and (c) the adjusted chi‐square tests seemed to yield acceptable results with appropriate sample sizes.  相似文献   

9.
This simulation study compared maximum likelihood (ML) estimation with weighted least squares means and variance adjusted (WLSMV) estimation. The study was based on confirmatory factor analyses with 1, 2, 4, and 8 factors, based on 250, 500, 750, and 1,000 cases, and on 5, 10, 20, and 40 variables with 2, 3, 4, 5, and 6 categories. There was no model misspecification. The most important results were that with 2 and 3 categories the rejection rates of the WLSMV chi-square test corresponded much more to the expected rejection rates according to an alpha level of. 05 than the rejection rates of the ML chi-square test. The magnitude of the loadings was more precisely estimated by means of WLSMV when the variables had only 2 or 3 categories. The sample size for WLSMV estimation needed not to be larger than the sample size for ML estimation.  相似文献   

10.
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   

11.
Given the relationships of item response theory (IRT) models to confirmatory factor analysis (CFA) models, IRT model misspecifications might be detectable through model fit indexes commonly used in categorical CFA. The purpose of this study is to investigate the sensitivity of weighted least squares with adjusted means and variance (WLSMV)-based root mean square error of approximation, comparative fit index, and Tucker–Lewis Index model fit indexes to IRT models that are misspecified due to local dependence (LD). It was found that WLSMV-based fit indexes have some functional relationships to parameter estimate bias in 2-parameter logistic models caused by violations of LD. Continued exploration into these functional relationships and development of LD-detection methods based on such relationships could hold much promise for providing IRT practitioners with global information on violations of local independence.  相似文献   

12.
Robust maximum likelihood (ML) and categorical diagonally weighted least squares (cat-DWLS) estimation have both been proposed for use with categorized and nonnormally distributed data. This study compares results from the 2 methods in terms of parameter estimate and standard error bias, power, and Type I error control, with unadjusted ML and WLS estimation methods included for purposes of comparison. Conditions manipulated include model misspecification, level of asymmetry, level and categorization, sample size, and type and size of the model. Results indicate that cat-DWLS estimation method results in the least parameter estimate and standard error bias under the majority of conditions studied. Cat-DWLS parameter estimates and standard errors were generally the least affected by model misspecification of the estimation methods studied. Robust ML also performed well, yielding relatively unbiased parameter estimates and standard errors. However, both cat-DWLS and robust ML resulted in low power under conditions of high data asymmetry, small sample sizes, and mild model misspecification. For more optimal conditions, power for these estimators was adequate.  相似文献   

13.
This study investigated the extent to which class-specific parameter estimates are biased by the within-class normality assumption in nonnormal growth mixture modeling (GMM). Monte Carlo simulations for nonnormal GMM were conducted to analyze and compare two strategies for obtaining unbiased parameter estimates: relaxing the within-class normality assumption and using data transformation on repeated measures. Based on unconditional GMM with two latent trajectories, data were generated under different sample sizes (300, 800, and 1500), skewness (0.7, 1.2, and 1.6) and kurtosis (2 and 4) of outcomes, numbers of time points (4 and 8), and class proportions (0.5:0.5 and 0.25:0.75). Of the four distributions, it was found that skew-t GMM had the highest accuracy in terms of parameter estimation. In GMM based on data transformations, the adjusted logarithmic method was more effective in obtaining unbiased parameter estimates than the use of van der Waerden quantile normal scores. Even though adjusted logarithmic transformation in nonnormal GMM reduced computation time, skew-t GMM produced much more accurate estimation and was more robust over a range of simulation conditions. This study is significant in that it considers different levels of kurtosis and class proportions, which has not been investigated in depth in previous studies. The present study is also meaningful in that investigated the applicability of data transformation to nonnormal GMM.  相似文献   

14.
When both model misspecifications and nonnormal data are present, it is unknown how trustworthy various point estimates, standard errors (SEs), and confidence intervals (CIs) are for standardized structural equation modeling parameters. We conducted simulations to evaluate maximum likelihood (ML), conventional robust SE estimator (MLM), Huber–White robust SE estimator (MLR), and the bootstrap (BS). We found (a) ML point estimates can sometimes be quite biased at finite sample sizes if misfit and nonnormality are serious; (b) ML and MLM generally give egregiously biased SEs and CIs regardless of the degree of misfit and nonnormality; (c) MLR and BS provide trustworthy SEs and CIs given medium misfit and nonnormality, but BS is better; and (d) given severe misfit and nonnormality, MLR tends to break down and BS begins to struggle.  相似文献   

15.
Empirical evidence from developed countries suggests that students' commitment to school is fundamental to their academic success. However, in developing countries, validated measures of student commitment to school do not exist. The current study helps fill this research gap by examining the validity and reliability of a commitment-to-school scale (CSS) adapted for the Ghanaian context. With a sample of 6,252 middle school–age students, the study employs exploratory and confirmatory factor analyses with weighted least squares means and variance adjusted (WLSMV) to establish and validate the construct as bidimensional. Measurement invariance tests confirm that the two-factor commitment model is generalizable across grade levels but not genders. Given its parsimony and good fit, the adapted CSS might be useful for future research in Ghana. Similarity of the model across grade levels suggests that the scale has potential uses in education research among diverse groups. We suggest that the CSS be developed further for better understanding of students' commitment to school.  相似文献   

16.
We investigate a method to estimate the combined effect of multiple continuous/ordinal mediators on a binary outcome: (a) fit a structural equation model with probit link for the outcome and identity/probit link for continuous/ordinal mediators, (b) predict potential outcome probabilities, and (c) compute natural direct and indirect effects. Step 2 involves rescaling the latent continuous variable underlying the outcome to address residual mediator variance and covariance. We evaluate the estimation of risk-difference- and risk-ratio-based effects (RDs, RRs) using the maximum likelihood (ML), mean-and-variance-adjusted weighted least squares (WLSMV) and Bayes estimators in Mplus. Across most variations in path-coefficient and mediator-residual-correlation signs and strengths, and confounding situations investigated, the method performs well with all estimators, but favors ML/WLSMV for RDs with continuous mediators, and Bayes for RRs with ordinal mediators. Bayes outperforms ML/WLSMV regardless of mediator type when estimating RRs with small potential outcome probabilities and in two other special cases. An adolescent alcohol prevention study is used for illustration.  相似文献   

17.
There is a need for effect sizes that are readily interpretable by a broad audience. One index that might fill this need is π, which represents the proportion of scores in one group that exceed the mean of another group. The robustness of estimates of π to violations of normality had not been explored. Using simulated data, three estimates of π (π? direct, r, and rrobust) were studied under varying conditions of sample size, distribution shape, and group mean difference. This study demonstrated that r and rrobust were biased estimates of π when data were nonnormal. We recommend that neither be used in estimating π unless data are normally distributed.  相似文献   

18.
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   

19.
The use of sample covariance matrices constructed with pairwise deletion for data missing completely at random (SPW) is addressed in a simulation study based on 3 sample sizes (n = 200, 500, 1,000) and 5 levels of missing data (%miss = 0, 1, 10, 25, and 50). Parameter estimates were unbiased, parameter variability was largely explicable in terms of the number of nonmissing cases, and no sample covariance matrices were nonpositive definite except when %miss was 50 and the sample size was 200. However, nominal χ2 test statistics (and, thus, fit indices based on χ2s) were substantially biased by %miss and its interaction with N. Corrected χ2s based on the minimum, mean, and maximum number of nonmissing cases per measured variables and cases per covariance term (NPC) reduced but did not eliminate the bias. Empirically derived power functions did substantially better but may not generalize to other situations. Whereas the minimum NPC (the default in the SPSS version of LISREL) is probably better than most simple alternatives in many applications, the problem of how to assess fit for models fit to SPWS has no simple solution; caution is recommended, and there is need for further research with more suitable methods for this problem.  相似文献   

20.
This article proposes 2 classes of ridge generalized least squares (GLS) procedures for structural equation modeling (SEM) with unknown population distributions. The weight matrix for the first class of ridge GLS is obtained by combining the sample fourth-order moment matrix with the identity matrix. The weight matrix for the second class is obtained by combining the sample fourth-order moment matrix with its diagonal matrix. Empirical results indicate that, with data from an unknown population distribution, parameter estimates by ridge GLS can be much more accurate than those by either GLS or normal-distribution-based maximum likelihood; and standard errors of the parameter estimates also become more accurate in predicting the empirical ones. Rescaled and adjusted statistics are proposed for overall model evaluation, and they also perform much better than the default statistic following from the GLS method. The use of the ridge GLS procedures is illustrated with a real data set.  相似文献   

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