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Date Issued. Date of Defense. July 5, Yet, unlike other robust corrections implemented in ML estimation, MLR implemented in M plus has not been systematically evaluated by means of a Monte Carlo study in previous studies, although its robust correction is similar, but not always equivalent, to other robust ML corrections e.
On the other hand, WLSMV has been specifically proposed to deal with ordinal data the default setting in M plus , mainly because it makes no distributional assumptions about the observed variables. When it comes to a CFA model with ordinal data, applied researchers tend to choose one or another estimator to perform data analysis in M plus. Some researchers prefer treating ordinal variables with more than five alternatives as if they were approximately continuous variables, and in turn they perform MLR in data analysis to adjust the violation of nonnormality, whereas others highly recommend using WLSMV as long as the observed variables are ordinally scaled.
An examination of the two estimators under varying empirical conditions is needed, since they are frequently used in research practice. However, the impact of the number of categories on parameter variability and adjusted standard error estimates has not yet been examined using MLR and WLSMV with larger numbers of categories e. The number of categories affects not only the distribution of the observed ordinal variables, but also the possibility of treating ordinal variables as approximately continuous.
Fourth, MLR was developed to permit the parameter estimation from nonnormality of continuous observed variables, whereas WLSMV has been implemented in CFA models with nonnormal observed data due to the categorical nature of measurement i.
Therefore, the latent distribution was manipulated by varying skewness and kurtosis in the study. It could be expected that the effect of the latent normality distribution violation would more likely be salient for the performance of WLSMV than for that of MLR, holding other experimental conditions equal.
Moreover, nonnormality, in the form of asymmetry observed in psychometric measurements, has not been uncommon in applied studies. Therefore, in order to be more realistic from an applied standpoint, this study also included asymmetric observed distributions of ordinal variables in the simulation design.
Finally, this study was designed to examine the effect of sample size on the parameter estimates produced while utilizing the two estimators, because researchers have noted that a desirable sample size is known to be an important factor in CFA models.
A small sample may cause inaccurate parameter estimates and unstable standard errors, and may result in nonconvergence and improper solutions, as well. A Monte Carlo simulation study was carried out to compare the effects of different configurations of latent response distributions, numbers of categories, and sample sizes on model parameter estimates, standard errors, and chi-square test statistics in a correlated two-factor model.
Marsh, Hau, Balla, and Grayson concluded that the accuracy of parameter estimates appeared to be optimal when the number of observed variables per factor was four, and marginally improved as the number of observed variables increased. Therefore, each factor was measured by five ordinal observed indicators in the study. For the first estimation procedure, ordinal observed indicators were treated as if they were approximately continuous variables in the data analysis.
The parameter estimates, standard errors, and chi-square statistics were obtained using MLR. The data analysis for MLR was based on a sample-based covariance matrix. Regarding the second estimation procedure, ordinal observed indicators were specified as categorical variables in the data analysis. A polychoric correlation matrix and the asymptotic covariance matrix of the polychoric correlation and threshold estimates were used in WLSMV to obtain the parameter estimates, standard errors, and chi-square statistics.
Reported standardized factor loadings range from. For the sake of facilitating interpretation, each factor loading was therefore held constant at. The interfactor correlation was set to. The factor variances were all set equal to 1 in the population.
The desired levels of skewness and kurtosis for the two latent distributions were simply specified in the EQS program. For the slightly nonnormal latent distribution, the asymmetric distributions of observed variables with skewness ranged from. The response probabilities of the ordinal observed indicators used in the study are displayed in Fig.
Note that Fig. In order to explore the impact of categorization, this study extended previous research by introducing two larger numbers of categories i. Four, six, eight, and ten categories were generated for each ordinal indicator within both the slightly and moderately nonnormal latent distributions. A sufficient sample size is highly associated with the amount of model complexity e.
A correlated two-factor CFA model with ten observed indicators in this study required a minimum sample size of Jackson, Gillaspy, and Purc-Stephenson systematically reviewed studies in CFA from to and reported that the median sample size was Five hundred data sets were generated per experimental condition using EQS 6 Bentler, , resulting in a total of 12, data sets.
Four outcome variables were examined in this study: rates of improper solutions or nonconvergence, parameter estimates i. The rate of improper solutions or nonconvergence for each experimental condition was defined as the proportions of replications for which the proposed model had a nonconverged solution or a solution that converged but had estimated interfactor correlations greater than 1 or negative residual variances under the M plus default setting.
The model rejection rates associated with the chi-square test statistic were calculated at an alpha level of. The difference between the estimated and true values of each parameter i. Since bias is highly dependent on the magnitude of the true parameter value, and a great number of parameter estimates and standard errors were involved in each experimental condition, ARB and ARMSE were calculated.
The root mean squared error can be regarded as a measure of the overall estimation quality, since it accounts for both the amount of bias and the sampling variability of estimates; this value was also averaged across replications i. A smaller ARMSE value is suggested as being favorable, reflecting better overall quality of the estimates.
Confidence interval coverage was determined as the percentage of confidence intervals containing the true parameter. A lower rate of coverage i. A higher rate of rejection i.
Inspecting Table 1 , the factor loadings were, on average, underestimated by MLR. In particular, they were substantially underestimated when the ordinal data had only four response categories. The negative bias was an inverse function of the number of categories. Thus, the factor loadings in WLSMV can be considered essentially unbiased, especially when the latent distribution is only slightly nonnormal.
The positive bias did not vary as a function of the number of categories. Regarding the overall quality of the estimated factor loadings, ARMSE varied as an inverse function of the sample size and the number of categories for both estimation methods. It is of particular interest that WLSMV was better than MLR in the overall quality of factor loading estimates from four to ten categories across different sample sizes, even when ordinal observed data were generated from a moderately nonnormal latent distribution.
Interfactor correlations were, on average, trivially biased either positively or negatively for both estimators. With respect to the overall quality of the estimated interfactor correlations, ARMSE varied as an inverse function of the sample size and number of categories for both estimation methods. This implies that for a moderately nonnormal latent distribution, MLR demonstrates better performance than WLSMV for estimating the interfactor correlations.
The standard errors exhibited, on average, a slight bias either positive or negative for both estimators. It is noteworthy that a moderately negative bias was produced by WLSMV when the sample size was small i. Generally, the performance of MLR surpasses that of WLSMV for estimating standard errors when the sample size is small and latent distributions are nonnormal.
Regarding the overall quality of the estimated standard errors of factor loadings, ARMSE was an inverse function of sample size for both estimation methods. The standard errors demonstrated, on average, a slight bias in either a positive or a negative direction with MLR, whereas they were moderately underestimated by that WLSMV estimator on average for a sample smaller than With respect to the overall quality of the estimated standard errors of interfactor correlations, ARMSE was an inverse function of sample size for both estimation methods.
The average coverage for factor loadings with MLR was adversely affected by the size of ARB, in particular for those indictors with four response categories. It is noteworthy that an increase in sample size appeared to exacerbate the problem of lower average coverage, partly because of the comparably smaller standard error estimates in larger sample sizes.
Furthermore, as the level of latent nonnormality increased, the average coverage decreased in the WLSMV estimation. This is thought to be due mainly to the relatively large bias with moderately nonnormal latent distributions. Moreover, the lower rate of coverage indicated lower power to capture the true factor loadings using MLR. With moderately nonnormal latent distributions, WLSMV, however, is not superior to MLR in the recovery of true factor loadings, except for the conditions with four response categories.
As is shown in Table 5 , the average coverage for interfactor correlations in MLR seemed to be stable and satisfactory across experimental conditions. Table 6 gives the chi-square rejection rates for the two estimators. The boldface numbers in the table indicate unacceptable rejection rates, implying that acceptable difference rates in the table are within the range [2.
In these exceptional conditions, the proposed model seemed to be over-rejected, producing slightly inflated Type I error rates. Generally speaking, both the corrected chi-square test statistics performed well in controlling for Type I error rates when the sample size was greater than This study was designed to compare the performance of MLR and WLSMV with regard to parameter estimates, standard errors, and chi-square test statistics in a correlated two-factor model with ordinal observed indicators under different experimental configurations of latent response distributions, numbers of categories, and sample sizes.
Several general findings are discussed, as follows. This study also revealed that the factor loadings obtained by WLSMV were more precise and accurate than those obtained by MLR when the latent normality assumption was moderately violated. Generally speaking, WLSMV was preferable to MLR across most of the conditions observed in this study, given its properties of being less biased and having small sampling variation in estimating factor loadings.
This implies that the polychoric correlation estimates may demonstrate robustness against violations of the latent normality assumption in estimating factor loadings rather than in interfactor correlations. This observation is consistent with that of Coenders, Satorra, and Saris , who concluded that Pearson product-moment correlations between ordinal observed indicators using ML perform badly in factor loading estimates due to the categorical nature of measurement. However, such lower measurement quality estimates of ordinal variables can lead to relatively unbiased point estimates of factor relationships.
The overall quality of parameter estimates i. It is interesting that the overall quality of the standard error estimates was quite sensitive to sample size, regardless of the number of categories and the level of the latent normality assumption violation.
Researchers need to exercise caution in the evaluation of model fits under a small sample size, and they should take into account the supplemental fit indices e. An assessment of supplemental fit indices was not included in the present study, mainly because 1 these fit indices, unlike chi-square statistics, do not follow a known sampling distribution and 2 they do not have coherent cutoff values for fit indices in applications, to use to evaluate their performance.
However, one can expect that RMSEA, for example, would exhibit adequate power in the model evaluation when a model had no specification error like the CFA model in this study. Future research assessing the effects of various fit indices is still suggested, specifically addressing the question of which fit indices are reliable and robust to detect model misspecification. Finally, as we may be aware, there are multitudinous combinations to manipulate in a simulation study, but researchers can only focus on some factors of particular interest to make the research design feasible.
Therefore, this study shares the same limitation as all simulation studies, in that the results cannot be generalized beyond the experimental conditions investigated in the study.
Although previous simulation studies have suggested that the estimation of ordinal CFA models is robust to slight model misspecification, a natural extension of this study would consider different levels of misspecified models e. It is worthwhile to point out that each estimator has its advantages and disadvantages, as was discussed above. This study does provide conclusive evidence that WLSMV performs uniformly better than MLR in factor loading estimates across all experimental conditions i.
Likewise, MLR has its unique strengths—for instance, generally less biased standard error estimates and good recovery of the population interfactor correlations. Beauducel, A. On the performance of maximum likelihood versus means and variance adjusted weighted least squares estimation in CFA.
Structural Equation Modeling, 13, — Article Google Scholar. Bentler, P. EQS 6 structural equations program manual. Google Scholar. Bollen, K. Structural equations with latent variables. New York, NY: Wiley. ML for categorical data in SEM hasn't been around for all that long. Add a comment. Active Oldest Votes. Improve this answer.
Aleksandr Blekh Aleksandr Blekh 7, 2 2 gold badges 27 27 silver badges 93 93 bronze badges. If data are binary, their mean is a proportion and the resulting model estimates proportion differences. However, other probability models do not require normal error assumptions, like a logit or quasilogit model. Are you saying that SEM will not use a mean variance relationship to improve estimation with categorical outcomes?
Erik Erik 91 1 1 silver badge 10 10 bronze badges. Sign up or log in Sign up using Google. Sign up using Facebook.
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