Investigation of bootstrap estimates of the parameters, their standard errors, and associated confidence intervals of structural equation models with ordered categorical variables
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This study investigates the performance of the bootstrap when it is applied to structural equation models with ordered categorical variables. The study focuses on the parameter estimates, their standard errors and the coverage rates of the associated bootstrap confidence intervals. Structural equation models are used widely in many disciplines and often the data analyzed involve ordered categorical variables. The performance of the bootstrap has been investigated through simulation, and it is also compared with the Maximum Likelihood estimator applied on both polychoric correlation matrices and Pearson's product moment correlation matrices. The bootstrap samples are generated randomly and transformed, so that they preserve the covariance structure of the model. Then the polychoric correlation matrix is computed and analyzed for each sample. The study involves three different models, and for each model different sample sizes have been analyzed. One of the models that has been analyzed is one that Muthen and Kaplan used in their research to investigate the performance of the Categorical Variable Methodology (CVM) estimator, so direct comparisons between the two methods have been made. The bootstrap compares well with the CVM estimator. The results of this research indicate that the bootstrap pro\ ides correct standard errors that are larger than the standard errors obtained from the Maximum Likelihood estimator when it is applied on the Pearson's product moment correlation matrices. The coverage rates of the bootstrap confidence intervals have also been investigated, using two methods: the percentile method and the bootstrap-t method. The results are not very encouraging, especially for the bootstrap-t method, since the coverage rates are in some cases far away from the prespecified confidence level. The percentile method seems to perform better than the bootstrap-t method with regard to coverage rates, though it presents problems also. The performance of the bootstrap is affected by the sample size, the complexity of the model and the parameter values. Overall, the bootstrap performs rather adequately and could provide a valid alternative to other estimation methods for structural equation models if the researchers are cautious on its application.