The concept of measurement invariance is essential in ensuring psychological and educational tests are interpreted consistently across diverse groups. This dissertation investigated the practical challenges associated with measurement invariance, specifically on how measurement invariance delta fit indexes are affected by non-normal data. Non-normal data distributions are common in real-world scenarios, yet many statistical methods and measurement invariance delta fit indexes are based on the assumption of normally distributed data. This raises concerns about the accuracy and reliability of conclusions drawn from such analyses. The primary objective of this research is to examine how commonly used delta fit indexes of measurement invariance respond under conditions of non-normality. The present research was built upon Cao and Liang (2022a)’s study to test the sensitivities of a series of delta fit indexes, and further scrutinizes the role of non-normal data distributions. A series of simulation studies was conducted, where data sets with varying degrees of skewness and kurtosis were generated. These data sets were then examined by multi-group confirmatory factor analysis (MGCFA) using the Satorra-Bentler scaled chi-square difference test, a method specifically designed to adjust for non-normality. The performance of delta fit indexes such as the Delta Comparative Fit Index (∆CFI), Delta Standardized Root Mean Square residual (∆SRMR) and Delta Root Mean Square Error of Approximation (∆RMSEA) were assessed. These findings have significant implications for professionals and scholars in psychology and education. They provide constructive information related to key aspects of research and practice in these fields related to measurement, contributing to the broader discussion on measurement invariance by highlighting challenges and offering solutions for assessing model fit in non-normal data scenarios.

The present paper develops a procedure based on multivariate copulas for simulating multivariate non-normal data that satisfies a pre-specified covariance matrix. The covariance matrix used, can comply with a specific moment structure form (e.g., a factor analysis or a general SEM model). So the method is particularly useful for Monte Carlo evaluation of SEM models in the context of non-normal data. The new procedure for non-normal data simulation is theoretically described and also implemented on the widely used R environment. The quality of the method is assessed by performing Monte Carlo simulations. Within this context a one-sample test on the observed VC-matrix is involved. This test is robust against normality violations. This test is defined through a particular SEM setting. Finally, an example for Monte Carlo evaluation of SEM modeling of non-normal data using this method is presented. (author's abstract) / Series: Research Report Series / Department of Statistics and Mathematics

This thesis, which consists of five papers, is concerned with various aspects of confirmatory factor analysis (CFA) of ordinal variables and the generation of non-normal data. The first paper studies the performances of different estimation methods used in CFA when ordinal data are encountered.  To take ordinality into account the four estimation methods, i.e., maximum likelihood (ML), unweighted least squares, diagonally weighted least squares, and weighted least squares (WLS), are used in combination with polychoric correlations. The effect of model sizes and number of categories on the parameter estimates, their standard errors, and the common chi-square measure of fit when the models are both correct and misspecified are examined. The second paper focuses on the appropriate estimator of the polychoric correlation when fitting a CFA model. A non-parametric polychoric correlation coefficient based on the discrete version of Spearman's rank correlation is proposed to contend with the situation of non-normal underlying distributions. The simulation study shows the benefits of using the non-parametric polychoric correlation under conditions of non-normality. The third paper raises the issue of simultaneous factor analysis. We study the effect of pooling multi-group data on the estimation of factor loadings. Given the same factor loadings but different factor means and correlations, we investigate how much information is lost by pooling the groups together and only estimating the combined data set using the WLS method. The parameter estimates and their standard errors are compared with results obtained by multi-group analysis using ML. The fourth paper uses a Monte Carlo simulation to assess the reliability of the Fleishman's power method under various conditions of skewness, kurtosis, and sample size. Based on the generated non-normal samples, the power of D'Agostino's (1986) normality test is studied. The fifth paper extends the evaluation of algorithms to the generation of multivariate non-normal data.  Apart from the requirement of generating reliable skewness and kurtosis, the generated data also need to possess the desired correlation matrices.  Four algorithms are investigated in terms of simplicity, generality, and reliability of the technique.