Previous research has individually assessed parallel analysis and minimum average partial for factor retention in exploratory factor analysis using ordinal variables. The current study is a comprehensive simulation study including the manipulation of eight conditions (type of correlation matrix, sample size, number of variables per factor, number of factors, factor correlation, skewness, factor loadings, and number of response categories), and three types of retention methods (minimum average partial, parallel analysis, and empirical Kaiser criterion) resulting in a 2 × 2 × 2 × 2 × 2 × 3 × 3 × 4 × 5 design that totals to 5,760 condition combinations tested over 1,000 replications each. Results show that each retention method performed worse when utilizing polychoric correlation matrices. Moreover, minimum average partials are quite sensitive to factor loadings and overall perform poorly compared to parallel analysis and empirical Kaiser criterion. Empirical Kaiser criterion performed almost identical to parallel analysis in normally distributed data; however, performed much worse under highly skewed conditions. Based on these findings, it is recommended to use parallel analysis utilizing principal components analysis with a Pearson correlation matrix to determine the number of factors to retain when dealing with ordinal data.

<p>This study investigates the most important reasons behind the rst-year students' decision to select University of the Western Cape (UWC) as higher education institution.<br /> These reasons were organized into a few factors for easy interpretation. The data to be analyzed for this project is a subsection of the data collected during the orientation period of 2008. During the orientation week of 2008, the questionnaires were completed on a voluntary basis by new rst-year students. All questionnaires were anonymously completed and therefore the data does not contain any information that could be linked to any individual. For the purpose of this study, only the black African and coloured students were considered. The other racial groups were not analyzed due to too small sample sizes. Questionnaires with missing information on the reasons for selecting UWC were not&nbsp / nalyzed. We ended up with a sample of size 600. The data were statistically analyzed, using descriptive statistics, bivariate analyses, factor analysis, coefficient of congruence and bootstrap factor analysis. The results indicated that the most important reasons aecting students to choose UWC were identied as good academic reputation, family member's advice, UWC graduates are successful and UWC graduates get good jobs. The least important reasons were found to be not accepted anywhere, parents / family members graduated from UWC, recruited by UWC and wanted to study near to home. The results also indicated that there were significant differences among students according to population groups, parent's monthly income and grade 12 average. Factor analysis of 12 variables yielded three extracted factors upon which student decisions were based. Similarities of these three factors were tested, and a high similarity among demographic characteristics and grade 12 average were found. Additional analyses were conducted to measure the accuracy of factor analyses models constructed using Spearman and Polychoric correlation matrices. The results indicated that both correlation matrices were&nbsp / nbiased, with higher variance and higher loadings when the Polychoric correlation matrix was used to construct a factor analysis model for categorical data.</p>

<p>This study investigates the most important reasons behind the rst-year students' decision to select University of the Western Cape (UWC) as higher education institution.<br /> These reasons were organized into a few factors for easy interpretation. The data to be analyzed for this project is a subsection of the data collected during the orientation period of 2008. During the orientation week of 2008, the questionnaires were completed on a voluntary basis by new rst-year students. All questionnaires were anonymously completed and therefore the data does not contain any information that could be linked to any individual. For the purpose of this study, only the black African and coloured students were considered. The other racial groups were not analyzed due to too small sample sizes. Questionnaires with missing information on the reasons for selecting UWC were not&nbsp / nalyzed. We ended up with a sample of size 600. The data were statistically analyzed, using descriptive statistics, bivariate analyses, factor analysis, coefficient of congruence and bootstrap factor analysis. The results indicated that the most important reasons aecting students to choose UWC were identied as good academic reputation, family member's advice, UWC graduates are successful and UWC graduates get good jobs. The least important reasons were found to be not accepted anywhere, parents / family members graduated from UWC, recruited by UWC and wanted to study near to home. The results also indicated that there were significant differences among students according to population groups, parent's monthly income and grade 12 average. Factor analysis of 12 variables yielded three extracted factors upon which student decisions were based. Similarities of these three factors were tested, and a high similarity among demographic characteristics and grade 12 average were found. Additional analyses were conducted to measure the accuracy of factor analyses models constructed using Spearman and Polychoric correlation matrices. The results indicated that both correlation matrices were&nbsp / nbiased, with higher variance and higher loadings when the Polychoric correlation matrix was used to construct a factor analysis model for categorical data.</p>

Magister Scientiae - MSc / This study investigates the most important reasons behind the rst-year students' decision to select University of the Western Cape (UWC) as higher education institution. These reasons were organized into a few factors for easy interpretation. The data to be analyzed for this project is a subsection of the data collected during the orientation period of 2008. During the orientation week of 2008, the questionnaires were completed on a voluntary basis by new rst-year students. All questionnaires were anonymously completed and therefore the data does not contain any information that could be linked to any individual. For the purpose of this study, only the black African and coloured students were considered. The other racial groups were not analyzed due to too small sample sizes. Questionnaires with missing information on the reasons for selecting UWC were not nalyzed. We ended up with a sample of size 600. The data were statistically analyzed, using descriptive statistics, bivariate analyses, factor analysis, coefficient of congruence and bootstrap factor analysis. The results indicated that the most important reasons a ecting students to choose UWC were identi ed as good academic reputation, family member's advice, UWC graduates are successful and UWC graduates get good jobs. The least important reasons were found to be not accepted anywhere, parents / family members graduated from UWC, recruited by UWC and wanted to study near to home. The results also indicated that there were significant differences among students according to population groups, parent's monthly income and grade 12 average. Factor analysis of 12 variables yielded three extracted factors upon which student decisions were based. Similarities of these three factors were tested, and a high similarity among demographic characteristics and grade 12 average were found. Additional analyses were conducted to measure the accuracy of factor analyses models constructed using Spearman and Polychoric correlation matrices. The results indicated that both correlation matrices were&nbsp; nbiased, with higher variance and higher loadings when the Polychoric correlation matrix was used to construct a factor analysis model for categorical data. / South Africa

摘要 皮爾森相關係數通常作為描述區間尺度變數間相關性的參考指標，然而在社會科學領域中，由於資料多數以順序尺度的形式呈現，因此藉由傳統的皮爾森相關係數來描述順序尺度資料間的相關性通常會導致某種程度的誤差。儘管如此，以往的文獻多數傾向支持以等距離分數來取代順序尺度資料，並直接計算皮爾森相關係數。藉由模擬實驗的結果，我們發現這樣的作法並非在所有情況下都合理。 此外本研究中也對多序類相關係數進行探討。就表示順序變數間相關性的準確程度而言，多序類相關係數明顯優於利用等距離分數來計算皮爾森相關係數的方法；但若以操作上的便利程度而言，後者仍具有其優勢。 關鍵字：順序尺度、皮爾森相關係數、多序類相關係數。 / Abstract Pearson correlation coefficient is typically used to describe the correlation between two interval-scaled variables. In social science, however, most of the data are represented in ordinal-scale, and hence describing the correlation between two ordinal-scaled variables in terms of Pearson correlation coefficient would inevitably result in certain errors. Though the practice is deemed acceptable and generally supported in literatures, we found, through intensive simulations, that it should be executed with care. Polychoric correlation coefficient was also investigated. In order to describe the correlation between two ordinal-scaled variables, we found, in terms of the degree of accuracy, that Polychoric correlation coefficient is definitely better than Pearson correlation coefficient with equal-distance scores. Pearson correlation coefficient, on the other hands, is much easier to calculate, and should not be totally ignored. Key words：Ordinal-scale、Pearson correlation coefficient、Polychoric correlation coefficient。

A modelagem de equações estruturais é uma técnica estatística multivariada que permite analisar variáveis que não podem ser medidas diretamente, mas que podem ser estimadas através de indicadores. Dado o poder que esta técnica tem em acomodar diversas situações em um único modelo, sua aplicação vem crescendo nas diversas áreas do conhecimento. Diante disto, este trabalho teve por objetivo avaliar a incorporação de indicadores categóricos ordinais em modelos de equações estruturais, fazendo um resumo dos principais procedimentos teóricos e subjetivos presentes no processo de estimação de um modelo, avaliando as suposições violadas quando indicadores ordinais são utilizados para estimar variáveis latentes e criando diretrizes que devem ser seguidas para a correta estimação dos parâmetros do modelo. Mostramos que as correlações especiais (correlação tetracórica, correlação policórica, correlação biserial e correlação poliserial) são as melhores escolhas como medida de associação entre indicadores, que estimam com maior precisão a correlação entre duas variáveis, em comparação à correlação de Pearson, e que são robustas a desvios de simetria e curtose. Por fim aplicamos os conceitos apresentados ao longo deste estudo a dois modelos hipotéticos com o objetivo de avaliar as diferenças entre os parâmetros estimados quando um modelo é ajustado utilizando a matriz de correlações especiais em substituição à matriz de correlação de Pearson. / The structural equation modeling is a multivariate statistical technique that allows us to analyze variables that cant be measured directly but can be estimated through indicators. Given the power that this technique has to accommodate several situations in a single model, its application has increased in several areas of the knowledge. At first, this study aimed to evaluate the incorporation of ordinal categorical indicators in structural equation models, making a summary of the major theoretical and subjective procedures of estimating the present model, assessing the assumptions that are violated when ordinal indicators are used to estimate latent variables and creating guidelines to be followed to correct estimation of model parameters. We show that the special correlations (tetrachoric correlation, polychoric correlation, biserial correlation and poliserial correlation) are the best choices as a measure of association between indicators, that estimate more accurately the correlation between two variables, compared to Pearsons correlation, and that they are robust to deviations from symmetry and kurtosis. Finally, we apply the concepts presented in this study to two hypothetical models to evaluate the differences between the estimated parameters when a model is adjusted using the special correlation matrix substituting the Pearsons correlation matrix.

A modelagem de equações estruturais é uma técnica estatística multivariada que permite analisar variáveis que não podem ser medidas diretamente, mas que podem ser estimadas através de indicadores. Dado o poder que esta técnica tem em acomodar diversas situações em um único modelo, sua aplicação vem crescendo nas diversas áreas do conhecimento. Diante disto, este trabalho teve por objetivo avaliar a incorporação de indicadores categóricos ordinais em modelos de equações estruturais, fazendo um resumo dos principais procedimentos teóricos e subjetivos presentes no processo de estimação de um modelo, avaliando as suposições violadas quando indicadores ordinais são utilizados para estimar variáveis latentes e criando diretrizes que devem ser seguidas para a correta estimação dos parâmetros do modelo. Mostramos que as correlações especiais (correlação tetracórica, correlação policórica, correlação biserial e correlação poliserial) são as melhores escolhas como medida de associação entre indicadores, que estimam com maior precisão a correlação entre duas variáveis, em comparação à correlação de Pearson, e que são robustas a desvios de simetria e curtose. Por fim aplicamos os conceitos apresentados ao longo deste estudo a dois modelos hipotéticos com o objetivo de avaliar as diferenças entre os parâmetros estimados quando um modelo é ajustado utilizando a matriz de correlações especiais em substituição à matriz de correlação de Pearson. / The structural equation modeling is a multivariate statistical technique that allows us to analyze variables that cant be measured directly but can be estimated through indicators. Given the power that this technique has to accommodate several situations in a single model, its application has increased in several areas of the knowledge. At first, this study aimed to evaluate the incorporation of ordinal categorical indicators in structural equation models, making a summary of the major theoretical and subjective procedures of estimating the present model, assessing the assumptions that are violated when ordinal indicators are used to estimate latent variables and creating guidelines to be followed to correct estimation of model parameters. We show that the special correlations (tetrachoric correlation, polychoric correlation, biserial correlation and poliserial correlation) are the best choices as a measure of association between indicators, that estimate more accurately the correlation between two variables, compared to Pearsons correlation, and that they are robust to deviations from symmetry and kurtosis. Finally, we apply the concepts presented in this study to two hypothetical models to evaluate the differences between the estimated parameters when a model is adjusted using the special correlation matrix substituting the Pearsons correlation matrix.

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.