This study investigated the item parameter recovery of two methods of factor analysis. The methods researched were a traditional factor analysis of tetrachoric correlation coefficients and an IRT approach to factor analysis which utilizes marginal maximum likelihood estimation using an EM algorithm (MMLE-EM). Dichotomous item response data was generated under the 2-parameter normal ogive model (2PNOM) using PARDSIM software. Examinee abilities were sampled from both the standard normal and uniform distributions. True item discrimination, a, was normal with a mean of .75 and a standard deviation of .10. True b, item difficulty, was specified as uniform [-2, 2]. The two distributions of abilities were completely crossed with three test lengths (n= 30, 60, and 100) and three sample sizes (N = 50, 500, and 1000). Each of the 18 conditions was replicated 5 times, resulting in 90 datasets. PRELIS software was used to conduct a traditional factor analysis on the tetrachoric correlations. The IRT approach to factor analysis was conducted using BILOG 3 software. Parameter recovery was evaluated in terms of root mean square error, average signed bias, and Pearson correlations between estimated and true item parameters. ANOVAs were conducted to identify systematic differences in error indices. Based on many of the indices, it appears the IRT approach to factor analysis recovers item parameters better than the traditional approach studied. Future research should compare other methods of factor analysis to MMLE-EM under various non-normal distributions of abilities.

The odds ratio is one of the most widely used measures of association for 2 × 2 tables. A generalized Yule coefficient transforms the odds ratio into a correlation-like scale with a range from -1 to 1. Yule's Y, Yule's Q, Digby's H, and a new coefficient are special cases of a generalized Yule coefficient. The new coefficient is shown to be similar in value to the phi coefficient. A confidence interval and sample size formula for a generalized Yule coefficient are proposed. The proposed confidence interval is shown to perform much better than the Wald intervals that are implemented in statistical packages.

The tetrachoric correlation describes the linear relation between two continuous variables that have each been measured on a dichotomous scale. The treatment of the point estimate, standard error, interval estimate, and sample size requirement for the tetrachoric correlation is cursory and incomplete in modern psychometric and behavioral statistics texts. A new and simple method of accurately approximating the tetrachoric correlation is introduced. The tetrachoric approximation is then used to derive a simple standard error, confidence interval, and sample size planning formula. The new confidence interval is shown to perform far better than the confidence interval computed by SAS. A method to improve the SAS confidence interval is proposed. All of the new results are computationally simple and are ideally suited for textbook and classroom presentations.

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 article explains what we mean by the unidimensionality of a measurement instrument, therefore we present some definitions and  theoretical contributions about this subject. Factor analysis is proposed as one of the many methods for assessing the unidimensionality of a measurement instrument. The use of Pearson correlations matrices on item-level factor analysis is identified as an important problem. Those correlations are problematic because items didn’t carry out the necessary assumptions in order to apply the Pearson correlation: interval-level measurement and normal distribution of the variable. As an alternative we propose and exemplify the use of tetrachoric and polychoric correlations. / Este artículo explica qué es lo que implica la unidimensionalidad de un instrumento de medición. Para ello se presentan algunas definiciones y aportes teóricos sobre el tema. Luego, el análisis factorial es propuesto como uno de los métodos para evaluar la dimensionalidad de un instrumento de medición. Se señala como un problema importante el uso de las matrices de correlaciones de Pearson en los análisis factoriales a nivel de ítems. Estas correlaciones son problemáticas porque los ítems no cumplen con los supuestos necesarios para aplicar la correlación de Pearson: nivel de medición de intervalo y distribución normal de la variable. Como alternativa se postula y ejemplifica el uso de las correlaciones tetracóricas y policóricas.