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Comparing Tests of Multinormality under Sparse Data Conditions a Monte Carlo Study
"... Mardia’s tests of multivariate skewness and kurtosis and von Eye and Gardiner’s and von Eye and Bogat’s sector and overall tests of multinormality are compared under sparse data conditions. A Monte Carlo study is reported in which five factors were varied: sample size, number of variables, type of d ..."
Abstract

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Mardia’s tests of multivariate skewness and kurtosis and von Eye and Gardiner’s and von Eye and Bogat’s sector and overall tests of multinormality are compared under sparse data conditions. A Monte Carlo study is reported in which five factors were varied: sample size, number of variables, type of distribution (normal, uniform, logtransformed, inverse Laplacetransformed, and cube roottransformed), magnitude of correlation among variables, and the number of segments used for the ÷tests. Results suggest that, even under sparse data conditions, (1) the Mardia tests are2 differentially sensitive to the violations they were designed to detect; (2) the new sector and overall tests are sensitive to all violations included in the simulations; (3) the effects of the small samples can be seen in an increased random component of the data structure; and (4) although the overall and the sector tests are still sensitive to a wide range of violations, the test statistics are not always distributed as ÷ , due to the well known inflation of X. These results replicate, in part,2 2 the results of von Eye’s (2005) study, in which larger samples had been used. However, they also show that sufficiently large samples are needed for valid statistical decisions about multinormality. Comparing Tests of Multinormality under Sparse Data Conditions a Monte Carlo Study
Characteristics of measures of directional dependence
"... Recent results (Dodge & Rousson, 2000; von Eye & DeShon, 2008) show that, in the context of linear models, the response variable will always have less skew than the explanatory variable. This applies accordingly to the kurtosis of the two variables. These facts can be used to determine the d ..."
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Recent results (Dodge & Rousson, 2000; von Eye & DeShon, 2008) show that, in the context of linear models, the response variable will always have less skew than the explanatory variable. This applies accordingly to the kurtosis of the two variables. These facts can be used to determine the direction of dependence. Specifically, using third and fourth order moments, and information concerning the deviation of variables from normality, it can be ascertained which of two variables is the response and which is the explanatory variable. In this article, the ratio of two skewness measures, the ratio of two kurtosis measures, and one omnibus test of deviation from normality are examined. Simulation studies are reported that allow one to answer the question whether, these measures (1) are sensitive to various data distributions, (2) sample size, and (3) a simple correlation structure. Results suggest that all three measures are highly sensitive to these factors. The ratio of two kurtosis measures is sensitive in particular to the correlation structure. It is concluded that (1) directional dependence can be based on various types of deviation from normality, (2) measures that respond to deviations based on skewness and kurtosis have characteristics that make them prime candidates for determining directional dependence, and (3) each of the measures proposed thus far
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"... Multinormal distributions are symmetric. The degree of deviations from axial symmetry can be assessed using the well known Bowker test. A recently proposed test (von Eye & Bogat, 2004; von Eye & Gardiner, 2004) is based on comparing the observed frequencies in sectors of the multivariate sp ..."
Abstract
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Multinormal distributions are symmetric. The degree of deviations from axial symmetry can be assessed using the well known Bowker test. A recently proposed test (von Eye & Bogat, 2004; von Eye & Gardiner, 2004) is based on comparing the observed frequencies in sectors of the multivariate space with the corresponding expected frequencies that were estimated based on multinormality. Because this test is an omnibus test of multinormality, it should also be sensitive to deviations from axial symmetry. In this article, we describe the results of simulations that were performed on four types of bivariate distributions: normal, uniform, inverse Laplacetransformed, and cuberoot transformed. As expected, the Bowker test showed that inverse Laplacetransformed distributions are likely to show deviations from axial symmetry. None of the other distributions was asymmetric. The new omnibus test of multinormality exhibited 100 % sensitivity to violations of axial symmetry, but was also sensitive to elevated skewness and kurtosis. Thus, it also flagged the uniform and the cube roottransformed distributions as deviating from multinormality. Results also show that the Bowker test is sensitive only to violations of axial symmetry.