The Robustness of O'Brien's r Transformation to Non-Normality

• Main Author: Gordon, Carol J. (Carol Jean)
• Other Authors: McCallon, Earl L.
• Format: Others
• Language: English
• Published: North Texas State University 1985
• Subjects: r transformation; non-normal distributions; Analysis of variance.; Robust statistics.
• Online Access: https://digital.library.unt.edu/ark:/67531/metadc332002/

A Monte Carlo simulation technique was employed in this study to determine if the r transformation, a test of homogeneity of variance, affords adequate protection against Type I error over a range of equal sample sizes and number of groups when samples are obtained from normal and non-normal distributions. Additionally, this study sought to determine if the r transformation is more robust than Bartlett's chi-square to deviations from normality. Four populations were generated representing normal, uniform, symmetric leptokurtic, and skewed leptokurtic distributions. For each sample size (6, 12, 24, 48), number of groups (3, 4, 5, 7), and population distribution condition, the r transformation and Bartlett's chi-square were calculated. This procedure was replicated 1,000 times; the actual significance level was determined and compared to the nominal significance level of .05. On the basis of the analysis of the generated data, the following conclusions are drawn. First, the r transformation is generally robust to violations of normality when the size of the samples tested is twelve or larger. Second, in the instances where a significant difference occurred between the actual and nominal significance levels, the r transformation produced (a) conservative Type I error rates if the kurtosis of the parent population were 1.414 or less and (b) an inflated Type I error rate when the index of kurtosis was three. Third, the r transformation should not be used if sample size is smaller than twelve. Fourth, the r transformation is more robust in all instances to non-normality, but the Bartlett test is superior in controlling Type I error when samples are from a population with a normal distribution. In light of these conclusions, the r transformation may be used as a general utility test of homogeneity of variances when either the distribution of the parent population is unknown or is known to have a non-normal distribution, and the size of the equal samples is at least twelve.