ANOVA on ranks

Context:
- It was designed for when a statistical population is not normally distributed.
See: ANOVA, Rank Transformation, Normal distribution, Nonparametric Statistical Model.

References

(Wikipedia, 2016) ⇒ https://www.wikiwand.com/en/ANOVA_on_ranks Retrieved 2016-07-31
- In statistics, one purpose for the analysis of variance (ANOVA) is to analyze differences in means between groups. The test statistic, F, assumes independence of observations, homogeneous variances, and population normality. ANOVA on ranks is a statistic designed for situations when the normality assumption has been violated. (...)ANOVA on ranks means that a standard analysis of variance is calculated on the rank-transformed data. Conducting factorial ANOVA on the ranks of original scores has also been suggested. However, Monte Carlo studies and subsequent asymptotic studies found that the rank transformation is inappropriate for testing interaction effects in a 4x3 and a 2x2x2 factorial design. As the number of effects (i.e., main, interaction) become non-null, and as the magnitude of the non-null effects increase, there is an increase in Type I error, resulting in a complete failure of the statistic with as high as a 100% probability of making a false positive decision. Similarly, it was found that the rank transformation increasingly fails in the two dependent samples layout as the correlation between pretest and posttest scores increase. It was also discovered that the Type I error rate problem was exacerbated in the context of Analysis of Covariance, particularly as the correlation between the covariate and the dependent variable increased.

(Conover & Iman, 1981) ⇒ Conover, W. J., & Iman, R. L. (1981). Rank transformations as a bridge between parametric and nonparametric statistics. The American Statistician, 35(3), 124-129. DOI: http://dx.doi.org/10.1080/00031305.1981.10479327
- Many of the more useful and powerful nonparametric procedures may be presented in a unified manner by treating them as rank transformation procedures. Rank transformation procedures are ones in which the usual parametric procedure is applied to the ranks of the data instead of to the data themselves. This technique should be viewed as a useful tool for developing nonparametric procedures to solve new problems