The rank transform procedure is often used in the ANOVA when observations are not consistent with normality. The data are ranked and the ANOVA is applied to the ranked data. Often the rank residuals will be consistent with normality and a valid analysis results even though the residuals of the raw data were not consistent with normality. Here we observe that the rank transform procedure is equivalent to applying the intended ANOVA analysis to what we call scaled first order orthonormal polynomials. Using higher order scaled orthonormal polynomials extends the analysis to higher order effects, detecting, roughly, dispersion, skewness etc. differences between treatment ranks. We discuss using unscaled orthonormal polynomials.

AUTHORS: J.C.W. Rayner and D.J. Best