Partitioning Anderson's statistic for tied data

Anderson (Biometrics 15 (1959) 582) proposed a χ²-type statistic for the nonparametric analysis of a randomized blocks design with no ties in the data. In this paper, we propose an Anderson statistic that allows for ties in the data. We show that the asymptotic distribution of the statistic under the null hypothesis of no treatment effect is a χ² distribution. Under weak assumptions on the tie structure it is shown that the degrees of freedom for the asymptotic distribution is unchanged compared to the untied case. An extended analysis based on a partition of the statistic into independent components is suggested. The first component is shown to equal the Friedman rank statistic corrected for ties. The subsequent components allow for the detection of dispersion effects, higher order effects and differences in distribution. A simulation study is given and the new analysis is applied to a sensory evaluation data set.