Bartels Rank Test

The Bartels rank test is both a nonparametric test and a rank test for serial correlation in an ordered sample. If is the statistical rank of the th observation among observations, its rank von Neumann ratio is

The numerator is the sum of squared differences between successive ranks and therefore measures local change along the given order. The denominator is the total squared deviation from the mean rank and measures overall statistical dispersion among the ranks. Thus RVN is the rank version of the ordinary von Neumann ratio, which uses the observations and their sample mean in the two sums. When there are no ties, the denominator is .

Under the null hypothesis that the observations occur in random order, significance can be assessed from the exact or asymptotic null distribution of RVN, or by a permutation test. Small values indicate that neighboring ranks tend to be similar, as occurs with positive serial correlation or a trend. Large values indicate alternating high and low ranks.