Skew-Dependent Detection of Change of Central Tendency in Hydrologic Data

Changes in central tendency in hydrologic data are commonly evaluated with nonparametric statistics because the assumptions of parametric alternatives are not met. The Wilcoxon signed-ranks (WSR) test can be used to infer changes in central tendency in paired comparisons. Standard critical values for the WSR test assume that the data samples are drawn from distributions with equal skew, such that the differences are symmetrically distributed. When the data are drawn from distributions with unequal skews, the WSR test with the zero-skew critical values can lead to incorrect or inaccurate research decisions. This work provides critical values that can be applied to paired data with unequal skews. Skew-dependent critical values were developed by simulation. Equations are presented enabling the estimation of critical values as a function of sample size, skew, and type I error for both positively and negatively skewed data. The new critical values are applied to compare the annual maximum discharge and runoff depths measured at two small agricultural watersheds. The results show that the skew-dependent critical values provide decisions with different statistical significance than those decisions made using the traditional critical values that assume zero skew. The implication is that more rational decisions will be made, using the skew-dependent critical values, when the WSR test is applied to data characterized by unequal skews.