Wilcoxon Rank-Sum

What is it? The Wilcoxon Rank-Sum is a method that determines whether two populations are statistically different from each other based on ranks rather than the original values of the measurements. In other words, it ranks all values to determine whether the values are or are not evenly distributed across both populations.

When is it used? This test is performed when 1) the samples are independent of each other and 2) are or are not normally distributed.

How does it work?

Wilcoxon Rank-Sum Example

We want to determine whether the concentration of Protein 1 in serum is significantly different between healthy and diseased patients. We first collect data for healthy and diseased patients (Figure 1).

  1. Wilcoxon Rank-Sum then ranks the values, and assigns the rank to the values (Figure 2). The average ranks from the groups are determined; these averages will be close if there is no difference between the groups.
  2. The rank mean of one group is compared to the overall rank mean to determine a test statistic called a z-score. If the groups are evenly distributed, then the z-score will be closer to 0. In this case, the z-score is 3.81, which is equal to a p-value < 0.001. (A p-value of ~0.05 is approximately equal to a z-score of 2.5.)

Is Protein 1 a potential biomarker of disease? Yes, the concentration of Protein 1 in healthy and diseased patients is very different from each other, which is indicated by a very low p-value.

Figure 1. Collected protein concentration data for healthy (blue) and diseased (green) patients.
Figure 2
Figure 2. Ranked protein concentrations from high to low collected for healthy (blue) and diseased (green) patients. The average concentration of Protein 1 in the healthy patient is clearly higher than the average concentration of Protein 1 in the diseased patient.

What does the data look like? Wilcoxon Rank-Sum produces a test statistic value (i.e., z-score), which is converted into a “p-value.” A p-value is the probability that the null hypothesis – that both populations are the same – is true. In other words, a lower p-value reflects a value that is more significantly different across populations. Biomarkers with significant differences between sample populations have p-values ≤ 0.05.

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