Wilcoxon Signed Rank Test

The Wilcoxon Signed Rank Test to compare the mean to a threshold as performed by VSP on the selected data values follows the procedure outlined in EPA's Data Quality Assessment: Statistical Methods for Practitioners (EPA. 2006b.)

 

A deviation for each observed measurement is calculated by subtracting the observation from the action level to produce \( n \) deviations \(d_i=AL-X_i\) . Any deviations of zero are discarded and the sample size is reduced accordingly. The observations are then assigned ranks, based on ordering the absolute deviations \(\vert d_i \vert \) from smallest to largest, with the average of the ranks being assigned to any ties. Each observation is also assigned a sign based on the sign of the corresponding deviation. The sum \( R \) of the ranks with a positive sign is computed and compared with the critical value \( w \), which for \(n\) > 20 is calculated using the following formula:

$$ w=\frac{n(n+1)}{4}+z_{1-\alpha}\sqrt{\frac{n(n+1)(2n+1)}{24}} $$

where \(z_{1-\alpha}\) is the (1- \(\alpha\)) percentile of the standard normal distribution.

 

The test statistic \(R\) is compared with the critical value \( w \) to determine whether there is enough evidence to reject the null hypothesis. If the null hypothesis is that the population mean is greater than or equal to the threshold (\(H_0:\mu \geq AL \)) , then the null hypothesis can be rejected if \(R > w \). If the null hypothesis is that the population mean is less than or equal to the threshold (\(H_0:\mu \leq AL \)), the null hypothesis is rejected if \(R < w \).

To perform the Wilcoxon Signed Rank test for 20 or fewer observations the procedure remains largely the same, except that the critical value \( w \) is obtained from Table A-6 in Appendix A of EPA's Data Quality Assessment: Statistical Methods for Practitioners (EPA. 2006b.) and the condition required to be satisfied to reject the null hypothesis \(H_0:\mu \leq AL \) is modified from \( R > w \) to \( R > \frac {n(n+1)}{2}-w \).

For an illustration of how to use the Wilcoxon Signed Rank test, please refer to the Compare Average to a Fixed Threshold section in chapter 3 of the VSP User's Guide.