The MARSSIM WRS (Wilcoxon Rank Sum) test is a two-sample test that compares the distribution of a set of measurements in a Survey Unit to that of a set of measurements in a Reference Area (i.e., background area). The MARSSIM WRS test is used to test whether the true median in a Survey Unit population is greater than the true median in a Reference Area population. The test compares medians of the two populations because the WRS is based on ranks rather than the measurements themselves. Note that if both the Survey Unit and Reference Area populations are symmetric, then the median and mean of each distribution are identical. In that special case the MARSSIM WRS test is comparing means. The assumption of symmetry and the appropriate use of the WRS test for final status surveys is discussed in Multi-Agency Radiation Survey and Site Investigation Manual (MARSSIM) (EPA 2000). This document is currently available at: https://www.epa.gov/radiation/multi-agency-radiation-survey-and-site-investigation-manual-marssim
This design assumes that you have more than one contaminant nuclide in the surface soil study area. To enter the nuclides of concern, use the MARSSIM button on the Analyte page. The nuclides will have standard DCGLw values which may need to be modified for your particular application.
If you are an expert in MARSSIM designs, you can choose to enter all your own design parameters including:
Confidence%: Minimum desired probability of concluding the site is dirty if the true sum-of-fractions exceed the DCGLw.
Beta%: Maximum desired probability of concluding the site is dirty if the true sum-of-fractions is less than the lower bound of the gray region.
Lower bound of the gray region (LBGR): A true sum-of-fractions value (below the DCGLw) above which you are willing to accept an increased risk of concluding the site is dirty.
Estimated standard deviation for the sum-of-fractions.
Estimated mean of the sum-of-fractions (in order to compute the power of the test).
As an alternative, you can have VSP assist you with the computation of the Sum of Fractions values. This approach will be emphasized here. Begin by entering an estimated mean and standard deviation for the survey unit and the reference area for each of the nuclides as illustrated below.
After entering all the values, VSP will calculate the Mean and Standard Deviation Sum-of-Fractions for you. VSP uses the following formulas:
\begin{equation} Mean_{\text{sof}} = \dfrac{Mean_\text{su1}-Mean_\text{ref1}}{DCGL_1} + \dfrac{Mean_\text{su2}-Mean_\text{ref2}}{DCGL_2} + etc.\end{equation}
\begin{equation} S_{\text{sof}} = \sqrt{\Bigg(\dfrac{\text{Max}[S_\text{su1},S_\text{ref1}]}{DCGL_1}\Bigg)^2 + \Bigg(\dfrac{\text{Max}[S_\text{su2}, S_\text{ref2}]}{DCGL_2}\Bigg)^2 + etc.}\end{equation}
where:
\(Mean_\text{su1}\) |
is the survey unit mean of radionuclide 1 |
\(Mean_\text{ref1}\) |
is the reference area mean of radionuclide 1 |
\(S_\text{su1}\) |
is the survey unit standard deviation for radionuclide 1 |
\(S_\text{ref1}\) |
is the reference area standard deviation for radionuclide 1 |
\(\text{Max}\) |
is the maximum of the two values |
For the example above, \(Mean_{\text{sof}} = \frac{\text{0.8 - 0.2}}{\text{2.1}} + \frac{\text{4 - 1}}{\text{11}} + \frac{\text{0.7 - 0.2}}{\text{1.7}} = \text{0.29 + 0.28 + 0.29} = \text{0.85}\)
and \(S_{\text{sof}} = \sqrt{(\frac{0.5}{2.1})^2 + (\frac{1.5}{11})^2 + (\frac{0.4}{1.7})^2 } = \sqrt{\text{0.2381}^2 + \text{0.1364}^2 + \text{0.2353}^2 } = \sqrt{0.1306} = 0.36\)
You will need to enter the remaining design parameters:
Confidence%: Minimum desired probability of concluding the site is dirty if the true sum-of-fractions exceed the DCGLw.
Beta%: Maximum desired probability of concluding the site is dirty if the true sum-of-fractions is less than the lower bound of the gray region.
Lower bound of the gray region (LBGR): A true sum-of-fractions value (below the DCGLw) above which you are willing to accept an increased risk of concluding the site is dirty.
A DCGLw of 1 is assumed for the unity rule calculations.
The number of samples is calculated using Equation (3) (EPA 2000, p. 5-28 and Gilbert et al. 2001, p. 3.12) and Equation (4) (Gilbert et al. 2001, p. 3.13).
\begin{equation} n+m = \dfrac{(z_{1-\alpha}+z_{1-\beta})^2}{3(P_{r}-0.5)^2} \end{equation}
\begin{equation} P_{r} = \Phi\Bigg(\dfrac{\Delta}{\sqrt{2}s_{\text{total}}}\Bigg) \end{equation}
where:
\(n+m\) |
is the sum of the minimum number of study-site and reference-area samples, assuming n = m. |
\(s_{\text{total}}\) |
is the estimated standard deviation due to both sampling and measurement variability. |
\(z_{1-\alpha}\) |
is the value of the standard normal distribution for which the proportion of the distribution to the left of \(z_{1-\alpha}\) is \(1-\alpha\). |
\(z_{1-\beta}\) |
is the value of the standard normal distribution for which the proportion of the distribution to the left of \(z_{1-\beta}\) is \(1-\beta\). |
\(\Delta\) |
is the width of the gray region. |
\(\alpha\) |
is the probability of rejecting the null hypothesis when the null hypothesis is true. |
\(\beta\) |
is the probability of not rejecting the null hypothesis when the null hypothesis is false. |
\({\Phi}_{\text{(x)}}\) |
is the probability that a standard normal variate takes on a value \(\le\) x (CDF). |
The assumptions associated with the formulas for computing the number of samples are:
1. although the population does not have to be normally distributed, the test statistic is approximately normally distributed,
2. the variances of the site and reference populations are equal,
3. the variance estimate, \(s^2\), is reasonable and representative of the populations being sampled,
4. the population values are not spatially or temporally correlated, and
5. the sampling locations will be selected randomly.
The first four assumptions will be assessed in a post data collection analysis. The last assumption is valid because the sample locations were selected using a random process.
EPA. 2000. Multi-Agency Radiation Survey and Site Investigation Manual (MARSSIM). NUREG-1575, Rev. 1, EPA 402-R-97-016, Rev.1, DOE/EH-0624, Rev. 1. Environmental Protection Agency, Office of Research and Development, Quality Assurance Division, Washington DC.
Gilbert, RO, JR Davidson, JE Wilson, BA Pulsipher. 2001. Visual Sample Plan (VSP) models and code verification. PNNL-13450, Pacific Northwest National Laboratory, Richland, Washington.
Lower Bound of the Gray Region (LBGR)
Estimated Standard Deviation (Sum of Fractions)
Estimated Mean (Sum of Fractions)