MARSSIM WRS One Analyte at a Time Design

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

Equations Used to Calculate Recommended Minimum Number of Samples

The number of samples is calculated using Eq. (1) (EPA 2000, p. 5-28 and Gilbert et al. 2001, p. 3.12). The value of \(P_{r}\) is calculated using Eq. (2) when the MQO option is not selected and Eq. (3) when the MQO option is selected (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}

\begin{equation} P_{r} = \Phi\Bigg(\dfrac{\Delta}{\sqrt{2}\sqrt{{s_{\text{sample}}}^2 + \dfrac{{s_{\text{analytical}}}^2}{r}}}\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).
MQO Specific:
\(s_{\text{sample}}\)	is the standard deviation due to the inherent variability in the sampling process alone, i.e., when the analysis error is zero.
\(s_{\text{analytical}}\)	is the standard deviation due to the inherent variability in the analysis process alone.
\(r\)	is the number of times an individual sample is analyzed.

Statistical Assumptions

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.

References:

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.