MARSSIM One Analyte at a Time Design

Background Information

The MARSSIM Sign test is a one-sample, nonparametric statistical test used to determine compliance with a release criterion (Action Level, i.e., DCGLw) when the radionuclide of concern is not present in background. The purpose of a MARSSIM sign test is to test a hypothesis involving the true mean or median of a population against an Action Level. This "One Analyte at a Time" sub-design is provided for backward compatibility and for use in Non MARSSIM-specific applications. The appropriate use of the sign 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. 5-33) when the MQO option is not selected. The number of samples is calculated using Equation (2) when the MQO option is selected (Gilbert et al. 2001, pp. 3.8 and 3.9).

\begin{equation} n = \frac{(z_{1-\alpha}+z_{1-\beta})^2}{4(\text{Sign} P-0.5)^2} \quad \mbox{where} \quad \text{Sign} P = \Phi\Big(\frac{\Delta}{s_{\text{Total}}}\Big) \end{equation}

\begin{equation} = \frac{(z_{1-\alpha}+z_{1-\beta})^2}{4(\text{Sign} P-0.5)^2} \quad \mbox{where} \quad \text{Sign} P = \Phi\Bigg(\frac{\Delta}{\sqrt{{s_{\text{sample}}}^2 + \frac{{s_{\text{analytical}}}^2}{r}}}\Bigg) \end{equation}

where:

\(n\)	is the recommended minimum sample size.
\(s_{\text{Total}}\)	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. The computed sign test statistic is normally distributed.

2. The variance estimate, \(s^2\), is reasonable and representative of the population being sampled.

3. The population values are not spatially or temporally correlated.

4. The sampling locations will be selected randomly.

The first three assumptions will be assessed in a post data collection analysis. The last assumptions 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.