Wilcoxon Signed Ranks Test Design

Background Information

The purpose of a Wilcoxon signed ranks (one-sample) test is to test a hypothesis involving the true mean or median of a population against an Action Level. Please consult EPA's guidance document, Guidance for the Data Quality Objectives Process (EPA 2006a), to put this test in the context of environmental decision-making.

Note that using this test for the mean assumes the distribution of the target population is symmetrical. Before deciding to develop a sampling plan based on using the Wilcoxon signed ranks test, consider the assumptions and limitations involved. For a discussion of these assumptions, limitations, and for the details of the test, please consult EPA's Guidance for Data Quality Assessment Practical Methods for Data Analysis (EPA 2006). This document, as well as the DQO guidance document, is currently available at: http://www.epa.gov/quality/qa_docs.html.

Equations Used to Calculate Recommended Minimum Number of Samples

The number of samples is calculated using Eq. (1) (EPA 2000b, p. 3-12) 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, p. 3-6).

\begin{equation}n = 1.16\Bigg[\frac{s_{\text{Total}}^2(z_{1-\alpha} + z_{1-\beta})^2}{\Delta^2}+0.5z_{1-\alpha}^2\Bigg]\end{equation}

\begin{equation}n = 1.16\Bigg[\frac{\Big(s_{\text{Sample}}^2 + \frac{s_{\text{Analytical}}^2}{r}\Big)(z_{1-\alpha} + z_{1-\beta})^2}{\Delta^2} + 0.5Z_{1-\alpha}^2\Bigg]\end{equation}

where:

\(n\)	is the recommended minimum sample size.
\(s_{\text{Total}}\)	is the estimated standard deviation due to both sampling and analytical 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.

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 data originate from a symmetric (but not necessarily normal) population,

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, and

4. the sample locations will be selected randomly

The first three 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.

Limitations for the Wilcoxon Test

1. The Wilcoxon test may produce misleading results if many measurements are the same value. When many values are the same, their relative ranks are the same, and this has the effect of diluting the test.

2. If the data is approximately symmetric the test should not be used, rather a t-test is more appropriate.

References:

EPA. 2006a. Guidance on Systematic Planning Using the Data Quality Objectives Process. EPA QA/G-4, EPA/240/B-06/001, U.S. Environmental Protection Agency, Office of Environmental Information, Washington DC.

EPA, February 2006. Data Quality Assessment: Statistical Methods for Practitioners, EPA QA/G-9S, Office of Environmental Information, U.S. Environmental Protection Agency, 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.