Index > 3 Characterisation of radioactively contaminated sites >

3.5.1 Contaminant present in background - determining numbers of data points for statistical tests

Contents
3.5.1.1 Calculate the Relative Shift
3.5.1.2 Determine probability Pr
3.5.1.3 Determine Decision Error Percentiles
3.5.1.4 Calculate Number of Data Points for WRS Test
3.5.1.5 Obtain Number of Data Points for WRS Test from Table 3.35

The comparison of measurements from the reference area and survey unit is made using the Wilcoxon Rank Sum test (WRS), which should be conducted for each survey unit. In addition, the elevated measurement comparison (EMC) is performed against each measurement to ensure that the measurement result does not exceed a specified investigation level. Decisions have to be defined if any measurement of a survey exceeds the specified investigation level. As an example, if it is a measurement of a remediated survey unit, then additional investigation is recommended, at least locally, regardless of the outcome of the WRS test.

The WRS test is most effective when residual radioactivity is uniformly present throughout a survey unit. The test is designed to detect whether or not this activity exceeds the DCGLW. The advantage of this non-parametric test is that it does not assume the data are normally or log-normally distributed. The WRS test also allows for “less than” measurements to be present in the reference area and the survey units. As a general rule, this test can be used with up to 40% “less than” measurements in either the reference area or the survey unit. However, the use of “less than” values in data reporting is not recommended. Wherever possible, the actual result of a measurement, together with its uncertainty, should be reported.

This section introduces several terms and statistical parameters that will be used to determine the number of data points needed to apply the non-parametric tests. An example is provided to better illustrate the application of these statistical concepts.

3.5.1.1 Calculate the Relative Shift

The lower bound of the gray region (LBGR) is selected during the DQO Process along with the target values for α and β. The width of the gray region, equal to (DCGL – LBGR), is a parameter that is central to the WRS test. This parameter is also referred to as the shift, Δ. The absolute size of the shift is actually of less importance than the relative shift, Δ/σ, where σ is an estimate of the standard deviation of the measured values in the survey unit. This estimate of σ includes both the real spatial variability in the quantity being measured and the precision of the chosen measurement system. The relative shift, Δ/σ, is an expression of the resolution of the measurements in units of measurement uncertainty.

The shift (Δ = DCGLW – LBGR) and the estimated standard deviation in the measurements of the contaminant (σr and σs) are used to calculate the relative shift, Δ/σ (see Appendix A). The standard deviations in the contaminant level will likely be available from previous survey data (e.g., scoping or characterization survey data for un-remediated survey units or remedial action support surveys for remediated survey units). If they are not available, it may be necessary to:

  • Perform some limited preliminary measurements to estimate the distributions, or
  • Make a reasonable estimate based on available site knowledge.

If the first approach above is used, it is important to note that the scoping or characterization survey data or preliminary measurements used to estimate the standard deviation should use the same technique as that to be used during the final status survey. When preliminary data are not obtained, it may be reasonable to assume a coefficient of variation on the order of 30%, based on experience.

The value selected as an estimate of σ for a survey unit may be based on data collected only from within that survey unit or from data collected from a much larger area of the site. Note that survey units are not finalized until the planning stage of the final status survey. This means that there may be some difficulty in determining which individual measurements from a preliminary survey may later represent a particular survey unit. For many sites, the most practical solution is to estimate σ for each area classification (i.e., Class 1, Class 2, and Class 3) for both interior and exterior survey units. This will result in all exterior Class 3 survey units using the same estimate of σ, all exterior Class 2 survey units using a second estimate for σ, and all exterior Class 1 survey units using a third estimate for σ. If there are multiple types of surfaces within an area classification, additional estimates of σ may be required. For example, a Class 2 concrete floor may require a different estimate of σ than a Class 2 cinder block wall, or a Class 3 unpaved parking area may require a different estimate of σ than a Class 3 lawn. In addition, EURSSEM recommends that a separate estimate of σ be obtained for every reference area.

The importance of choosing appropriate values for σr and σs must be emphasized. If the value is grossly underestimated, the number of data points will be too few to obtain the desired power level for the test and a resurvey may be recommended (see Section 3.10.8). If, on the other hand, the value is overestimated, the number of data points determined will be unnecessarily large.

Values for the relative shift that are less than one will result in a large number of measurements needed to demonstrate compliance. The number of data points will also increase as Δ becomes smaller. Since the DCGL is fixed, this means that the lower bound of the gray region also has a significant effect on the estimated number of measurements needed to demonstrate compliance. When the estimated standard deviations in the reference area and survey units are different, the larger value should be used to calculate the relative shift (Δ/σ).

3.5.1.2 Determine probability Pr

The probability that a random measurement from the survey unit exceeds a random measurement from the background reference area by less than the DCGLW when the survey unit median is equal to the LBGR above background is defined as Pr. Pr is used in Equation 3-16 for determining the number of measurements to be performed during the survey. Table 3.33 lists relative shift values and values for Pr. Using the relative shift calculated in the preceding section, the value of Pr can be obtained from Table 3.33.
.

Δ/σ Pr Δ/σ Pr
0.1 0.528182 1.4 0.838864
0.2 0.556223 1.5 0.855541
0.3 0.583985 1.6 0.871014
0.4 0.611335 1.7 0.885299
0.5 0.638143 1.8 0.898420
0.6 0.664290 1.9 0.910413
0.7 0.689665 2.0 0.921319
0.8 0.714167 2.25 0.944167
0.9 0.737710 2.5 0.961428
1.0 0.760217 2.75 0.974067
1.1 0.781627 3.0 0.983039
1.2 0.801892 3.5 0.993329
1.3 0.820978 4.0 0.997658

Table 3.33 Values of Pr for given values of the relative shift, Δ/σ when the contaminant is present in background (example: if Δ/σ > 4.0, use Pr = 1.000000).
.

If the actual value of the relative shift is not listed in Table 3.33, always select the next lower value that appears in the table. For example, Δ/σ = 1.67 does not appear in Table 3.20. The next lower value is 1.6, so the value of Pr would be 0.871014.

3.5.1.3 Determine Decision Error Percentiles

The next step in this process is to determine the percentiles Z1-α and Z1-β, represented by the selected decision error levels, α and β, respectively (Table 3.34). Z1-α and Z1-β are standard statistical values.
.

α and β Z1-α (or Z1-β) α and β Z1-α (or Z1-β)
0.005 2.576 0.10 1.282
0.01 2.326 0.15 1.036
0.015 2.241 0.20 0.842
0.025 1.960 0.25 0.674
0.05 1.645 0.30 0.524

Table 3.34 Percentiles represented by selected values of α and β.
.

3.4.1.4 Calculate Number of Data Points for WRS Test

The number of data points, N, to be obtained from each reference area/survey unit pair for the WRS test is next calculated using:

N = (Z1-α + Z1-β)2 / ( 3 (Pr – 0.5)2) ………………………………………………….. 3-16

The value of N calculated using Equation 3-16 is an approximation based on estimates of σ and Pr, so there is some uncertainty associated with this calculation. In addition, there will be some missing or unusable data from any survey. The rate of missing or unusable measurements, R, expected to occur in survey units or reference areas and the uncertainty associated with the calculation of N should be accounted for during survey planning. The number of data points should be increased by 20%, and rounded up, over the values calculated using Equation 3-16 to obtain sufficient data points to attain the desired power level with the statistical tests and allow for possible lost or unusable data. The value of 20% is selected to account for a reasonable amount of uncertainty in the parameters used to calculate N and still allow flexibility to account for some lost or unusable data. The recommended 20% correction factor should be applied as a minimum value. Experience and site-specific considerations should be used to increase the correction factor if required. If the user determines that the 20% increase in the number of measurements is excessive for a specific site, a retrospective power curve should be used to demonstrate that the survey design provides adequate power to support the decision (see Appendix E).

N is the total number of data points for each survey unit/reference area combination. The N data points are divided between the survey unit, n, and the reference area, m. The simplest method for distributing the N data points is to assign half the data points to the survey unit and half to the reference area, so n=m=N/2. This means that N/2 measurements are performed in each survey unit, and N/2 measurements are performed in each reference area. If more than one survey unit is associated with a particular reference area, N/2 measurements should be performed in each survey unit and N/2 measurements should be performed in the reference area.

3.5.1.5 Obtain Number of Data Points for WRS Test from Table 3.35

Table 3.35 provides a list of the number of data points used to demonstrate compliance using the WRS test for selected values of α, β and Δ/σ. The values listed in Table 3.35 represent the number of measurements to be performed in each survey unit as well as in the corresponding reference area. The values were calculated using Equation 3-16 and increased by 20% for the reasons discussed in the previous section.

Example 3.14: Calculation of the number of data points for a survey unit and reference area when the contaminant is present in background

A site has 14 survey units and 1 reference area, and the same type of instrument and method is used to perform measurements in each area. The contaminant has a DCGLW which when converted to cpm equals 160 cpm. The contaminant is present in background at a level of 45 ± 7 (1σ) cpm. The standard deviation of the contaminant in the survey area is ± 20 cpm, based on previous survey results for the same or similar contaminant distribution. When the estimated standard deviation in the reference area and the survey units are different, the larger value, 20 cpm in this example, should be used to calculate the relative shift. During the DQO process the LBGR is selected to be one-half the DCGLW (80 cpm) as an arbitrary starting point for developing an acceptable survey design1, and Type I and Type II error values (α and β) of 0.05 have been selected. Determine the number of data points to be obtained from the reference area and from each of the survey units for the statistical tests.
The value of the relative shift for the reference area, Δ/σ, is (160-80)/20 or 4. From Table 3.20, the value of Pr is 0.997658. Values of percentiles, represented by the selected decision error levels, are obtained from Table 3.34. In this case Z1-α(for α = 0.05) is 1.645 and Z1-β (β = 0.05) is also 1.645.

The number of data points, N, for the WRS test of each combination of reference area and survey units can be calculated using Equation 3-16:

N = (1.645 + 1.645)2 / ( 3 (0.997658 – 0.5)2 ) = 14.6

Adding an additional 20% gives 17.5 which is then rounded up to the next even number, 18. This yields 9 data points for the reference area and 9 for each survey unit.
Alternatively, the number of data points can be obtained directly from Table 3.35. For α = 0.05, β = 0.05, and Δ/σ = 4.0 a value of 9 is obtained for N/2. The table value has already been increased by 20% to account for missing or unusable data.

Table 3.35 Values of N/2 for use with the Wilcoxon Rank Sum Test
Table 3.35 Values of N/2 for use with the Wilcoxon Rank Sum Test

.

1 Section 3.10 provides more detailed guidance on the selection of the LBGR.