Index > 3 Characterisation of radioactively contaminated sites >

3.10.4 Example of statistical data processing: Contaminant present in background

Contents
3.10.4.1 Two-Sample statistical test
3.10.4.2 Applying the Wilcoxon Rank Sum Test
3.10.4.3 Wilcoxon Rank Sum test example: Class 2 interior drywall survey unit
3.10.4.4 Class 1 interior concrete survey unit
3.10.4.5 Multiple radionuclides

The two-sample statistical test (Wilcoxon Rank Sum test, discussed in Section 3.5) should be used when the radio-nuclide of concern appears in background or if measurements are used that are not radio-nuclide specific, e.g., as at a final status survey. The two-sample Wilcoxon Rank Sum (WRS) test (Section 3.10.4.1) assumes the reference area and survey unit data distributions are similar except for a possible shift in the medians. When the data are severely skewed, the value for the mean difference may be above the DCGLW, while the median difference is below the DCGLW. In such cases, the survey unit does not meet the release criterion regardless of the result of the statistical test. On the other hand, if the difference between the largest survey unit measurement and the smallest reference area measurement is less than the DCGLW, the WRS test will always show that the survey unit meets the release criterion.

The statistical tests discussed in this section will be used to compare each survey unit with an appropriately chosen, site-specific reference area. Each reference area should be selected on the basis of its similarity to the survey unit, as discussed in Section 3.3.5.

3.10.4.1 Two-Sample statistical test

The comparison of measurements from the reference area and survey unit is made using the Wilcoxon Rank Sum (WRS) test (also called the Mann-Whitney test). The WRS test should be conducted for each survey unit. In addition, the elevated measurement comparison (EMC) is performed against each measurement to ensure that it does not exceed a specified investigation level. If any measurement in the re-mediated survey unit exceeds the specified investigation level, 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 the non-parametric WRS test is that it does not assume that 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, the WRS test can be used with up to 40 percent ‘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 as discussed in Section 3.11. When possible, report the actual result of a measurement together with its uncertainty.

The hypothesis tested by the WRS test is:

  • Null Hypothesis
    H0: The median concentration in the survey unit exceeds that in the reference area by more than the DCGLW.

versus

  • Alternative Hypothesis
    Ha: The median concentration in the survey unit exceeds that in the reference area by less than the DCGLW.

The null hypothesis is assumed to be true unless the statistical test indicates that it should be rejected in favour of the alternative. One assumes that any difference between the reference area and survey unit concentration distributions is due to a shift in the survey unit concentrations to higher values (i.e., due to the presence of residual radioactivity in addition to background). Note that some or all of the survey unit measurements may be larger than some reference area measurements, while still meeting the release criterion. Indeed, some survey unit measurements may exceed some reference area measurements by more than the DCGLW. The result of the hypothesis test determines whether or not the survey unit as a whole is deemed to meet the release criterion. The elevated measurement comparison (EMC) is used to screen individual measurements.

Two assumptions underlying this test are:

  1. Samples from the reference area and survey unit are independent, identically distributed random samples, and
  2. Each measurement is independent of every other measurement, regardless of the set of samples from which it.

3.10.4.2 Applying the Wilcoxon Rank Sum Test

The WRS test is applied as outlined in the following six steps and further illustrated by the examples in Section 3.10.4.3:

  1. Obtain the adjusted reference area measurements, Zi , by adding the DCGLW to each reference area measurement, Xi . Zi = Xi +DCGLW
  2. The m adjusted reference sample measurements, Zi , from the reference area and the n sample measurements, Yi , from the survey unit are pooled and ranked in order of increasing size from 1 to N, where N = m+n.
  3. If several measurements are tied (i.e., have the same value), they are all assigned the average rank of that group of tied measurements.
  4. If there are t ‘less than’ values, they are all given the average of the ranks from 1 to t. Therefore, they are all assigned the rank t(t+1)/(2t) = (t+1)/2, which is the average of the first t integers. If there is more than one detection limit, all observations below the largest detection limit should be treated as ‘less than’ values1.
  5. Sum the ranks of the adjusted measurements from the reference area, Wr. Note that since the sum of the first N integers is N(N+1)/2, one can equivalently sum the ranks of the measurements from the survey unit, Ws, and compute Wr = N(N+1)/2 -Ws.
  6. Compare Wr with the critical value given in Appendix E Table E.4 for the appropriate values of n, m, and α. If Wr is greater than the tabulated value, reject the hypothesis that the survey unit exceeds the release criterion.

3.10.4.3 Wilcoxon Rank Sum test example: Class 2 interior drywall survey unit

In this example, the gas-flow proportional counter measures total beta-gamma activity (see Appendix E) and the measurements are not radio-nuclide-specific. The two-sample non-parametric test is appropriate for the Class 2 interior drywall survey unit because gross beta-gamma activity contributes to background even though the radio-nuclide of interest does not appear in background.

Table 3.53 shows that the DQOs for this survey unit include α = 0.025 and β = 0.05. The DCGLW is 8,300 Bq/m2 (5,000 dpm per 100 cm2) and the estimated standard deviation of the measurements is about σ = 1,040 Bq/m2 (625 dpm per 100 cm2). The estimated standard deviation is 8 times less than the DCGLW. With this level of precision, the width of the gray region can be made fairly narrow. As noted earlier, sample sizes do not decrease very much once Δ/σ exceeds 3 or 4. In this example, the lower bound for the gray region was set so that Δ/σ is about 4.

If Δ/σ = (DCGLW – LBGR)/σ = 4

then LBGR = DCGLW – 4σ
. . . . . . . . .= 8,300 – (4 × 1,040)
. . . . . . . . .= 4,100 Bq/m2 (2,500 dpm per 100 cm2).

In Table 3.35, one finds that the number of measurements estimated for the WRS test is 11 in each survey unit and 11 in each reference area (α = 0.025, β = 0.05, and Δ/σ = 4). (Table E.3 in Appendix E also lists the number of measurements estimated for the WRS test.) This survey unit was classified as Class 2, so the 11 measurements needed in the survey unit and the 11 measurements needed in the reference area were made using a random-start triangular grid2.

Table 3.56 lists the data obtained from the gas-flow proportional counter in units of counts per minute. A reading of 160 cpm with this instrument corresponds to the DCGLW of 8,300 Bq/m2 (5,000 dpm per 100 cm2). Column A lists the measurement results as they were obtained. The average and standard deviation of the reference area measurements are 44 and 4.4 cpm, respectively. The average and standard deviation of the survey unit measurements are 98 and 5.3 cpm, respectively.

A
Data (cpm)
B
Area
C
Adjusted data
D
Ranks
E
Reference area ranks


1 49 R 209 22 22
2 35 R 195 12 12
3 45 R 205 17.5 17.5
4 45 R 205 17.5 17.5
5 41 R 201 14 14
6 44 R 204 16 16
7 48 R 208 21 21
8 37 R 197 13 13
9 46 R 206 19 19
10 42 R 202 15 15
11 47 R 207 20 20
12 104 S 104 9.5 0
13 94 S 94 4 0
14 98 S 98 6 0
15 99 S 99 7 0
16 90 S 90 1 0
17 104 S 104 9.5 0
18 95 S 95 5 0
19 105 S 105 11 0
20 93 S 93 3 0
21 101 S 101 8 0
22 92 S 92 2 0
Sum = 253 187

Table 3.56 WRS test for Class 2 interior drywall survey unit

In column B, the code ‘R’ denotes a reference area measurement, and ‘S’ denotes a survey unit measurement. Column C contains the adjusted data. The adjusted data are obtained by adding the DCGLW to the reference area measurements (see Section 3.10.4.2, Step 1). The ranks of the adjusted data appear in Column D. They range from 1 to 22, since there is a total of 11+11 measurements (see Section 3.10.4.2, Step 2).

Note that there were two cases of measurements tied with the same value, at 104 and 209. Each tied measurement is always assigned the average of the ranks. Therefore, both measurements at 104, are assigned rank (9+10)/2 = 9.5 (see Section 3.10.4.2, Step 3). Also note that the sum of all of the ranks is still 22(22+1)/2 = 253. Checking this value with the formula in Step 5 of Section 3.10.4.2 is recommended to guard against errors in the rankings.

Column E contains only the ranks belonging to the reference area measurements. The total is 187. This is compared with the entry for the critical value of 156 in Table E.4 of Appendix E for α = 0.025, with n = 11 and m =11. Since the sum of the reference area ranks is greater than the critical value, the null hypothesis (i.e., that the average survey unit concentration exceeds the DCGLW) is rejected.

The analysis for the WRS test is very well suited to the use of a computer spreadsheet. The spreadsheet formulas used for the example above are given in Appendix E.4, Table E.6.

3.10.4.4 Class 1 interior concrete survey unit

As in the previous example, the gas-flow proportional counter measures total beta-gamma activity (see Appendix E) and the measurements are not radionuclide specific. The two-sample non-parametric test is appropriate for the Class 1 interior concrete survey unit because gross beta-gamma activity contributes to background even though the radionuclide of interest does not appear in background.

3.10.4.5 Multiple radionuclides

The use of the unity rule when there is more than one radionuclide to be considered is discussed in Appendix E.4. An example application appears in Section E.4.4.

Footnote’s

1 If more than 40 percent of the data from either the reference area or survey unit are ‘less than’, the WRS test cannot be used. Such a large proportion of non-detects suggest that the DQO process be re-visited for this survey to determine if the survey unit was properly classified or the appropriate measurement method was used. As stated previously, 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.

2 A random start systematic grid is used in Class 2 and 3 survey units primarily to limit the size of any potential elevated areas. Since areas of elevated activity are not an issue in the reference areas, the measurement locations can be either random or on a random start systematic grid (see Section 3.5.1).