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3.5.2 Contaminant not present in background - determining numbers of data points for statistical tests

Contents
3.5.2.1 Calculate the Relative Shift
3.5.2.2 Determine Sign p
3.5.2.3 Determine Decision Error Percentiles
3.5.2.4 Calculate Number of Data Points for Sign Test
3.5.2.5 Obtain Number of Data Points for Sign Test from Table 3.24

For the situation where the contaminant is not present in background or is present at such a small fraction of the DCGLW (release criteria) as to be considered insignificant, a background reference area is not necessary. Instead, the contaminant levels are compared directly with the DCGL value. The general approach closely parallels that used for the situation when the contaminant is present in background as described above. However, the statistical tests differ slightly. The one-sample Sign test replaces the two-sample Wilcoxon Rank Sum test described below.

3.5.2.1 Calculate the Relative Shift

The initial step in determining the number of data points in the one-sample case is to calculate the relative shift, Δ/σs = (DCGL-LBGR)/σs, from the DCGL value, the lower bound of the gray region (LBGR), and the standard deviation of the contaminant in the survey unit, σs, as described above. Also as described above, the value of σs may be obtained from earlier surveys, limited preliminary measurements, or a reasonable estimate. Values of the relative shift that are less than one will result in a large number of measurements needed to demonstrate compliance.

3.5.2.2 Determine Sign p

Sign p is the estimated probability that a random measurement from the survey unit will be less than the DCGLW when the survey unit median is actually at the LBGR. The Sign p is used to calculate the minimum number of data points necessary for the survey to meet the DQOs. The value of the relative shift calculated in the previous section is used to obtain the corresponding value of Sign p from Table 3.36.
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Δ/σ Sign p Δ/σ Sign p
0.1 0.539828 1.2 0.884930
0.2 0.579260 1.3 0.903199
0.3 0.617911 1.4 0.919243
0.4 0.655422 1.5 0.933193
0.5 0.691462 1.6 0.945201
0.6 0.725747 1.7 0.955435
0.7 0.758036 1.8 0.964070
0.8 0.788145 1.9 0.971284
0.9 0.815940 2.0 0.977250
1.0 0.841345 2.5 0.993790
1.1 0.864334 3.0 0.998650

Table 3.36 Values of Sign p for given values of the relative shift, Δ/σ, when the contaminant is not present in background (example: if Δ/σ > 3.0, use Sign p = 1.000000).
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3.5.2.3 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 (see Table 3.34).

3.5.2.4 Calculate Number of Data Points for Sign Test

The number of data points, N, to be obtained for the Sign test is next calculated using the following formula:

N = (Z1-α + Z1-β)2 / (4 (Sign p – 0.5)2) ………………………………………………. (3-17)

Finally, the number of anticipated data points should be increased by at least 20% as discussed before to ensure sufficient power of the tests and to allow for possible data losses.

3.5.2.5 Obtain Number of Data Points for Sign Test from Table 3.24

Table 3.24 provides a list of the number of data points used to demonstrate compliance using the Sign test for selected values of α, β, and Δ/σ. The values listed in Table 3.37 represent the number of measurements to be performed in each survey unit. These values were calculated using Equation 3-17 and increased by 20% to account for missing or unusable data and uncertainty in the calculated value of N.
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Table 3.37 Values of N for use with the Sign test
Table 3.37 Values of N for use with the Sign test

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Example 3.15: Calculation of the number of data points for a survey unit and a reference area when the contaminant is not present in background

A site has 1 survey unit. The DCGL level for the contaminant of interest is 140 Bq/kg (3.9 pCi/g) in soil. The contaminant is not present in background; data from previous investigations indicate average residual contamination at the survey unit of 3.7 ± 3.7 (1σ) Bq/kg. The lower bound of the gray region was selected to be 110 Bq/kg. A value of 0.05 is next selected for the probability of Type I decision errors (α) and a value of 0.01 is selected for the probability of Type II decision errors (β) based on the survey objectives. Determine the number of data points to be obtained from the survey unit for the statistical tests.
The value of the shift parameter, Δ/σ, is (140-110)/3.7 or 8. From Table 3.36, the value of Sign p is 1.0. Since Δ/σ > 3, the width of the gray region can be reduced. If the LBGR is raised to 125, then Δ/σ is (140-125)/3.7 or 4. The value of Sign p remains at 1.0. Thus, the number of data points calculated will not change. The probability of a Type II error is now specified at 125 Bq/kg (3.4 pCi/g) rather than 110 Bq/kg (3.0 pCi/g). As a consequence, the probability of a Type II error at 110 Bq/kg (3.0 pCi/g) will be even smaller.
Values of percentiles represented by the selected decision error levels are obtained from Table 3.21. Z1-α (for α = 0.05) is 1.645, and Z1-β (β = 0.01) is 2.326.
The number of data points, N, for the Sign test can be calculated using Equation 3-17:

N = (1.645 + 2.326)2 / (4 (1.0 – 0.5)2)

Adding an additional 20% gives 19.2 and rounding up yields 20 data points for the survey unit.
Alternatively, the number of data points can be obtained directly from Table 3.37. For α = 0.05, β = 0.01, and Δ/σ > 3.0 a value of 20 is obtained for N. The table value has already been increased by 20% to account for missing or unusable data and uncertainty in the calculated value of N.