- Email: [email protected]
Calculation of the detection limits by explicit expressions
Applied Radiation and Isotopes (xxxx) xxxx–xxxx
Contents lists available at ScienceDirect
Applied Radiation and Isotopes journal homepage: www.elsevier.com/locate/apradiso
Calculation of the detection limits by explicit expressions ⁎
D. Glavič-Cindro, M. Korun , B. Vodenik, B. Zorko “Jožef Stefan” Institute, Jamova cesta 39, 1000 Ljubljana, Slovenia
A R T I C L E I N F O
A B S T R A C T
Keywords: Detection limit Decision threshold Null measurement uncertainty
A method for calculating the approximate value of the detection limit for measurements of ionizing radiation is presented. The method can be applied when the indication corresponding to the detection limit and its uncertainty are given as explicit functions. Then also the detection limit can be calculated explicitly, which means that the iteration procedure for its calculation can be avoided. The advantage of the method becomes apparent when the iteration process for calculating the detection limit is difficult to apply.
1. Introduction According to the standard ISO 11929 (ISO 11929, 2010) the decision threshold and the detection limit are expressed in terms of the value of the measurand and its uncertainty. Whereas the decision threshold is calculated from the null-measurement uncertainty, which is the uncertainty of the measured value if the conventional value of the measurand is zero, the detection limit is defined by the uncertainty of the measurand having the value of the detection limit, i.e., the value of the measurand that can be recognized with a predefined probability. It is easy to see that from this definition an implicit equation follows, that in general cannot be solved in a closed form, but by iteration only. This difficulty does not appear in the case of the decision threshold, since it is proportional to the null-measurement uncertainty. Two sources of uncertainty contribute to the uncertainty of the measurand: the uncertainty of the indication, this is the quantity value provided by the measuring system (ISO/IEC, 2007), and the uncertainty of the conversion factor, which converts the indication into the value of the measurand. However, in the methods of calculation of the decision threshold and the detection limit the two uncertainties do not play equivalent roles. Namely, the uncertainty of the indication is treated as a function of its value, whereas the value of the conversion factor and its uncertainty are assumed to be fixed. Therefore, by separating both sources of uncertainty and treating them differently, expressions can be derived that can be used in calculations of detection limits instead of the iteration process. The calculation can be divided into two steps: in the first step the indication corresponding to the detection limit is calculated and in the second the detection limit itself is calculated. To do this, from the uncertainty of the indication as a function of its value, approximate expressions can be derived that make it possible to express the detection limit in an explicit, although approximate, form. The
⁎
method is validated by applying it to two cases where the detection limit can be expressed in an exact explicit form by comparing the approximate results with the results obtained from exact expressions: for the case when the uncertainty of the indication does not depend on its value and for the case elaborated by Currie (Currie, 1968). 2. Methods The standard ISO 11929 defines the detection limit as
y#
= y* + k1− β ·u (y # ), #
(1) #
where y denotes the detection limit, u(y ) is the uncertainty of the measurand at the detection limit and k1-β is the quantile of the standardized normal distribution corresponding to the predefined probability β for making errors of type-II. y* denotes the decision threshold, i.e., the value of the measurand for which, when exceeded, the probability that the true value of the measurand is zero, is less than a predefined probability α, the probability of making type-I errors
y* = k1− α⋅u (0).
(2)
Here, u(0) denotes the null-measurement uncertainty, i.e., the uncertainty of the observed value of the measurand that is equal to zero, and k1-α is the quantile of the standardized normal distribution corresponding to the probability α. In radiation measurements the measuring function is (ISO 2010)
y = w·nn
(3)
is used. Here nn = ng – n0, where ng and n0 are the gross and background indications respectively, denotes the net indication and w denotes the conversion factor converting the net indication into the observed value of the measurand. Since dy/dnn = w and dy/dw = nn,
Corresponding author. E-mail address: [email protected] (M. Korun).
http://dx.doi.org/10.1016/j.apradiso.2017.02.014 Received 1 August 2016; Received in revised form 20 December 2016; Accepted 11 February 2017 0969-8043/ © 2017 Elsevier Ltd. All rights reserved.
Please cite this article as: Glavic-Cindro, D., Applied Radiation and Isotopes (2017), http://dx.doi.org/10.1016/j.apradiso.2017.02.014
Applied Radiation and Isotopes (xxxx) xxxx–xxxx
D. Glavič-Cindro et al. 2 2 1 + k1− β u rel (w )
according to the propagation of uncertainties (ISO/IEC, 1995), u(y) can be expressed as
k1− β ′ = k1− β
u (y ) 2
and the detection limit to the well-known expression
=
w 2u (n
n
)2
+ nn
2u (w )2 .
(4)
Substituting Eqs. (3) and (4) into Eq. (1) and by dividing it by w, the relation
nn#
= nn* + k1− β
u (nn# )2
+
= nn* + k1− β u (nn# ) 1 +
y# =
2 nn#2 urel (w ) 2 nn#2 urel (w ) u (nn# )2
(5a)
2 y* . 2 2 1 − k1− β u rel (w )
is presented. By eliminating u(nn ) from Eqs. (5) and (13), for the detection limit it follows
nn# = k1− α u (nn = 0) + 1+
(6)
k1− β 2 = k1− β 2·[1 + urel (w )2 ·(k1− α + k1− β )2], ’ ’
(14)
(15)
2
k1− β ′ =
urel (w ) 2 u (w ) 2 − 3 k 2 k 2 k1− α k1− β rel 1− α 1− β
n (nn = 0) urel (w )2 2 u (w ) 2 − k 2 1 − k1− k α rel 1− β 1− β u (nn = 0) 2
2
+
urel (w ) 2 − k 2 ) u (w ) 2 + k 2 k1− β 1 + (k1− 1− α k1− β α 1− β rel
u (nn = 0)
4
urel (w ) 3 k2 − 3 k1− α 1− β
u (nn = 0)
2
urel (w ) 2 u (w ) 2 − k 2 1 − k1− 1− α k1− β β rel
u (nn = 0)
(16)
Finally, the detection limit of the indication can be calculated from Eq. (5) by inserting expressions from Eqs. (13), (14) and (16).
(7)
3. Example and discussion When the uncertainty of the indication is described by Eq. (13), the detection limit can be derived in an exact, explicit form. Rewriting Eq. (4) into
u (y # ) 2 =
u (nn# )2 #2 y nn#2
+ urel (w )2 y #2
= [u (nn = 0)2 + nn# ] w 2 + urel (w )2y #2 = u (0)2 + wy # + urel (w )2 y #2
(17)
it is easy to see that from Eq. (1) the exact solution
(8)
y# =
2 2 2 2 2 k1− β k1− α u rel (w ) + k1− β 1 + (k1− α − k1− β ) u rel (w )
2 w/2 k1− α u (0) + k1− β 2 u (w ) 2 1 − k1− β rel
+ (9)
2 w / 2]2 − (k 2 − k 2 ) u (0)2 [1 − k 2 u (w )2] [k1− α u (0) + k1− 1− α 1− β 1− β rel β 2 u (w ) 2 1 − k1− β rel
(18)
follows. To present the difference between the results of the calculations of the detection limit obtained with the exact expression and the calculation according to Eqs. (13), (14) and (16) a numerical example is presented. It is supposed that the background is 102 counts and that the factor converting the net counts registered during the acquisition time to the measurand is w =10−2 s−1. The decision threshold is calculated
follows. Consequently, according to Eqs. (3) and (5), the detection limit is
y # = w·u (nn = 0)·(k1− α + k1− β ). ’
2 k1− k1− α β′ + u (nn = 0) 4u (nn = 0)2
follows in the lowest order of approximation. It can be observed that in Eq. (15) the first term corresponds to the expression nn#/u(nn#) for the case when the u(nn#) does not depend on nn#; therefore, only the second term introduces this dependence. It is important to note that in the second term k1-β’3 does not appear, since it cancels out in the derivation. This offers the opportunity to arrive at an explicit expression for k1-β’ by substituting nn#/u(nn#) from Eq. (15) into Eq. (6) and by solving it for k1-β’:
from where
2 2 1 − k1− β u rel (w )
+ k1− β ′ u (nn = 0)
2 3 2 (k1− nn#2 α + 3k1− α k1− β ′ + k1− α k1− β ′) 2 − = ( k + k ) α β 1− 1− ′ u (nn = 0) u (nn# )2
where nB denotes the total indication originating in all sources of bias. If the indication is expressed by the number of counts, then u(ng) = ng1/2. In Eq. (7) u(nB) comprises all the sources of uncertainty that depend on the presence of the background radiation, which may be poorly characterized and unstable over time. So u(nB) may greatly exceed the statistical uncertainty of the gross indication and then the uncertainty of the net indication is almost independent of its value, i.e., u(nn#) = u(nn=0). In this case the indication corresponding to the decision threshold is n*= k1-α u(nn#) and Eq. (6) simplifies to
α
2
and then its uncertainty follows from Eq. (13). It can be shown that by expanding nn#/u(nn#) on powers of 1/u(nn=0) the expression
This quantile corresponds to the probability β’ of the indication, which takes into account the uncertainty of the conversion factor and in Eq. (5b) replaces the quantile corresponding to the predefined probability β. It is clear that Eq. (6) presents an implicit equation for k1-β’, since the relative uncertainty u(nn#)/nn# depends on it. However, as soon as n# and u(n#) are known as functions of the null measurement uncertainty of the indication and the probabilities α and β’ of making type-I and type-II errors, the detection limit in terms of the indication nn# can be calculated explicitly, introducing a suitable approximation from Eqs. (5) and (6). Consequently, the application of Eq. (6) and the expression for u(nn#) represents an approximative method for calculating the detection limit, as an alternative to the method using iteration. The simplest case for applying the method occurs when the uncertainty of the background indication dominates the uncertainty budget of the net indication. This occurs frequently in gamma-ray spectrometry. Here, besides the continuous and peaked background usually occurring in the measurements of naturally occurring radionuclides in environmental samples, contributions from other radionuclides (ISO 18589-3, 2015) and the contribution of the blank sample (Korun et al., 2016a) can also contribute to the bias. The uncertainty of the net indication is
When k1-
2 k1− β′
(5b)
u (nn )2 = u (ng )2 + u (nB )2 ,
(13) #
where k1-β’ is given by
k1− β ′ =
(12)
u (nn# )2 = u (nn = 0)2 + nn# ,
follows, where nn and nn* denote the net indications corresponding to the detection limit and the decision threshold respectively, and urel(w) denotes the relative uncertainty of w. The relation between nn# and nn* can be simplified to
k1− β 2 = k1− β 2·[1 + urel (w )2 ·nn #2 / u (nn # )2 ] ’
(11)
To illustrate the method on a more complex example, the case considered by Currie (Currie, 1968), where the uncertainty of the indication given by
#
nn # = nn* + k1− β ,·u (nn # ),
2 2 1 − k1− β u rel (w )
(10)
= k1-β, the expression for k1-β’ simplifies to 2
Applied Radiation and Isotopes (xxxx) xxxx–xxxx
D. Glavič-Cindro et al.
region. It follows then, that the detection limit depends on an additional criterion and therefore is not unique. Another example occurs when in gamma-ray spectrometry when peak-analysis results are used to calculate the decision thresholds and detection limits (Korun et al., 2016b). In the iteration process the indication (the peak area) is varied to solve the implicit Eq. (1) on y#. Therefore, also its uncertainty u(y#) is varied, but the modified uncertainty is not possible to calculate by the peak analysis, because the spectra with modified peak areas are not available. Usually, k1-α = k1-β is supposed; however, their values may be reflected in the data-collection objectives. The supposition k1-α = k1-β is justified in determinations of material properties, but in environmental applications this may not be justified. If the measurements are performed for the purpose of dose assessment, errors of type I are more easily tolerated than errors of type II, because the doses should be assessed conservatively. Thus, here the probability for reporting radionuclides not present in the sample is not so harmful as overlooking radionuclides; therefore, k1-α < k1-β can be used. On the other hand, in samples belonging to monitoring programs of radiological control, such as, e.g., monitoring carried out by the CTBT organization, type-I errors cannot be tolerated, because they might lead to unjustified actions. So here, k1-α > k1-β can be used.
Table 1 The exact values of the detection limits, the values of the quantile k1-β’ and the approximate values of the detection limit, obtained from Eqs. (4), (13), (14) and (16). u (w)/w
y#, Eq. (18) / s−1
k1-β’
y#, Approximate / s−1
0.02 0.05 0.10 0.20 0.40 0.80
0.765 0.772 0.795 0.898 1.569 −1.543
1.651 1.686 1.810 2.349 6.132 –
0.765 0.772 0.795 0.896 1.630 –
for a probability of 1% for making errors of the first kind, i.e., k1-α =3.47. The detection limit is calculated supposing a probability of 5% for making errors of type-II, i.e., k1-β =1.65. The uncertainty of the background is u(nB) =10; therefore, the null measurement uncertainty is u(0) =0.14. A relatively small number of background counts and a large difference in k1-α and k1-β are used in the example in order to increase the influence of the net indication and the difference of the quantiles on the detection limit. The detection limits obtained from Eq. (18) and from Eqs. (4), (13), (14) and (16) are compared in Table 1. It can be observed that the values for the detection limit agree at the relative uncertainties of the conversion factor below 0.20. At higher relative uncertainties they do not agree because the first-order approximation is not accurate enough to achieve full agreement. At the value 0.80 the relative uncertainty urel(w) exceeds 1/k1-β; therefore, Eq. (18) yields an unacceptable value and the approximate value cannot be calculated. Larger relative uncertainties of the conversion factor, where the approximated value deviates from the exact value, typically occur when the counting conditions are poorly characterized. In gamma-ray spectrometric measurements examples when the conversion factor is poorly known include measurements of inhomogeneous samples when the inhomogeneity is poorly known, e.g., in measurements of 131I in charcoal filters, and in in-situ measurements. The application of the approximate method is advantageous in cases when the expression for the uncertainty of the indication as a function of the indication is not known or the iteration is difficult to apply. This occurs when u(nn=0) is obtained with the method of least squares by fitting a spectral region. Then, in successive approximations the uncertainties u(nn#i) are obtained by the method of least squares as well. To perform the calculation, the number of counts in the i-th step of the iteration corresponding to the detection limit, nn#i, representing the response of the detection system to the measured quantity having the value corresponding to the detection limit, has to be inserted into the spectral region. Since the uncertainty of the nn#i counts depends on the spectral shape, additional suppositions have to be used in order to define how the additional nn#i counts are distributed over the spectral
4. Conclusion It can be concluded that the method for calculating the detection limit via the quantile of the distribution of indications simplifies the calculation of detection limits when the application of the iteration procedure is not convenient. Since approximate expressions are used in the calculations, it is expected that the results are less reliable at high values of the relative uncertainty of the conversion factor. References Currie, L.A., 1968. Limits of qualitative detection and quantitative determination. Anal. Chem. 40, 586–593. ISO 11929, 2010. ISO 11929:2010(E) Determination of the characteristic limits (decision threshold, detection limit and limits of the confidence interval) for measurements of ionizing radiation – Fundamentals and application, ISO, 2010. ISO 18589-3, 2015. ISO 18589-3 Measurement of the radioactivity in the environment – Soil – Part 3: Test method of gamma-emitting radionuclides using gamma-ray spectrometry, ISO 2015. ISO/IEC, 1995. Guide to expression of uncertainty in measurement, ISO, 1995. ISO/IEC, 2007. ISO/IEC Guide 99 International vocabulary of metrology – Basic and general concepts and associated terms (VIM), ISO/IEC, 2007. Korun, M., Vodenik, B., Zorko, B., 2016a. Measurement function for the activities of multi-gamma-ray emitters in gamma-ray spectrometric measurements. Appl. Radiat. Isot. 109, 518–521. Korun, M., Vodenik, B., Zorko, B., 2016b. Calculation of the decision thresholds for radionuclides identified in gamma-ray spectra by post-processing peak analysis results. Nucl. Instrum. Methods A 813 (216), 102–110.
3
Contents lists available at ScienceDirect
Applied Radiation and Isotopes journal homepage: www.elsevier.com/locate/apradiso
Calculation of the detection limits by explicit expressions ⁎
D. Glavič-Cindro, M. Korun , B. Vodenik, B. Zorko “Jožef Stefan” Institute, Jamova cesta 39, 1000 Ljubljana, Slovenia
A R T I C L E I N F O
A B S T R A C T
Keywords: Detection limit Decision threshold Null measurement uncertainty
A method for calculating the approximate value of the detection limit for measurements of ionizing radiation is presented. The method can be applied when the indication corresponding to the detection limit and its uncertainty are given as explicit functions. Then also the detection limit can be calculated explicitly, which means that the iteration procedure for its calculation can be avoided. The advantage of the method becomes apparent when the iteration process for calculating the detection limit is difficult to apply.
1. Introduction According to the standard ISO 11929 (ISO 11929, 2010) the decision threshold and the detection limit are expressed in terms of the value of the measurand and its uncertainty. Whereas the decision threshold is calculated from the null-measurement uncertainty, which is the uncertainty of the measured value if the conventional value of the measurand is zero, the detection limit is defined by the uncertainty of the measurand having the value of the detection limit, i.e., the value of the measurand that can be recognized with a predefined probability. It is easy to see that from this definition an implicit equation follows, that in general cannot be solved in a closed form, but by iteration only. This difficulty does not appear in the case of the decision threshold, since it is proportional to the null-measurement uncertainty. Two sources of uncertainty contribute to the uncertainty of the measurand: the uncertainty of the indication, this is the quantity value provided by the measuring system (ISO/IEC, 2007), and the uncertainty of the conversion factor, which converts the indication into the value of the measurand. However, in the methods of calculation of the decision threshold and the detection limit the two uncertainties do not play equivalent roles. Namely, the uncertainty of the indication is treated as a function of its value, whereas the value of the conversion factor and its uncertainty are assumed to be fixed. Therefore, by separating both sources of uncertainty and treating them differently, expressions can be derived that can be used in calculations of detection limits instead of the iteration process. The calculation can be divided into two steps: in the first step the indication corresponding to the detection limit is calculated and in the second the detection limit itself is calculated. To do this, from the uncertainty of the indication as a function of its value, approximate expressions can be derived that make it possible to express the detection limit in an explicit, although approximate, form. The
⁎
method is validated by applying it to two cases where the detection limit can be expressed in an exact explicit form by comparing the approximate results with the results obtained from exact expressions: for the case when the uncertainty of the indication does not depend on its value and for the case elaborated by Currie (Currie, 1968). 2. Methods The standard ISO 11929 defines the detection limit as
y#
= y* + k1− β ·u (y # ), #
(1) #
where y denotes the detection limit, u(y ) is the uncertainty of the measurand at the detection limit and k1-β is the quantile of the standardized normal distribution corresponding to the predefined probability β for making errors of type-II. y* denotes the decision threshold, i.e., the value of the measurand for which, when exceeded, the probability that the true value of the measurand is zero, is less than a predefined probability α, the probability of making type-I errors
y* = k1− α⋅u (0).
(2)
Here, u(0) denotes the null-measurement uncertainty, i.e., the uncertainty of the observed value of the measurand that is equal to zero, and k1-α is the quantile of the standardized normal distribution corresponding to the probability α. In radiation measurements the measuring function is (ISO 2010)
y = w·nn
(3)
is used. Here nn = ng – n0, where ng and n0 are the gross and background indications respectively, denotes the net indication and w denotes the conversion factor converting the net indication into the observed value of the measurand. Since dy/dnn = w and dy/dw = nn,
Corresponding author. E-mail address: [email protected] (M. Korun).
http://dx.doi.org/10.1016/j.apradiso.2017.02.014 Received 1 August 2016; Received in revised form 20 December 2016; Accepted 11 February 2017 0969-8043/ © 2017 Elsevier Ltd. All rights reserved.
Please cite this article as: Glavic-Cindro, D., Applied Radiation and Isotopes (2017), http://dx.doi.org/10.1016/j.apradiso.2017.02.014
Applied Radiation and Isotopes (xxxx) xxxx–xxxx
D. Glavič-Cindro et al. 2 2 1 + k1− β u rel (w )
according to the propagation of uncertainties (ISO/IEC, 1995), u(y) can be expressed as
k1− β ′ = k1− β
u (y ) 2
and the detection limit to the well-known expression
=
w 2u (n
n
)2
+ nn
2u (w )2 .
(4)
Substituting Eqs. (3) and (4) into Eq. (1) and by dividing it by w, the relation
nn#
= nn* + k1− β
u (nn# )2
+
= nn* + k1− β u (nn# ) 1 +
y# =
2 nn#2 urel (w ) 2 nn#2 urel (w ) u (nn# )2
(5a)
2 y* . 2 2 1 − k1− β u rel (w )
is presented. By eliminating u(nn ) from Eqs. (5) and (13), for the detection limit it follows
nn# = k1− α u (nn = 0) + 1+
(6)
k1− β 2 = k1− β 2·[1 + urel (w )2 ·(k1− α + k1− β )2], ’ ’
(14)
(15)
2
k1− β ′ =
urel (w ) 2 u (w ) 2 − 3 k 2 k 2 k1− α k1− β rel 1− α 1− β
n (nn = 0) urel (w )2 2 u (w ) 2 − k 2 1 − k1− k α rel 1− β 1− β u (nn = 0) 2
2
+
urel (w ) 2 − k 2 ) u (w ) 2 + k 2 k1− β 1 + (k1− 1− α k1− β α 1− β rel
u (nn = 0)
4
urel (w ) 3 k2 − 3 k1− α 1− β
u (nn = 0)
2
urel (w ) 2 u (w ) 2 − k 2 1 − k1− 1− α k1− β β rel
u (nn = 0)
(16)
Finally, the detection limit of the indication can be calculated from Eq. (5) by inserting expressions from Eqs. (13), (14) and (16).
(7)
3. Example and discussion When the uncertainty of the indication is described by Eq. (13), the detection limit can be derived in an exact, explicit form. Rewriting Eq. (4) into
u (y # ) 2 =
u (nn# )2 #2 y nn#2
+ urel (w )2 y #2
= [u (nn = 0)2 + nn# ] w 2 + urel (w )2y #2 = u (0)2 + wy # + urel (w )2 y #2
(17)
it is easy to see that from Eq. (1) the exact solution
(8)
y# =
2 2 2 2 2 k1− β k1− α u rel (w ) + k1− β 1 + (k1− α − k1− β ) u rel (w )
2 w/2 k1− α u (0) + k1− β 2 u (w ) 2 1 − k1− β rel
+ (9)
2 w / 2]2 − (k 2 − k 2 ) u (0)2 [1 − k 2 u (w )2] [k1− α u (0) + k1− 1− α 1− β 1− β rel β 2 u (w ) 2 1 − k1− β rel
(18)
follows. To present the difference between the results of the calculations of the detection limit obtained with the exact expression and the calculation according to Eqs. (13), (14) and (16) a numerical example is presented. It is supposed that the background is 102 counts and that the factor converting the net counts registered during the acquisition time to the measurand is w =10−2 s−1. The decision threshold is calculated
follows. Consequently, according to Eqs. (3) and (5), the detection limit is
y # = w·u (nn = 0)·(k1− α + k1− β ). ’
2 k1− k1− α β′ + u (nn = 0) 4u (nn = 0)2
follows in the lowest order of approximation. It can be observed that in Eq. (15) the first term corresponds to the expression nn#/u(nn#) for the case when the u(nn#) does not depend on nn#; therefore, only the second term introduces this dependence. It is important to note that in the second term k1-β’3 does not appear, since it cancels out in the derivation. This offers the opportunity to arrive at an explicit expression for k1-β’ by substituting nn#/u(nn#) from Eq. (15) into Eq. (6) and by solving it for k1-β’:
from where
2 2 1 − k1− β u rel (w )
+ k1− β ′ u (nn = 0)
2 3 2 (k1− nn#2 α + 3k1− α k1− β ′ + k1− α k1− β ′) 2 − = ( k + k ) α β 1− 1− ′ u (nn = 0) u (nn# )2
where nB denotes the total indication originating in all sources of bias. If the indication is expressed by the number of counts, then u(ng) = ng1/2. In Eq. (7) u(nB) comprises all the sources of uncertainty that depend on the presence of the background radiation, which may be poorly characterized and unstable over time. So u(nB) may greatly exceed the statistical uncertainty of the gross indication and then the uncertainty of the net indication is almost independent of its value, i.e., u(nn#) = u(nn=0). In this case the indication corresponding to the decision threshold is n*= k1-α u(nn#) and Eq. (6) simplifies to
α
2
and then its uncertainty follows from Eq. (13). It can be shown that by expanding nn#/u(nn#) on powers of 1/u(nn=0) the expression
This quantile corresponds to the probability β’ of the indication, which takes into account the uncertainty of the conversion factor and in Eq. (5b) replaces the quantile corresponding to the predefined probability β. It is clear that Eq. (6) presents an implicit equation for k1-β’, since the relative uncertainty u(nn#)/nn# depends on it. However, as soon as n# and u(n#) are known as functions of the null measurement uncertainty of the indication and the probabilities α and β’ of making type-I and type-II errors, the detection limit in terms of the indication nn# can be calculated explicitly, introducing a suitable approximation from Eqs. (5) and (6). Consequently, the application of Eq. (6) and the expression for u(nn#) represents an approximative method for calculating the detection limit, as an alternative to the method using iteration. The simplest case for applying the method occurs when the uncertainty of the background indication dominates the uncertainty budget of the net indication. This occurs frequently in gamma-ray spectrometry. Here, besides the continuous and peaked background usually occurring in the measurements of naturally occurring radionuclides in environmental samples, contributions from other radionuclides (ISO 18589-3, 2015) and the contribution of the blank sample (Korun et al., 2016a) can also contribute to the bias. The uncertainty of the net indication is
When k1-
2 k1− β′
(5b)
u (nn )2 = u (ng )2 + u (nB )2 ,
(13) #
where k1-β’ is given by
k1− β ′ =
(12)
u (nn# )2 = u (nn = 0)2 + nn# ,
follows, where nn and nn* denote the net indications corresponding to the detection limit and the decision threshold respectively, and urel(w) denotes the relative uncertainty of w. The relation between nn# and nn* can be simplified to
k1− β 2 = k1− β 2·[1 + urel (w )2 ·nn #2 / u (nn # )2 ] ’
(11)
To illustrate the method on a more complex example, the case considered by Currie (Currie, 1968), where the uncertainty of the indication given by
#
nn # = nn* + k1− β ,·u (nn # ),
2 2 1 − k1− β u rel (w )
(10)
= k1-β, the expression for k1-β’ simplifies to 2
Applied Radiation and Isotopes (xxxx) xxxx–xxxx
D. Glavič-Cindro et al.
region. It follows then, that the detection limit depends on an additional criterion and therefore is not unique. Another example occurs when in gamma-ray spectrometry when peak-analysis results are used to calculate the decision thresholds and detection limits (Korun et al., 2016b). In the iteration process the indication (the peak area) is varied to solve the implicit Eq. (1) on y#. Therefore, also its uncertainty u(y#) is varied, but the modified uncertainty is not possible to calculate by the peak analysis, because the spectra with modified peak areas are not available. Usually, k1-α = k1-β is supposed; however, their values may be reflected in the data-collection objectives. The supposition k1-α = k1-β is justified in determinations of material properties, but in environmental applications this may not be justified. If the measurements are performed for the purpose of dose assessment, errors of type I are more easily tolerated than errors of type II, because the doses should be assessed conservatively. Thus, here the probability for reporting radionuclides not present in the sample is not so harmful as overlooking radionuclides; therefore, k1-α < k1-β can be used. On the other hand, in samples belonging to monitoring programs of radiological control, such as, e.g., monitoring carried out by the CTBT organization, type-I errors cannot be tolerated, because they might lead to unjustified actions. So here, k1-α > k1-β can be used.
Table 1 The exact values of the detection limits, the values of the quantile k1-β’ and the approximate values of the detection limit, obtained from Eqs. (4), (13), (14) and (16). u (w)/w
y#, Eq. (18) / s−1
k1-β’
y#, Approximate / s−1
0.02 0.05 0.10 0.20 0.40 0.80
0.765 0.772 0.795 0.898 1.569 −1.543
1.651 1.686 1.810 2.349 6.132 –
0.765 0.772 0.795 0.896 1.630 –
for a probability of 1% for making errors of the first kind, i.e., k1-α =3.47. The detection limit is calculated supposing a probability of 5% for making errors of type-II, i.e., k1-β =1.65. The uncertainty of the background is u(nB) =10; therefore, the null measurement uncertainty is u(0) =0.14. A relatively small number of background counts and a large difference in k1-α and k1-β are used in the example in order to increase the influence of the net indication and the difference of the quantiles on the detection limit. The detection limits obtained from Eq. (18) and from Eqs. (4), (13), (14) and (16) are compared in Table 1. It can be observed that the values for the detection limit agree at the relative uncertainties of the conversion factor below 0.20. At higher relative uncertainties they do not agree because the first-order approximation is not accurate enough to achieve full agreement. At the value 0.80 the relative uncertainty urel(w) exceeds 1/k1-β; therefore, Eq. (18) yields an unacceptable value and the approximate value cannot be calculated. Larger relative uncertainties of the conversion factor, where the approximated value deviates from the exact value, typically occur when the counting conditions are poorly characterized. In gamma-ray spectrometric measurements examples when the conversion factor is poorly known include measurements of inhomogeneous samples when the inhomogeneity is poorly known, e.g., in measurements of 131I in charcoal filters, and in in-situ measurements. The application of the approximate method is advantageous in cases when the expression for the uncertainty of the indication as a function of the indication is not known or the iteration is difficult to apply. This occurs when u(nn=0) is obtained with the method of least squares by fitting a spectral region. Then, in successive approximations the uncertainties u(nn#i) are obtained by the method of least squares as well. To perform the calculation, the number of counts in the i-th step of the iteration corresponding to the detection limit, nn#i, representing the response of the detection system to the measured quantity having the value corresponding to the detection limit, has to be inserted into the spectral region. Since the uncertainty of the nn#i counts depends on the spectral shape, additional suppositions have to be used in order to define how the additional nn#i counts are distributed over the spectral
4. Conclusion It can be concluded that the method for calculating the detection limit via the quantile of the distribution of indications simplifies the calculation of detection limits when the application of the iteration procedure is not convenient. Since approximate expressions are used in the calculations, it is expected that the results are less reliable at high values of the relative uncertainty of the conversion factor. References Currie, L.A., 1968. Limits of qualitative detection and quantitative determination. Anal. Chem. 40, 586–593. ISO 11929, 2010. ISO 11929:2010(E) Determination of the characteristic limits (decision threshold, detection limit and limits of the confidence interval) for measurements of ionizing radiation – Fundamentals and application, ISO, 2010. ISO 18589-3, 2015. ISO 18589-3 Measurement of the radioactivity in the environment – Soil – Part 3: Test method of gamma-emitting radionuclides using gamma-ray spectrometry, ISO 2015. ISO/IEC, 1995. Guide to expression of uncertainty in measurement, ISO, 1995. ISO/IEC, 2007. ISO/IEC Guide 99 International vocabulary of metrology – Basic and general concepts and associated terms (VIM), ISO/IEC, 2007. Korun, M., Vodenik, B., Zorko, B., 2016a. Measurement function for the activities of multi-gamma-ray emitters in gamma-ray spectrometric measurements. Appl. Radiat. Isot. 109, 518–521. Korun, M., Vodenik, B., Zorko, B., 2016b. Calculation of the decision thresholds for radionuclides identified in gamma-ray spectra by post-processing peak analysis results. Nucl. Instrum. Methods A 813 (216), 102–110.
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