Download APPLICATION OF THE POWER-MODERATE WEIGHTED MEAN

APPLICATION OF THE POWER-MODERATE WEIGHTED MEAN
(PMM) CONCEPT IN THE CALCULATION OF REFERENCE VALUES
OF INTERLABORATORY COMPARISONS*
M. SAHAGIA**,1, A. LUCA1, A. ANTOHE1, M.-R. IOAN1,
E. GARCIA-TORAÑO2
1
“Horia Hulubei” National Institute for R&D in Physics and Nuclear Engineering, IFIN-HH,
P.O Box MG-6, RO-077125, Bucharest, Romania
2
Centro de Investigaciones Energéticas, Medioambientales y Tecnológicas (CIEMAT),
Madrid, Spain
* Paper presented at the 5th International Proficiency Testing Conference, Timisoara, Romania,
September 16–18, 2015
** Corresponding author: [email protected]
Received June 25, 2015
Recently, the Consultative Committee for Ionizing Radiations Section II,
Measurement of Radionuclides, of the International Committee of Weights and
Measures [CIPM-CCRI(II)], decided to apply the new concept of power-moderate
weighted mean (PMM) in calculation of the Key Comparison Reference Values
(KCRV), standard uncertainties and degrees of equivalence, Document CCRI(II)/13–
37. The paper will exemplify the calculations of 226Ra activity concentration of a slag
sample, using the PMM mode, for which we had recommended before the use of the
weighted mean. This value will also be compared with a recently published one.
Key words: power-moderate weighted mean, interlaboratory comparison (ILC),
comparison reference value (CRV).
1. INTRODUCTION
The problem of the most adequate method for treatment of experimental data
in the calculation of some representative parameters describing the whole
population is of prime importance in many situations. It includes two essential
steps in judgement:
(i) The critical evaluation of individual results, in order to decide which of
them are part of the distribution and elimination of the outliers;
(ii) The choice of the most convenient method to combine the individual
results, in order to obtain the best evaluation of the mean value of the
final result and its uncertainty.
Rom. Journ. Phys., Vol. 61, Nos. 3–4, P. 697–706, Bucharest, 2016
698
M. Sahagia et al.
2
The most frequent situations found in our field of work are the following:
– Evaluation of Nuclear Decay Data for radionuclides within the
international programs, as the Decay Data Evaluation Program (DDEP), the
Evaluated Nuclear Structure Data File (ENSDF), etc. from published experimental
data, such as: half-life, emitted radiation energies and their intensities, which are
then published in Data Tables [1–3] and recommended to be used by the entire
scientific community in the field.
– Calculation of the Key Comparison Reference Value (KCRV) within the
key and supplementary comparisons, organized within the CIPM-MRA (Mutual
Recognition Arrangement) in order to establish: (a) The reference value for the
response of the International System of Reference (SIR), existing at BIPM and (b)
The degree of equivalence of the National and Designated Metrology Institutes,
signatories of the CIPM-MRA, in order to support their claims for Calibration and
Measurement Capabilities (CMCs) [4]. In these particular cases, many efforts were
done in order to calculate the best KCRV, recognizing its importance on
international scale. Up to now, the responsible persons for evaluation of data
usually processed them in several steps [5]: analysis of the data from the national
participants, application of criteria for identifying outliers, calculation of the
indicators: arithmetic mean, weighted mean, median, weighted median and their
uncertainties. The external and internal uncertainties and their ratio, Birge ratio
[6], were calculated, in order to detect some underestimation or overestimation of
uncertainties. This evaluation, although very comprehensible, involves much effort
and consequently an equilibrated, simplified, method was looked for. The final
calculation of the KCRV was based on the elimination of outliers and calculation
of the arithmetic mean [7]. Recently, the Consultative Committee for Ionizing
Radiations Section II, Measurement of Radionuclides, of the International
Committee of Weights and Measures [CIPM-CCRI(II)], decided to apply the new
concept of power-moderate weighted mean (PMM) in calculation of the Key
Comparison Reference Values (KCRV), standard uncertainties and degrees of
equivalence, Document CCRI(II)/13–37 [8], to be found at the address
http://www.bipm.org. The concept, introduced in the publication EUR 25355 of the
JRC-IRMM, 2012, by S. Pommé [9], takes the advantage of balancing between the
arithmetic and the weighted mean calculations, as an improvement of the MandelPaule mean calculation [10], which in some cases tends to be similar to the
arithmetic mean. The new method was already applied in calculation of the KCRV
in the case of some new key or supplementary comparisons [11, 12].
– Calculation of the consensus value of a reference material as a result of an
intercomparison, on international as well as national scale, such as presented in
[13].
The paper focuses on a similar situation, a comparison between two
Romanian and three Spanish laboratories, aimed to determine the 226Ra activity
concentration of a slag sample, proposed to be used for spiking with a standard
3
Application of the power-moderate weighted mean (PMM) concept
699
226
Ra solution, for which in the published paper [14] we had recommended the use
of the weighted mean for calculation of the reference values. This result will also
be compared with a recently published one regarding the subject [15].
2. PRESENTATION OF THE POWER-MODERATE WEIGHTED MEAN (PMM)
CALCULATION
According to the literature data, the most significant quantities used in
describing a set of N experimental data, with individual values xi, having individual
uncertainties ui, is described by the parameters presented in Table 1 [8].
Table 1
Quantities describing a data set and their expressions
No.
Quantity
Expression

xi
1
Arithmetic mean
x  N
2
Weight of a result
wa i 
3
Standard deviation of the mean
(External)
4
Internal uncertainty of the mean value
5
Weighted mean
6
Weight
7
Standard deviation of the weighted mean
(Internal)
8
Reduced observed chi-squared value
1
N


( x  x )2
u ( x)  N ( Ni 1)

u ( x) 
1
N
u
2
i
x weigh   1i/ u 2i or xweigh  u 2 ( xweigh ) xi / ui2
 i
x / u2
wweigh i 
1/ ui2
u( xweigh )  1 /
~2
 
1
N 1
 (x  x
i
1 / ui2
1 / u
weigh
2
i
) 2 / ui2
Comments on Table 1
When reporting a result as arithmetic (unweighted) mean, it is advisable to
compare the two quantities, rows (3) and (4) in Table 1 and take into account the
maximum of them;
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Normally, the reduced observed chi-squared value, row (8) in Table 1,
should be equal to unity; in the contrary case, mainly when the uncertainties are
underestimated, the set is non consistent.
Application of the exclusion criteria
In the case of discrepant data, it is necessary to reject outlier values, i.e.
those values in the set which do not belong to the distribution, supposed to be
normal, and which could distort the calculation of the mean value. The most used
exclusion criterion is known as the Chauvenet’s criterion [16], [17]. It is based on
the calculation of the ratio:


p  ( xi  x) / u ( x)
(1)
for which an upper limit of acceptability is defined. This parameter is calculated
from the arithmetic mean and its standard deviation. The limit value depends on the
number of data, N. In [16] p is approximated by the relation:
(2)
p  0.91772  1.0168log N .
In the paper [17] it is given as an equivalent table.
Both the arithmetic and weighted means were considered as extreme cases of
calculation and consequently other intermediate calculations were taken into
account. In paper [16] several alternatives are proposed for application, when
discrepant data are to be evaluated, one of them being to use weights according to
relation:
wweigh i /(  wweigh i )  0.5 .
(3)
A well-known, recommended calculation method is that of Mandel-Paule
(M-P) [10], situated between the two extreme calculations, mainly applied when
the set is non consistent, that is when the reduced observed chi-squared value
~2

is higher than unity. In this method, an additional uncertainty “s” is calculated
such as that the new chi squared value should become:
~2
 
1
N 1
 (x  x
i
ref
) 2 /(ui2  s 2 )  1 .
(4)
A new development of the M-P method was done by S. Pommé and Y.
Spasova [18] and published in the concise document EUR 25355 [9]. It consists in
the following: after determining the value of s, the new M-P weight, weighted
mean and its uncertainty are calculated as:
w( M P ) i  [1 /(ui2  s 2 )] /
1 /(u
2
i
 s2 ) ;
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Application of the power-moderate weighted mean (PMM) concept
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x M  P   1i/(u 2i  s2 )  u 2 ( xM P ) u 2xi s 2
i
 i
x /( u 2  s 2 )
and
u ( xM  P )  1 /
1 /(u
2
i
 s2 )
(5)
The concept of power-moderated weighted mean, introduced in [9], is based
on the use of a new parameter, “α”, describing the degree of confidence in the
reported uncertainties “ui”, and reflected in different types of mean evaluation, as
indicated in Table 2.
Table 2
Parameter α values versus evaluation type
α – values
α=0
α=2
α = 2–3/N
Type of evaluation
ui are “noninformative”, nonconfident and not considered –
arithmetic mean
ui are “informative”, fully confident and taken into account –
weighted and
Mandel-Paule mean
ui are only “partially informative”, partially confident and are
considered as being underestimated – calculation of an
intermediary weighted mean.
Further one, Pommé and Spasova propose to calculate the parameter
“characteristic uncertainty per datum” – S, allowing to choose which is the most
convenient uncertainty evaluation mode, according to the relation:

S  N  Max[u 2 ( x), u 2 ( xM P ) ] .
(6)
The relations for calculation of the new proposed calculation mean and
uncertainty, denominated as power-moderate weighted mean (PMM) were deduced
as:
wPMMi  1 / (ui2  s 2 ) S 2 / 1 /(  ui2  s 2 ) S 2
xPMM  u 2 ( xPMM )
xi
( ui2  s 2 ) S 2
and
u( xPMM )  1 /
(
ui2  s 2 )  S  2 .
(7)
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Relations (7) reduce to the arithmetic, respectively weighted means, for the
particular cases: s = 0, α = 0 and respectively s = 0, α = 2.
3. CALCULATION OF THE POWER-MODERATE WEIGHTED MEAN
FOR THE DETERMINATION OF 226Ra CONCENTRATION IN FURNACE SLAG
The data set, which is presented in Table 3, refers to the content of 226Ra in a
slag sample from a Spanish foundry, which was measured by gamma-ray
spectrometry by two Romanian and three Spanish laboratories, with a total of six
results. The individual values were presented in paper [14] and seem to be rather
discrepant and having very different uncertainties, which are due to the difficulty to
measure such low activities
Table 3
Radionuclide activity concentration of the sample, in Bq kg-1, reported by participants
LABORATORY
IFINHH,
μBq
IFINHH,
RML
CIEMAT,
CIEMAT, LMRI
CIEMAT, LMPR
LRA
*
Prompt
222
Rn
emission
Equili
brium
226
Ra222
Rn
Prompt
222
Rn
emission
Equili
brium
226
Ra222
Rn
Equili
brium
226
Ra22
Rn
17.1
14.0
23.5
18.9
–
16.0
13.8
±2.7
±7.0
±14.3
±11.0
±9.0
±0.6
Radionuclide
226
Ra
Method
*
xi, Bq/kg
ui, Bq/kg
* Both laboratories measured the activity from the 186.21 keV Full Absorption Peak (FAP),
corrected for the contribution of the 185.72 keV FAP from 235U.
Applying the Chauvenet criterion for the arithmetic mean
The quantity xi = 23.5 Bq/kg seems to be suspect. Indeed, the value p = 4.67
is superior to p = 1.78 from relation (2) for N = 6 and the quantity is an outlier.
The estimators: mean values and their uncertainties for all modes of
calculation – arithmetic, weighted, Mandel-Paule, power-moderate weighted means
and medians- are compared in Table 4 and Figure 1, for two situations: use of all
reported values and exclusion of the outlier.
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Application of the power-moderate weighted mean (PMM) concept
703
Table 4
Mean values and their uncertainties
No.
Quantity, 226Ra activity concentration,
Bq/kg
Calculated with all
reported values
Exclusion of the value
23.5 Bq/kg
1
Arithmetic mean
17.2
15.96
2
Median
17.1
16.0
3
Standard deviation of the arithmetic
mean
±1.4
±0.96
4
Weighted mean
14.0
14.0
5
Standard deviation of the weighted
mean “internal error (uncertainty)”
±0.6
±0.6
6
Reduced observed chi-squared value
0.42
0.42
7
Paul Mandel mean
s = 0; 14.0
s = 0; 14.0
8
Standard deviation of the Mandel-Paule
mean
±0.6
±0.6
9
S-value;
α-value
3.43; 1.5
2.15;
10
Power-moderated weighted mean
14.3
14.3
11
Standard deviation of the powermoderated weighted mean
0.8
1.1
a.
Fig. 1
1.4
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M. Sahagia et al.
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b.
Fig. 1 (continued) – Graphical representation of data and mean values (a. all reported results;
b. result 3 removed).
By analyzing Table 4 and Fig. 1, the following conclusions can be obtained:
– In all cases the individual uncertainties of the experimental data were not
underestimated.
– When calculating the arithmetic mean, the differences between the mean
of all six experimental data and that without the outlier, xi = 23.5 Bq/kg, are
relatively high. The median values are very close of the arithmetic means.
– In the case of weighted means the difference vanishes as the reported
uncertainty of the value x = = 23.5 Bq/kg, with ui = 14.3 Bq/kg, is high. Both
weighted means are smaller than the arithmetic ones, due to the strong influence of
the lowest value xi = 13.8 Bq/kg, reported with a very low uncertainty, ui = = 0.6
Bq/kg. The two uncertainties are equal, but they seem to be rather small, among
others due to the small predominant uncertainty.
– Due to the correct evaluation of the uncertainties, the reduced observed
chi-squared value was lower than unity and there was not necessary to adjust the
“s” parameter when applying the Mandel-Paule model.
– Application of the power-moderated weighted mean resulted in the
mitigation of the influence of the lowest value, xi = 13.8 Bq/kg, and a small rise of
both mean values. The two calculations, using all results and rejecting the
“outlier”, are equal. The uncertainties are moderately higher as compared with the
weighted means.
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Application of the power-moderate weighted mean (PMM) concept
705
4. COMPARISON WITH OTHER MEASURED VALUES
The calculation of the power moderate weighed mean, due to the moderation
in the use of the weights, provided a value a little higher than the weighted mean,
recommended in the paper [14] to be used as reference value of 226Ra concentration
in slag, namely: x(weigh) = (14.0 ± 0.6) Bq/kg. The new x(PMM) = (14.3 ± 0.8) Bq/kg
proves to be a better estimate, as it accounts for the smaller influences of extreme
values; however, it is not statistically different from the former one.
After the publication of our result, a new determination of the 226Ra
concentration, using another sample prepared from the same batch, was performed
in the underground laboratory HADES in Mol (Belgium), by a common team from
CIEMAT-Spain and JRC-IRMM-Belgium, and was reported in [15]. The new
determined and reported value was (12.8 ± 2.0) Bq/kg, a concentration value in
good agreement with our measured and reported values.
5. CONCLUSIONS
– The new method, power-moderate weighted mean, adopted for the
calculation of the key comparison reference value (KCRV) in international
comparisons, was applied to calculate the 226Ra concentration in a slag sample
within an interlaboratory comparison.
– The method provided satisfactory results in a situation of using discrepant
results, due to the moderation in taking into account of the individual reported
uncertainties.
Acknowledgements. This paper is a continuation of the work deployed within the EURAMET
EMRP Joint Research Project IND04 “MetroMetal” and was supported by the IFIN-HH Research
Project 09 37 02 05/ 2015.
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