ISO 11843-6:2013
(Main)Capability of detection — Part 6: Methodology for the determination of the critical value and the minimum detectable value in Poisson distributed measurements by normal approximations
Capability of detection — Part 6: Methodology for the determination of the critical value and the minimum detectable value in Poisson distributed measurements by normal approximations
ISO 11843-6:2013 presents methods for determining the critical value of the response variable and the minimum detectable value in Poisson distribution measurements. It is applicable when variations in both the background noise and the signal are describable by the Poisson distribution. The conventional approximation is used to approximate the Poisson distribution by the normal distribution consistent with ISO 11843-3 and ISO 11843-4. The accuracy of the normal approximation as compared to the exact Poisson distribution is discussed.
Capacité de détection — Partie 6: Méthodologie pour la détermination de la valeur critique et de la valeur minimale détectable pour les mesures distribuées selon la loi de Poisson approximée par la loi Normale
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INTERNATIONAL ISO
STANDARD 11843-6
First edition
2013-03-15
Corrected version
2014-08-01
Capability of detection —
Part 6:
Methodology for the determination
of the critical value and the
minimum detectable value in Poisson
distributed measurements by normal
approximations
Capacité de détection —
Partie 6: Méthodologie pour la détermination de la valeur critique et
de la valeur minimale détectable pour les mesures distribuées selon la
loi de Poisson approximée par la loi Normale
Reference number
ISO 11843-6:2013(E)
©
ISO 2013
---------------------- Page: 1 ----------------------
ISO 11843-6:2013(E)
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© ISO 2013
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ii © ISO 2013 – All rights reserved
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ISO 11843-6:2013(E)
Contents Page
Foreword .iv
Introduction .v
1 Scope . 1
2 Normative references . 1
3 Terms and definitions . 1
4 Measurement system and data handling . 1
5 Computation by approximation. 2
5.1 The critical value based on the normal distribution . 2
5.2 Determination of the critical value of the response variable . 4
5.3 Sufficient capability of the detection criterion . 4
5.4 Confirmation of the sufficient capability of detection criterion . 5
6 Reporting the results from an assessment of the capability of detection .6
7 Reporting the results from an application of the method . 6
Annex A (informative) Symbols used in ISO 11843-6 . 7
Annex B (informative) Estimating the mean value and variance when the Poisson distribution is
approximated by the normal distribution . 9
Annex C (informative) An accuracy of approximations .10
Annex D (informative) Selecting the number of channels for the detector .14
Annex E (informative) Examples of calculations .15
Bibliography .20
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ISO 11843-6:2013(E)
Foreword
ISO (the International Organization for Standardization) is a worldwide federation of national standards
bodies (ISO member bodies). The work of preparing International Standards is normally carried out
through ISO technical committees. Each member body interested in a subject for which a technical
committee has been established has the right to be represented on that committee. International
organizations, governmental and non-governmental, in liaison with ISO, also take part in the work.
ISO collaborates closely with the International Electrotechnical Commission (IEC) on all matters of
electrotechnical standardization.
International Standards are drafted in accordance with the rules given in the ISO/IEC Directives, Part 2.
The main task of technical committees is to prepare International Standards. Draft International
Standards adopted by the technical committees are circulated to the member bodies for voting.
Publication as an International Standard requires approval by at least 75 % of the member bodies
casting a vote.
Attention is drawn to the possibility that some of the elements of this document may be the subject of
patent rights. ISO shall not be held responsible for identifying any or all such patent rights.
ISO 11843-6 was prepared by Technical Committee ISO/TC 69, Application of statistical methods,
Subcommittee SC 6, Measurement methods and results.
ISO 11843 consists of the following parts, under the general title Capability of detection:
— Part 1: Terms and definitions
— Part 2: Methodology in the linear calibration case
— Part 3: Methodology for determination of the critical value for the response variable when no calibration
data are used
— Part 4: Methodology for comparing the minimum detectable value with a given value
— Part 5: Methodology in the linear and non-linear calibration cases
— Part 6: Methodology for the determination of the critical value and the minimum detectable value in
Poisson distributed measurements by normal approximations
— Part 7: Methodology based on stochastic properties of instrumental noise
This corrected version of ISO 11843-6:2013 incorporates the following correction: in the key of Figure 1,
the meanings of X and Y have been transposed.
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ISO 11843-6:2013(E)
Introduction
Many types of instruments use the pulse-counting method for detecting signals. X-ray, electron and
ion-spectroscopy detectors, such as X-ray diffractometers (XRD), X-ray fluorescence spectrometers
(XRF), X-ray photoelectron spectrometers (XPS), Auger electron spectrometers (AES), secondary ion
mass spectrometers (SIMS) and gas chromatograph mass spectrometers (GCMS) are of this type. These
signals consist of a series of pulses produced at random and irregular intervals. They can be understood
statistically using a Poisson distribution and the methodology for determining the minimum detectable
value can be deduced from statistical principles.
Determining the minimum detectable value of signals is sometimes important in practical work. The
value provides a criterion for deciding when “the signal is certainly not detected”, or when “the signal is
[1-8]
significantly different from the background noise level” . For example, it is valuable when measuring
the presence of hazardous substances or surface contamination of semi-conductor materials. RoHS
(Restrictions on Hazardous Substances) sets limits on the use of six hazardous materials (hexavalent
chromium, lead, mercury, cadmium and the flame retardant agents, perbromobiphenyl, PBB, and
perbromodiphenyl ether, PBDE) in the manufacturing of electronic components and related goods sold
in the EU. For that application, XRF and GCMS are the testing instruments used. XRD is used to measure
the level of hazardous asbestos and crystalline silica present in the environment or in building materials.
The methods used to set the minimum detectable value have for some time been in widespread use in the
field of chemical analysis, although not where pulse-counting measurements are concerned. The need
[9]
to establish a methodology for determining the minimum detectable value in that area is recognized.
In this part of ISO 11843 the Poisson distribution is approximated by the normal distribution, ensuring
consistency with the IUPAC approach laid out in the ISO 11843 series. The conventional approximation
is used to generate the variance, the critical value of the response variable, the capability of detection
[10]
criteria and the minimum detectability level.
In this part of ISO 11843:
— α is the probability of erroneously detecting that a system is not in the basic state, when really it is
in that state;
— β is the probability of erroneously not detecting that a system is not in the basic state when the
value of the state variable is equal to the minimum detectable value(x ).
d
This part of ISO 11843 is fully compliant with ISO 11843-1, ISO 11843-3 and ISO 11843-4.
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INTERNATIONAL STANDARD ISO 11843-6:2013(E)
Capability of detection —
Part 6:
Methodology for the determination of the critical value
and the minimum detectable value in Poisson distributed
measurements by normal approximations
1 Scope
This part of ISO 11843 presents methods for determining the critical value of the response variable and
the minimum detectable value in Poisson distribution measurements. It is applicable when variations in
both the background noise and the signal are describable by the Poisson distribution. The conventional
approximation is used to approximate the Poisson distribution by the normal distribution consistent
with ISO 11843-3 and ISO 11843-4.
The accuracy of the normal approximation as compared to the exact Poisson distribution is discussed
in Annex C.
2 Normative references
The following documents, in whole or in part, are normatively referenced in this document and are
indispensable for its application. For dated references, only the edition cited applies. For undated
references, the latest edition of the referenced document (including any amendments) applies.
ISO Guide 30, Reference materials - Selected terms and definitions
ISO 3534-1, Statistics — Vocabulary and symbols — Part 1: General statistical terms and terms used in
probability
ISO 11843-1, Capability of detection — Part 1: Terms and definitions
ISO 11843-2, Capability of detection — Part 2: Methodology in the linear calibration case
ISO 11843-3, Capability of detection — Part 3: Methodology for determination of the critical value for the
response variable when no calibration data are used
ISO 11843-4, Capability of detection — Part 4: Methodology for comparing the minimum detectable value
with a given value
3 Terms and definitions
For the purposes of this document, the terms and definitions given in ISO 3534-1, ISO 11843-1,
ISO 11843-2, ISO 11843-3, ISO 11843-4, and ISO Guide 30 apply.
4 Measurement system and data handling
The conditions under which Poisson counts are made are usually specified by the experimental set-up.
The number of pulses that are detected increases with both the time and with the width of the region
over which the spectrum is observed. These two parameters should be noted and not changed during
the course of the measurement.
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ISO 11843-6:2013(E)
The following restrictions should be observed if the minimum detectable value is to be determined
reliably:
a) Both the signal and the background noise should follow the Poisson distributions. The signal is the
mean value of the gross count.
b) The raw data should not receive any processing or treatment, such as smoothing.
c) Time interval: Measurement over a long period of time is preferable to several shorter measurements.
A single measurement taken for over one second is better than 10 measurements over 100 ms each.
The approximation of the Poisson distribution by the normal distribution is more reliable with
higher mean values.
d) The number of measurements: Since only mean values are used in the approximations presented
here, repeated measurements are needed to determine them. The power of test increases with the
number of measurements.
e) Number of channels used by the detector: There should be no overlap of neighbouring peaks. The
number of channels that are used to measure the background noise and the sample spectra should
be identical (Annex D, Figure D.1).
f) Peak width: The full width at half maximum (FWHM) is the recommended coverage for monitoring a
single peak. It is preferable to measurements based on the top and/or the bottom of a noisy peak. The
appropriate FWHM should be assessed beforehand by measuring a standard sample. An identical
value of the FWHM should be used for both the background noise and the sample measurements.
Additional factors are: the instrument should work correctly; the detector should be operating within
its linear counting range; both the ordinate and the abscissa axes should be calibrated; there should
be no signal that cannot be clearly identified as not being noise; degradation of the specimen during
measurement should be negligibly small; at least one signal or peak belonging to the element under
consideration should be observable.
5 Computation by approximation
5.1 The critical value based on the normal distribution
The decision on whether a measured signal is significant or not can be made by comparing the arithmetic
mean y of the actual measured values with a suitably chosen value y . The value y , which is referred
g c c
to as the critical value, satisfies the requirement
Py()>=yx 0 ≤α (1)
gc
where the probability is computed under the condition that the system is in the basic state (x = 0) and α
is a pre-selected probability value.
Formula (1) gives the probability that yy> under the condition that:
gc
11
yy=±z σ + (2)
cb 1b−α
JK
where
is the (1 − α)-quantile of the standard normal distribution where 1 − α is the confidence
z
1−α
level;
σ is the standard deviation under actual performance conditions for the response in the
b
basic state;
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ISO 11843-6:2013(E)
y is the arithmetic mean of the actual measured response in the basic state;
b
J
is the number of repeat measurements of the blank reference sample. This represents the
value of the basic state variable;
K
is the number of repeat measurements of the test sample. This gives the value of the actual
state variable.
The + sign is used in Formula (2) when the response variable increases as the state variable increases.
The − sign is used when the opposite is true.
The definition of the critical value follows ISO 11843-1 and ISO 11843-3. Its relationship to the measured
values in the active and basic states is illustrated in Figure 1.
Y
α
y
g
y
c
y
b
β
0
x x , x
c g
X
Key
X state variable
Y response variable
α the probability that an error of the first kind has occurred
β the probability that an error of the second kind has occurred
Figure 1 — A conceptual diagram showing the relative position of the critical value and the
measured values of the active and basic states
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ISO 11843-6:2013(E)
5.2 Determination of the critical value of the response variable
If the response variable follows a Poisson distribution with a sufficiently large mean value, the standard
deviation of the repeated measurements of the response variable in the basic state is estimated as y .
b
This is an estimate of σ . The standard deviation of the repeated measurements of the response variable
b
in the actual state of the sample is y , giving an estimate of σ (see Annex B).
g g
The critical value, y , of a response variable that follows the Poisson distribution approximated by the
c
normal distribution generally satisfies:
11 11
yy=+z σ +≈ yz++y (3)
cb 1−−ααbb 1b
JK JK
where
is the arithmetic mean of the actual measured response in the basic state.
y
b
5.3 Sufficient capability of the detection criterion
The sufficient capability of detection criterion enables decisions to be made about the detection of a
signal by comparing the critical value probability with a specified value of the confidence levels, 1−β .
If the criterion is satisfied, it can be concluded that the minimum detectable value, x , is less than or
d
equal to the value of the state variable, x . The minimum detectable value then defines the smallest
g
value of the response variable, η , for which an incorrect decision occurs with a probability, β . At this
g
value, there is no signal, only background noise, and an ‘error of the second kind’ has occurred.
If the standard deviation of the response for a given value x is σ , the criterion for the probability to
g g
be greater than or equal to 1−β is set by inequality (4), from which inequalities (5) and (6) can be
derived:
11
2 2
ησ≥+yz + σ (4)
gc 1−β b g
JK
11
If y is replaced by yz=+ησ + , defined in Formulae (2) and (3), then:
c cb 1b−α
JK
11 11
2 2
ηη−≥z σσ++z + σ (5)
gb 11−−αβbb g
JK JK
where
α
is the probability that an error of the first kind has occurred;
β
is the probability that an error of the second kind has occurred;
η is the expected value under the actual performance conditions for the response in the basic
b
state;
η is the expected value under the actual performance conditions for the response in a sample
g
with the state variable equal to x .
g
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ISO 11843-6:2013(E)
With βα= and KJ= , the criterion simplifies to:
1
2 2
ηη−≥ z 2σσ++σ (6)
gb 1−α bb g
J
If σ is replaced with an estimate of y following 5.2 and similarly σ is replaced with an estimate
b b g
of y (see Annex B), the criterion becomes inequality (7).
g
1
ηη−≥ z 2yy++ y (7)
()
gb 1−α bb g
J
NOTE When validating a method, the capability of detection is usually determined for KJ 1 in accordance
with ISO 11843-4.
5.4 Confirmation of the sufficient capability of detection criterion
The standard deviations and expected values of the response are usually unknown, so an assessment
using criterion inequality (6) has to be made from the experimental data. The expression on the left-
hand side of the simplified criterion inequality (6) is unknown, whereas that on the right-hand side is
known.
A confidence interval of ηη− is provided by N repeated measurements in the basic state and N
gb
repeated measurements of a sample with the state variable equal to x . A 100 12−α/% confidence
()
g
interval for ηη− is:
gb
11 11
2 2 2 2
()yy−−z σσ+≤ηη−≤()yy−+z σ + σσ (8)
gb (/12−−αα)(b g gb gb 12/) b g
NN NN
where z is the 100 12−α/ quantile of the standard normal distribution.
()
(/12−α )
To confirm the sufficient capability of detection criterion, a one-sided test is used. With βα= ,
100 1−α % of the one-sided lower confidence bound on ηη− is:
()
gb
11
2 2
ηη−≥()yy−−z σσ+ (9)
gb gb ()1−α b g
NN
where
is the number of replications of measurements of each reference material used to assess the
N
capability of detection;
y is the arithmetic mean of the actual measured response in a sample with the state variable
g
equal to x ;
g
η is the expected value under actual performance conditions for the response in the basic state;
b
η is the expected value under actual performance conditions for the response in a sample with
g
the state variable equal to x .
g
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= =
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ISO 11843-6:2013(E)
The one-sided lower confidence bound on η − η of inequality (9) is compared to the right-hand side of
g b
inequality (6), giving:
11 1
2 2 2 2
ηη−=()yy−−z σσ+≥z 2σσ++σ (10)
gb gb ()11−−ααb g () bb g
NN J
An approximate 100 1−α % lower confidence limit T for ηη− is obtained by replacing σ and σ
()
0 gb b g
with y and y , respectively, as defined in Formula (3) and inequality (7):
b g
1
Ty=−()yz−+yy (11
...
DRAFT INTERNATIONAL STANDARD ISO/DIS 11843-6
ISO/TC 69/SC 6 Secretariat: JISC
Voting begins on Voting terminates on
2010-11-22 2011-04-22
INTERNATIONAL ORGANIZATION FOR STANDARDIZATION • МЕЖДУНАРОДНАЯ ОРГАНИЗАЦИЯ ПО СТАНДАРТИЗАЦИИ • ORGANISATION INTERNATIONALE DE NORMALISATION
Capability of detection —
Part 6:
Methodology for the determination of the critical value and the
minimum detectable values in Poisson distribution
measurements
Capacité de détection —
Partie 6: Méthodologie de détermination de la valeur critique et des valeurs minimales détectables dans les
mesures de distribution selon la loi de Poisson
ICS 03.120.30; 17.020
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secretariat. ISO Central Secretariat work of editing and text composition will be undertaken at
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© International Organization for Standardization, 2010
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ISO/DIS 11843-6
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ii © ISO 2010 – All rights reserved
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ISO/WD 11843-6
Contents Page
Foreword .v
Introduction.vi
1 Scope.1
2 Normative references.1
3 Terms and definitions .1
4 Symbols.1
5 Measurement system and data Handling .1
6 Computation by the simple approximation(Annex B.1) .2
6.1 General
6.2 Computation of the critical value of the response variable by the simple approximation.2
6.3 The Sufficient capability of detection criterion by the simple approximation.4
6.4 Confirmation of the sufficient capability of detection criterionby the simple
approximation .5
7 Computation by the square root transformation approximation(Annex B.2) .6
7.1 Computation of the critical value of the response variable by the square root
transformation approximation .6
7.2 The Sufficient capability of detection criterion by the square root transformation
approximation.6
7.3 Confirmation of the sufficient capability of detection criterion by the square root
transformation approximation .6
8 Directions for use two approximations.7
9 Reporting of results from an assessment of the capability of detection .7
10 Reporting of results from an application of the method .8
Annex A (normative) Symbols used in ISO 11843, Part 6 .9
Annex B (informative) Approximations of the Poisson distribution to the Normal distribution .10
B.1 Simple aapproximation .10
B.2 Square root transformation approximation.10
Annex C (informative) Application limits-the accuracy of the approximations.12
Annex D (informative) Measurement conditions for pulse counting method.13
Annex E (informative) Examples of calculations .13
Bibliography.17
Figures
Figure 1 Relation between the critical value of the response variable and the minimum
detectable value of the state variable.4
Figure B.1 Variance produced by square-root transformations…………………………………………….11
Figure C.1(a) Difference between the Poisson distribution and Normal distribution.……….………….12
Figure C.1(b) Difference in probability between the Poisson distribution and Normal distribution ….12
Figure D.1 Specification of Signal Regions … .……………………………………………………………….13
Figure E.1 Carbon 1s spectrum and measured response variables.………….………………………….…14
Figure E.2 A powder XRD scan from Chrysotile asbestos as a plot of scattering intensity vs. the
scattering angle 2θ .……………………………………….…………………………………………….15
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ISO/WD 11843-6
iv © ISO 2002 – All rights reserved
DRAFT 2010
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ISO/WD 11843-6
Foreword
ISO (the International Organization for Standardization) is a worldwide federation of national standards bodies
(ISO member bodies). The work of preparing International Standards is normally carried out through ISO
technical committees. Each member body interested in a subject for which a technical committee has been
established has the right to be represented on that committee. International organizations, governmental and
non-governmental, in liaison with ISO, also take part in the work. ISO collaborates closely with the
International Electrotechnical Commission (IEC) on all matters of electrotechnical standardization.
International Standards are drafted in accordance with the rules given in the ISO/IEC Directives, Part 2.
The main task of technical committees is to prepare International Standards. Draft International Standards
adopted by the technical committees are circulated to the member bodies for voting. Publication as an
International Standard requires approval by at least 75 % of the member bodies casting a vote.
Attention is drawn to the possibility that some of the elements of this document may be the subject of patent
rights. ISO shall not be held responsible for identifying any or all such patent rights.
ISO 11843-6 was prepared by Technical Committee ISO/TC 69, Application of statistical methods,
Subcommittee SC 6, Measurement methods and results.
ISO 11843 consists of the following parts, under the general title Capability of detection:
⎯ Part 1: Terms and definitions
⎯ Part 2: Methodology in the linear calibration case
⎯ Part 3: Methodology for determination of the critical value for the response variable when no calibration
data are used
⎯ Part 4: Methodology for comparing the minimum detectable value with given value
⎯ Part 5: Methodology in the linear and non-linear calibration cases
⎯ Part 6: Methodology for the determination of the critical value and the minimum detectable values in
Poisson distribution measurements
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ISO/WD 11843-6
Introduction
Many types of instruments use the pulse-counting method for detecting signals. X-ray, electron and ion-
spectroscopy detectors, such as X-ray diffractometer (XRD), X-ray fluorescence spectrometer (XRF), X-ray
photoelectron spectrometer (XPS), Auger electron spectrometer (AES), and secondary ion mass spectrometer
(SIMS), are of this type. These signals consist of a series of pulses produced at random and irregular intervals
that can be understood statistically using a Poisson distribution. Since the variation in a signal follows a
Poisson distribution, a methodology for determining the minimum detectable value can be deduced from
statistical principles.
Determining the minimum detectable value of signals is sometimes important in practical work. It provides a
criterion for deciding when “the signal is certainly not detected”, or when “the signal is significantly different
from the background noise level" [1-8]. For example, it is valuable when measuring the presence of hazardous
substances or surface contamination of a semi-conductor materials. It also allows the condition of an
analyzer to be quantified when assessing the limiting performance of an instrument.
The methods used to set the minimum detectable value have for some time been in widespread use in the
field of chemical analysis, although not where pulse-counting measurements are involved. The need to
establish a methodology to determine the minimum detectable value in that area is recognized [9].
This part of ISO 11843 considers how to approximate the Poisson distribution to the Normal distribution,
ensuring consistency with the IUPAC approach and, particularly, with the series of ISO 11843 [10] standards.
Two types of approximation, the Simple approximation and Square Root approximation, are used for this
purpose. Definitions are given for the variance, for the critical value of the response variable, for the capability
of detection criteria and for the minimum detectability level [17].
The methods considered in this standard should be applied only when the mean value of Poisson counts is
not too small.
In this part of ISO 11843
the probability isα of detecting (erroneously) that a system is not in the basic state when it is in the basic
state;
the probability isβof (erroneously) not detecting that a system, for which the value of the net state
variable is equal to the minimum detectable value( x ) is not in the basic state.
d
This standard is fully compliant with ISO 11843, part 1, 3 and 4.
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WORKING DRAFT I ISO/WD 11843-6
Capability of detection —
Part 6:
Methodology for the determination of the critical value and the
minimum detectable values in Poisson distribution
measurements
1 Scope
This part of ISO 11843 specifies statistical values for assessing the capability of detection in Poisson
distribution measurements and whether a specific signal is significant or not. It is applicable when variations in
both the background noise and the signal are describable by the Poisson distribution. Two types of
approximation, the Simple approximation and the Square Root approximation, are used to approximate the
Poisson distribution to the Normal distribution, and are used to determine the variance, the critical value of the
response variable, and the minimum detectable value. This part of ISO 11843 should be applied to a sufficient
number of counts of the response variable.
2 Normative references
The following documents are indispensable for the application of this document. For dated references, only
the edition cited is applicable. For undated references, the latest edition of the reference document (including
any amendments) is applicable.
ISO 11843-1:1997, Capability of detection - Part 1: Terms and definitions
ISO11843-3: 2003, Capability of detection - Part 3: Methodology for the determination of the critical value for
the response variable when no calibration data are used
ISO11843-4: 2003, Capability of detection - Part 4: Methodology for comparing the minimum detectable value
with a given value
3 Terms and definitions
For the purposes of this document, the terms and definitions given in ISO 3534 (all parts), ISO 5479, ISO
5725-2, ISO 11095, ISO 11843-1 and ISO Guide 30 apply.
4 Symbols
The meanings of the symbols used here are given in Annex A.
5 Measurement system and data handling
The conditions under which Poisson distribution measurement is made are usually specified by the
experimental set-up. The number of pulses that are detected increases with both the time and with the width
of the region over which the spectrum is being observed. These two parameters should therefore be noted
and not changed during the course of the measurement.
The following conditions are essential for determining the minimum detectable value:
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(1) Both the signal and the background noise should have Poisson distributions.
(2) The spectra under consideration should not have received any data treatment, such as smoothing.
(3) Time interval: A single measurement over a longer period of time is preferable to several shorter
measurements. A measurement taken for over one second is better than 10 measurements over 100
milliseconds each. The approximation of the Poisson distribution by the Normal distribution is more reliable
with higher mean values.
(4) The number of measurements: Since only mean values are used in the approximations presented here,
repeated measurements are needed to determine them. Moreover, the detectability of the minimum value
increases with the number of measurements.
(5) Number of channels used by the detector: There should be no overlap of neighboring peaks. The number
of channels used to measure the background noise and sample spectra should be identical ( Annex D, Figure
4).
(6) Peak width: The full width at half maximum (FWHM) is the recommended coverage for monitoring a
single peak, and is preferable to measurements based on the top and/or bottom of the peak with noise. The
appropriate FWHM should be determined beforehand by measurements on a standard sample. An identical
value of the FWHM width should be adopted for both background noise and sample measurements.
NOTE Additional conditions for the interpretation of the data are: The instrument works correctly. The
detector works in the linear counting range. Both the ordinate and the abscissa axes are calibrated. There is
no signal that cannot be identified, except statistical fluctuation. Degradation of the specimen during
measurement is negligibly small. At least one signal or peak that belongs to an element can be observed.
6 Computation by the simple approximation (Annex B.1)
6.1 General
The critical value, y ,is defined as the value of the response variable, y ,if the system is considered to move
c
from a dormant or ‘basic’ state to an ‘active’ state. Its value is chosen so that, when it is in the ‘basic’ state, the
system is considered to be ‘active’ with only a small probability,α. In other words, y is the lowest significant
c
value of the measurement or signal that can be discriminated from the background noise [5,6,10].
The decision about whether the measured signal is ‘significant’ or ‘not significant’ is made by comparing the
arithmetic mean of the actual measured values, y ,with the critical value, y . The probability that y exceeds
a c a
the critical value, y , for the distribution in the basic state should be less than or equal to an appropriate pre-
c
selected probability,α .
The critical value, y , of the response variable generally satisfies
c
P( y > y x = 0 ) ≤ α , (1)
a c
where
x a value of net state variable;
y a value of response variable;
y arithmetic mean of measured responses of an actual states;
a
α the probability that an error of the first kind has occurred : the probability of rejecting the null
hypothesis η =η when the null hypothesis is true;
g b
η the expected value under the actual performance conditions for the responses of the basic state;
b
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η the expected value under the actual performance conditions for the responses of a sample with
g
the net state variable equal to x .
g
6.2 Computation of the critical value of the response variable by the simple
approximation
Eq. (1) gives the probability that y > y under the condition that x = 0 . If the response variable follows a
c
a
Poisson distribution with a large mean value,λ ,and the standard deviation,σ ,equals to λ , and then the
0
critical value of the response variable is given by the following simplified version of Eq. (1):
1 1
y = y ± z σ + . (2)
c b 1−α 0
J K
Also, ifσ is replaced by y , and then
0 b
1 1
y = y ± z y + , (3)
c b 1-α b
J K
where
z (1−α)-quantile of the standard normal distribution;
1−α
σ actual standard deviation at zero level of state variable;
0
y observed mean response of the basic state;
b
J number of replication of measurements on the reference material representing the value of the net
state variable (blank sample) in an application of the method;
K number of replication of measurements on the actual state (test sample) n an application of the
method.
Note that the + sign is used when the response variable increases with increasing level of the net state
variable and the - sign is used when the response variable decreases with increasing level of the net state
variable. The + sign is preferable in Poisson distribution measurements as the response variable increases
with increasing level of the net state variable. In general, when the value of the net state variable in the basic
state (the baseline for the response variable) is known without a significant error, then σ =σ . The latter is
0 b
estimated through y , the standard deviation of the repeated measurements of the response variable in the
b
basic state. As mentioned in the simple approximation section, y gives an estimate of σ . The concept of
b b
the critical value is illustrated in Figure 1.
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ErErErrorororrr o o offf t t thhheee
fififirst kindrst kindrst kind
ααα
yyy
ggg
CriCriCriticaticaticalll val val valuuue e e
of the respof the respof the respooonnnse se se
yyy
ccc
varvarvariiiababablelele
yyy
ErrErrErrooor or or of f f ttthhhe e e
bbb
βββ
sssecoecoecondndnd k k kiiindndnd
MMMiiinnniiimumumum dm dm deeettteeeccctttaaabbbllleee
value of stavalue of stavalue of stattte vae vae varrriiiablablableee
CriCriCriticaticaticalll v v vaaallluuue e e ooofff
stastastattte vare vare variiiaaabbblelele
000
xxx
StateStateState va va variririababablelele
xxx
ccc xx ,,
gg ddd
Figure 1 Relation between the critical value of the response variable and the minimum detectable
value of the state variable [10].
6.3 Sufficient capability of the detection criterion by the simple approximation
The sufficient capability of detection criterion enables decisions to be made about the detection of a signal by
comparing the critical value probability with a specified value,1−β . If the criterion is satisfied, it may be
concluded that the minimum detectable value, ,is less than or equal to the value of the net state
x
d
variable, x .The minimum detectable value then defines the smallest expectation of the response
g
variable,η ,for which an incorrect decision occurs with a probability,β, that there is no signal, but only
g
background noise. This is termed an ‘error of the second kind’ [10].
If the standard deviation of the responses for a given value x is σ , the criterion for the probability to be
g g
greater than or equal to1−β is defined by inequality (4), and it is actually given by inequalities (5) and (6):
1 1
2 2
η ≥ y + z σ + σ , (4)
g c 1−β b g
J K
1 1
If y is replaced byη + z σ + , defined in Eq. (2), then
c b 1−α b
J K
1 1 1 1
2 2
η − η ≥ z σ + + z σ + σ , (5)
g b 1−α b 1− β b g
J K J K
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where1−α and1−β are the confidence levels.
Withβ =α , K = J and under the assumption thatσ ≥σ (it is unusual for the standard deviation to decrease
g b
as the net state variable increases), the criterion is simplified to
η − η
2 z
g b
1−α
≥ . (6)
2 2
J
σ +σ
b g
The relation between the critical value of the response variable and the minimum detectable value of net state
variable is shown in Figure 1.
6.4 Confirmation of the sufficient capability of detection criterion by a simple
approximation
The standard deviations and expected values of the responses are usually unknown, so an assessment using
criterion inequality (6) has to be made for the experimental data. The expression on the left-hand side of the
simplified criterion inequality (6) is unknown, whereas that on the right-hand side is known.
From a validation experiment with N measurements of the basic state and N measurements of a sample with
the net state variable equal to x , one can obtainη − η on the left hand side of inequality (6). The
g g b
difference in the means is then represented by:
1 1 1 1
2 2 2 2
( y − y ) − z σ + σ ≤ η − η ≤ ( y − y ) + z σ + σ . (7)
g b 1−α b g g b g b 1−α b g
N N N N
With β =α ,also under the assumption thatσ ≥σ , inequality (7) is rearranged as
g b
y − y η − η y − y
1 1
g b g b g b
. (8)
− z ≤ ≤ + z
1−α 1−α
2 2 2 2 2 2
N N
σ +σ σ +σ σ +σ
b g b g b g
The smaller expression on the left-hand side of inequality (8) is chosen and compared to the right -hand side
of inequality (6), giving
η − η y − y
2 z
1
g b g b
1−α
≥ − z ≥ . (9)
1− α
2 2 2 2
N
N
σ +σ σ +σ
b g b g
2 2
An approximate 100(1−α) % lower confidence limit (CL) for (η −η ) / σ +σ is given by
b g b g
y − y
z
g b
1−α
CL = − , (10)
y + y N
g b
where
N number of repeated measurements on each reference material in assessment of the capability of
detection;
y observed mean response of a sample with the net state variable equal to x .
g g
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2 2
If the lower confidence limit of satisfies Criterion inequality (6), it is concluded that the
(η −η )/ σ +σ
b g b g
minimum detectable value, x , is less than or equal to x . Note that, for relatively large values of N , the lower
d g
confidence limit Eq. (10) will suffice.
7 Computation by the square-root transformation approximation (Annex B.2)
7.1 Computation of the critical value of the response variable by the square-root
transformation approximation
If 2 y is a realization of a normal variable with standard deviationσ ,then the critical value of the response
i 0
y
variable, , is given by the following simplified version of Eq. (2):
c
1 1 1 1
2
2 y = 2 y ± z σ + = 2 y ± z + (since σ ≅ 1,σ ≅ 1). (11)
c b 1−α 0 b 1−α 0 0
J K J K
Note that the + sign is used when the response variable increases with increasing level of the net state
variable and the - sign is used when the response variable decreases with increasing level of the net state
variable. The + sign is preferable in Poisson distribution measurements as the response variable increases
with increasing level of the net state variable.
7.2 Sufficient capability of detection criterion by the square-root transformation
If the standard deviation of the response for a given value x of the net state variable isσ , the criterion for the
g g
probability to be greater than or equal to1−β is given by:
1 1
2 2
2 η ≥ 2 y + z σ + σ . (12)
g c 1− β b g
J K
1 1
If 2 y is replaced by 2 η + z σ + defined in Eq. (2), then
b 1−α b
c
J K
1 1 1 1
2 2
2 η − 2 η ≥ z σ + + z σ + σ . (13)
g b 1 − α b 1 − β b g
J K J K
Withβ =α , K = J ,and under the assumption thatσ ≥σ , the criterion is simplified to
g b
1
η − η ≥ 2 z . (14)
g b 1 − α
J
7.3 Confirmation of the sufficient capability of detection criterion by the square-root
transformation
An assessment using criterion inequality (14) has to be made of the experimental data in the same way as for
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the Simple approximation.
Using a validation experiment with N measurements of both the basic state and a sample with the net state
variable, one gets a range for the interval on the left-hand side of criterion inequality (14) within
which z andα can be expected to lie with a pre-specified probability,1−α .
1−α
1 1 1 1
2 2 2 2
(2 y − 2 y ) − z σ + σ ≤ 2 η − 2 η ≤ (2 y − 2 y ) + z σ + σ (15)
g b 1−α b g g b g b 1−α b g
N N N N
2
Withβ =α and under the assumptions thatσ ≥σ and that the variance is stabilized to 1 (σ ≅ 1 in the
g b 0
square root transformation), then the bound of interval estimate becomes also
1 1
(2 y − 2 y ) − z (1+1) ≤ 2 η − 2 η ≤ (2 y − 2 y ) + z (1+ 1). (16)
g b 1−α g b g b 1−α
N N
The smaller expression on the left-hand side of inequality (16) is chosen and compared to the constant
criterion value of inequality (14) as follows:
1 1
η − η ≥ y − y − z ≥ 2z . (17)
g b g b 1−α 1−α
2N N
An approximate 100(1−α)% lower confidence l
...
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