Documenting uncertainty dependent on analyte level

E.4: Documenting uncertainty dependent on analyte level

E.4.1 Introduction

approximation can be obtained by simple linear regression through four or more

E.4.1.1 It is often observed in chemical

combined uncertainties obtained at different

measurement that, over a large range of analyte

analyte concentrations. This procedure is

levels, dominant contributions to the overall

consistent with that employed in studies of

uncertainty vary approximately proportionately to

reproducibility and repeatability according to

the level of analyte, that is u(x) ∝ x. In such cases

ISO 5725:1994. The relevant expression is

it is often sensible to quote uncertainties as

then u x () ≈ s ' 0 + xs .' 1

relative standard deviations or, for example,

E.4.2.2 The figure can be divided into coefficient of variation (%CV).

approximate regions (A to C on the figure):

E.4.1.2 Where the uncertainty is unaffected by

A : The uncertainty is dominated by the term s 0 , level, for example at low levels, or where a and is approximately constant and close to s 0 . relatively narrow range of analyte level is

involved, it is generally most sensible to quote an

B : Both terms contribute significantly; the absolute value for the uncertainty.

resulting uncertainty is significantly higher than either s 0 or xs 1 , and some curvature is

E.4.1.3 In some cases, both constant and

visible.

proportional effects are important. This section sets out a general approach to recording

C : The term xs 1 dominates; the uncertainty rises uncertainty information where variation of

approximately linearly with increasing x and uncertainty with analyte level is an issue and

is close to xs 1 .

reporting as a simple coefficient of variation is

E.4.2.3 Note that in many experimental cases the inadequate. complete form of the curve will not be apparent.

Very often, the whole reporting range of analyte

E.4.2 Basis of approach

level permitted by the scope of the method falls

E.4.2.1 To allow for both proportionality of within a single chart region; the result is a number uncertainty and the possibility of an essentially

of special cases dealt with in more detail below. constant value with level, the following general

expression is used: E.4.3 Documenting level-dependent

uncertainty data

E.4.3.1 In general, uncertainties can be documented in the form of a value for each of s 0

where and s 1 . The values can be used to provide an

u (x) is the combined standard uncertainty in the uncertainty estimate across the scope of the result x (that is, the uncertainty expressed

method. This is particularly valuable when as a standard deviation)

calculations for well characterised methods are s 0 represents a constant contribution to the

implemented on computer systems, where the overall uncertainty

general form of the equation can be implemented s 1 is a proportionality constant.

independently of the values of the parameters (one of which may be zero - see below). It is

The expression is based on the normal method of accordingly recommended that, except in the combining of two contributions to overall

special cases outlined below or where the

dependence is strong but not linear * , uncertainties constant and one (xs 1 ) proportional to the result.

uncertainty, assuming one contribution (s 0 ) is

Figure E.4.1 shows the form of this expression.

* An important example of non-linear dependence is N OTE : The approach above is practical only where it

the effect of instrument noise on absorbance is possible to calculate a large number of

measurement at high absorbances near the upper limit values. Where experimental study is

of the instrument capability. This is particularly employed, it will not often be possible to

pronounced where absorbance is calculated from establish the relevant parabolic relationship. In

transmittance (as in infrared spectroscopy). Under such circumstances, an adequate

these circumstances, baseline noise causes very large uncertainties in high absorbance figures, and the

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Quantifying Uncertainty Appendix E – Statistical Procedures

are documented in the form of values for a estimates, when required, can then be produced

on the basis of the reported result. This remains term represented by s 1 .

constant term represented by s 0 and a variable

the recommended approach where practical.

E.4.4. Special cases

N OTE : See the note to section E.4.2.

b) Applying a fixed approximation analyte (s 0 dominant)

E.4.4.1. Uncertainty not dependent on level of

An alternative which may be used in general The uncertainty will generally be effectively

testing and where

independent of observed analyte concentration when:

• the dependence is not strong (that is, • evidence for proportionality is weak)

The result is close to zero (for example,

or

within the stated detection limit for the • the range of results expected is moderate method). Region A in Figure E.4.1

• leading in either case to uncertainties which do The possible range of results (stated in the

not vary by more than about 15% from an method scope or in a statement of scope for

average uncertainty estimate, it will often be the uncertainty estimate) is small compared to

reasonable to calculate and quote a fixed value of the observed level.

uncertainty for general use, based on the mean

value of results expected. That is, recorded as zero. s 0 is normally the calculated

Under these circumstances, the value of s 1 can be

either

standard uncertainty.

a mean or typical value for x is used to

E.4.4.2. Uncertainty entirely dependent on

calculate a fixed uncertainty estimate, and this

analyte (s 1 dominant)

is used in place of individually calculated estimates

Where the result is far from zero (for example, above a ‘limit of determination’) and there is

or

clear evidence that the uncertainty changes

a single standard deviation has been obtained, proportionally with the level of analyte permitted

based on studies of materials covering the full within the scope of the method, the term xs 1 range of analyte levels permitted (within the

dominates (see Region C in Figure E.4.1). Under scope of the uncertainty estimate), and there is these circumstances, and where the method scope

little evidence to justify an assumption of does not include levels of analyte near zero, s 0 proportionality. This should generally be

treated as a case of zero dependence, and the simply the uncertainty expressed as a relative

may reasonably be recorded as zero and s 1 is

relevant standard deviation recorded as s 0 . standard deviation.

E.4.5. Determining s 0 and s 1

E.4.5.1. In the special cases in which one term In intermediate cases, and in particular where the

E.4.4.3. Intermediate dependence

dominates, it will normally be sufficient to use the situation corresponds to region B in Figure E.4.1,

uncertainty as standard deviation or relative two approaches can be taken:

standard deviation respectively as values of s 0 and s 1 . Where the dependence is less obvious,

however, it may be necessary to determine s 0 and The more general approach is to determine,

a) Applying variable dependence

s 1 indirectly from a series of estimates of

record and use both s 0 and s 1 . Uncertainty

uncertainty at different analyte levels.

E.4.5.2. Given a calculation of combined

uncertainty from the various components, some of

uncertainty rises much faster than a simple linear

which depend on analyte level while others do

estimate would predict. The usual approach is to

not, it will normally be possible to investigate the

reduce the absorbance, typically by dilution, to bring

dependence of overall uncertainty on analyte level

the absorbance figures well within the working range;

by simulation. The procedure is as follows:

the linear model used here will then normally be adequate. Other examples include the ‘sigmoidal’ response of some immunoassay methods.

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Quantifying Uncertainty Appendix E – Statistical Procedures

1: Calculate (or obtain experimentally) where the uncertainty as standard deviation is uncertainties u(x i ) for at least ten levels x i of

calculated as above, and if necessary expanded analyte, covering the full range permitted.

(usually by a factor of two) to give increased

2 2. Plot u(x 2

i ) against x i confidence. Where a number of results are reported together, however, it may be possible,

3. By linear regression, obtain estimates of m

2 2 and is perfectly acceptable, to give an estimate of and c for the line u(x) = mx +c uncertainty applicable to all results reported.

4. Calculate s 0 and s 1 from s 0 = √ c, s 1 = √ m

E.4.6.2. Table E.4.1 gives some examples. The

5. Record s 0 and s 1 uncertainty figures for a list of different analytes

E.4.6 . Reporting may usefully be tabulated following similar

principles.

E.4.6.1.

The approach outlined here permits estimation of a standard uncertainty for any single

N OTE : Where a ‘detection limit’ or ‘reporting limit’

result. In principle, where uncertainty information

is used to give results in the form “<x” or

is to be reported, it will be in the form of

“nd”, it will normally be necessary to quote the limits used in addition to the uncertainties

[result] ± [uncertainty]

applicable to results above reporting limits.

Table E.4.1: Summarising uncertainty for several samples Situation

Dominant term

Reporting example(s)

Uncertainty essentially constant

Standard deviation: expanded across all results

s 0 or fixed approximation

(sections E.4.4.1. or E.4.4.3.a) uncertainty; 95% confidence

interval

Uncertainty generally xs 1 relative standard deviation; proportional to level

coefficient of variance (%CV) Mixture of proportionality and

(see section E.4.4.2.)

quote %CV or rsd together with lower limiting value for

Intermediate case

lower limit as standard uncertainty

(section E.4.4.3.)

deviation.

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Quantifying Uncertainty Appendix E – Statistical Procedures

Figure E.4.1: Variation of uncertainty with observed result

Uncertainty u(x)

1.6 Uncertainty significantly

Uncertainty

greater than

approximately 1.4 either s 0 or x.s 1 equal to x.s 1