Uncertainties from linear least squares calibration

E.3 Uncertainties from linear least squares calibration

E.3.1 An analytical method or instrument is often

var( x pred ) =

calibrated by observing the responses, y, to

var( y ) + x 2 obs pred ⋅ var( b 1 ) + 2 ⋅ x pred ⋅ covar ( b 0 , b 1 ) + var( b 0 )

different levels of the analyte, x. In most cases

this relationship is taken to be linear viz: Eq. E3.3

y=b 0 +b 1 x

Eq. E3.1

and the corresponding uncertainty u(x pred , y) is This calibration line is then used to obtain the

√ var(x pred ).

concentration x pred of the analyte from a sample which produces an observed response y obs from

From the calibration data.

x pred = (y obs –b 0 )/b 1 Eq. E3.2

The above formula for var(x pred ) can be written in terms of the set of n data points, (x i ,y i ), used to

determine the calibration function: weighted or un-weighted least squares regression

It is usual to determine the constants b 1 and b 0 by

on a set of n pairs of values (x

E.3.2 There are four main sources of uncertainty pred − )

2  to consider in arriving at an uncertainty on the

i i ) − ( ∑ w i x i ) ∑ w i )   estimated concentration x pred :

Eq. E3.4 • Random variations in measurement of y,

i ( y i − y f ) affecting both the reference responses y i and

i − y fi ) is the the measured response y obs .

2 where i S = , ( y

• th Random effects resulting in errors in the

residual for the i point, n is the number of data assigned reference values x i .

points in the calibration, b 1 the calculated best fit gradient, w i the weight assigned to y i and

• Values of x i and y i may be subject to a ( x pred − x ) the difference between x pred and the constant unknown offset, for example arising

when the values of x are obtained from serial mean x of the n values x 1 ,x 2 .... dilution of a stock solution

For unweighted data and where var(y obs ) is based • The assumption of linearity may not be valid

on p measurements, equation E3.4 becomes Of these, the most significant for normal practice 2 S  1 1 ( x

pred − x ) 2  x ) = 2 ⋅ + +

 are random variations in y, and methods of

b  p n ( ( x 2 ) − ( x ) 1 2  ∑ i ∑ i n )   estimating uncertainty for this source are detailed here. The remaining sources are also considered

var( pred

Eq. E3.5 briefly to give an indication of methods available.

This is the formula which is used in example 5 x 2 2 with S 2 xx = [ ( i ) − ( x

E.3.3 The uncertainty u(x pred , y) in a predicted

From information given by software used to

value x pred due to variability in y can be estimated

derive calibration curves.

in several ways: Some software gives the value of S, variously

From calculated variance and covariance.

described for example as RMS error or residual

If the values of b 1 and b 0 , their variances var(b 1 ),

standard error. This can then be used in equation

E3.4 or E3.5. However some software may also determined by the method of least squares, the

var(b 0 ) and their covariance, covar(b 1 ,b 0 ), are

give the standard deviation s(y c ) on a value of y variance on x, var(x), obtained using the formula

calculated from the fitted line for some new value in Chapter 8. and differentiating the normal

of x and this can be used to calculate var(x pred ) equations, is given by

since, for p=1

1 ( x − x ) pred 2

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Quantifying Uncertainty Appendix E – Statistical Procedures

giving, on comparison with equation E3.5, terms in the calibration function. Methods of

var(x pred ) = [ s(y c )/ b 1 ]

2 calculating var(x) in these cases are given in

Eq. E3.6

standard texts. It is also possible to make a

E.3.4 The reference values x i may each have judgement based on the size of the systematic uncertainties which propagate through to the final

trend.

result. In practice, uncertainties in these values

E.3.6 The values of x and y may be subject to a are usually small compared to uncertainties in the constant unknown offset (e.g. arising when the system responses y i , and may be ignored. An values of x are obtained from serial dilution of a approximate estimate of the uncertainty u(x pred ,x i ) stock solution which has an uncertainty on its in a predicted value x pred due to uncertainty in a certified value). If the standard uncertainties on y particular reference value x i is and x from these effects are u(y, const) and

u (x, const), then the uncertainty on the where n is the number of x

u (x pred ,x i ) ≈ u(x i )/n

Eq. E3.7

i values used in the

interpolated value x pred is given by:

calibration. This expression can be used to check

2 (x 2

pred ) = u(x, const) +

the significance of u(x pred ,x i ). (u(y, const)/b 2

1 ) + var(x) Eq. E3.8

E.3.5 The uncertainty arising from the assumption

E.3.7 The four uncertainty components described of a linear relationship between y and x is not

in E.3.2 can be calculated using equations normally large enough to require an additional Eq. E3.3 to Eq. E3.8. The overall uncertainty estimate. Providing the residuals show that there arising from calculation from a linear calibration is no significant systematic deviation from this can then be calculated by combining these four assumed relationship, the uncertainty arising from components in the normal way. this assumption (in addition to that covered by the

resulting increase in y variance) can be taken to

be negligible. If the residuals show a systematic trend then it may be necessary to include higher

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Quantifying Uncertainty Appendix E – Statistical Procedures