New calibration procedure in
analytical chemistry in agreement
to VIM 3
Miloslav Suchanek
ICT Prague and EURACHEM
Czech Republic
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Prague castle
and
Vltava river
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Overview
- New definition of calibration
- Theoretical backround of various
calibration methods
- Practial calculation with MS Excel
- Do we need measurement uncertainty?
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Terminology
x, independent variable
y, dependent variable


c, concentration, content
y, Y, indication, signal
Measurement in chemistry:
calibration of a measurement procedure
not calibration of an instrument
Result :
quantity value ± expanded measurement uncertainty
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ISO/IEC Guide 99:2008
International vocabulary of metrology (VIM 3)
2.39 calibration
operation that, under specified conditions, in a first step,
1) established a relation between the quantity values with
measurement uncertainties provided by measurement
standards and corresponding indications with associated
measurement uncertainties and, in a second step,
2) uses this information to establish a relation for obtaining a
measurement result from an indication
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Calibration models
x
u(x)
y
u(y)
Ordinary linear regression


Bivariate regression




Monte Carlo simulation




Bracketing




x – concentration, content; y – indication, signal
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Ordinary regression cannot be used!

underestimation of measurement
uncertainty
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Solution:
1. Least square analysis with uncertainties in both variables bivariate (bilinear) regression
2. Monte Carlo simulation (regression) (MCS)
3. Bracketing calibration
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Bivariate (bilinear) regression – theory
(J.M. Lisy et.all: Computers Chem. 14, 189, 1990)
Task:
Estimate the parameters of linear equation y = b1 + b2.x
providing that experimental data have a structure:
xi  u(xi)
and
yi  u(yi)
(u(xi) and u(yi) are standard uncertainties)
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Solution:
U   wRi ( yi  f ( xi , b j ))2
N
j = 1,2; N is the number of experimental points
wRi  1/ u
2
Ri
Ri  yi  f ( xi ,bj )
uRi2  uyi2  b22 .uxi2
U   wRi .Ri2
See EXCEL calculations
N
Parameters of linear model are estimated iteratively
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The Monte Carlo steps
1. Each calibration point is characterised by
{xi  u(xi), yi  u(yi) } assumed to be normally distributed {N(xi, u2(xi)),
N(yi, u2(yi)}
2. Replace each calibration point by a randomly selected point (j) {xi(j),
yi(j)}
3. Perform a (simple) Linear Regression using the « new » calibration
dataset (j)
4. Derive the slope and intercept of calibration (j): b2(j), b1(j)
5. Repeat the sequence (e.g. 1000 times)
6. Compute the average and standard deviation of all b2(j), b1(j) to obtain
the slope b2 and intercept b1, respectively.
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The Monte Carlo calculation
provides reliable results
compliant with GUM (ISO/IEC Guide 98-3:2008)
easy to implement in a spreadsheet
See EXCEL calculations
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Bracketing calibration
Model equation
c2 .(Yx  Y1 )  c1.(Y2  Yx )
cx 
(Y2  Y1 )
concentration of analyte in sample
concentration of analyte in standards
(one below and one above concn. in sample)
signals corresponding to the analyte concns.
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See EXCEL calculations
cx
c1, c2
Y1, Y2, Yx
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c[mg/L]
10
20
30
40
50
sample
u( c)
0,3
0,6
0,9
1,2
1,5
A
0,117
0,208
0,304
0,403
0,506
0,252
u(A)
0,005
0,005
0,007
0,005
0,006
Rsc
3,0%
3,0%
3,0%
3,0%
3,0%
RsA
4,3%
2,4%
2,3%
1,2%
1,2%
0,007
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5 points calibration
Absorbance (a.u.)
0,5
0,4
0,3
0,2
0,1
10
20
30
40
50
concentration (mg/L)
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BIVARIATE REGRESSION
GOTO EXCEL
RESULT
X(sample)
24,25
u (k=1)
0,75
Rsu
3,10%
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Monte Carlo simulation
GOTO EXCEL
RESULT
X(sample)
24,28
u (k=1)
0,83
Rsu
3,40%
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The simulated dataset
0,6
Absorbance (a.u.)
0,5
0,4
0,3
0,2
0,1
0,0
0
10
20
30
40
50
60
concentration (mg/L)
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Bracketing
GOTO EXCEL
RESULT
X(sample)
24,58
u (k=1)
1,00
Rsu
4,05%
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Conclusions
Sample
value, c u(c)
Rsu
Ordinary linear regression 24,29
0,48
2,0%
Bivariate linear regression 24,25
0,75
3,1%
Monte Carlo simulation
24,28
0,84
3,4%
Bracketing
24,58
1,00
4,1%
Measurement uncertainty is the most important in decision
making process!
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L
Measurement result
with 95% probability
below limit
Measurement result
with 95% probability
over limit
5%
u
u is the procedure characterization!
u
5%
results
L-1.64*u
acceptance area
L+1.64*u
¿ grey zone ?
rejection area
3.28 * u
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Thank you!
miloslav.suchanek@vscht.cz
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