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 3T&M Conference 2010, SA

Overview - New definition of calibration - Theoretical backround of various calibration methods - Practial calculation with MS Excel - Do we need measurement uncertainty? 4T&M Conference 2010, SA

Terminology x, independent variable  c, concentration, content y, dependent variable  y, Y, indication, signal Measurement in chemistry: calibration of a measurement procedure not calibration of an instrument 5T&M Conference 2010, SA Result : quantity value ± expanded measurement uncertainty

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 6T&M Conference 2010, SA

xu(x)yu(y) Ordinary linear regression  Bivariate regression  Monte Carlo simulation  Bracketing  Calibration models x – concentration, content; y – indication, signal 7T&M Conference 2010, SA

Ordinary regression cannot be used!  underestimation of measurement uncertainty Ordinary regression cannot be used!  underestimation of measurement uncertainty 8T&M Conference 2010, SA

Solution: 1. Least square analysis with uncertainties in both variables - bivariate (bilinear) regression 2. Monte Carlo simulation (regression) (MCS) 3. Bracketing calibration 9T&M Conference 2010, SA

Bivariate (bilinear) regression – theory (J.M. Lisy et.all: Computers Chem. 14, 189, 1990) Task: Estimate the parameters of linear equation y = b 1 + b 2.x providing that experimental data have a structure: x i  u(x i ) and y i  u(y i ) (u(xi) and u(yi) are standard uncertainties) 10T&M Conference 2010, SA

Solution: j = 1,2; N is the number of experimental points Parameters of linear model are estimated iteratively 11T&M Conference 2010, SA See EXCEL calculations

1.Each calibration point is characterised by {x i  u(x i ), y i  u(y i ) } assumed to be normally distributed {N(x i, u 2 (x i )), N(y i, u 2 (y i )} 2.Replace each calibration point by a randomly selected point (j) {x i (j), y i (j)} 3.Perform a (simple) Linear Regression using the « new » calibration dataset (j) 4.Derive the slope and intercept of calibration (j): b 2 (j), b 1 (j) 5.Repeat the sequence (e.g times) 6.Compute the average and standard deviation of all b 2 (j), b 1 (j) to obtain the slope b 2 and intercept b 1, respectively. The Monte Carlo steps 12T&M Conference 2010, SA

The Monte Carlo calculation provides reliable results compliant with GUM (ISO/IEC Guide 98-3:2008) easy to implement in a spreadsheet 13T&M Conference 2010, SA See EXCEL calculations

Bracketing calibration Model equation concentration of analyte in samplecxcx concentration of analyte in standardsc 1, c 2 (one below and one above concn. in sample) signals corresponding to the analyte concns.Y 1, Y 2, Y x 14T&M Conference 2010, SA See EXCEL calculations

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5 points calibration 16T&M Conference 2010, SA

BIVARIATE REGRESSION GOTO EXCEL 17T&M Conference 2010, SA X(sample)u (k=1)Rsu 24,250,753,10% RESULT

18T&M Conference 2010, SA Monte Carlo simulation GOTO EXCEL X(sample)u (k=1)Rsu 24,280,833,40% RESULT

19T&M Conference 2010, SA The simulated dataset

20T&M Conference 2010, SA GOTO EXCEL Bracketing X(sample)u (k=1)Rsu 24,581,004,05% RESULT

21T&M Conference 2010, SA Conclusions Sample value, cu(c)Rsu Ordinary linear regression24,290,482,0% Bivariate linear regression24,250,753,1% Monte Carlo simulation24,280,843,4% Bracketing24,581,004,1% Measurement uncertainty is the most important in decision making process!

T&M Conference 2010, SA22 L uu L-1.64*uL+1.64*u u is the procedure characterization! acceptance arearejection area Measurement result with 95% probability below limit Measurement result with 95% probability over limit ¿ grey zone ? 5 % results 3.28 * u

23T&M Conference 2010, SA Thank you!