Uncertainty of Measurement (Analytical) Maré Linsky 14 October 2015

Transcription

1 Uncertainty of Measurement (Analytical) Maré Linsky 14 October 015

2 Introduction Uncertainty of Measurement GUM (Guide to the Expression of Uncertainty in Measurement ) (Analytical) Method Validation and Quality Control Uncertainty Estimation using method validation and quality control data (Nordtest Guide)

3 What is a measurement? A set of operations to determine the value of a quantity. Measurements are made using a measuring instrument. It tells us about a property of a thing. The result of a measurement has two parts: Number Unit Uncertainty

4 Accuracy (ISO 3534) Closeness of agreement between a test result or measurement result and the true value.

5 True value (VIM 3) Quantity value consistent with the definition of a quantity. Conventional true value A value attributed to a particular quantity and accepted, sometimes by convention, as having an uncertainty appropriate for a given purpose.

6 Precision (ISO 3534) Closeness of agreement between independent test / measurement results obtained under stipulated conditions

7 Metrological traceability (VIM 3) Result can be related to a reference through a documented unbroken chain of calibrations, each contributing to the measurement uncertainty

8 Calibration hierarchy establishing measurement traceability to the reference unit

9 Analytical Results Vary Variations are always present. If there seems to be none, the resolution is not high enough ,3 3,5 3,3 3, 31,7 31,8 31,7 3, 31,6 3,4 3,4 3,9 3,49 3,8 3,17 31,67 31,76 31,75 3,17 31,58 3,36 3,40

10 Histogram

11 Bell shaped Completely determined by µ and σ The curve is symmetrical about µ Normal Distribution The greater the value of σ the greater the spread of the curve y = σ 1 π e ( x µ ) σ

12 Distribution Characteristics The data distribution can be described with the following characteristics: Measures of location Measures of dispersion Skewness

13 Arithmetic Mean: Best estimation of the population mean µ for random samples from a population. Location x n i = = 1 n x i

14 Robust estimates: Not affected by extreme values or outliers: Median: Sort data by value, median is the central value of this series. Mode: Most frequent value Huber mean, etc. Location

15 Dispersion Variance: s = n ( xi x) i= 1 n 1 The mean of the squares of the deviations of the individual values from the population mean. Standard Deviation: s = n ( xi x) i= 1 n 1 The positive square root of the variance.

16 Skewness The skewness of a distribution represents the degree of symmetry Analytical data close to the detection limit often are skewed because negative concentrations are not possible Microbiology

17 Normal Distribution: Important Properties Approximately 68% (68,7%) of the data lie within µ±1σ Approximately 95 % (95,45%) of the data lie within µ±σ Approximately 99,7 % (99,73%) of the data lie within µ±3σ

18 Confidence Limits / Intervals The confidence limits describe the range within which we expect with given confidence the true value to lie.

19 Uncertainty of measurement Parameter, associated with the result of a measurement, that characterises the spread of values that could reasonably be attributed to the measurand (GUM). OR Non-negative parameter characterizing the dispersion of the quantity values being attributed to a measurand, based on the information used (VIM 3).

20 What is uncertainty of measurement? It tells us something about the quality of the measurement i.e. how much you can trust the measurement. We need two numbers to quantify uncertainty: The width of the margin of doubt, the confidence interval, and The confidence level, how sure we are that the true value is within the margin of doubt.

21 Standard Uncertainty u x LOC = ( ) 68% Combined Standard Uncertainty Expanded Uncertainty u U = k u ( y) k = Coverage factor associated with: Level of Confidence Degrees of Freedom c ( x ) c ( y ) = (ci u i ) Basic concepts

22 Basic concepts Uncertainty (of measurement) Y = y ± U m = ± g Level of Confidence Coverage factor Effective degrees of freedom

23 Guide to the Expression of Uncertainty in Measurement (GUM)

24 Introduction Purpose of the GUM Establish general rules and procedures for evaluating and expressing uncertainty of measurement To provide a basis for international comparison of measurement results Applicable to a broad spectrum of measurements at various levels of accuracy Bottom Up approach

25 Method of evaluation: Analytical measurement Step 1: Specification and modeling Step : Identify the uncertainty sources Step 3: Quantify the uncertainty sources Step 4: Calculate the total uncertainty (combined standard uncertainty) Step 5: Calculate the expanded uncertainty Step 6: Reporting the uncertainty

26 Step 1: Specification and modeling A clear and unambiguous statement of what is being measured. Measurand Matrix Method Mathematical Model y = f ( x,x,..., ) 1 xn

27 Step : Identify uncertainty sources M(gross) M(tare) d(etoh) Volume d ( EtOH ) = M gross V M tare

28 Accuracy, Trueness & Precision Improving trueness Improving accuracy Error Improving precision from: LGC, vamstat II

29 Step : Identify uncertainty sources Sampling Storage conditions Environmental conditions Reagent purity, blank correction Sample, matrix effects Operator effects Instrument effects Calibration Standards Calibration effects Random effects Constants

30 Cause and Effect / Fishbone diagram M(gross) Linearity Bias M(tare) Linearity Bias Calibration Calibration Precision Precision d(etoh) Precision Temperature Volume Calibration d ( EtOH ) = M gross V M tare

31 Step 3: Quantifying uncertainty sources Two categories based on method of evaluation Type A: Estimate and associated uncertainty are directly determined by the current measurement (Statistics) Type B: Estimate and associated uncertainty are brought into the measurement from external sources (Other sources) Both are based on probability distributions Standard uncertainty of each input estimate is obtained from a distribution of possible values for the input quantity Based on the state of our knowledge

32 Type A evaluation For component of uncertainty arising from random effects. Applied when multiple independent observations are made under the same (repeatability) conditions. Usually obtained from a normal (Gaussian) probability density function µ-kσ µ µ+kσ

33 Type A evaluation Best estimate of the expected value of an input quantity: Arithmetic mean x = 1 n n i= 1 x i Distribution of the quantity: Experimental standard deviation (Reproducibility) s n 1 ( ) ( ) x = x x i n 1 i= 1 i Type A standard uncertainty: Spread of the distribution of the means: Experimental standard deviation of the mean (Repeatability) u ( x ) i = s ( x ) i n Degrees of freedom ν i = n 1

34 Example The potassium concentration in a tap water sample was analysed 5 times (mg/l) by AAS: 34.6; 36.78; 35.9; 34.17; Type of Uncertainty: Probability Distribution: Best estimate of Value: Standard Uncertainty: Degrees of Freedom: A Normal C P = mg/l s = mg/l u(c P ) = mg/l 4

35 Type B evaluation Based on other sources of information available: Calibration certificates Manufacturer s specifications Previous measurement data, e.g. control charts Experience of the behaviour of instruments or materials, i.e. scientific/professional judgment Reference data from textbooks

36 Type B evaluation: Normal µ-kσ µ µ+kσ Best estimate Standard uncertainty Degrees of freedom µ = µ u u ( x ) i ( x ) = s( x ) i i i i = = U k ν = n 1,or ν =,or ν Specified i

37 The certificate of analysis for the Cadmium calibration standard has a certified value of ± 1.9 mg/l at a LOC = 90%, ν = 31 Example Type of Uncertainty: Probability Distribution: Best estimate of Value: Standard Uncertainty: Degrees of Freedom: B Normal C[Cd] = mg/l U = 1.9 mg/l k = 1.70 u(c[cd]) = mg/l 31

38 a a Type B evaluation: Rectangular a - µ a + Best estimate Half range Standard uncertainty Degrees of freedom µ = a = a ( x ) u i = ν i a = a + + a + a 3

39 Example The purity of KHP (Potassium hydrogen phthalate) is quoted in the supplier s catalogue to be within the limit of 99,90% and 100,00%. Type of Uncertainty: Probability Distribution: Best estimate of Value: Standard Uncertainty: Degrees of Freedom: B Rectangular P = 99.95% a = 0.05% u(p) = 0.089%

40 a a Type B evaluation: Triangular distribution a - µ a + Best estimate µ = a a + + Half range Standard uncertainty a+ a a = a u( x i ) = 6 Degrees of freedom ν i =

41 The manufacturer s specification for the pipette is: 10.0 ± 0 C. Example Type of Uncertainty: Probability Distribution: Best estimate of Value: Standard Uncertainty: Degrees of Freedom: B Triangular V = 10.0 ml a = 0.3 ml u(v) = ml

42 Step 4: Calculating the combined uncertainty All uncertainty components must be in the same unit of measurement Sensitivity coefficients, c i describes how the output estimate, y vary with changes in the input quantities x 1, x,, x n Calculate sensitivity coefficients, c i f Partial derivatives: ci = xi f A numerical estimation: ci x i

43 Step 4: Calculating the combined Calculate uncertainty contributions: uncertainty u( y i ) = c i u( x i ) Combined standard uncertainty: u ( x ) c ( y ) = (ci u i )

44 Step 5: Determine the expanded uncertainty U = k u c ( y) k = coverage factor chosen from the t-distribution table, depending on: the desired level of confidence, and ν eff the effective degrees of freedom,, calculated from the Welch-Satterthwaite formula: 4 uc ( y) ν eff = N 4 u y i= 1 i ν ( ) i

45 Concluding remarks: GUM approach Provides a framework for assessing uncertainty. Helps to identify and quantify uncertainty sources their contribution to the total uncertainty.

46 Method validation and Quality Control

47 Overview Method Validation Precision Trueness (Bias / Recovery) Linearity Limit of Detection (LOD) & Limit of Quantification (LOQ) Selectivity / Specificity Traceability Quality Control Mean (x) control chart Range (R) control chart

48 Method validation Method validation is required to establish the fitness for purpose of a method for the specific requirements of customers. Method validation studies produce data on: Overall performance Individual influence quantities Representative Sample Matrix Concentration levels

49 s Validation parameters: 1 n 1 n ( ) ( ) x = x x k k = 1 Repeatability standard deviation (s r ) Precision Short term: 1 lab, 1 day, 1 analyst, 1 instrument, etc. Reproducibility standard deviation (s R ) Long term: Interlaboratory, months, different analysts, etc. Intermediate precision (s zi ) Variation of specific variable(s) only k

50 Validation parameters: Trueness (Bias/Recovery) Typically studied through the use of reference materials or spiking studies. Analytical Recovery: Cmeas % Re cov ery = 100 C Bias: Cmeas CCRM % Bias = 100 C CRM CRM Expected to be negligible or accounted for.

51 Validation parameters: Checked for by: Inspection Significance tests for non-linearity (e.g. correlation coefficient, r ) Non-linearity corrected for by: Non-linear calibration Restricted operating range Remaining deviations from linearity accounted for by: Calibration uncertainty Linearity

52 Validation parameters: Linearity

53 Validation parameters: Checked for by: Inspection Significance tests for non-linearity (e.g. correlation coefficient, r ) Non-linearity corrected for by: Non-linear calibration Restricted operating range Remaining deviations from linearity accounted for by: Calibration uncertainty Linearity

54 Validation parameters: Detection limit (LOD) Lowest concentration that can be reliably detected, but not quantified. y = a + 3 s LOD y x Uncertainties near the detection limit may require careful consideration and special treatment.

55 Validation parameters: Quantification limit (LOQ) Lowest concentration that can be accurately quantified. y = a + 10 s LOQ y Determined to establish the lower end of the practical operating range of a method. Uncertainties near the LOQ may require careful consideration and special treatment. x

56 Validation parameters: Selectivity/specificity The degree to which a method responds uniquely to the required analyte. Investigate the effects of likely interferents by adding the potential interferent to both blank and fortified samples. Can use the data to estimate the uncertainty associated with potential interferences.

57 Traceability in Analytical Chemistry Traceability basis for establishing comparability of measurement results Calibrated equipment Certified calibration standards Validated methods Uncertainty is part of the definition Uncertainty of a traceable result = Uncertainty (reference) + Uncertainty (measurement)

58 Control of the quality of measurements over longer time period. Trueness Precision Different control charts Mean / X-control chart: Stable homogeneous sample / standard Range Control chart: Duplicate samples Quality Control: Control Charts

59 Requirements: Representative Matrix Concentrations Sufficient quantities available Long term stability (if possible) Homogeneous Quality Control: Control Sample(s)

60 Quality Control: Mean Control Chart Warning limit: Mean ± SD Action limit: Mean ± 3SD

61 Quality Control: Range Control Chart Real samples analysed in duplicate/triplicate/ etc. Calculate %Absolute Range Mean %Range Standard deviation s = MeanRange d Number of replicate d measurements (n)

62 Quality Control: Range Control Chart Warning limit: Mean ±.83SD Action limit: Mean ± 3.69SD

63 Analytical Process: Quality Control: Control Charts R Control chart Duplicate samples Sampling Sample preparation Measurement Matrix Control sample Synthetic Control sample X Control chart

64 Estimation of Uncertainties in Analytical Measurement Using Method validation and Quality control data to estimate Measurement Uncertainty

65 Challenges in implementation of GUM in Analytical Laboratories Chemical and Microbiological analyses often are very complex testing procedures. Large number of uncertainty sources that are challenging to quantified separately. Determining a complete mathematical model to describe method. Large number of routine tests for various measurands (analytes), concentration levels and in variety of matrices.

66 Cause and Effect diagram: Pour Plate method sample preparation Mgross prec cal sampling sample bottles pipettes petri dishes air Mtare aliquot temp volume prec cal medium homogeneity temp time humidity media chlorine removal target colonies selectivity purity weighing ph and conductivity mixing moisture content cfu/ml sterilisation incubation counting

67 Simplification of GUM-approach for Analytical Laboratories Bottom-Up (GUM) vs Top Down approach: Using method validation and quality control data NORDTEST - Handbook for calculation of measurement uncertainty in environmental laboratories Goal: Provide a practical, understandable and common way of measurement uncertainty calculations, based on existing quality control and validation data. Report TR 537:

68 Combination of: NORDTEST-Approach Reproducibility within the laboratory Estimation of the method and laboratory bias Provided that reproducibility and bias data is representative: Different stock standard solutions Different batches of reagents Re-calibration of instruments Representative period of time ideally 1 year Minimum number of results: 50

69 Cause and Effect diagram: GUM M(gross) Linearity Bias M(tare) Linearity Bias Calibration Calibration Precision Precision d(etoh) Precision Temperature Calibration Volume

70 Cause and Effect diagram: NORDTEST Calibration (Bias) Temperature M(tare) Volume M(gross) M(tare) M(gross) Temperature Volume d(etoh) Precision (Reproducibility)

71 Combination of: NORDTEST-Approach Reproducibility within the laboratory Estimation of the method and laboratory bias Provided that reproducibility and bias data is representative: Different stock standard solutions Different batches of reagents Re-calibration of instruments Representative period of time ideally 1 year Minimum number of results: 50

72 NORDTEST Step 3 Quantify uncertainty components 3.1: Within laboratory reproducibility 3.: Method and laboratory bias 3.3: Additional factors u( xi ) % u = 100 x i

73 3.1: Within-Laboratory Reproducibility Calibration (Bias) Temperature M(tare) Volume M(gross) M(tare) M(gross) Temperature Volume d(etoh) Precision (Reproducibility)

74 NORDTEST Step 3.1 Within laboratory reproducibility, R w Control sample covering the whole analytical process Control sample not covering the whole process, matrix different Unstable samples no control sample

75 3.1: Reproducibility within the laboratory: R w Control sample covering the whole analytical process If Control sample covers the whole analytical process, and Control samples have a matrix similar to the samples, then: The within-laboratory reproducibility at that concentration level can be estimated from the analyses of the control sample. If the analyses performed cover a wide range of concentration levels, control samples of other concentration levels should also be used.

76 Analytical Process: Sampling Sample preparation Matrix Control sample X Control chart Measurement

77 3.1: Reproducibility within the laboratory: R w Mean Control Chart SD % R w = 100 x

78 3.1: Reproducibility within the laboratory: R w Control sample covering the whole analytical process Example: Reproducibility within the lab R w Mean Mean Control chart s Rw - value % Relative Uncertainty (% R w ) Comments Control sample 1: X = 0.01 µg/l Control sample : X = 50.3 µg/l Standard deviation: s Rw = 0.5 µg/l Standard deviation: s Rw = 3.7 µg/l.5% from 75 measurements in % from 50 measurements in 00

79 3.1: Reproducibility within the laboratory: R w Control sample not covering the whole process, matrix different If: A synthetic control solution is used for quality control, and The matrix type of the control sample is not similar to the natural samples, then: Mean Control Chart: Long term reproducibility contribution, but not does not include uncertainties arising from different matrices and sample preparation procedures. Range control chart: Estimate the repeatability from different matrices and sample preparation processes.

80 Analytical Process: Sampling R Control chart Duplicate samples Sample preparation Measurement Synthetic Control sample X Control chart

81 3.1: Reproducibility within the laboratory: R w Mean Control Chart SD % R w = 100 x

82 3.1: Reproducibility within the laboratory: R w Range control data: Duplicate analyses (n=) Sample No Result 1 Result Mean Range ABS (%Range) %Mean (ABS).119

83 3.1: Reproducibility within the laboratory: R w Range control chart s = MeanRange d

84 3.1: Reproducibility within the laboratory: R w Control sample not covering the whole process, matrix Example: Reproducibility within the lab R w different Low level: -15 µg/l High level: >15 µg/l s Rw - value Calculation %Uncertainty (%R w ) Mean control chart: s Rw = 1.5% Range control chart: s rw = 5.18 / 1.18 = 4.6% Mean control chart: s Rw = 1.5% Range control chart: s rw = 4.06 / 1.18 = 3.6% u( x ) = 1.5% u( x ) = 1.5% + 4.6% + 3.6% 4.8% 3.9% Note: The repeatability component is included twice

85 3.1: Reproducibility within the laboratory: R w Unstable control samples If The laboratory does not have access to stable control samples, then: It is only possible to estimate uncertainty components from repeatability via the range control chart. The long-term uncertainty component (from batch to batch) has to be estimated e.g. by expert judgement.

86 Analytical Process: Sampling R Control chart Duplicate samples Sample preparation? Measurement

87 3.1: Reproducibility within the laboratory: R w Range control chart s = MeanRange d

88 3.1: Reproducibility within the laboratory: R w Unstable control samples Example: Reproducibility within the lab R w s - value Range Control of natural samples (n=) Estimated variation from differences in calibration over time Combined uncertainty for R w s = 0.04 mg/l mean: 7.53 mg/l Relative Uncertainty (R w ) 0.3% s = 0.5 % 0.5% Repeatability + Reproducibility in calibration 0.3% + 0.5% = 0.59%

89 3.: Laboratory and Method Bias Calibration (Bias) Temperature M(tare) Volume M(gross) M(tare) M(gross) Temperature Volume d(etoh) Precision (Reproducibility)

90 NORDTEST Step 3. Method and Laboratory bias: Sources of bias should always be eliminated if possible. According to GUM a measurement result should always be corrected if the bias is significant, constant and based on reliable data, such as a CRM. In many cases the bias can vary depending on changes in matrix. This can be identified when analysing several matrix CRMs.

91 Method and Laboratory Bias: %u Bias Consists of components: %Bias: %Difference from the reference value Uncertainty of the reference value: %u(c ref ) Method and laboratory bias, %u Bias, can be estimated from: Use of several Certified Reference Materials Use of one Certified Reference Material Use of Proficiency Testing / Interlaboratory Comparison data Use of Recovery Data

92 3.: Method and Laboratory bias: u bias Use of several certified reference material When several CRMs are analysed: Bias: RMS bias ( bias Uncertainty of the Reference Values: Employ GUM-approach for individual uncertainties Combine: u( Cref ( i ) ) u( Cref ) = n Uncertainty of the Bias: = n u + bias = RMSbias u( Cref ) i )

93 Example: 3.: Method and Laboratory bias: u bias Use of several certified reference material % Bias % Standard deviation n %u(c ref ) CRM %.% 1.1% CRM -0.9%.0% 7 1.8% CRM 3.4%.8% %

94 3.: Method and Laboratory bias: u bias Use of several certified reference material Quantification of the %Bias: RMS bias ( biasi ) 3.48% + ( 0.9%) +.4% = = n 3 Uncertainty of the certified value u(c ref ): u( C ref ) u = n ( C ) ref ( i) Then the Uncertainty of the Bias is: = = 1.9% =.5% u bias = RMSbias + u( Cref ) =.5% + 1.9% = 3.1%

95 3.: Method and Laboratory bias: u bias Use of one certified reference material If only one CRM is analysed: sbias ( bias) + u( Cref u bias = + n The reference material should be analysed in at least 5 different analytical series. )

96 3.: Method and Laboratory bias: u bias Use of one certified reference material Example: A CRM was analysed 1 times. The mean is 11.9 mg/l with a standard deviation of.%. Certified value: 11.5 ± 0.5 mg/l (95% confidence interval) Quantify the %Bias: % Bias = 100 ( ) / 11.5 = 3.48% and % s bias =.% with n = 1

97 3.: Method and Laboratory bias: u bias Use of one certified reference material Example (cont): Quantify the uncertainty of the reference material: Uncertainty component from the uncertainty of the certified value Certified value: 11.5 ± 0.5 mg/l (95% confidence interval) Convert the confidence Interval: Convert to %Relative uncertainty: %u(c ref ) Expanded Uncertainty (Type B, Normal distribution): U = 0.5 mg/l k = 1.96 Standard uncertainty: 0.5/1.96 = 0.6 mg/l (0.6/11.5)x100 =.1%

98 3.: Method and Laboratory bias: u bias Use of one certified reference material Therefore the Bias uncertainty is: sbias ( bias) + u( Cref u bias = + n ) = (3.48%).% % = 4.%

99 3. Method and Laboratory bias: u bias Use of Proficiency Testing (PT) results Laboratory should participate at least 6 times within a reasonable time interval. ( biasi ) Bias: RMSbias = n Uncertainty of PT Reference Value: Individual PT results: Arithmetic means u( C ) = ref i s R, i n T, i Robust means Combined Uncertainty of Reference value: u( Cref ) i i u( Cref ) = n R u(c ref ) i = 1,5 s R,i n T,i

100 Example: 3. Method and Laboratory bias: u bias Use of Proficiency Testing (PT) results Uncertainty of the reference value u(c ref ) Data from the PT/ILC: For robust means: u( C ) = 1,5 ref i s R, i n T, i s R,1 = 3,1%, n T,1 = 8 s R, = 4,8%, n T, = 8 s R,3 = 7,6%, n T,3 = 8 s R,4 = 5,3%, n T,4 = 35 s R,5 = 6,9%, n T,5 = 35 s R,6 = 8,4%, n T,6 = 35 u( C ref ) = i u( C n R ref ) i u(c ref ) 1 = u(c ref ) = u(c ref ) 3 = u(c ref ) 4 = u(c ref ) 5 = u(c ref ) 6 = 0,73% 1,13% 1,80% 1,1% 1,46% 1,77% u(c ref ) = 1,34%

101 3. Method and Laboratory bias: u bias Use of Proficiency Testing (PT) results Quantification of the %Bias: In the 6 participations the biases have been: %, 7%, -%, 3%, 6% and 5% RMS bias = ( bias n i ) = % + 7% + ( %) + 3% 6 + 6% + 5% = 4.6%

102 3. Method and Laboratory bias: u bias Use of Proficiency Testing (PT) results Then the Uncertainty of the Bias is: u + bias = RMSbias u( Cref ) = 4.6% % = 4.8%

103 3.: Method and Laboratory bias: u bias From Recovery Tests Recovery tests can be used to estimate bias: Bias: RMS bias = Uncertainty of Reference Value: Use GUM-approach to quantify Uncertainty of Bias u + ( bias bias = RMSbias u( Cref ) n i )

104 3.: Method and Laboratory bias: u bias From Recovery Tests Example: In an experiment the recoveries for an added spike were 95 %, 98 %, 97 %, 96 %, 99 % and 96 % for 6 different sample matrices. The spike of 0.5 ml was added with a micropipette. Uncertainty of reference value uncertainty of the concentration of the spike u(conc) uncertainty of the added volume u(vol) from the certificate: 95% confidence interval = ± 1. % u ( conc ) = 0.6% from the manufacturer of the micro pipette: max. bias: 1% (rectangular interval), repeatability: max. 0.5% (standard dev.) u( vol ) = 1% % = 0.76% uncertainty of the reference value u(c recovery ) u( conc ) + u( vol ) = 0.6% % = 1.0%

105 3.: Method and Laboratory bias: u bias From Recovery Tests Example (cont): Quantification of the bias: RMS bias = 5% + % + 3% + 6 4% + 1% + 4% = 3.44% Then the Uncertainty of the Bias is: u bias = RMSbias + u( Cre cov ery ) = 3.44% + 1.0% = 3.6%

106 3.3 Additional factors: u(f i ) Uncertainty contributions not incorporated into precision and bias data. GUM-approach: Type A Type B Experimentally: Study of the effect of a variation of a single parameter on the result. Robustness studies, systematically examining the significance of moderate changes in parameters. Systematic multifactor experimental designs.

107 NORDTEST: Step 4 Calculate Combined Standard Uncertainty (u c ) Reproducibility (R w ): From control samples and other estimations Bias (u bias ): From CRM, PT or recovery tests Additional factors (f i ) ( ) %u %u( f % u + c = %u( Rw ) + bias i ) All expressed as % relative uncertainty

108 NORDTEST: Step 5 Determine the expanded uncertainty: U = k u c ( y) The NORDTEST-approach adopts k= for an approximate level of confidence of 95% with assumed effective degrees of freedom > 30.

109 NORDTEST: Step 6 Reporting final result (with measurement unit): Result (Y) Expanded Uncertainty (U) Coverage factor (k) Level of confidence (LOC) Estimation method used, e.g. Measurement uncertainty derived from interlaboratory comparison data

110 Nordtest approach: Summary u(bias) = f(bias, u(c ref )) Several CRMs One CRM Proficiency testing Recovery tests Matrix Control Sample Synthetic control sample, duplicate samples Reproducibility Unstable samples d(etoh)

111 Conclusions Bottom Up (GUM) Mathematical model needed Complex calculations Smaller uncertainties Top Down (Nordtest, Eurachem/CITAC) No model needed Simpler combination of data already available in accredited laboratory Uncertainties are larger, but perhaps more realistic? Fit for purpose?

112 NORDTEST: Example Analysis of Cd in aqua regia extract of soil The laboratory has a X control chart for routine analysis of Cd in soil using a real soil sample. From 30 measurements of the control sample: Mean = 0.41 mg/kg Standard deviation = 0.04 mg/kg. BCR-14R (light sandy soil CRM) was analysed during method validation, with the following results obtained: No. BCR14R Results (mg/kg)

113 Nordtest approach: Cd in Soil u(bias) = f(bias, u(c ref )) Several CRMs One CRM Proficiency testing Recovery tests Matrix Control Sample Synthetic control sample, duplicate samples Reproducibility Unstable samples d(etoh)

114 Nordtest approach: Cd in Soil u(bias) = f(bias, u(c ref )) Several CRMs One CRM Proficiency testing Recovery tests Matrix Control Sample Synthetic control sample, duplicate samples Reproducibility Unstable samples d(etoh)

115 Analysis of Cd in aqua regia extract of soil Within laboratory reproducibility, s Rw Control sample covering the whole analytical process: Mean = 0.41 mg/kg Standard deviation = 0.04 mg/kg SD % sr w = 100 x 0.04 = = 9.76%

116 Analysis of Cd in aqua regia extract of soil Method and Laboratory bias: u bias Use of one certified reference material x = 0.337mg / s bias %s sbias ( bias) + u( Cref u bias = + n = 0.011mg bias = kg / kg 100 = ) 4.80% No. BCR14R Results (mg/kg)

117 Analysis of Cd in aqua regia extract of soil Method and Laboratory bias: u bias Use of one certified reference material µ = x = 0.337mg / %Bias 0.49mg sbias ( bias) + u( Cref u bias = + n Bias: Certificate of analysis BCR-14R / kg kg = 0.49 ) 100 = 6.14% No. BCR14R Results (mg/kg)

118 Analysis of Cd in aqua regia extract of soil Method and Laboratory bias: u bias Use of one certified reference material µ = u ( C ) %u sbias ( bias) + u( Cref u bias = + n u(c ref ): Certificate of analysis BCR-14R 0.49mg ref / kg = / 3.18 = mg / ( C ) = 100 = 1.63% ref ) kg

119 Analysis of Cd in aqua regia extract of soil Method and Laboratory bias: u bias Use of one certified reference material sbias ( bias) + u( Cref u bias = + n ) = ( 6.14 ) ( 1.63) = 6.45%

120 Analysis of Cd in aqua regia extract of soil Combined Standard uncertainty Reproducibility: %R w = 9.76% Method and Laboratory bias: %u bias = 6.45% %u Expanded Uncertainty ( ) ( ) % c = = Assume k=, degrees of freedom > 30 U = k u = c 3.4%

121 Analysis of Nitrate-N in waste water NORDTEST: Example A laboratory routinely analyses a standard nitrate quality control solution. Duplicate analysis of real waste water samples are used to construct a range control chart. The laboratory doesn t have a CRM, but participates in a PT scheme on nitrate in waste water.

122 Nordtest approach: Nitrate-N u(bias) = f(bias, u(c ref )) Several CRMs One CRM Proficiency testing Recovery tests Matrix Control Sample Synthetic control sample, duplicate samples Reproducibility Unstable samples d(etoh)

123 Analysis of Nitrate-N in waste water Reproducibility The laboratory routinely analyse a standard nitrate quality control solution: Concentration = 0 mg/l nitrate-n. From 40 measurements of the solution: Mean = 19.5 mg/l Standard deviation = 0.45 mg/l SD % sr w = 100 x 0.45 = =.31%

124 Analysis of Nitrate-N in waste water Reproducibility Duplicate analysis of real waste water samples gives a range control chart with the following data: Rpt No x(1) x() Rpt No x(1) x()

125 Analysis of Nitrate-N in waste water Reproducibility Duplicate analysis of real waste water samples gives a range control chart with the following data: %s = = r w = % %MeanRange d

126 Analysis of Nitrate-N in waste water Reproducibility Reproducibility contribution to overall uncertainty: Mean control Chart: %s Rw =.31% Range control Chart: %s rw = 4.96% %S ( ) ( ) % 5.47% R = =

127 PT no. Analysis of Nitrate-N in waste water Bias The laboratory doesn t have a CRM, but participates in a PT scheme on nitrate in waste water with the following results (all values in mg/l N): Assigned value (median) Reproducibility SD No. Participants Lab result L L L L L L u(c ref ) i = 1,5 s R,i n T,i

128 Analysis of Nitrate-N in waste water Bias The laboratory doesn t have a CRM, but participates in a PT scheme on nitrate in waste water with the following results (all values in mg/l N): PT No. s R (%) %u(c ref ) %Bias L L L L L L ( bias RMSbias = n = 1.% i ) u(c ref ) i = 1,5 s R,i n T,i u(c ref ) = = 0.67% i u(c n R ref ) i

129 Analysis of Nitrate-N in waste water Bias contribution to overall uncertainty: Bias: %RMS Bias = 1.% Uncertainty of Reference value: %u Cref = 0.67% Bias u bias = RMS bias + u( C ref ) = = ( ) 1. + ( 0.67) 1.39%

130 Analysis of Nitrate-N in waste water Combined Standard uncertainty Reproducibility: %R w = 5.47% Method and Laboratory bias: %u bias = 1.39% %u Expanded Uncertainty ( ) ( ) % c = = Assume k=, degrees of freedom > 30 U = k u = c 11.3%

131 Acknowledgements NLA NMISA dti Dr. Michael Koch