Estimation of measurement uncertainty
Measurement uncertainty according to GUM
The estimationm of uncertainty is described in various documents. The most important one certainly is the ISO-Guide to the expression of uncertainty in measurement. Its application is described in many other documents. For the application in Chemical Analysis this has been done in the EURACHEM/CITAC-Guide Quantifying Uncertainty in Analytical Measurement.
The process of measurement uncertainty estimation can be divided into four steps:
- Specification of the measurand
- Identification of uncertainty sources
- Quantification of uncertainties
- Conversion and combination of uncertainties as well as calculation of expanded uncertainties
The procedure aims to first acquire all uncertainty sources and to quantify them as standard uncertainties (i.e. standard deviations). These components are then combined to a combined uncertainty. Finally an expanded uncertainty is calculated to increase the confidence level to a relevant level.
Step 1 - Specification of the measurand
"In the context of uncertainty estimation, “specification of the measurand” requires both a clear and unambiguous statement of what is being measured, and a quantitative expression relating the value of the measurand to the parameters on which it depends" (Eurachem/CITAC Guide). This is the only way to ensure that all uncertainty sources can be identified. The "quantitative expression" should describe as a formula how the different parameters are linked to the measurand, e.g.
Parameters that do not directly influence the measurement results, but contribute with their uncertainty to the overall uncertainty of the result, can be included with a value of 1 which again has an uncertainty to be quantified. From the connection between the single parameters it is possible to quantify the influence of the respective quantity on the total uncertainty.
Step 2 - Identification of uncertainty sources
In the previous step the parameters were investigated on which the mesaurand is depending. Now we have to find out, which uncertainty sources the respctive parameters have. Examples are:
- for weighings: e.g. calibration uncertainty of the balance, precision of the balance, buoyancy correction,
- for volumetric measurements: e.g. calibartion uncertainty of the volumetric equipment, accuracy of reading and filling, temperature,
- purity of chemicals
- for high accuracy measurements: uncertainty of tabulated values of densities, atomic weights etc.
For the graphical display of the whole context so-called fish bone (or Ishikawa-) diagrams can be made, e.g. for a acid-base-titration:
Step 3 - Quantification of uncertainties
Now the uncertainty contribution all have to be quantified on the confidence level of the standard deviation. If contributions are determined from statistical data or if we can assume for other reasons that they are normally distributed we directly use the standard deviation as standard uncertainty. If mean values from multiples measurements are used, we need to use the standard deviation of the mean (i.e. s/√n). If nothing is known about the distribution of the possible values, e.g. for the specification of the volume of a volumetric equipment or for the purity of a substance, we may assume a rectangular distribution. In this case the standard deviation is calculated by dividing the half width by root 3. If the probability for th value to be in the middle of the reported range, we may use a triangular distribution where the standard deviation is the half width divided by root 6.
Step 4 - Conversion and combination of uncertainties as well as calculation of expanded uncertainties
Now the standard uncertainties of all the parameters have to be converted in uncertainty components of the result, caused by the respective parameter. Here we have to take into consideration how strongly the uncertainty of the result is influenced by the uncertainty of the parameter. These sensitivity coefficients can be determined for all parameters of the formula in step 1 by partial derivatisation of the formula to the respective parameter. A numerical calculation is decribed in the Eurachem/CITAC-Guide. After multiplying the quantified uncertainty component for the parameter P with its sensitivity coefficient we get the standard uncertainty component for the measurement result caused by the parameter P.
These standard uncertainty components are brought together to the combined standard uncertainty according to the law of propagation of errors:
To get a higher level of confidence we finally have to multiply the combined standard uncertainty with a coverage factor. The - mostly used - factor k=2 delivers a confidence level of approx. 95%.
The NORDTEST approach
The "NORDTEST-Handbook for calculation of measurement uncertainty in environmental laboratories" descirbes a procedure that uses data from routine quality control and from validation for the evaluation of the measurement uncertainty. Here the analytical procedure is not divided in steps as small as possible to quantify all the uncertainty contributions from each step. In this alternative approach it is tried, to determine from these data all contribution to a method and laboratory bias and all contribution that affect the precision. This approach is also followed in the new ISO/DIS 11352.
The fundamental procedure is shown in the following diagram:
Flow chart of the measurement uncertainty evaluation according to ISO/DIS 11352
Determination of the uncertainty component for random variations uRw (reproducibility within laboratory)
The determination of this component shpould be done under conditions that are also valid during routine analysis. So neither repeatability conditions, nor reproducibility conditions apply, but intermediate conditions (between batches). The same conditions apply for control charts.
Possibility 1: from control charts covering the whole analytical process
If the control sample covers the whole analytical process (including sample preparation) and if the matrix of the control sample is similar to routine samples, Rw can be estimated directly from the results of the analysis of the control samples (control chart). If the concentration range is large, several control samples with different concentartions should be used.
uRw = sRw
Possibility 2: from control samples with differing matrices and/or concentrations
If a synthetic control sample is used and the matrix of the control sample is not similar from that of the routine samples, then uncertainty contributions originating from changes in the matrix have to be considered in addition. This additional contribution can be estimated from the repeatability with changing matrices (range control chart).
Possibility 3: from unstable control samples
If no stable control samples are available (e.g. measurement of oxygen in water), in a first step only uncertainty contributions from repeatability can be estimated. The long term component (between batches) have to be added based on the basis of expert judgement (qualified guess).
Determination of the uncertainty component for method and laboratory bias ubias
This component can be estimated from:
- analysis of certified reference materials
- participation in proficiency tests or
- recovery experiments.
Wherever possible, sources for bias should be eliminated. If this is npot possible and the bias is significant and its quantification is based on reliable data (e.g. CRM), the measurement result should be corrected for this bias. Biases might be dependent on the matrix. This can be checked by using several reference materials different in the matrix.
The uncertainty component is consisting of more sub-components:
- the bias itself (e.g. as % difference from the assigned or certified value) and eventually the uncertainty of the determination of this bias
- the uncertainty of the assigned or certified value uCref
ubias may be estimated from the combination of these two components:
or with the inclusion of the uncertainty of the determination of the bias
Possibility 1: Use of one certified reference material
bias - from the mean of biases of multiple analyses of the reference material
sbias - the standard deviation of the multiple analyses of the reference material
nM - number of measurements of the reference materials
uCref - standard uncertainty of the reference material (taken from the certificate)
Possibility 2: use of several certified reference materials
The analyses of several reference materials delivers average biases for each reference material i. From these deviations biasi, the number of analysed reference materials and the standard uncertainty of the certified value it is possible according to the equation shown above to calculate ubias.
Possibility 3: Use of PT results
If the assigned values is calculated from the consensus mean of the participants, its uncertainty is:
or if the median or a robust estimation method was used to calculate then mean (according to IOS 13528):
If other methods were used to determine the assigned values, the Pt provider has to be asked for the respective uncertainty.
To get reasonably clear picture of the biases of a laboratory, the laboratory should have been participating in at least six PT samples within a adequate time interval.
For all analysed PT samples the respective biases biasi from the assigned value are deteremined. The uncertainties of the assigned values are averaged for the subsequent calculations.
ubias then is:
where nR is the number of used PT samples.
Possibility 4: Use of results from recovery experiments
Recovery experiments often are made during validation or verification of analytcial procedures. A sample is spiekd with analyte and measured before and after spiking. From the difference of the results and the spiked amount the recovery rate can be calculated. If the results are not biased the average recovery should be around 100%
The spiked amount of analyte of course also has an uncertainty.
As an example we look on the spiking of a real sample with a standard solution (with a manufacturer's certificate) using a micro pipette. The uncertainty of the concentration of the standard solution is talken directly from the manufacturer's certificate. There we find a statement: ± 1,2 % (confidence level 95%). To convert it to a standard uncertainty we have to divided this value by 1.96 and we get (approcimately):
uconc = 0,6 %
For the estimation of the uncertainty of the added volume we use the information provided by the manufacturer of the micro pipette. He is stating a max. bias of 1% and for the repeatability (random effects) a standard deviation of max. 0.5%. The repeatability statement can be used directly, the max. bias still has to be converted. Since we assume all values in the range ± 1 % to be of equal probability, we assume a rectangular distribution of the possibel values. For a rectangular distribution with a half width of a the standard deviation is a/√3. Therefore we get for the standard uncertainty of the added volume:
The total uncertainty of the spike therefore is:
To calculate the total uncertainty due to bias we now include the biases in the recovery experiments. For recovery rates of e.g. 95%, 98%, 97%, 96%, 99% and 96% wie get biases of 5%, 2%, 3%, 4%, 1% and 4%.
As already shown above we calculate:
or for our example:
Determination of the total standard uncertainty
The total standard uncertainty or combined standard uncertainty uc is calcuölated from the combination of the uncertainty component describing the random variations uRw with the uncertainty component describing the method and laboratory bias ubias.
This values describes the estimated uncertainty of the measurement result on a level of confidence of the standard deviation (approx. 68 %). To convert this value to a higher level of confidence it is multiplied with a coverage factor k. The choice of k determines the level of confidence.
Usually a coverage factor k=2 is used. this corresponds to a level of confidence of about 95 %.
Alternative approach - direct use of reproducibility standard deviations
If the data for the calculations shown above are not available and if the requirements on the uncertainty a relatively low, then the measurement uncertainty roughly can be estimated directly from the reproducibility in interlaboratory comparisons. we set
uc = sR
Here sR is the reproducibility standard deviation in an interlaboratory compaorison.
Then we get:
U = 2 ∙ SR
This estimate - depending on the quality of the lab - could be too high ("worst case"). But because of more inhomogeneity in real samples compared to the interlaboratory samples it also could be too low.