# Characteristic functions from German PT data

## The function

This website is intended to show the results of characteristic function calculations from German Water PT data. PT data from the German Water PT providers (mainly AQS Baden-Württemberg) are used to show the concenctration dependence of the relative reproducibility standard deviation^{1}. A characteristic function^{2-4} of the following form is used to describe this dependence.

, where y(hat) is the relative standard deviation and x the concentration of the analyte.

The function is calculated for the various data sets by weighted least-square regression.

For low concentrations this function implies that the absolute standard deviation is constant and equal to α. For high concentrations the relative standard deviation is assumed to be constant and equal to β. The concentration that descirbes the transition from constant absolute to constant relative standard deviation (the break point) can be calculated from α/β.

So α and β are estimated from the PT data. Due to scoring and assessment problems proficiency tests tend to avoid concentrations close to the quantification limit. So in most cases the estimation of the parameter β is possible, but the estimation of α from PT data often is not reliable enough. In the table below α-values are reported only if the break point is above the 1^{st} quartile of the data set, i.e. at least 25% of the data are below the breakpoint. In the same way β-values are reported only, if the break point is below the 3rd quartile of the data set, so 25% of the data are above the break point.

## Possible uses of the characteristic function

### The break point

For many instrumental methods it is current practice to report relative uncertainties at high concentrations and absolute uncertainty at low concentrations. Many laboratories wonder how to select the limit concentration. The break point calculated from the characteristic functions may be used for that, assuming that the concentration dependence of the intra-laboratory precision is comparable to that of the inter-laboratory precision.

### Constant absolute standard deviation α at low concentrations

A commmon way to calculate the limit of quantification LoQ is from LoQ = 10 ·s_{r,0}, with s_{r,0} being the repeatability standard deviation of a blank. If we assume - as often is done - that the repeatability standard deviation is half of the reproducibility standard deviation, we can estimate an average LoQ value of the participating laboratories of LoQ=10·s_{R,0}/2=5·α Wherever a valid α could be calculated from the PT data, the LoQ calculated in this way is given below. In all other cases it can be stated, that LoQ is below 5 times the absolute standard deviation at the lowest concentration

Please be aware, that the calculated LoQ values are strongly dependent on the concentrations of the samples analysed in the PT. High concentrations lead to high LoQs. In the reverse way laboratory specific LoQ values also can be used to calculate α-values for the characteristic function.

### Constant relative standard deviation β at high concentrations

In ISO 11352 and in NORDTEST technical report TR537 an estimation approach for the measurement uncertainty is described, which quantifies precision and bias components separately and combines the uncertainties afterwards. For the precision component the reproducibility-within-lab has to be estimated and quantified as a standard deviation u_{R,w}. If we assume the repeatability standard deviation to be half of the reproducibility standard deviation, the average reproducibility-within-lab standard deviation can roughly be estimated to be 0.8·s_{R}, so u_{R,w}=0.8·β. This value is also given in the table below. Whenever no valid β could be estimated a value of < 0.8 times the function value at the highest concentration is reported instead. The reported average values for u_{R,w} may be used by laboratories to check the plausibilty of their own estimates.

### Use of the characteristic function to estimate measurement uncertainties over the whole concentration range

According to ISO 21748 the expanded uncertainty, U equals 2·s_{R}. With α and β estimated the uncertainty can now be evaluated over the entire range using the ‘characteristic function’.

Legal regulations, such as the European Drinking Water Directive or the European Water Framework Directive, require analytical methods to be used that fulfill certain requirements on the uncertainty. The ‘characteristic function’ - with its parameters estimated in the way shown above – gives guidance on what measurement uncertainty at a specified level is possible on average in reality.

^{1} The data originate from the following sources:

- AQS Baden-Württemberg, Stuttgart
- Institut für Hygiene und Umwelt der Freien und Hansestadt Hamburg (http://www.hamburg.de)
- Niedersächsiches Landesgesundheitsamt, Standort Aurich (http://www.nlga.niedersachsen.de)
- Landesamt für Natur, Umwelt und Verbraucherschutz Nordrhein-Westfalen (http://www.lanuv.nrw.de)
- Bayerisches Landesamt für Umwelt (http://www.lfu.bayern.de)
- Landesbetrieb Hessisches Landeslabor (http://www.lhl.hessen.de)
- Niedersächsischer Landesbetrieb für Wasserwirtschaft, Küsten- und Naturschutz (http://www.nlwkn.niedersachsen.de)
- Landesamt für Umwelt- und Arbeitsschutz (http://www.saarland.de)
- Staatliche Betriebsgesellschaft für Umwelt und Landwirtschaft Sachsen (http://www.smul.sachsen.de)

^{1} Thompson, M., Mathieson, K., Damant, A.P., and Wood, R.: A general model for interlaboratory precision accounts for statistics from proficiency testing in food analysis. Accred. Qual. Assur. (2008) 13:223-230).

^{2} Thompson, M, and Coles, B.J.: Examples of the â€˜characteristicâ€™ function applied to instrumental precision in chemical measurement. Accred. Qual. Assur. (2009) 14:147-150.

^{3} Thompson, M, and Coles, B.J.: Use of â€˜characteristic functionâ€™ for modelling repeatability precision. Accred. Qual. Assur. (2011) 16:13-19.

#### Last update: 22.07.2013