amc recommendation
Analytical Methods Committee Editor: Michael Thompson AMCR No 2. March 2005

General and specific fitness functions for proficiency
tests and other purposes—clarifying an old idea
A fitness function is an algebraic relationshipbetween the concentration of an analyte and theuncertainty that is regarded as fit for a specificpurpose. It is a requirement based on the needs ofthe end-users of analytical data, and therefore
determined without any reference to thecharacteristics of individual analytical methods.
Such functions are widely used in proficiency
tests (in various guises) as fitness for purpose
criteria that define the value of the standard
deviation for proficiency (the ‘target value’) usedin the calculation of the z-score from theparticipant’s result and the assigned value. They
are also useful for the selection of a validated
analytical method for a particular purpose: thevalidated characteristics of the method are simplycompared with the fitness function.
The particular usefulness of the fitness function in
proficiency testing resides in its capacity to enable
the scheme provider to specify a requirement ofanalytical performance, in terms of the optimal levelof uncertainty, without imparting to the participants
any information about the concentration of theanalyte.
The general fitness function in proficiency tests
The providers of many proficiency testing schemes
have adopted the practice of using a general fitness
function u f = g (c) , based on end-user needs, that isappropriate for a wide range of test materials,
analytes and concentrations (c) within the analyticalsector. A fitness function widely used in this general
way is simple proportionality. An example might
be g (c) = 0.05c , implying that a relative standard
uncertainty of 5 % is required. Another frequentlyused fitness function is the Horwitz
relationship g (c ) = 0.02c 0.8495 , in which both g (c )and c are expressed as mass fractions (i.e., 1 mg kg-1
= 1 ppm = 10-6). Both of these functions show therequired standard uncertainty increasing as a smoothfunction of concentration. (Scheme providers, withthe assistance of their advisory committees, areresponsible for the appropriate choice of function.)c\data\amc\amctb\statsc\fitness function\ reccn v3.doc

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Valuable though general fitness functions may be,they are unsatisfactory in instances where theconcentration of the analyte is small and the implied
uncertainty approaches (or is actually lower than) thelowest level of interest to the end-user. Such a level
would tend to vary with the analyte and theapplication, so a general fitness function would haveto be customised for each specific case.
An example fitness function
As an illustrative example, consider thedetermination of a hypothetical food constituent for
which there was a statutory upper limit ofconcentration of 1 % m/m. A proficiency test
provider specifies a general fitness-for-purpose
requirement by g (c) = 0.1c . According to thatspecification, a concentration of 5 % m/m of theconstituent implies that an uncertainty of 0.5 % m/m
defines fitness for purpose at that level. That issatisfactory in relation to the statutory limit: wewould be quite happy that a measurement result
falling in the range 4.0-6.0 % m/m with 95 %confidence was actually greater than 1.0. Moreover,the uncertainty would be small enough to include theresult in exposure surveys.
If the concentration of the constituent were 0.05 %m/m, however, the corresponding uncertainty of0.005 would not represent fitness for purpose. The
concentration is well below what is regarded as ahealth hazard, but the implied accuracy is muchhigher than necessary to make the correct decision inrelation to the statutory limit (or to use the result forexposure studies). In addition, the implied accuracy
would probably be unduly expensive to achieve—thelaboratories might well have to use special methods
to achieve the implied accuracy.
Customising the simple fitness function
There is a simple method of modifying the fitness
function to incorporate information about the lowest
level of interest and provide suitable u f -values atany point in the concentration range. Such a level of
uncertainty is clearly related to, and somewhat© The Royal Society of Chemistry 2005

smaller than, any statutory or other decision limit,
and its value would have to be determined by theprovider of the proficiency scheme. In thehypothetical example above, we might decide that anuncertainty of lower than 0.05 would never berequired. This minimal uncertainty u L could then be
combined with the general fitness function g (c )suitable for higher concentrations.
A simple way of doing that is just to use the higherof the two uncertainties, so that we have a fitness
functionu f = max(u L , g (c )) .
Alternatively, combining the terms in the mannernormal for independent uncertainties we might use
u f = u L2 + g (c) 2 .
These two customised functions give similar results(Figure 1) and both are dominated by g (c ) at highconcentrations, and by u L at low concentrations.
Either function clearly describes fit-for-purpose
uncertainties over the whole concentration range.

Recommendation
1. Scope
This recommendation applies to situations thatdemand a fitness function, that is, aconcentration–dependent specification of theuncertainty regarded as fit for purpose.
2. General fitness function
Specify a concentration-dependent uncertainty
g(c) that is regarded as fit for purpose at highconcentrations of the analyte.
(Note. This might take the form of a simple
proportionality or a more complex function suchas a power law. A simple example is g(c) = 0.1c,
which specifies a constant relative standard
uncertainty of 10 %.)3. Lower level of interest
Specify a constant standard uncertainty uL,
which is the uncertainty regarded as appropriate
for the lowest concentration of interest in theapplication sector.
4. Combined standard uncertainty
Combine the two uncertainties to give the fitness
function u f = u 2L + g (c ) 2 .

Figure 1. Fitness functions for the hypothetical example, with aminimal uncertainty u L of 0.05 % m/m. The upper (solid) line
shows the function u f = u 2L + g (c) 2 , while the lower (dashed)line shows u f = max( uL , g (c ))This Recommendation was prepared for theAnalytical Methods Committee by theStatistical Subcommittee (Chairman MThompson), which is supported by the FoodStandards Agency).

(Note: an alternative combination is given by
uf = max( uL, g(c)) )5. Evaluation of σ p
In a proficiency test, uf can be evaluated bysubstituting the assigned value x a for c. Then zscores z = (x - x a )/σp can be calculated fromresults x by identifying the standard deviation
for proficiency (‘target value’) σp with uf .

AMC Recommendations provide advice on good practice in evolving areas of analytical chemistrywhere there is insufficient help in reference documents or where the help in such documents has
been superseded by improved techniques. They are produced under the same procedure as that
established for AMC Technical Briefs.

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