amc technical brief
Analytical Methods Committee

Editor: Michael Thompson

No. 18 Dec 2004

GMO Proficiency testing: Interpreting z-scoresderived from log-transformed data
In some proficiency tests concerned with measuring
the proportion of genetically modified organism(GMO) in food the results produced are logtransformed (converted into logarithms) before zscores are calculated [1]. The transformation can bejustified both theoretically and practically. However,
the transformation gives rise to z-scores that are noton the same type of scale as the original data, and are
therefore less readily interpreted. A certain amount ofbackground in logarithmic transformation may behelpful.
What is a lognormal distribution?
Figure 1 shows the density of a lognormally distributedvariable. It is asymmetric, with a positive skew and allvalues of x necessarily greater than zero. If alternatively
we plot the density against the logarithm of x, we see thefamiliar shape of the normal distribution (Figure 2). (Notethat logarithms base ten are implied throughout thisBrief.)

PCR is log-linear in form and this will tend to produce alognormal distribution of results from a normal input.
Finally there is the usual normal distribution of errors
from the instrumental readout system.

Figure 1. A lognormal distribution with an RSD of 0.3.

Definition: a variable x is lognormally
distributed if log x is normally distributed.
While all normal distributions are essentially the same
shape, the shape of a lognormal distribution depends on
its RSD (relative standard deviation, here expressed as afraction). For example, the highly-skewed distribution in
Figure 1 has an RSD of 0.3, while Figure 3, also a
lognormal but with an RSD of 0.1, shows only a slightskew. (For reference, results from a round of a GMO
proficiency test commonly have an RSD of about 0.7.)
Data from GMO proficiency testing
At present, nearly all quantitative measurements of agenetically modified species in a food are based on thepolymerase chain reaction (PCR). In interlaboratory
studies such as the proficiency test, the results almost
invariably show a strongly skewed distribution of results,
Figure 4 for example. There are a priori reasons forexpecting this outcome. Firstly the procedure may startwith a small number of copies of the gene, so that there isa binomial distribution of copies in the sample taken for
PCR. Binomial distributions are positively skewed forsmall number of copies. If the DNA is associated with asmall number of particles, sampling these particles could
give rise to a skewed result even if the number of copiesof the gene is reasonably large. The calibration function inAMC\AMCTB\STATSSC\LOGNORMAL\log1_v6.DOC

06/12/2004

Figure 2. The same distribution as Figure 1, with the density plottedagainst log x.

Figure 3. A lognormal distribution with an RSD of 0.1.

© The Royal Society of Chemistry 2004

Substituting the corresponding values z = 2 and x = 2.0 x ain the equation above gives us
2 = (log 2 xa − log xa ) σ p , or
σ p = 12 log 2 = 0.1505 .

(Using z = −2 and x = 0.5 x a gives the same result: try it!)
Generalising, if limits given by x a q and qx a arerequired, we need σ p = 12 log q . Furthermore, we caneasily toggle between a z-score and the corresponding
value of r = x x a (the factor by which a result differs fromzσ

the assigned value) by using the equations r = 10 p andz = log r σ p . For instance, if z = 3.5 and σ p = 0.1505 ,Figure 4. Results from a single round of a proficiency test involving
measuring the concentration of GMO soya.

As an outcome of all this, the distribution of errors isexpected to be a complex convolution of distribution
types, but with a tendency towards a positive skew. It istherefore tempting to suggest that log-transformation ofparticipants’ results may be appropriate before theformation of z-scores. A detailed study of proficiency test
data has justified this action in practice [2]. But how are
we to relate such z-scores to everyday requirements andthe performance of an individual laboratory?
Z-scoring in GMO proficiency testing
What we have to bear in mind is that, in quantitative
GMO testing, errors seem to be largely multiplicative,
rather than additive as in most other analytical work. In
that context, a very useful property of log-transformationis that various datasets with the same relative standard
deviation in the original scale have the same absolute
standard deviation in the log-scale. In the instance of
GMO proficiency testing, this enables providers to set asingle σ p value for the scheme, regardless of theconcentration of the analyte (except, of course, where theconcentration is zero, or very close to it). A z-score canthen be calculated from a result x and an assigned value
x a (both in the original scale) according to the equationz = (log x − log x a ) σ p = log(x x a ) σ p .

log xa will usually be the robust mean of the log x values.
The σ p value (the standard deviation for proficiency,
previously called the ‘target value’) should be a fitnessfor-purpose criterion, if at all possible.
So if fitness for purpose demanded that a satisfactory
result should be within limits of (say) 0.5 x a and 2.0 x a ,we would need to set σ p such that these limiting results
produced z-scores of –2 and +2 respectively.

we have r = 10 0.527 = 3.36 : the result exceeds the assigned
value by a factor of 3.36.
In conclusion
The essential point here is that the major errors seem to be
multiplicative in quantitative GMO testing based on PCR.
As a consequence, the uncertainty on the original
measurement scale is not symmetrically disposed aroundthe result. Regardless of this, z-scores based on logtransformed data can still be treated as symmetric: a zscore of –3.5 has the same importance as one of +3.5.
Similar considerations might apply to any measurement
system (such as quantitative microbiology) based on amultiplicative procedure.
This Technical Brief was prepared for the Analytical
Methods Committee by the Statistical Subcommittee(Chairman M Thompson) with support from the FoodStandards Agency.
References
1. J Powell and L Owen, Accred Qual Assur, 2002, 7,392-402.
2. M Thompson et al. (in press).

AMC Health Warning: inappropriate logtransformation could damage your statistics.
Positively skewed datasets, with a greater or lesser
resemblance to lognormal, often occur in nature (forexample, concentrations of a trace element in
randomly selected soils). However, datasets withlognormal distribution of error are rarely
encountered in chemical measurement. Logtransformation should be used with some caution: inthe wrong context, it could provide misleading
statistics (as can most other practices). An AMC
Report on broader aspects of log-transformation is inpreparation and will cover this topic.

Other AMC Technical Briefs can be found on:
www.rsc.org/lap/rsccom/amc/amc_index.htm
AMC\AMCTB\STATSSC\LOGNORMAL\log1_v6.DOC

06/12/2004© The Royal Society of Chemistry 2004