statistics and data analysis in proficiency testing 2007_tcm18-87000.pdfstatistics and data analysis...
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Statistics and Data Analysisin
Proficiency Testing
Michael ThompsonSchool of Biological and Chemical Sciences
Birkbeck College (University of London)Malet Street
London WC1E [email protected]
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Organisation of a proficiency test
“Harmonised Protocol”. Pure Appl Chem. 2006, 78, 145-196.
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Where do we use statistics inproficiency testing?
• Finding a consensus and its uncertainty touse as an assigned value
• Assessing participants’ results• Assessing the efficacy of the PT scheme• Testing for sufficient homogeneity and
stability of the distributed test material• Others
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Criteria for an ideal scoringmethod
• Adds value to raw results.• Easily understandable, based on the
properties of the normal distribution.• Has no arbitrary scaling transformation.• Is transferable between different
concentrations, analytes, matrices, andmeasurement principles.
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How can we construct a score?
• An obvious idea is to utilise the propertiesof the normal distribution to interpret theresults of a proficiency test.
BUT…
We do not makeany assumptionsabout the actualdata.
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Example dataset A• Determination of protein nitrogen in a meat
product.
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A weak scoring method
• On average, slightly more than 95% of laboratoriesreceive z-score within the range ±2.
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Robust mean and standarddeviation
• Robust statistics is applicable to datasets thatlook like normally distributed samplescontaminated with outliers and stragglers (i.e.,unimodal and roughly symmetric.
• The method downweights the otherwise largeinfluence of outliers and stragglers on theestimates.
• It models the central ‘reliable’ part of the dataset.
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Can I use robust estimates?
Measurement axis
Skewed
Bimodal
Heavy-tailed
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References: robust statistics
• Analytical Methods Committee,Analyst,1989, 114, 1489
• AMC Technical Brief No 6, 2001(download from www/rsc.org/amc)
• P J Rousseeuw, J. Chemomet, 1991, 5, 1.
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Is that enough?
• On average, slightly less than 95% oflaboratories receive a z-score between ±2.
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What more do we need?
• We need a method that evaluates the datain relation to its intended use, rather thanmerely describing it.
• This adds value to the data rather thansimply summarising it.
• The method is based on fitness forpurpose.
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Fitness for purpose
• Fitness for purpose occurs when the uncertaintyof the result uf gives best value for money.
• If the uncertainty is smaller than uf , the analysismay be too expensive.
• If the uncertainty is larger than uf , the cost andthe probability of a mistaken decision will rise.
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Fitness for purpose
• The value of uf can sometimes be estimatedobjectively by decision theoretic methods, but ismost often simply agreed between thelaboratory and the customer by professionaljudgement.
• In the proficiency test context, uf should bedetermined by the scheme provider.
Reference: T Fearn, S A Fisher, M Thompson,and S L R Ellison, Analyst, 2002, 127, 818-824.
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• If we now define a z-score thus:
we have a z-score that is both robustified againstextreme values and tells us something about fitnessfor purpose.
• In an exactly compliant laboratory, scores of 2<|z|<3will be encountered occasionally, and scores of |z|>3rarely. Better performers will receive fewer of theseextreme z-scores.
A score that meets all of thecriteria
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Example data A again• Suppose that the fitness for purpose criterion set
for the analysis is an RSD of 1%. This gives us:021.01.201.0 p
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Finding a consensus fromparticipants’ results
• The consensus is not theoretically the bestoption for the assigned value but is usuallythe only practicable value.
• The consensus is not necessarily identicalwith the true value. PT providers have tobe alert to this possibility.
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What is a ‘consensus’?
• Mean? - easy to calculate, but affected byoutliers and asymmetry.
• Robust mean? - fairly easy to calculate, handlesoutliers but affected by asymmetry.
• Median? - easy to calculate, more robust forasymmetric distributions, but larger standarderror than robust mean.
• Mode? - intuitively good, difficult to define,difficult to calculate.
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• The robust mean provides a useful consensusin the great majority of instances, where theunderlying distribution is roughly symmetricand there are 0-10% outliers.
• The uncertainty of this consensus can besafely taken as
The robust mean as consensus
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When can I use robust estimates?
Measurement axis
Skewed
Bimodal
Heavy-tailed
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Skewed distributions
• Skews can arise when the participants’results come from two or moreinconsistent methods.
• They can also arise as an artefact at lowconcentrations of analyte as a result ofdata recording practice.
• Rarely, skews can arise when thedistribution is truly lognormal.
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Possible use of a trimmed dataset?
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Can I use the mode?How many modes? Where are they?
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The normal kernel density foridentifying a mode
where Φ is the standard normal density,
AMC Technical Brief No. 4
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A normal kernel
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A kernel density
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Another kernel density
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Graphical representation of sample data
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Kernel density of the aflatoxin data
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Uncertainty of the mode
• The uncertainty of the consensus can beestimated as the standard error of themode by applying the bootstrap to theprocedure.
• The bootstrap is a general procedurebased on resampling for estimatingstandard errors of complex statistics.
• Reference: Bump-hunting for the proficiencytester – searching for multimodality. P JLowthian and M Thompson, Analyst, 2002,127,1359-1364.
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The normal mixture model
AMC Technical Brief No 23, and AMC Software.Thompson, Acc Qual Assur, 2006, 10, 501-505.
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Mixture models found by the maximumlikelihood method (the EM algorithm)
• The M-step
• The E-step
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Kernel density and fit of 2-componentnormal mixture model
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Kernel density and variance-inflatedmixture model
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Useful References
• Mixture modelsM Thompson. Accred Qual Assur. 2006, 10, 501-505.AMC Technical Brief No. 23, 2006. www/rsc.org/amc
• Kernel densitiesB W Silverman, Density estimation for statistics and dataanalysis. Chapman and Hall, London, 1986.AMC Technical Brief, no. 4, 2001 www/rsc.org/amc
• The bootstrapB Efron and R J Tibshirani, An introduction to thebootstrap. Chapman and Hall, London, 1993AMC Technical Brief, No. 8, 2001 www/rsc.org/amc
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• Use z-scores based on fitness forpurpose.
• Estimate the consensus as the robustmean and its uncertainty asif the dataset is roughly symmetric.
• If the dataset is skewed and plausiblycomposite, use kernel density methodsor mixture models
Conclusions—scoring
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Homogeneity testing
• Comminute and mix bulk material.• Split into distribution units.• Select m>10 distribution units at random.• Homogenise each one.• Analyse 2 test portions from each in
random order, with high precision, andconduct one-way analysis of variance onresults.
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Design for homogeneity testing
2,
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Problems with simple ANOVAbased on testing
• Analytical precision too low—methodcannot detect consequential degree ofheterogeneity.
• Analytical precision too high—methodfinds significant degree of heterogeneitythat may not be consequential.
(Everything is heterogeneous!)
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• Material passes homogeneity test if
• Problems are:– ssam may not be well estimated;– too big a probability of rejecting
satisfactory test material.
“Sufficient homogeneity”:original definition
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Fearn test
• Test by rejecting when
Ref: Analyst, 2001, 127, 1359-1364.
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Problems with homogeneitydata
• Problems with data are common:e.g., no proper randomisation, insufficientprecision, biases, trends, steps,insufficient significant figures recorded,outliers.
• Laboratories need detailed instructions.• Data need careful scrutiny before
statistics.• HP1 is incorrect in saying that all outlying
data should be retained.
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General references
• The Harmonised Protocol (revised)M Thompson, S L R Ellison and R WoodPure Appl. Chem., 2006, 78, 145-196.
• R E Lawn, M Thompson and R F Walker,Proficiency testing in analytical chemistry. TheRoyal Society of Chemistry, Cambridge, 1997.
• ISO Guide 43. International StandardsOrganisation, Geneva, 1997.