jan.hrouzek@hermeslab.sk

Uncertainty Uncertainty EEstimation of stimation of AAnalytical nalytical RResultsesults

ininForensic AnalysisForensic Analysis

Ing. Ján Hrouzek, Ph.D.* Ing. Svetlana Hrouzková, Ph.D.Hermes Labsystems, Púchovská 12, SK-831 06 Bratislava *Department of Analytical Chemistry, FChFT, Slovak University of Technology in Bratislava, Radlinského 9, SK-812 37 Bratislava

7th. International Symposium on Forensic Sciences, Papiernička, Slovakia, September 30, 2005

f o r

f o rISO 17025

f o r

EN45001

f o r

Uncertainty Estimation

jan.hrouzek@hermeslab.sk

Quality

• method validation

– am I measuring what I set out to measure?

• uncertainty

– how well do I know the result of what I’ve measured?

• traceability of result

– can I compare this result with other results?

jan.hrouzek@hermeslab.sk

Quality vs. Time

SHALL I RUSH YOUR

RUSH JOB BEFORE I

START THE

RUSH JOB I WAS

RUSHING WHEN YOU

RUSHED IN ?

jan.hrouzek@hermeslab.sk

Uncertainty

• how well do you know the result?

– essential part of being able to compare!

– are these two results the same???

• are these results good enough?

– fit-for-purpose

result = value ± uncertainty

Quality

jan.hrouzek@hermeslab.sk

Uncertainty Estimation

Specify Measurand

Identify all Sources of ux

Quantify ux components

Calculate Combined uc

jan.hrouzek@hermeslab.sk

The Uncertainty Estimation Process

Specify Measurand

Identify Sources of ux

Quantify ux

Calculate uc and U

Simplify, Group by existing data

Quantify Group of ux

Quantify remaining ux

Convert to SD

Calculate uc

Re-evaluatelarge components

Calculate U

jan.hrouzek@hermeslab.sk

jan.hrouzek@hermeslab.sk

Normal distribution

k p % (µ±kσ)

1 68.27

1.645 90

1.960 95

2 95.45

2.576 99

3 99.73

µ

σ

+1σ +2σ +3σ-3σ -2σ -1σ

jan.hrouzek@hermeslab.sk

Specify Measurand

• Write down a clear statement of what is being measured, including the relationship between the measurand and the input quantities (e.g. measured quantities, constants, calibration standard values etc.) upon which it depends.

• Where possible, include corrections for known systematic effects.

• The specification information should be given in the relevant Standard Operating Procedure (SOP) or other method description.

jan.hrouzek@hermeslab.sk

Identify Uncertainty Sources

• List the possible sources of uncertainty. This will include sources that contribute to the uncertainty on the parameters in the relationship specified in Step 1, but may include other sources and must include sources arising from chemical assumptions.

• Tool for forming a structured list is the Cause and Effect diagram.

• Appendix D. Analysing Uncertainty Sources based on S. L. R. Ellison, V. J. Barwick; Accred. Qual. Assur. 3 101-105 (1998)

jan.hrouzek@hermeslab.sk

Formulasamp

OPOPOP

OPrefOP m

VCP

I

ICC

RecRef

jan.hrouzek@hermeslab.sk

Cause and effect diagram

jan.hrouzek@hermeslab.sk

Cause and effect diagram - rearrangement

jan.hrouzek@hermeslab.sk

Quantify Uncertainty Components

• Measure or estimate the size of the uncertainty component associated with each potential source of uncertainty identified.

• It is often possible to estimate or determine a single contribution to uncertainty associated with a number of separate sources.

• It is also important to consider whether available data accounts sufficiently for all sources of uncertainty. If necessary plan additional experiments and studies carefully to ensure that all sources of uncertainty are adequately accounted for.

jan.hrouzek@hermeslab.sk

How to quantify grouped components

• Uncertainty estimation using prior collaborative method development and validation study data

• Uncertainty estimation using in-house development and validation studies

• Evaluation of uncertainty for empirical methods

• Evaluation of uncertainty for ad-hoc methods

jan.hrouzek@hermeslab.sk

Uncertainty components

• Standard uncertainty ux

– estimated from repeatability experiments

– estimated by other means

• Combined standard uncertainty uc(y)

• Expanded uncertainty U

U = k · uc coverage factor k = 2, level of confidence α = 95%

• Result = x ± U (units) e.g.: nitrates = 7,25 ± 0,06 % (weight)

n

ki ki

n

ii

iji ikx

x

y

x

yx

x

yxy

1,

2

1

2

...,C ,suu

jan.hrouzek@hermeslab.sk

Standard uncertainty ux• Experimental variation of input variables

– often measured from repeatability experiments and is quantified in terms of the standard deviation

– study of the effect of a variation of a single parameter on the result

– robustness studies

– systematic multifactor experimental designs

• From standing data such as measurement and calibration certificates

– Proficiency Testing (PT) schemes

– Quality Assurance (QA) data

– suppliers' information

• By modelling from theoretical principles

• Using judgement based onexperience or informed bymodelling of assumptions

1

s 1

2

n

xxn

ii

x

jan.hrouzek@hermeslab.sk

Combined standard uncertainty uc(y)

• In general

• Assumption: y = f(x) is linear OR u(xi) << xi

n

ki ki

n

ii

iji ikx

x

y

x

yx

x

yxy

1,

2

1

2

...,C ,suu

i

iii

i x

xyxxy

x

y

u

u

reduce by u(xi)

jan.hrouzek@hermeslab.sk

Combined standard uncertainty uc(y)

• In general

• Assumption: y = (x1+x2+...+x3)

• Assumption: y = (x1 · x2 · ... · x3)

n

ki ki

n

ii

iji ikx

x

y

x

yx

x

yxy

1,

2

1

2

...,C ,suu

222

21...,C uuuu nji xxxxy

2

3

3

2

2

2

2

1

1...,C

uuuu

x

x

x

x

x

xyxy ji x

jan.hrouzek@hermeslab.sk

Uncertainty – numerical calculation

x

xy

xux

xuxy

xyxuxyyu

yu

12

12

xx

yyGradient

yu

xu1x 2x

1y

2y

jan.hrouzek@hermeslab.sk

Terms

n

xx

n

ii

1

11

2

n

xxs

n

ii

x

sRSD

Arithmetic mean

Standard deviation

Relative standard deviation

jan.hrouzek@hermeslab.sk

Uncertainty y = f (p, q, r, s)

A B C D E

1 u(p) u(q) u(r) u(s)

2

3 p p + u(p) p p p

4 q q q + u(q) q q

5 r r r r + u(r) r

6 s s s s s + u(s)

7

8 y=f(p,q,...) y=f(p’, ...) y=f(..,q’,..) y=f(..,r’,..) y=f(..,s’,..)

9 u(y,p) u(y,q) u(y,r) u(y,s)

10 u(y) u(y,p)2 u(y,q) 2 u(y,r) 2 u(y,s) 2

Eurachem

jan.hrouzek@hermeslab.sk

Standard uncertainty estimation rectangular 3 /triangular 6 distribution

• Uncertainty component was evaluated experimentally u(x)=s

• limits of ±a are given with confidence level – assume rectangular distribution (e.g. ±0.2 mg 95%; ux = 0.2/1.96 = 0.1 mg)

• limits of ±a are given without confidence level – assume rectangular distribution (e.g. 1000 ± 2 mg.l-1 ux = 2/3 = 1,2 mg.l-1)

• limits of ±a are given without confidence level and extreme values are unlikely (volumetric glassware)

ux = a/3

ux = a/6

ux = s

ux = a/(tabelated value)

jan.hrouzek@hermeslab.sk

Standard uncertainty estimation normal 9 distribution

• evaluated experimentally from the dispersion of repeated measurements

• uncertainty given as s OR σ, RSD, CV%, without information about distribution

• uncertainty given as 95% (OR other)

confidence band I without information about distribution

ux = s

ux = sux = x.(s/x)ux = CV/100.x

ux = I/2confidence level for I = 95%

jan.hrouzek@hermeslab.sk

Uncertainies from linear calibration

bxay

bayx obspred /

calibration of the responses y to different level of analytes x

to obtain predicted concentration x from a sample giving observed response y

uncertainty in xpred due to variability in y for n pairs of values (xi, yi) and p meassurements

iiiii

pred

i

iii

predwxwxw

xx

wbn

bxayw

yxu 22

2

2

2

12,

nxx

xx

npbn

bxay

yxuii

pred

ii

pred 22

2

2

2

112,

jan.hrouzek@hermeslab.sk

Uncertainty Uncertainty EEstimation of stimation of AAnalytical nalytical RResultsesults

ininForensic AnalysisForensic Analysis

Ing. Ján Hrouzek, Ph.D.* Ing. Svetlana Hrouzková, Ph.D.Hermes Labsystems, Púchovská 12, SK-831 06 Bratislava *Department of Analytical Chemistry, FChFT, Slovak University of Technology in Bratislava, Radlinského 9, SK-812 37 Bratislava

7th. International Symposium on Forensic Sciences, Papiernička, Slovakia, September 30, 2005