Uncertainty in measurement - Introduction and

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Uncertainty in measurement Introduction and examples from laboratory medicine

All modern c omputers and c alc ulators have predefined routines for the c alc ulation of the mean and SD. However, the c alc ulated properties are only estimates of the true standard deviation and mean of the distribution. If these estimates are made from too small a number of observations the estimate will be non - reliable i.e. the unc ertainty of the standard deviation and mean will be

Author

large.

Anders Kallner, Dept. of Clinical Chemistry, Karolinska H ospital, S - 171 76 Stockholm, Sweden

When c harac terizing a result we do not only want to desc ribe how prec ise we have managed to perform the measurement ( repeatable or reproduc ible) but also how c orrec t it is i.e. the deviation of the result from the “true value”. We never know the “true

T he concept uncertainty in measurement

value” and it is therefore not possible to desc ribe the trueness of the result. This dilemma is resolved by assigning a value and c all it the “true value”. This is normally c alled “c onventional true

It is unavoidable that all dec isions, all ac tions and therefore all measurements harbour an inherent unc ertainty. For the lay man/woman the term unc ertainty is used in many c onnec tions and is a frequently used c onc ept with no sc ientific meaning.

value” ( c onvention is here used in the meaning of agreement) or ‘assigned value’. The differenc e between the mean of many measurements of the same quantity and the c onventional true value is the “ b ia s “. The statistic “ tru e n e ss “ will therefore describe the systematic error.

However, metrologists have defined unc ertainty and given it a sc ientific c ontent that is useful for all measurements. The definitio n o f unce r ta inty in me a sur e me nt ac c ording to ISO is:

The prec ision of a result ( standard error of the mean) c an be improved by inc reasing the number of observations whereas the trueness of the result c annot be influenc ed this way. In our role as analysts we are interested to find not only the random error but

‘parameter, associated with a result of a measurement that characterises the dispersion of the values that could be reasonably attributed to the measurand’.

also the systematic error. However, as a c onsumer of results of measurements we would prefer a c onc ept that would inc lude both the random and the systematic error for a single measurement i.e. both the prec ision and the trueness. This c onc ept is c alled a c c u ra c y and in laboratory medic ine often equivalent to

This definition may be diffic ult to apply to prac tic al work and we

total error . The c o n c e pt o f u n c e rta i n ty as defined above will

shall therefore expand on the definition and explain how it c an

offer a simple and tangible alternative to total error and will also

be used for desc ribing how well a measurement proc edure

give us a possibility to avoid the use of “error” to desc ribe the

performs’.

variation inherent in measurements.

The result of a measurement shall always give information about

W hy uncertainty?

an interval within whic h the results c an be expec ted with a given probability. This c an be reported as an interval, as a standard deviation ( SD) , c oeffic ient of variation or standard error of the mean ( SEM) . Most c ommonly the distribution of results is given as one SD. The SD is a desc ription of the prec ision and c an always be c alc ulated from 2 or more results but the interpretation of this value requires knowledge of the properties of the distribution of values. The most probable value within a Gaussian distribution is represented by the mean. Together with the mean, the SD fully defines the shape of a Gaussian distribution. In these distributions the mean defines the position of the distribution on the X axis and the SD its width and height. The SD is therefore a measure of the random error ( the width of the distribution) . In a Gaussian distribution about 2 /3 of all results will be found

down into smaller entities and study eac h of them. This approac h c an be used also to estimate the unc ertainty in measurements and is usually not too c omplic ated. However, as in all sc ienc es the synthesis of results that should be performed is the diffic ult part when sc ientists may make mistakes and end up with grossly erroneous c onc lusions. The c onc ept of unc ertainty as defined by ISO requires that all steps involved in a measurement are defined and evaluated with regard to their unc ertainty. This is a great help in c harac terizing and optimising the performanc e of a measurement proc edure bec ause the sourc es of unc ertainties will be revealed. The c hemist c an review the result and selec t the steps that require improvement in an orderly and reproduc ible

between -1 SD and + 1 SD c ounted from the mean of the distribution. Further, about 9 5 % will be found within the interval -2 SD to + 2 SD and about 9 9 % between -3 SD and + 3 SD.

An ac c epted sc ientific approac h to solve a problem is to break it

manner. All steps should be inc luded, preanalytic al, e.g. sampling and sample preparation, as well as analytic al, e.g. c alibrations,

Page 16 eJIFCC2001Vol13No1pp016-024

dilutions and postanalytic al e.g. transformations and c orrec tions.

rec ording is made one must c onsider the position of the eye in

The ac c umulated data shall then be c ombined. The following text

relation to the sc ale, i.e. one sourc e of unc ertainty that c an be

will desc ribe how these steps c an be systematized and performed

referred to the operator. When a c alibration func tion is estab-

in a pragmatic manner.

lished one must estimate the unc ertainty of the value assigned to the c alibrator. Possible sourc es that might influenc e the c alibration func tion are for instanc e the assigned value of the c alibrator,

Formal ac c reditation of laboratories and measurement proc edures or methods ac c ording to the ISO standards 1 5 1 8 9 ( 1 ) and 1 7 0 2 5 ( 2 ) requires that the unc ertainty in measurements is estimated. The preferred method for estimation of unc ertainties is desc ribed in ‘Guide to the Expression of unc ertainties n measure-

the fitting of the c urve, the dilution of the c alibrators etc . When an extrac tion is inc luded one must estimate the yield. Even if all sourc es of bias were eliminated an unc ertainty of the suc c ess of that proc edure remains and should be estimated.

ments’ ( GUM) ( 3 ) . All these unc ertainty sourc es must be evaluated and brought into the unc ertainty budget. An experienc ed c hemist may direc tly

Besides estimating the unc ertainty in measurements to identify

disregard some of the unc ertainty sourc es bec ause the experienc e

areas in whic h improvements should be foc used, there is a point

is that they are very small or bec ause those remaining are muc h

in estimating the unc ertainty in all measurement that produc e results for the diagnosis and management of diseases. The reason

larger.

is that the unc ertainty in measurements will inc lude a c ontribution from c alibration etc that affec t the bias as well as estimates of

The proc edure of measurement shall then be desc ribed in

the imprec ision and any pre - and postanalytic al unc ertainties.

mathematic al terms to illustrate how the different variables affec t eac h other and how their unc ertainties c an be joined in a c ombined unc ertainty. This proc edure is tric ky and requires

U ncertainty of measurement

experienc e from establishing stoc hiometric c alc ulations i.e. the qualitative and quantitative desc ription of a c hemic al reac tion

The c onc ept of unc ertainty is not a statistic al c onc ept in tradi-

and c alc ulations of c onc entrations and amounts involved.

tional sense i.e. it needs not be assoc iated with a known distribution of the data. The unc ertainty shall rather be understood as an interval within whic h the result c an be found with a given

The estimation of the size of the sourc es of unc ertainty c an be done in two different ways c alled type A and type B . The value and

probability. Thus, the result will be within the interval but all

results of both approac hes are treated similarly but the type A is

values within the interval have the same probability to represent the result. This is quite different from other distributions, e.g. the Gaussian distribution where the mean is the most probable value. Therefore, there is not nec essarily any symmetry in the c onc ept of unc ertainty. There is no value that is more probable or c ommon than any other. It is c onvenient, though, to let a number represent the distribution and one c an c hoose any representative value from

rec ommended whenever possible. In both c ases the ‘standard unc ertainty’ is estimated. The standard unc ertainty is abbreviated

u( x i ) .

Estimation of the uncertainty in measurement, type A

the interval. One is as good as the other but the c entral value has advantages from a presentation point of view. B y these statements

In this model the estimation is based on the standard deviations

we have in fac t desc ribed the essential features of a ‘rec tangular

derived from repeated measurements. The SD and the standard

distribution’

unc ertainty will therefore have the same size. A word of warning: sometimes the variation is given as a c oeffic ient of variation ( CV)

The c alc ulation or estimation of the unc ertainty in measurements

or c onfidenc e interval rather than SD. That information c an also

is princ ipally very simple. This is done by estimating the unc er-

be used but it must then be c orrec ted for what the numbers

tainty in eac h single step of a proc edure ( input variables, X i ) and

represent e.g. multiply with the result of measurement and

c o mbining them in an ‘unc ertainty budget’. The c o mbinatio n o f

divided with 1 0 0 if the CV is given in perc ent or multiplied with

the individual unc ertainties follows similar rules as the propaga-

the square root of the number of observations if the c onfidenc e

tion of errors in normal statistic s giving the ‘c ombined unc er-

interval or SEM ( standard error of the mean) is given. One must

tainty’ for the proc edure. The c ombined unc ertainty is the

also c larify how many SD the information refers to. As a rule the

c onc ept that is c losest to ‘the total error’.

number is one but sometimes multiples are given, for instanc e two SD.

In the examples in this doc ument we will use a worksheet from Exc el 5 .0 that simplifies these c alc ulations and do not require any knowledge of the mathematic s behind. The worksheet c an be used for evaluation of all unc ertainty budgets that deal with

Estimation of the uncertainty in measurement, type B

independent observations. Sometimes, and partic ularly in c omplex measurement proc edures, we do not have ac c ess to or we are not able to estimate the

Sources of uncertainties

variation from repeated experiments, whic h is a prerequisite for the estimation of the SD. The professional experienc e, informa-

It is essential when establishing a realistic unc ertainty budget to

tion in the literature or spec ific ations from a manufac turer will

identify the variables that give rise to the unc ertainty and their

usually allow the demarc ation of an interval within whic h a result

sizes. Therefore one must have detailed knowledge about the

c an reasonably be expec ted. For instanc e, it is fairly safe to

proc edure of measurement to allow identific ation and quantific a-

assume that the European woman is between 1 3 0 c m and 2 1 0

tion of all reasonable sourc es. Examples of sourc es of unc ertainty

c m tall, that a litre of milk c osts between USD 0 ,1 and 2 and that

in our field are the measurement of volumes, weighing, reac tion

one litre of water has a mass between 9 9 4 g and 1 0 0 4 g. The

temperature, purity of reagents, and value assigned to the

more one knows about the proc edure of measurement and the

c alibrator. Also properties of the instrument used e.g. the size and

items measured the better will the estimate of interval be. In

toleranc e of the c uvette, if the instrument uses more than one

laboratory medic ine it is often important to find a reasonable

c uvette they may be different, the wavelength etc . If a visual

interval for the volume, the mass or the reading of an instrument.

Page 17 eJIFCC2001Vol13No1pp016-024

For instanc e a 1 mL pipette might deliver between 0 .9 0 mL and

now have eac h volume defined to its size and ac c ompanying

1 .0 5 mL, the body mass of a grown up man without exc essive fat

unc ertainty. Observe that the unc ertainty of the measuring c ylinder

might be between 6 5 and 8 5 kg; based on your experienc e your

is given as an interval whereas the unc ertainty for the pipette is

might even assume that it is between 6 9 kg and 7 6 kg etc .

given as a relative standard unc ertainty.

The interval that is defined in this way will not identify any result

The volumes V 1 and V 2 should be added and thus the ‘reac tion’

within the interval as being more reasonable than any other in the

c ould be written

same interval. Thus, the interval represents a rec tangular distribution of possible results.

Throwing a single dic e will give results belonging to a rec tangular distribution. One c annot foresee the number of dots that will c ome up but a reasonable ( the largest possible) interval is 1 - 6 .

The standard unc ertainties are added but not as suc h but as their

In fac t no other results c an be obtained. The probability for eac h

squares:

result within the interval is the same, provided the dic e is fair. The mean of the interval ( 3 .5 ) , however, c an never be obtained!

Given an interval there is a somewhat smaller interval within whic h the probability that a result will oc c ur is inc reased. This new interval is c reated from the half - width of the initial interval divided by the square root of 3 ( about 1 .7 3 ) in both direc tions from the middle of the initial interval. It is about twic e as c ommon that results will be within this interval as outside these limits. Using the example with the dic e, the inner interval will be

The c ombined unc ertainty is then obtained by taking the square root from the sum, in this example 8 .7 mL. Perform the c alc ulations yourself and find that the c ontribution from the pipette is of minor importanc e in the c ombined unc ertainty!

Coverage factor

from 2 to and inc luding 5 . For the dic e it means that between 2 and 5 there are four alternatives ( 2 , 3 , 4 and 5 ) , whereas outside

The standard unc ertainty will thus demarc ate an interval where we

there are only two, 1 and 6 . It is therefore more probable that any

c an estimate that 2 /3 of all the results will be found. This is also

of the four dots 2 to 5 will show up than any of the two outside

the usual way to give the variation ( usually the standard deviation)

the smaller interval. The probability that any given number of dots

of results in sc ientific literature. If we want to find the interval

within the new interval will be obtained is the same as outside

that is large enough to give a 9 5 % probability to c over the results

and one c annot predic t whic h. I.e. it is more probable than one

in a Gaussian distribution the SD should be multiplied with about

of four given alternatives is obtained than one of two.

2 for a two - tailed distribution ( the size of the fac tor is depending on the number of observations) . To reac h a c orresponding

The width of this inner interval is twic e the standard unc ertainty 2

probability for the c ombined unc ertainty it shall be multiplied by

u ( x i ) . The standard unc ertainty will c orrespond approximately

2 . If we wish a larger probability then a larger fac tor, for instanc e

to the probability within the interval mean ± 1 SD in a Gaussian

3 for about 9 9 % probability, should be used. This fac tor is c alled

distribution where about 2 /3 of all observations will be found.

the co ve ra ge fa cto r ( k ) and the result e xpa n de d u n c e rta i n ty

You c an c onvinc e yourself about this by playing with a good dic e

U y . The c overage fac tor shall always be given in the answer

and you would expec t 2 /3 of all answers to be found between and

together with the unc ertainty. In c ase no c overage fac tor is given

inc lude 2 and 5 dots.

then the c ombined standard unc ertainty c overing about 2 /3 of the result is given ( k = 1 ) .

An example from our own profession is the estimation of the unc ertainty of a measured volume using a two - litre measure-

U ncertainty budget

ment c ylinder. Suppose we want to measure 5 0 0 mL, and assume a reasonable interval to be ± 3 % or ( 4 8 5 -5 1 5 ) mL. The

When estimating the c ombined unc ertainty ( u

standard unc ertainty is then 1 5 ( half the interval) divided by the

point is to define how the different parts of the proc edure

square root of 3 i.e. 8 .7 mL.

interac t. In our model we assume that the unc ertainty sourc es are

c

( y) ) the starting

independent. In the example above the two volumes were added

Combined standard uncertainty (uc-(y))

to reac h the total volume. For additions ( subtrac tions) , the c ombined unc ertainty is the square root of the sum of the squares of the ingoing standard unc ertainties. In c ase the variables shall

Onc e the sourc es of unc ertainty have been identified and the sizes

be multiplied ( divided) the squares of the ingoing relative

of the unc ertainties estimated then we fac e the problem to

standard unc ertainties shall be added. When the square root is

c ombine the c ontributions to an unc ertainty for the entire

drawn the relative c ombined unc ertainty is ac hieved.

proc edure i.e. the c ombined unc ertainty in the result y. This is abbreviated u c ( y) .

The mathematic al expressions that desc ribe how the standard unc ertainties shall be c ombined bec ome rather c omplic ated

Let us begin with an example:

when the unc ertainty budget for a proc edure c ontains all four operators and a variable may partic ipate as a logarithm or

We want to make a dilution of a sample and are only interested in

exponent. The generic rule for the c ombination is based on

the final volume. Ac c ording to the proc edure we shall take 1 0 mL

partial derivatives but we will use an approximate numeric al

of the sample solution and dilute that to 5 0 0 mL. We use the

method for solving partial derivatives. This c an c onveniently be

measuring c ylinder from the example above and measure the

ac hieved by using a spreadsheet program like Exc el® .

sample with a pipette with a relative unc ertainty of 5 % . Thus, we

Page 18 eJIFCC2001Vol13No1pp016-024

Kragten ( 4 ) originally desc ribed the numeric al approximation of

Examples

partial derivatives. The doc ument from Eurac hem ( 5 ) desc ribes how this method c an be used in a worksheet.

The following examples are c hosen to illustrate an inc reasing c omplexity in the c alc ulation and originate in routine laboratory

Estimation of the uncertainty in measurements using an Excel® worksheet

work. The unc ertainties and other numbers do not nec essarily represent reality but have been c hosen to illustrate their influenc es. Eac h example is solved on the attac hed diskette where also you will also find a template for the c alc ulations.

General

1)

All c ells in the worksheet that will be filled by c alc ulations are

Let us repeat the example above dealing with adding two volumes

protec ted to avoid unintentional c hanges in the program. Cells that c an be c hanged are marked with a blue border. The number

and use the template.

of dec imals is fixed to three that will give a suffic ient presentation of results within laboratory medic ine even if it sometimes

Enter the nominations “Sample” and “Dilution” in the c orrec t

exaggerates the prec ision. You c an perform c alc ulations direc tly

c ells in row 3 and the volumes 1 0 and 4 9 0 respec tively in row 4 .

in that c ell where the value finally will be plac ed. Never c opy the

The standard unc ertainties are entered in row 5 and 6 ( absolute

c ontents of a c ell into another, delete and input it again or

and relative standard unc ertainty, respec tively) . Move the c ursor

unforeseeable errors may oc c ur.

to c ell C2 1 under the label “Nominal” and enter the formula ‘= C9 + C1 0 ’. Copy this formula two c olumns to the right. The c ombined unc ertainty will be shown in c ell C2 5 , the relative in

It is suggested that you enter different data into the given

E2 5 and the interval c alc ulated with a c overage fac tor 2 in the

examples to see how the outc ome c hanges.

c ells K2 5 - L2 5 . The expanded unc ertainty is given in c ell I2 5 . c ompare the result with our manually worked example.

D ata entry and calculations

2) Enter the names of the input variables in row 3 of the worksheet. The input variables c an be entered in any order. The c orresponding names will appear in c olumn B , rows 1 0 to 1 8 . Enter ac tual

We want to weigh NaCl on an old fashioned sc ale. The empty vessel weighs 1 2 g. We add NaCl until 1 2 7 g. Weighing within the

or representative values in row 4 and the standard unc ertainty of the results in row 5 or 6 depending on if the standard unc ertainty is given in absolute ( row 5 ) or relative terms ( row 6 ) . Relative

given interval c an be made with an unc ertainty of 1 ,5 % . Calc ulate the mass of the NaCl.

unc ertainties shall be given as parts of 1 , i.e. 1 .5 % should be In c ell C2 1 the formula will be:’ = final weight-weight of empty

0 .0 1 5 .

vessel ( tare) ’. Test the result if the unc ertainty is 0 .2 g! Then the interrelation between the various c omponents of the budget shall be entered in c ell C2 1 ( ‘Nominal’) , i.e. how the final

3)

result shall be c alc ulated from the input variables. Often this formula presents itself easily but it c an sometimes be rather

We shall c alc ulate the volume of a water bath whic h has a length

c omplic ated and require deep thoughts. It may be helpful to

of 3 5 c m, width 2 5 c m and height 2 0 c m. The edges are

establish and solve an equation that desc ribes how to c alc ulate

measured using a ruler with an unc ertainty of 2 % .

the result. In the dilution example above the expression will be simply the volume of the original sample plus the dilution

The formula in C2 5 will be: ‘= the length x width x height’. Chec k

volume. Compare example 1 .

the result if the unc ertainty of measurement in the interval is 0 .5 c m!

The expression whic h goes into c ell C2 1 will be a mathematic al expression and shall therefore, with the Exc el® nomenc lature,

4)

begin with “= ” ( “equal sign”) . Then, enter the algorithm and drag the c ontents of C2 1 as far to the right as variables have been entered. Finally press “enter” and all c alc ulations will be c arried

Let us add a slight c omplic ation to example 3 . Let us assume that

out. Note that the c ells to the right of the last used c olumn in row

the water bath is half - filled ( 1 0 c m) and you want to fill it up to

2 1 shall be empty. If not, it is safe to mark them and press delete.

3 c m ( the margin) from the upper rim. How muc h water should be added? We make the same assumptions regarding the unc ertainty of the lengths as above, but sinc e the vessel walls are

Depending on how your c omputer has been c onfigured you may

not quite even we must add the unc ertainty this c ontributes. Let

need to press the F9 key to trigger the c alc ulations. If nec essary

us assume that this unc ertainty is multiplic ative and optimally 1

you c an c hange the c onfiguration to ‘automatic ’ under ‘Tools-

but with the unc ertainty 3 % .

Options-Calc ulation’ and c hec k ‘Automatic ’.

The formula in C2 1 will be ‘“length x width x ( the height -

The c ombined unc ertainty will be given on the last row in both absolute and relative terms. Also the c overage fac tor ( default = 2 ) and the unc ertainty interval will be given. Contributions from the

marginal - the height at half - filled water bath) x unevenness fac tor’. Test the results if the unc ertainty in all lengths is 0 .2 c m!

different sourc es of unc ertainty are shown on row 2 3 and graphic ally in the inserted diagram. The diagram c an be freely

5)

moved within the surfac e if you want to study the underlying c ontents of the table. The sc ale of the Y - axis c an also be c hanged to improve the presentation.

Page 19 eJIFCC2001Vol13No1pp016-024

Let us inc rease the c omplexity of example 1 above. Assume that

spec ific absorbanc e at 5 6 0 nm will therefore be ( 0 .1 2 5 ± u a -2 0

the sample c onc entration is 3 2 0 mmol/L with a standard

x 0 .0 0 1 ± u l ) . We c an then formulate the absorbanc e at 5 8 0 nm

unc ertainty of 3 % . What is the c onc entration in the final solution

± 5 nm. It will be ( 0 .1 2 5 ± u a -2 0 x 0 .0 0 1 ± u l ) + 5 8 0 nm ±

and what is its c ombined unc ertainty? Enter the numbers to the

5 nm-5 6 0 ) x ( 0 .0 0 1 ± u l ) .

template as in example 1 but add the c onc entration of the sample solution in a c olumn of its own.

Although the wavelength has a small unc ertainty this will be the dominating sourc e of unc ertainty. Disc uss what happens if the

The formula will be: ‘= sample x c onc entration/( sample +

slope inc reases or if the unc ertainty in the volume measurements

dilutio n) ’.

is c hanged.

6)

10)

We want to estimate the unc ertainty in an HPLC - method. The

The c onc ept of unc ertainty c an also be used for other purposes

sample is diluted 1 + 9 with a so lutio n c o ntaining 0 .2 5 mmo l/L

than measurements and it c an inc lude for instanc e preanalytic al

internal standard ( IS) . We injec t 2 0 µL of the dilution. The

and post analytic al sourc es of unc ertainty. Let us examine this

sample peak is 2 4 7 mm, the IS - peak is 2 3 5 mm. How muc h

example that is a simplific ation of an artic le in “Clinic al Chemis-

substanc e did the sample c ontain and whic h is the c ombined

try and Laboratory Medic ine” ( 6 ) .

unc ertainty of the result? The standard unc ertainty in the sample vo lume is ± 3 % and fo r the dilutio n vo lume ± 2 % . The standard unc ertainty in weighing of IS is ± 1 % . Measuring of the peaks is assoc iated with an interval of ± 2 % . The same amounts of IS and sample give the same response on the printer with an

B - Gluc ose is used in the primary health c are for diagnosis and c ontrol of diabetes. Whic h is the smallest differenc e in results that, with a given probability c an be assumed to be different?

unc ertainty of ± 3 % . The patient shall be fasting. The patient shall be c alm and relaxed to avoid that c atec holamines and other hormones will give a

7)

falsely elevated gluc ose c onc entration. The c onc entration of gluc ose in erythroc ytes is less than in plasma and therefore one

The relation between signal and c onc entration ( i.e. the c alibra-

also has to c ontrol the intake of fluids to avoid falsely low results

tion func tion) is: signal = c onc entration x 2 -1 ,5 ; ( Y= 2 x X-1 ,5 )

and inc reased hematoc rit. The c apillary sampling is diffic ult. One

in the c onc entration interval 0 ,5 -5 units. The standard unc er-

c annot avoid a varying addition of interstitial fluid etc . There are

tainty in the slope ( b) is 2 % in the interc ept ( a) 5 % and in the

instruments on the market that measure P - Gluc ose and use an

signal 0 ,0 1 units. Estimate the c ombined unc ertainty in the

algorithm to transfer the value to B - Gluc ose. The algorithm is

middle of the interval, i.e. 2 ,7 5 units.

based on the assumption that P - Gluc ose is 1 5 % higher than B Gluc ose, whic h however is depending on the hematoc rit of the patient. Finally we also have unc ertainty c ontributions from the

8)

measurement itself. All these unc ertainties shall be c onsidered in

We c ompare the results from measuring the same samples on two different instruments. The results are assumed to be identic al,

estimating the c ombined unc ertainty that shall be the basis for estimating the smallest signific ant differenc e between two results.

that is the regression func tion is assumed to be Y = X. Whic h deviation in Y results c an we expec t if the standard unc ertainty in

Let us selec t 6 .0 mmol/L as an interesting c onc entration.

the estimation of the c oeffic ient is 5 % and the interval of the interc ept is -0 ,5 - + 0 ,8 at the c ritic al limit 1 7 mmo l/L? The

Fa sti n g a n d f lu i d i n ta k e . Just standing up c an c ause to up to

standard unc ertainty in the referenc e method is 3 % .

1 5 % c hanges of the plasma volume. B etween day variation of the plasma volume is given in the literature ( 7 ) to ± 6 .5 % based on repeated measurements. Let us assume that this inc ludes the

9)

variations in stress, fluid intake and fasting but exc luding whether the patient is sitting, standing or resting.

The absorbanc e of a sample is measured at 5 8 0 nm. The spec ific absorbanc e for the substanc e at this wavelength is 1 2 ,5 , the absorbanc e c urve is linear between 5 4 0 and 6 2 0 nm with a slope of 0 .0 0 1 absorbanc e units/nm. The sample was diluted 1 + 9 and the absorbanc e 1 .3 5 . Estimate the c onc entration and its

Sa m pli n g: Literature ( 8 ) postulates an unc ertainty of ± 3 .2 % for S - Gluc ose also based on repeated measurements. For c apillary sampling it is reasonable to add a little, ± 5 % .

unc ertainty if the standard unc ertainties in the volume measurements are 2 % and 1 ,5 % per sample and dilution, respec tively,

Me a su re m e n t: The prec ision of the measurement is related to the

the standard unc ertainty of the spec ific absorbanc e ( u a ) 0 ,5 % ,

instrument. An instrument like Hem c ue c an give an imprec ision

in the reading 1 % , in the slope ( u l ) 5 % and the wavelength of

of ± 3 - 5 % , many of the other instruments the same order and

the filter is given as an interval of 5 8 0 nm ± 5 nm.

magnitude if one uses the same batc h of reagent strips. B etween batc hes the manufac turers may allow as muc h as up to 8 %

There are many fac tors involved in this example and let us argue like this: We must somehow translate the unc ertainty in wave-

imprec ision. Let us assume that we c an manage an interval of ± 5 % unc ertainty that also inc ludes the unc ertainty of c alibration.

length to unc ertainty in spec ific absorbanc e. This c an be done by c alc ulating the spec ific absorbanc e at another wavelength. Let us

Po st a n a lyti c a l If the measurement is made with an instrument

assume 5 6 0 nm. This wavelength has no unc ertainty bec ause we

that really measures B - Gluc ose this sourc e of unc ertainty is not

assume it is without unc ertainty. At exac tly 5 8 0 nm the spec ific

interesting. If, however the instrument measures something

unc ertainty is 0 .1 2 5 . We c an inc lude the unc ertainty in the slope

different then one must divide with a fac tor. Some literature

of the absorbanc e c urve by ( 5 8 0 - 5 6 0 ) x 0 .0 0 1 ± u

suggests 1 .1 7 but it is c loser to 1 .1 1 ( 6 ) . Regardless of the value

l .

The

of the fac tor it is attac hed to an unc ertainty, let us assume 5 % .

Page 20 eJIFCC2001Vol13No1pp016-024

T hree important definitions by ISO :

The smallest differenc e that c an be c alled signific ant is

where D is the differenc e, u c ( y) is the c ombined unc ertainty and k the c overage fac tor. The square root is bec ause two values are

Trueness, agreement between the average value from many observations and the true value

c ompared. A c onvenient abbreviation is D= 3 x u c ( y) .

Precision, agreement between independent results of measurement

References 1 . ISO/DIS 1 5 1 8 9 . Quality management in the medic al laboratory, ISO, Geneva 2 0 0 0 .

Accuracy, agreement between the result of a measurement and a true value of the measurand

2 . ISO/IEC IS 1 7 0 2 5 . General requirements for the c ompetenc e of testing and c alibration laboratories. Geneva 1 9 9 9 . 1 . Guide to the Expression of Unc ertainty in Measurement ( GUM) . ISO Geneva, Sc hweiz 1 9 9 3 ( ISB N 9 2 - 6 7 - 1 0 1 8 8 9)

Note the difference between Trueness and ; the latter refers to one measurement whereas the former to the mean of many measurements. Therefore inaccuracy will include both the bias and the imprecisio n inherent in a specific result. Accuracy

2 . J Kragten. Calc ulating standard deviations and c onfidenc e intervals with a universally applic able spreadsheet tec hnique. Analyst 1 9 9 4 ; 1 1 9 : 2 1 6 1 - 2 1 6 6 . 3 . Eurac hem, Quantifying Unc ertainty in Analytic al Measurement, LGC Information Servic e, Teddington Middlesex 1 9 9 5 ISB N 0 - 948926 - 08 - 2 4 . A. Kallner, J Waldenstrom Does the Unc ertainty of Commonly Performed Gluc ose Measurements Allow Identific ation of Individuals at High Risk for Diabetes? Clin Chem Lab Med 1 9 9 9 ;3 7 :9 0 7 - 9 1 2

Terminology and nomenclature are cornerstones of science and a citation from L Carroll ‘Alice through the looking glass’ might be appropriate to consider:

5 . C Ric os et al. Current databases on biologic al variation: pros, c ons and progress. Sc and J Clin Lab Invest 1 9 9 9 ;5 9 ( 7 ) :4 9 1 500. 6 . X. Fuentes - Arderiu et al. Pre - metrologic al ( Pre - Analytic al)

‘When I use a word’ Humpty Dumpty said in rather a scornful tone, ‘it means just what I choose it to mean ? neither more nor less’

Variation of some B ioc hemic al Quantities. Clin Chem Lab Med 1 9 9 9 ;3 7 :9 8 7 - 9 8 9

‘The question is,’ said Alice, ‘whether you can make words mean different things’.

‘The question is, said Humpty Dumpty, ‘ which is to be master ? that’s all’.

Page 21 eJIFCC2001Vol13No1pp016-024

2VlNHUKHWVEXGJHW

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Independent variables:

Variable: Value: Stand.unc., constant: Stand.unc., relative:

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1.000



0.800

Relativae contributions of uncertatinty

0.900

0.700

 

0.600

0.500

0.400

0.300

0.200

0.100

Nominal  Part:

 

 

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Page 22 eJIFCC2001Vol13No1pp016-024

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Independent variables 7DUD 7RWZHLJKW    

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Stand.unc.: Variable: 7DUD 7RWZHLJKW

Value:  



 

Relativae contributions of uncertatinty

 

1.000 0.900 0.800 0.700 0.600 0.500 0.400 0.300 0.200 0.100 0.000 D

Nominal  Part:

 

E

F

G

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Comb unc.:  Rel. unc.:  k=  Exp.unc.:  Unc.interval: &RS\FHOO&WRWKHULJKWDVIDUDVWKHUHDUHHQWULHV'RQRWFRS\FHOOVGHOHWHWKHFRQWHQWVDQGUHHQWH '

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Page 23 eJIFCC2001Vol13No1pp016-024

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Independent variables

Variable: Value: Stand.unc., constant: Stand.unc., relative:

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+HLJKW  

Stand.unc.:







Variable: /HQJWK :LGWK +HLJKW

Value:   

  

  

Relativae contributions of uncertatinty

  

1.000 0.900 0.800 0.700 0.600 0.500 0.400 0.300 0.200 0.100 0.000 D

Nominal     Part:   

E

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k=  Exp.unc.:  Unc.intervall   Comb unc.:  Rel. unc.:  &RS\FHOO&WRWKHULJKWDVIDUDVWKHUHDUHHQWULHV'RQRWFRS\FHOOVGHOHWHWKHFRQWHQWVDQGUHHQWH '

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Page 24 eJIFCC2001Vol13No1pp016-024

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