Calibration and Detection Limits

In: Wenclawiak, Koch, Hadjicostas (eds.) Quality Assurance in Analytical Laboratories – Teaching Material

Author: TitelKaus, R., Nagel, T.: Calibration and Detection Limits © Springer-Verlag Berlin Heidelberg 2003

In: Wenclawiak, Koch, Hadjicostas (eds.) Quality Assurance in Analytical Chemistry – Training and Teaching

Calibration and Detection Limits

Rüdiger KausThomas Nagel




Contents

1 Introduction

2 Basics of Calibration

3 Excel-charts for calibration

4 Limits of detection, determination




1 Introduction

Calibration is an important process in

establishing traceability

method validation to get the

performance characteristics

the routine use of modern analytical

equipment




What is Calibration?

Calibration is the process of establishing how the response of a measurement process varies with respect to the parameter being measured.

The usual way to perform calibration is to subject known amounts of the parameter (e.g. using a measurement standard or reference material) to the measurement process and monitor the measurement response.




Calibration has Two Major Aims:

Establishing a mathematical function which describes the dependency of the system’s parameter (e. g. concentration) on the measured value Gaining statistical information of the analytical system, e. g. sensitivity, precision




Calibration Concepts

External standard

Internal standard

Standard addition

Definitive calibration methods




Goals of Calibration

“Ability to calculate a (measurement) result in a secure (safe) working range”

Funk, W., Dammann, V., and Donnevert, G., “Quality Assurance in Analytical Chemistry”,

VCH Weinheim 1995




First Steps to the Goal:

Establishing the calibration function Choosing the working range Measuring several calibration standards Linear regression Test of non linear regression Test of variance homogeneity Calculate performance characteristics




Next Steps to the Goal:

Calculating the (measurement) results Conversion of the calibration function Reporting the measurement results

Calculating the statistical limits Securing the lower working range critical value of detection limit calculation of the quantitation limit Securing the higher working range




2 Basics of CalibrationMathematical Functions

simple linear function with intercept in zero originy = m x

linear function with intercept a and slope by = ax + b

quadratic functiony = ax2 + bx + c

cubic functiony = ax3 + bx2 + ax + d

exponential functiony = a ebx




Principle of Linear Regression




Simple Example of Linear Regression

No. x y x2 y2 yx

1 0,5 0,47 0,25 0,221 0,235 0,491

2 1,0 1,01 1,00 1,020 1,010 0,997

3 1,5 1,54 2,25 2,372 2,310 1,503

4 2,0 1,98 4,00 3,920 3,960 2,009

Xm=1,25 ym=1,25 Σ=7,5 Σ=7,533 Σ=7,515

a = 1,012 b = -0,015




Simple Example of Linear RegressionNo. x

mg/Ly

1 1,0 37

2 2,0 120

3 3,0 170

4 4,0 205

Xm=2,50 ym=133

Calibration

0,000

50,000

100,000

150,000

200,000

250,000

0,000 0,500 1,000 1,500 2,000 2,500 3,000 3,500 4,000 4,500 5,000

mg/L

regression curve confidence interval




International Standards

ISO 8466 “Water quality– Calibration and evaluation of analytical methods and estimation of performance characteristics

- Part 1: Statistical evaluation of the linear calibration function”

- Part 2: Calibration strategy for non-linear second order calibration functions”




3 Excel-charts for CalibrationUniversity of Applied SciencesDepartment of Physical ChemistryProf Dr. R. Kaus

QA-guideline: QA_04_N10

Calibration and process data for the determination of Test data

method

Calibration data measured at

x y_1

mg/L Area / 1000 Process data of a calibration function ... of 1. order ... 2. ordervalue 1 0,050 3,060 Testing linearity: regression 2. Ordnung is significant not better

value 2 0,100 3,522 correlation coefficient 0,995 c = 0,362

value 3 0,150 3,707 slope 0,342 b = 9,487 b = 9,288

value 4 0,200 4,280 intercept 0,106 a = 2,508 a = 2,528

value 5 0,250 5,058 residual standard deviation s(y) = 0,155 s(y) = 0,166

value 6 0,300 5,510 process standard deviation s(x0) = 0,016 s(x0) = 0,017

value 7 0,350 5,703 process variation coeff icient V(x0) = 0,060 E = 9,487

value 8 0,400 6,205 0,206

value 9 0,450 6,950 auxiliary value for the determination of x_P y_P = 2,858 t(95%, single-sided) 1,86

value 10 0,500 7,178 testing value to secure the low er range limit x_P = 0,071 t(95%, single-sided) 1,86

number mean MDL = LC XN = 0,057 t(99%, single-sided) 2,90 sums10 0,275 5,117 detection limit (DIN 32645) XE = 0,115 t(99%, double-sided) 3,36

quantitation limit (DIN 32645) XB = 0,177k = 3

Calibration function 1.O: y =2,508 + 9,487 x

SampleData factor of Results

confidence interval for

resultsStandard uncertainty

no. Area / 1000 dilution mg/L ± mg/L rel. in % e.g. double probelowest 3,06 0,0581 1 0,058 ± 0,035 0,0351 60,3% M = 2

mean 5,12 0,275 1 0,275 ± 0,032 0,0319 11,6% Standard relative expanded

highest 7,18 0,4922 1 0,492 ± 0,035 0,0351 7,1% uncertainty S.U. S.U.

1 5,00 0,2626 1 0,263 ± 0,032 0,0319 12,2% 0,0127 4,8% 0,0252 1 ±

3 1 ±

4 1 ±

5 1 ±

6 1 ±

7 1 ±

8 1 ±

9 1 ±

10 1 ±

Quality Management Manual

Page: 1 von 3 PagesPart B

Version:Date : proved:

released:

2.0

Ka

13.04.2001

Calibration

x_P

XB

0,000

1,000

2,000

3,000

4,000

5,000

6,000

7,000

8,000

0,000 0,100 0,200 0,300 0,400 0,500 0,600mg/L





3 Excel-charts for CalibrationUniversity of Applied Sciences Version: 2.0

Department of Physical Chemistry Date : 13.04.01

Prof Dr. R. Kaus Part B proved:

QA-guideline: QA_04_N10 Page: 1 von 3 Pages released: Ka

data sheet with the complete data

x y residuals

mg/L Area / 1000

i_01 0,05 3,06i_02 0,1 3,522i_03 0,15 3,707i_04 0,2 4,28i_05 0,25 5,058i_06 0,3 5,51i_07 0,35 5,703i_08 0,4 6,205i_09 0,45 6,95i_10 0,5 7,178

number mean

10 0,275 5,117

Outlier testing data pair with suspected outlier i_03 0,15 3,707F(f1=1,f2=N-2)= 5,317645 no outlier above above above

F-Test PW = 1 no outlier for testing: line 33 should be eliminatedt-Test VB(yA) = 4,245976 Process data of the new linear calibration function (no outlier)

3,616836 slope b = 9,487intercept a = 2,508residual standard deviation s(y)_a = 0,155

data sheet with the reduced data process standard deviation s(x0) _a= 0,016x y process variation coefficient V(x0)_a = 6,0 %

i_01 0,05 3,06i_02 0,1 3,522i_03 0,15 3,707i_04 0,2 4,28i_05 0,25 5,058i_06 0,3 5,51i_07 0,35 5,703i_08 0,4 6,205i_09 0,45 6,95i_10 0,5 7,178

number mean

10 0,28 5,117test datas: y_P = 2,858 x_P = 0,07138Relevance: Comparing the slopes 0,00% deviation


residual analysis

-0,25

-0,2

-0,15

-0,1

-0,05

0

0,05

0,1

0,15

0,2

0,05 0,1 0,15 0,2 0,25 0,3 0,35 0,4 0,45 0,5mg/L

residual analysis

-0,3

-0,2

-0,2

-0,1

-0,1

0,0

0,1

0,1

0,2

0,2

0,05 0,1 0,15 0,2 0,25 0,3 0,35 0,4 0,45 0,5mg/L

0

1

2

3

4

5

6

7

8

0,000 0,100 0,200 0,300 0,400 0,500 0,600mg/L

Are

a / 1

000 x_P




3 Excel-charts for CalibrationUniversity of Applied Sciences Quality Management Manual Version: 2.0




Validation of an analytical method

for the determination of Test data

Principal calibrationProcess data of a calibration function of ...1. Order ... 2. Ordernumber of data N = 10 c = 0,362slope b = 9,487 b = 9,288intercept a = 2,508 a = 2,528residual standard deviation s(y) = 0,155 s(y) = 0,166process standard deviation s(x0) = 0,016 s(x0) = 0,017process variation coefficient V(x0) = 5,95% E = 9,487

calibration function 1.O: y = 2,508 + 9,487 x

Testing linearity: regression 2. order is not significant better

Securing the lower range limit range statistically secure

auxiliary value for the determination of x_P y_P = 2,8580 t(95%, einseitig) 1,86

testing value to secure the lower range limit x_P = 0,071 t(95%, einseitig) 1,86

capability of detection (DIN 32465) NG = 0,057 t(99%, einseitig) 2,90

quantitation limit BG = 0,115 t(99%, zw eiseitig) 3,36

k = 3

VB_x1 = 0,19

Relative analytical imprecision VB_x1(rel) = 381,73 %

1 0,025 770,12 0,05 381,73 0,075 252,34 0,1 187,65 0,125 148,76 0,15 122,87 0,175 104,38 0,2 90,59 0,225 79,7

10 0,25 71,111 0,275 64,012 0,3 58,113 0,325 53,114 0,35 48,915 0,375 45,216 0,4 41,917 0,425 39,118 0,45 36,619 0,475 34,320 0,5 32,221 0,525 30,422 0,55 28,723 0,575 27,224 0,6 25,825 0,625 24,5

confidence interval VB(x)rel in %

0,0

100,0

200,0

300,0

400,0

500,0

600,0

700,0

800,0

900,0

0 0,1 0,2 0,3 0,4 0,5 0,6 0,7mg/ L




Calibration data

x y_1

mg/L Area / 1000

value 1 0,050 3,060value 2 0,100 3,522value 3 0,150 3,707value 4 0,200 4,280value 5 0,250 5,058value 6 0,300 5,510value 7 0,350 5,703value 8 0,400 6,205value 9 0,450 6,950value 10 0,500 7,178number mean

10 0,275 5,117

3 Excel-charts for calibration3.1 Insert Data

University of Applied SciencesDepartment of Physical ChemistryProf Dr. R. Kaus



method


x y_1

mg/l Area / 1000 Process data of a calibration function ... of 1. order ... 2. Order

value 1 0,050 3,060 Testing linearity: regression 2. Ordnung is significant not better


value 3 0,150 3,707 slope 0,342 b = 9,487 b = 9,288




value 7 0,350 5,703 process variation coefficient V(x0) = 0,060 E = 9,487

value 8 0,400 6,205 0,206


value 10 0,500 7,178 testing value to secure the lower range limit x_P = 0,071 t(95%, single-sided) 1,86

number mean MDL = LC XN = 0,057 t(99%, single-sided) 2,90 sums

10 0,275 5,117 detection limit (DIN 32645) XE = 0,115 t(99%, double-sided) 3,36






Nr. Area / 1000 dilution mg/l ± ... mg/l rel. in % e.g. double probelowest 3,06 0,0581 1 0,058 ± 0,035 0,0351 60,3% M = 2

mean 5,12 0,275 1 0,275 ± 0,032 0,0319 11,6% Standard- relative expanded


1 5,00 0,2626 1 0,263 ± 0,032 0,0319 12,2% 0,0127 4,8% 0,0252 1 ±

3 1 ±

4 1 ±

5 1 ±

6 1 ±

7 1 ±

8 1 ±

9 1 ±

10 1 ±

Ka

13.04.2001




released:

2.0

Calibration

x_P

XB

0,000

1,000

2,000

3,000

4,000

5,000

6,000

7,000

8,000

0,000 0,100 0,200 0,300 0,400 0,500 0,600mg/l

Are

a /

10

00


part from sheet: Insert data & presenting result




3 Excel-charts for calibration3.2 Linear regression and performance characteristics

t(95%, single-sided) 1,86



t(99%, double-sided) 3,36




method


x y_1




value 3 0,150 3,707 slope 0,342 b = 9,487 b = 9,288





value 8 0,400 6,205 0,206













1 5,00 0,2626 1 0,263 ± 0,032 0,0319 12,2% 0,0127 4,8% 0,0252 1 ±

3 1 ±

4 1 ±

5 1 ±

6 1 ±

7 1 ±

8 1 ±

9 1 ±

10 1 ±

Ka

13.04.2001




released:

2.0

Calibration

x_P

XB

0,000

1,000

2,000

3,000

4,000

5,000

6,000

7,000

8,000

0,000 0,100 0,200 0,300 0,400 0,500 0,600mg/l

Are

a /

10

00


Process data of a calibration function of 1. order

correlation coefficient 0,995

slope b = 9,487intercept a = 2,508residual standard deviation s(y) = 0,155process standard deviation s(x0) = 0,016process variation coefficient V(x0) = 0,060

0,206

auxiliary value for the determination of x_P y_P = 2,858testing value to secure the lower range limit x_P = 0,071MDL = LC XN = 0,057detection limit (DIN 32645) XE = 0,115quantitation limit (DIN 32645) XB = 0,177

k = 3




22

2

2

2

)(

)(112

)(

11

xxb

yy

Ntsx

xx

x

Ntsay

i

PxoP

iYP

22

2

2

2

)(

)(11

)(

)(11

xxb

xXNk

NtskXB

xx

xx

NtsXN

ixo

ixo

3 Excel-charts for Calibration3.2 Linear Regression and Performance Characteristics

slope b = =SLOPE(C4:C13;B4:B13)

intercept a = =INTERCEPT(C4:C13;B4:B13)

residual standard deviation s(y) = =STEYX(C4:C13;B4:B13)

process standard deviation s(x0) = =s_y/b

process variation coefficient V(x0) = =s_x0/x_m

Q= =SUMQUADABW(B4:B12)

auxillary value for the determination of x_P y_P = =a+t*s_y*SQRT(1+1/N+x_m^2/Q)

testing value to secure the lower range limit x_P = =2*s_x0*t*SQRT(1/N+1+(y_P-y_m)^2/b^2/Q)

detection limit (DIN 32645) XN ==s_x0*t99e*SQRT(1+1/N+x_m^2/Q)

quantiation limit (DIN 32645) XB ==k*s_x0*t99z*SQRT(1+1/N+(k*NG-x_m)^2/Q)

k = 3

Performance characteristics of the calibration function (DIN 38402 Teil 51)

y a s tN

x

x x

x s tN

y y

b x x

P Y

i

P xoP

i

11

2 11

2

2

2

2 2

( )

( )

( )








t(99%, double-sided) 3,36

Do we need tables of statistics?

No, in EXCEL are a lot of functions integrated

Example: =TINV(0,1;N-2)= 1,86

3 Excel-charts for Calibration3.2 Linear Regression and Performance Characteristics




3 Excel-charts for Calibration3.3 Variance Homogeneity (Heteroscedasticity)

from sheet: Calibration (10 values) with test for homogeneity of the variances

Calibration and process dataunit: mg/L extinction

Values no. x y_1 0,05 0,14 y_4 y_5 y_6 y_7 y_8 y_9 y_10

1 0,05 0,140 0,143 0,140 0,146 0,144 0,145 0,144 0,146 0,145 0,148 0,14412 0,1 0,2813 0,15 0,4054 0,2 0,5355 0,25 0,6626 0,3 0,7897 0,35 0,9168 0,4 1,0589 0,45 1,17310 0,5 1,303 1,302 1,303 1,304 1,300 1,296 1,295 1,301 1,296 1,306 1,3006

10 0,275 0,7262

test for homogeneity of the variances

o.k.





3 Excel-charts for calibration3.4 Comparing function of 1. order with function of 2. order




method


x y_1




value 3 0,150 3,707 slope 0,342 b = 9,487 b = 9,288





value 8 0,400 6,205 0,206













1 5,00 0,2626 1 0,263 ± 0,032 0,0319 12,2% 0,0127 4,8% 0,0252 1 ±

3 1 ±

4 1 ±

5 1 ±

6 1 ±

7 1 ±

8 1 ±

9 1 ±

10 1 ±

Ka

13.04.2001




released:

2.0

Calibration

x_P

XB

0,000

1,000

2,000

3,000

4,000

5,000

6,000

7,000

8,000

0,000 0,100 0,200 0,300 0,400 0,500 0,600mg/l

Are

a /

10

00


Process data of a calibration function ... of 1. order ... 2. orderTesting linearity: regression of 2. order is not significantly better

correlation coefficient 0,995 c = 0,362

slope 0,342 b = 9,487 b = 9,288

intercept 0,106 a = 2,508 a = 2,528

residual standard deviation s(y) = 0,155 s(y) = 0,166

process standard deviation s(x0) = 0,016 s(x0) = 0,017

process variation coefficient V(x0) = 0,060 E = 9,487




Calibration function: y = a+ b *x (+ c* x2)

3 Excel-charts for calibration3.5 Calculating the function




method


x y_1




value 3 0,150 3,707 slope 0,342 b = 9,487 b = 9,288





value 8 0,400 6,205 0,206













1 5,00 0,2626 1 0,263 ± 0,032 0,0319 12,2% 0,0127 4,8% 0,0252 1 ±

3 1 ±

4 1 ±

5 1 ±

6 1 ±

7 1 ±

8 1 ±

9 1 ±

10 1 ±

Ka

13.04.2001




released:

2.0

Calibration

x_P

XB

0,000

1,000

2,000

3,000

4,000

5,000

6,000

7,000

8,000

0,000 0,100 0,200 0,300 0,400 0,500 0,600mg/l

Are

a /

10

00







3 Excel-charts for calibration3.6 Graphically representation the data

part from sheet: Insert data & presenting resultUniversity of Applied SciencesDepartment of Physical ChemistryProf Dr. R. Kaus



method


x y_1




value 3 0,150 3,707 slope 0,342 b = 9,487 b = 9,288





value 8 0,400 6,205 0,206













1 5,00 0,2626 1 0,263 ± 0,032 0,0319 12,2% 0,0127 4,8% 0,0252 1 ±

3 1 ±

4 1 ±

5 1 ±

6 1 ±

7 1 ±

8 1 ±

9 1 ±

10 1 ±

Ka

13.04.2001




released:

2.0

Calibration

x_P

XB

0,000

1,000

2,000

3,000

4,000

5,000

6,000

7,000

8,000

0,000 0,100 0,200 0,300 0,400 0,500 0,600mg/l

Are

a /

10

00


Calibration

x_P

XB

0,000

1,000

2,000

3,000

4,000

5,000

6,000

7,000

8,000

0,000 0,100 0,200 0,300 0,400 0,500 0,600mg/l





part from sheet: Outlier

3 Excel-charts for calibration3.7 Calculating outliers3.8 Graphically representation the outliers

University of Applied Sciences Version: 2.0





x y residuals

mg/L Area / 1000

i_01 0,05 3,06i_02 0,1 3,522i_03 0,15 3,707i_04 0,2 4,28i_05 0,25 5,058i_06 0,3 5,51i_07 0,35 5,703i_08 0,4 6,205i_09 0,45 6,95i_10 0,5 7,178

number mean

10 0,275 5,117





i_01 0,05 3,06i_02 0,1 3,522i_03 0,15 3,707i_04 0,2 4,28i_05 0,25 5,058i_06 0,3 5,51i_07 0,35 5,703i_08 0,4 6,205i_09 0,45 6,95i_10 0,5 7,178

number mean



residual analysis

-0,25

-0,2

-0,15

-0,1

-0,05

0

0,05

0,1

0,15

0,2

0,05 0,1 0,15 0,2 0,25 0,3 0,35 0,4 0,45 0,5mg/L

residual analysis

-0,3

-0,2

-0,2

-0,1

-0,1

0,0

0,1

0,1

0,2

0,2

0,05 0,1 0,15 0,2 0,25 0,3 0,35 0,4 0,45 0,5mg/L

0

1

2

3

4

5

6

7

8

0,000 0,100 0,200 0,300 0,400 0,500 0,600mg/L

Are

a / 1

000 x_P

residual analysis

-0,25

-0,2

-0,15

-0,1

-0,05

0

0,05

0,1

0,15

0,2

0,05 0,1 0,15 0,2 0,25 0,3 0,35 0,4 0,45 0,5

mg/L

data pair with suspected outlier i_03 0,15 3,707no outlier above above aboveno outlier for testing: line 33 should be eliminated




part from sheet: Outlier

3 Excel-charts for calibration3.7 Calculating outliers3.8 Graphically representation the outliers

University of Applied Sciences Version: 2.0





x y residuals

mg/L Area / 1000

i_01 0,05 3,06i_02 0,1 3,522i_03 0,15 3,707i_04 0,2 4,28i_05 0,25 5,058i_06 0,3 5,51i_07 0,35 5,703i_08 0,4 6,205i_09 0,45 6,95i_10 0,5 7,178

number mean

10 0,275 5,117





i_01 0,05 3,06i_02 0,1 3,522i_03 0,15 3,707i_04 0,2 4,28i_05 0,25 5,058i_06 0,3 5,51i_07 0,35 5,703i_08 0,4 6,205i_09 0,45 6,95i_10 0,5 7,178

number mean



residual analysis

-0,25

-0,2

-0,15

-0,1

-0,05

0

0,05

0,1

0,15

0,2

0,05 0,1 0,15 0,2 0,25 0,3 0,35 0,4 0,45 0,5mg/L

residual analysis

-0,3

-0,2

-0,2

-0,1

-0,1

0,0

0,1

0,1

0,2

0,2

0,05 0,1 0,15 0,2 0,25 0,3 0,35 0,4 0,45 0,5mg/L

0

1

2

3

4

5

6

7

8

0,000 0,100 0,200 0,300 0,400 0,500 0,600mg/L

Are

a / 1

000 x_P

test datas: y_P = 2,86 x_P = 0,07138Relevance: Comparing the slopes 0,00% deviation

residual analysis

-0,3

-0,2

-0,2

-0,1

-0,1

0,0

0,1

0,1

0,2

0,2

0,05 0,1 0,15 0,2 0,25 0,3 0,35 0,4 0,45 0,5mg/L

0

1

2

3

4

5

6

7

8

0,000 0,100 0,200 0,300 0,400 0,500 0,600mg/L

Are

a /

1000

0

1

2

3

4

5

6

7

8

0,000 0,100 0,200 0,300 0,400 0,500 0,600

mg/L

Are

a /

1000

x_P




The analytical results will be calculated by the inverse of the calibration function: x = (y-a)/b

3 Excel-charts for calibration3.9 Calculating analytical results




method


x y_1




value 3 0,150 3,707 slope 0,342 b = 9,487 b = 9,288





value 8 0,400 6,205 0,206













1 5,00 0,2626 1 0,263 ± 0,032 0,0319 12,2% 0,0127 4,8% 0,0252 1 ±

3 1 ±

4 1 ±

5 1 ±

6 1 ±

7 1 ±

8 1 ±

9 1 ±

10 1 ±

Ka

13.04.2001




released:

2.0

Calibration

x_P

XB

0,000

1,000

2,000

3,000

4,000

5,000

6,000

7,000

8,000

0,000 0,100 0,200 0,300 0,400 0,500 0,600mg/l

Are

a /

10

00





no. Area / 1000 dilution mg/L ± mg/L

lowest 3,06 0,0581 1 0,058 ± 0,035 0,035mean 5,12 0,275 1 0,275 ± 0,032 0,032

highest 7,18 0,4922 1 0,492 ± 0,035 0,0351 5,00 0,2626 1 0,263 ± 0,032 0,0322 1 ±

3 1 ±

4 1 ±

5 1 ±

6 1 ±

7 1 ±

8 1 ±

9 1 ±

10 1 ±






3 Excel-charts for calibration3.10 Estimating the uncertainty




method


x y_1




value 3 0,150 3,707 slope 0,342 b = 9,487 b = 9,288





value 8 0,400 6,205 0,206













1 5,00 0,2626 1 0,263 ± 0,032 0,0319 12,2% 0,0127 4,8% 0,0252 1 ±

3 1 ±

4 1 ±

5 1 ±

6 1 ±

7 1 ±

8 1 ±

9 1 ±

10 1 ±

Ka

13.04.2001




released:

2.0

Calibration

x_P

XB

0,000

1,000

2,000

3,000

4,000

5,000

6,000

7,000

8,000

0,000 0,100 0,200 0,300 0,400 0,500 0,600mg/l

Are

a /

10

00


Standard uncertainty

e.g. double measurementM = 2

Standard- relative expanded

uncertainty S.U. S.U.

0,0127 4,8% 0,025




4 Limits of Detection, Determination

4.1 Limit of Detection

4.2 Limit of Determination (Quantitation)




4. Limits of Detection, Determination4.1Limit of Detection

Detection limit

DIN 32645 from blanks

from calibration data

Funk dynamic model

IUPAC

Coleman recursive formula

explicit formula

2

D x 0 f;αx

1 1 xx s t

m n Q

LD f;α

s 1 1x t

b m n

2

c y n2

ii 1

0 x1y b s t 1

nx x

2

y cD n

22i

i 1

s t y y1x 2 1

a na x x

c 1 α,v 0S t sαβ,v 0 1 α,v 0

D

δ σ 2t σK Kx

A I A I

B A1 α,v

0

σ σK 1 r(B,A) t

σ A

2

A1 α,v

σI 1 t

A

11

2 22 2D

D n 2,1 α n 2,1 βxx xx

x xs 1 x 1x t 1 t 1

a n S n S

1

2 2

D H V

J J 4HKx DL

2H

n 2,1 β

aA

s t

12 2

n 2,1 α

n 2,1 β xx

t 1 xB 1

t n S

2xx

1F B 1 S

n

xxG 2AB S 2 1xxH A S J G 2x 2K F x





4.2Limit of Determination (Quantitation)

A) DIN 32645

Detection limit

by fast estimation:

Capability limit

Determination limit

by fast estimation

Factor for fast estimation

2k

D x0 f;ax

y - a 1 1 xx = = s t + +

b m n Q

n;α x0Dx = 1,2 Φ s

2

C NG x0 f;βx

1 1 xx = x + s t + +

m n Q

2

DDT x0 f;α

x

x - x1 1x = k s t + +

m n Q

n;α x0DTx = 1,2 k Φ s

n;α f;α

1Φ = t 1+

n






B) Funk

Detection limit dynamic model

Determination limit dynamic model

2

c y n2

ii 1

0 x1y b s t 1

nx x

2

y cD n

22i

i 1

s t y y1x 2 1

a na x x

2y

c n2

ii 1

s t 1 xx 1

a nx x

2

ch y n

2

ii 1

x x1y b 2 s t 1

nx x

2

y hhDT n

22i

i 1

s t y yy b 1x 1

a a na x x






C) IUPAC

Detection limitc 1 α,v 0S t s

αβ,v 0 1 α,v 0D

δ σ 2t σK Kx

A I A I

B A1 α,v

0

σ σK 1 r(B,A) t

σ A

2

A1 α,v

σI 1 t

A






D) Coleman/HUBAUX-VOS model

Detection limit recursive formula

explicit formula

1

2 2

D H V

J J 4HKx DL

2H

n 2,1 β

aA

s t

12 2

n 2,1 α

n 2,1 β xx

t 1 xB 1

t n S

2xx

1F B 1 S

n

xxG 2AB S 2 1xxH A S J G 2x 2K F x

11

2 22 2D

D n 2,1 α n 2,1 βxx xx

x xs 1 x 1x t 1 t 1

a n S n S




FUNK et. al. #BEZUG!

Prüfgröße y_P = 2914 t(95%,onesided) 1,86 detection limit x_P = 0,0862 t(95%,onesided) 1,86

quantitation limit BG = 0,1250 mg/l

Figure 1: The values are calculated with the formulas from Funk’s book [6] in an Excel-

sheet

Figure 2: The values are calculated with the formulas from DIN 32645 [5] in an Excel-sheet

4 Limits of Detection, Determination4.3 Critical Discussion

The values are calculated with the formulas from Funks book in an EXCEL sheet

Figure 1: The values are calculated with the formulas from Funk’s book [6] in an Excel-

sheet

DIN 32645

Nachweisgrenze detection limit NG = 0,070 mg/l t(99%,einseitig)2,90

Bestimmungsgrenze BG = 0,212 mg/l t(99%,zweiseitig)3,36 quantitation limit k = 3

Figure 2: The values are calculated with the formulas from DIN 32645 [5] in an Excel-sheet The values are calculated with the formulas from DIN 32645 in an EXCEL sheet




Which Values of Detection limits and Quantitation limits are correct?

Choosing a confidence range for the quantitation limit:

recommendation of the DIN 32645: k=3 +/- 33% ;

recommendation of the IUPAC: 1/k=0,1 +/- 10%





Detection limits from blanks - problems with the normal distribution

detection limits from blanks give very low values, but

- blanks don’t belong to the same statistically population as the calibration and measuring data

- often they are normally distributed





Determination of the detection limit from blanks for: Naphthalene(with test of normal distribution)

Sample-Nr. Blank IS Blank/ISBlank/IS sorted x-x_m u=(x-x_m)/s uexpected

1 339 515592 0,0006575 0 -0,00031 -1,068 -1,5382 193 518421 0,00037228 0 -0,00031 -1,068 -0,9993 196 515572 0,00038016 0 -0,00031 -1,068 -0,6594 203 522514 0,00038851 0 -0,00031 -1,068 -0,385 376 529887 0,00070959 0,000372 0,00007 0,233 -0,1216 0 527819 0 0,000380 0,00007 0,261 0,1217 288 525571 0,00054798 0,000389 0,00008 0,290 0,388 0 527309 0 0,000548 0,00024 0,847 0,6599 0 563544 0 0,000657 0,00035 1,230 0,999

10 0 517630 0 0,000710 0,00040 1,412 1,538number mean 0,000306

10 standard deviation 0,000286

standard deviation 0,000286

slope from the clal ibration curve 0,762

detection limit 0,000654 mg/l

referred to the sample weight 0,00245 mg/kg TS

-2,000

-1,500

-1,000

-0,500

0,000

0,500

1,000

1,500

2,000

-2 -1,5 -1 -0,5 0 0,5 1 1,5 2

u expected

u m

ea

su

re

d





Quantitation limits and working range

Quantitation limits are often higher than some of the calibration data

(in the procedure suggested by Funk the quantitation limit is always higher than the 1st calibration point).

Now there is the difficulty:

Which is the lowest concentration I'm allowed to record?





Fachhochschule Niederrhein Quality management Version: V_1.0

Fachbereich Chemie

Prof. Dr. R. Kaus Date: 30.10.97

Calibration Page: 1 von 4 pages prepared by : Kaus

Calibration and process data for the destination of Carbonx y_1 in water

Calibration data mg/l Area

Wert 1 0,05 3060 (measured data are taken from DIN 32 645)Wert 2 0,1 3522Wert 3 0,15 3707Wert 4 0,2 4280Wert 5 0,25 5058Wert 6 0,3 5510Wert 7 0,35 5703Wert 8 0,4 6205Wert 9 0,45 7156Wert 10 0,5 7178number means

10 0,275 5137,9

Process data of a linear calibration function (DIN 38402 Teil 51)

slope b = 9662intercept a = 2481residual standard deviation s(y) = 192process standard deviation s(x0) = 0,0199process variation coef ficient V(x0) = 7,24%

0,20625aux iliary value for the determination of x_P y_P = 2914 t( 95%, single-s ided) 1,86

testing value to secure the lower range limit x_P = 0,0862 t( 95%, single-s ided) 1,86

capability of detection (DIN 32465) XN = 0,070 t( 99%, single-s ided) 2,90

quantitation limit XB = 0,212 t(99%, double-s ided) 3,36

k = 3

Calibration

x_ P

XB(DIN)

__ XB(FUNK)

0

1 000

2 000

3 000

4 000

5 000

6 000

7 000

8 000

0 0,1 0,2 0,3 0,4 0,5 0,6mg/l

Are

a



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Calibration and Detection Limits