1

Linearity,linearrange,sensi.vity

Linearity

Methodsabilitytoobtainsignals

whicharedirectlypropor.onaltotheconcentra1onofanalyteinthesample.

2

Linearrange

Rangeofconcentra.onswherethesignalsaredirectlypropor1onaltothe

concentra1onoftheanalyteinthesample.

Sensi.vity

Thechangeininstrumentresponsewhichcorrespondstoachangein

themeasuredquan1ty.

Thegradientoftheresponsecurve.

3

Experimentplanningforevalua.onoflinearity

•  Typeofcalibra1onsamples•  Concentra1onrange•  Numberofconcentra1onlevels•  Measurementprotocol

Typeofcalibra.onsamples•  Standardsolu1ons– matrix-free

•  Matrix-containingsamples– Matrix-matchedcalibra1on– Blankmatrixextract(preferablythesametypeasthesample)

4

Concentra.onrange•  Appropriateforthemethod•  Expectedworkingrange±10%…20%•  70%…130%ofexpectedanalyteconcentra1on•  Blanksample

Numberofconcentra.onlevels•  Minimum6concentra1onlevelsareacceptable– Suggested10– Evenlyplaced– Preparedfromindependentdilu1ons

5

Measurementprotocol•  Assimilaraspossibletothereallifesitua1on– Randomorder– Betweenthesamples

•  Analysedatleasttwice

Evalua.onoflinearity

•  Visualevalua.on

•  Residuals

6

Visualevalua.on

0

20000000

40000000

60000000

80000000

100000000

120000000

0 2 4 6 8 10

Signal(a

u)

c(mg/kg)

0

10000000

20000000

30000000

40000000

50000000

60000000

70000000

80000000

0 1 2 3 4 5 6

Signal(a

u)

c(mg/kg)

LINEAR NOTLINEAR

Residual

εi = yi − yi

•  Differencebetweenexperimentalsignalandcalculatedsignal

yi

yi

yi = slope ⋅ xi +

yi

0

20000000

40000000

60000000

80000000

100000000

120000000

0 2 4 6 8 10

Signal(a

u)

c(mg/kg)

intercept

7

Visualevalua.on

0

20000000

40000000

60000000

80000000

100000000

120000000

0 2 4 6 8 10

Signal(a

u)

c(mg/kg)

0

10000000

20000000

30000000

40000000

50000000

60000000

70000000

80000000

0 1 2 3 4 5 6

Signal

c(mg/kg)

LINEAR NOTLINEAR

0

20000000

40000000

60000000

80000000

100000000

120000000

140000000

0 2 4 6 8 10 12

Signal(a

u)

c(mg/kg)

LINEAR?

Plotofresiduals

8

Linear,homoscedas.c

-6000000

-4000000

-2000000

0

2000000

4000000

6000000

8000000

10000000

0 0.2 0.4 0.6 0.8 1 1.2

Residu

al(a

u)

c(mg/kg)

Linear,heteroscedas.c

-30000000

-20000000

-10000000

0

10000000

20000000

30000000

0 2 4 6 8 10 12

Residu

al(a

u)

c(mg/kg)

9

Notlinear,heteroscedas.c

-40000000

-30000000

-20000000

-10000000

0

10000000

20000000

30000000

0 2 4 6 8 10 12

Residu

al(a

u)

c(mg/kg)

Rela.veresiduals

•  ±20%isacceptable

Yi =yi − yiyi

10

Correla.oncoefficientrCoefficientofdetermina.onr2

•  Totalvariabilityintheresponsethatisaccountedforbythemodel

•  Notameasureoflinearity!

Linearrange

11

“range”

0

20000000

40000000

60000000

80000000

100000000

120000000

0 2 4 6 8 10 12

Signal(a

u)

c(mg/kg)

Dynamicrange,Analy.calrange

12

0

20000000

40000000

60000000

80000000

100000000

120000000

0 2 4 6 8 10 12

Signal(a

u)

c(mg/kg)

Dynamicrange,Analy1calrange

Workingrange,Calibra.onrange

*LOQ

0

20000000

40000000

60000000

80000000

100000000

120000000

0 2 4 6 8 10 12

Signal(a

u)

c(mg/kg)

Dynamicrange,Analy1calrange

Workingrange,Calibra1onrange

Linearrange,Lineardynamicrange

*LOQ

13

Sensi.vity

•  gradientofthecalibra1ongraph

Sensi.vity

0

20,000,000

40,000,000

60,000,000

80,000,000

100,000,000

120,000,000

0 2 4 6 8 10

Signal(a

u)

c(mg/kg)

Slope = ΔSignalΔc

ΔSignal

Δc

Signal = 7610033⋅c+32270418

14

Sensi.vity•  Methodop1miza1on

•  Qualityassurance

•  Rou1nemonitoringoftheinstrument

Connectedtootherparameters•  Calcula1ngmatrixeffectcalibra1ongraphmethod

•  Calcula1ngLoDandLoQ

%ME = Slopematrix−matchedSlopesolvent

⋅100%

LoD = 3.3×Sy.xSlope

LoQ =10×Sy.xSlope