Detection Limits and

Measurements Uncertainties

From FTIR Instruments

Sylvie Bosch-Charpenay,

Peter Zemek, Barbara Marshik

MKS Instruments

On-Line Product Group

Terminology Detection Limit

EPA Method Detection Limit MDL– “Minimum change in concentration of analyte that the

method can detect with 99% confidence limit that a change has occurred”

ASTM D6348 Minimum Detection Limit MDC

Method 320 Detection limit MAU– “Lowest concentration limit for which its overall

fractional uncertainty is less than the analytical uncertainty (e.g. 5%) chosen”

Factors Influencing

Detection Limit

White noise

Biases (due to interferences, poor background)

NOT (or minimally) influenced by calibration,

instrument parameters, etc

MKS Confidential 3

How do we report

Detection Limits?

“Industry-accepted standard” is 3*stdev in N2– Does not account for effect of potential interferent

EPA Method 301??? MDL– Requires 7 low (non-zero) concentration samples +

Student’s t table for statistics

– Requires capability to create low level concentrations

ASTM D6348 MDC– 3 different definitions: which one to use ?

Method 320 MAU

MKS Confidential 4

EPA 40 CFR Part 63 Method 301 - Field Validation of Pollutant Measurements Methods From Various Waste Media

EPA 40 CFR Part 136 - Appendix B

Hubaux, A., Vos, G., Anal. Chem., 42, 8 (1970) 849-855

Terminology (NIST)

Repeatability = precision– “Closeness of agreement between the results of successive

measurements of the same measurand (e.g., cylinder) carried out under

the same conditions of measurement”

Reproducibility– “Closeness of agreement between the results of measurements of the

same measurand (e.g., cylinder) carried out under different conditions of

measurement (e.g., different instrument, different time, different location)”

Accuracy– “Closeness of the agreement between the result of a measurement and

the value of the measurand” (qualitative concept)

Error– “Result of a measurement minus the value of the measurand”

Confidence limit = Uncertainty– “A pair of numbers used to estimate a characteristic of a population, such

that the numbers can be stated with a specified probability that the

population characteristic is included between them”MKS Confidential 5

Terminology Uncertainty NIST

Total Uncertainty – sum of systematic error and random error

– Law of Propagation of Uncertainty: square root of the sum of the squares

Standard uncertainty u – 1-sigma

Expanded uncertainty U = u * k– k =coverage factor for stated level of confidence (k =

1.96 for 95%, k = 2.576 for 99%, k = 3 for 99.76%)

Confidence limit (or confidence interval) – Term used interchangeably with expanded

uncertainty

Random vs Systematic Error

MKS Confidential 7

8

Factors Influencing Uncertainty

Estimated uncertainty with

“good” spectral fit

Assuming “good” fit– i.e. no issues with interfering compounds, noise

much smaller than signal

Using released recipes

Assuming reading >> detection limit– i.e., within ~ 20-100 % of range

Combined effect of T, P, matrix broadening, resolution, frequency– 3% estimated uncertainty

After calibration span – 2% estimated uncertainty

Estimates do not hold at low level and poor spectral fit

What if the spectral fit is not

very good ?

How does goodness-of-fit

relate to overall uncertainty ?

One of the factors only

Loose correlation

Reading with a poor spectral fit might still have

high accuracy (low error)– but it is less likely

Reading with a great spectral fit might have a low accuracy (high error) if P, T, span factor etc… are not correct

However, if we assume all other factors are in spec (P, T, resolution, laser frequency, span factor, etc…), goodness-of-fit gives an indication of uncertainty

ASTM D6348, EPA m320

ASTM 6348 provides 3 different “Minimum

Detection Limits” calculations – MDC1, MDC2, MDC3

Method 320– MAU = detection limit

– FRU, FCU, FAU, FMU = fractional uncertainties

– Uses the largest as the overall uncertainty

Calculations across methods are in some cases similar, but not equivalent

How do ASTM D6348 and EPA

m320 compare ?

MDC1 and MAU are similar but not equivalent – MDC1 is using RMS and MAU is using the surface

area in the normalizing denominator

MDC2 has no equivalent

FRU, FCU, FAU have no equivalent– They are always smaller than MAU and FMU

MDC3 and FMU*reading are similar but not

equivalent – MDC3 is using RMS and FMU is using the surface

area in the normalizing denominator

14

EPA Method 320

Name What it does What it needs Level

MAU ??????

Analytical

Uncertainty

Goodness-of-fit parameter

accounting for noise

Noise spectrum Can be

large

FRU Fractional

Reproducibility

Uncertainty

Goodness-of-fit parameter

accounting for errors in

reproducibility of CTS spectra

CTS spectra taken

before and after

reference spectra

Small 1

FCU Fractional

Calibration

Uncertainty

Accounts for errors in Beer’s Law Reference Spectra Small 2

FAU Fractional

Analytical

Uncertainty

Accounts for errors due to different

pathlengths, T, P between CTS

spectra taken at under different

conditions

CTS done with

different PL, T, P

Small 3

FMU Fractional

Model

Uncertainty

Goodness-of-fit parameter

accounting for errors in the model

to extract multiple concentrations

from overlapping compounds

Sample Can be

large

OFU Overall

Fractional

Uncertainty

Maximum of all Uncertainties

(1) Because Multigas instruments have very similar alignment, and are very stable

(2) Because Multigas calibrations have multi points

(3) Because Multigas instruments have the same pathlength and are run at the same T, P as the reference spectra

15

ASTM D6348

Name What it does What it needs Level

MDC1 Noise-limited

Minimum

Detection Limit

Goodness-of-fit parameter

accounting for noise

Noise spectrum Can be

large

MDC2 Analytical

Algorithm Error

Calculated as 3 * stdev in

spectra with no analyte but with

interferents

Spectra with

interferents but no

analyte

Can be

large

MDC3 Analytical

Algorithm Error

Goodness-of-fit parameter

accounting for errors in the

model to extract multiple

concentrations from

overlapping compounds

Sample spectrum Can be

large

Graphical Goodness-of-fit

Representation

16

Black = calibration spectrum

Red = sample spectrum

Green = noise-only spectrum

Yellow filling:MDC3 or FMU*RS

Green filling:MDC1 or MAU

Calculations

17

Assume same pathlength, T and P between reference and sample

This assumption highlights absolute calculation differences

ASTM MDC1 and m320 MAU

18

NEAi = Noise Equivalent Absorbance at wavenumber i(spectrum in N2)

Aref i = Absorbance of reference spectrum at wavenumber i

Cref = Concentration of reference spectrum

N = number of points in analysis region

For constant Aref i = A, MDC1=MAU

refN

iref

N

i

C

A

NEA

MDC

0

2

0

2

)(

)(

1#refN

i

ref

N

i

C

A

NEA

NMAU

i

)(

)(0

2

NAA

NN 1)()(

00

2

MDC1 vs MAU

MKS Confidential 19

MDC1 = MAU

Absorbance

Wavenumber

Absorbance

MDC1 MAU

Wavenumber

ASTM MDC2

20

P spectra containing interferents but no analyte

P = number of measurements (spectra), minimum 8.

Cave = average concentration for analyte (= analytical bias)

Cp = measured concentration (reading) on spectrum p

P

pave CCP

MDC0

2)(1

32#

ASTM MDC3 and m320 FMU

21

REAi = Residual Equivalent Absorbance at wavenumber i

Aref i = Absorbance of reference spectrum at wavenumber i

Cref = Concentration of reference spectrum

N = number of points in analysis region

RS = reading of sample

For constant Aref i = A, MDC1=FMU*RS

refN

iref

N

i

C

A

REA

MDC

0

2

0

2

)(

)(

3#RS

C

A

REA

NFMUref

N

i

ref

N

i

i

)(

)(0

2

SEC = standard error of estimated

concentration

22

N

ref

ref

iA

CSEC

0

2)(

)1(

)(0

2

2

N

REAN

i

REAi = Residual Equivalent Absorbance at wavenumber i

Aref i = Absorbance of reference spectrum at wavenumber i

Cref = Concentration of reference spectrum

N = number of points in analysis region

refN

iref

N

i

C

A

REA

NSEC

0

2

0

2

)(

)(

)1(

1

Comparison ASTM MDC3 and SEC

23

refN

iref

N

i

C

A

REA

MDC

0

2

0

2

)(

)(

3# refN

iref

N

i

C

A

REA

NSEC

0

2

0

2

)(

)(

)1(

1

SEC calculation is similar to MDC3, except that it includes an additional factor of 1/sqrt(N-1), with N = number of points in analysis region

The additional factor is because the error is assumed to be random instead of systematic

SEC values are much smaller than MDC3, and are loosely correlated to the precision (standard deviation in N2)

Where does the +/- from MG2000

fit in?

24

Similar to FMU*RS except smaller by a factor or sqrt(N)

Similar in value to SEC (but not equivalent, again due to different normalizing denominator)

Closer to a precision value than an uncertainty value

refN

i

ref

N

i

C

A

REA

MG

i

)(

)(

/20000

2

RS

C

A

REA

NFMUref

N

i

ref

N

i

i

)(

)(0

2

What do I need for DL

calculation and if possible

uncertainty calculation ??

MKS Confidential 25

MKS Analysis Validation Utility

provides needed parameters

MDC1, MDC2 and MDC3 calculated

MAU and FMU calculated

FRU, FCU, FAU not calculated – They are smaller than MAU and FMU

– FRU small because Multigas instruments have very

similar alignment and are very stable

– FCU small because of multi-points calibrations

– FAU small because Multigas instruments have the

same pathlength and are run at the same T, P as the

reference spectra

SEC and +/- ??? MKS Confidential 26

27

Parameters in MKS’s Analysis

Validation Utility

MDC1, MAU MDC2 MDC3, FMU*Reading

Purpose Quantify

goodness-of-fit

for noise

Quantify detection

limit from precision

Quantify goodness-of-

fit for sample

Required

spectra

One noise-only

spectrum

At least eight

interference-only

spectra

Sample Spectrum

Comments on

required

spectra

Easy to obtain,

can be the first

spectrum in N2

taken after a

background

Can be difficult to

obtain for each

instrument

Available

Drawbacks Assumes all

error is

systematic

Combination of

mostly random

(noise) and some

systematic

(interferent) error

Assumes all error is

systematic

Example

28

similar similar

Typically lower than MDC3 or FMU*R

Which MDC to use for

Detection Limit?

MDC2 likely slightly lower than “real” DL – It does not include a measure of bias under sample

conditions

– Best option for DL

MDC3 likely significantly higher than “real” DL – It assumes all error is systematic.

– MDC3 provides more information on the uncertainty

than detection limit

MDC1 not directly related to “real” DL – Uses noise-only data

– Assumes all error is systematicMKS Confidential 29

Estimated Detection Limit

~ DL = MDC2 + bias– Provides “rule of thumb” values

– Value slightly more conservative than MDC2

– No goodness-of-fit parameters used in calculation

MDC2 – Estimate of mostly random error (due to noise) and

some systematic error (due to interferents) at zero

Bias – Average bias for analyte in the interferents-only spectra

used for MDC2)

– Estimate of systematic error due to interferents

MKS Confidential 30

Can any MDC be used for

Confidence Limit?

MDC2 likely lower than “real” CL – It does not include a measure of bias under sample

conditions

MDC3 likely higher than “real” CL – It assumes all the error from the goodness-of-fit yields

a systematic error in the reading

MDC1 not directly related to “real” CL – Uses noise-only data

MKS Confidential 31

Can we calculate a “Rule of

Thumb” Confidence Limit ?

Systematic Error = Max (MDC3-MDC1-C, bias) – MDC3-MDC1 is a goodness-of-fit estimate of sample

systematic error without the effect of noise

– C is typical value for goodness-of-fit for a “good fit”

– Bias is estimate of systematic error due to interferents

Random Error = MDC2

Best level for uncertainty is 2%– “Good fit”

– Readings in 20-100% of range

MKS Confidential 32

“Rule of Thumb” Confidence Limit

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~CL = max [max (MDC3-MDC1-C, bias) + MDC2, 2%]

Estimated Systematic Error fromsample goodness-of-fit

Estimated Systematic Error frominterferentspectra

Estimated Systematic Error from Sample goodness-of-fit

Estimated Random (+ some Systematic) Error

Estimated Error With “good” goodness-of-fit (but errors in T, P,…)

Estimated Error from sample goodness-of-fit and interferents

Estimated CL and DL

MKS Confidential 34

Color-coded for faster evaluation

Recommendations

MDC2 is best value to use for detection limit– Choice of interference spectra is critical

– Nitrogen spectra would only give DL due to noise

A more conservative value for DL is MDC2 + bias

MDC3 is a rough estimate for uncertainty– Because it assumes that all error is systematic

– Does not account for significant reduction of random

error in spectral analysis

– Does not account for possible interferent bias

~CL is a more reliable estimate for uncertainty– Based on goodness-of-fit, precision, and typical

parameters uncertainties

– Truly only an estimate ! Could still be “off” MKS Confidential 35

Big Question:

which spectra for MDC2

should be acceptable ?

Best choice– Interferents at (multiple) levels covering levels seen in

sample

– Different interferents for each analyte

– Collect a minimum of 8 (consecutive, non-

consecutive??) spectra

Interference spectra taken on other instrument ?

H2O and CO2 interference spectra taken on other

instrument ?

N2 spectra taken on instrument?

MKS Confidential 36

Recommended Procedure to

Generate DLs

Setup instrument – Allow instrument to equilibrate

– Load recipe (calibrations + instrument parameters)

Run interferent gases (no analyte) – Interferents at levels same as in sample

– Collect a minimum of 8 (consecutive??) spectra

– Calculate stdev and bias (average concentration of

analyte, which should be close to 0)

Calculate DL=MDC2 (3 * stdev) and ~DL=MDC2

+ bias

Determine if DL and ~DL under those conditions

meets regulatory requirements MKS Confidential 37