Data Analysis:Quantitative Statements about Instrument and Method Performance

Quantitative Description of Instrument Performance:

Why is it required

• To (quantitatively) describe instrument performance so that analytical measurements of unknowns can result in a quantitative result.

• For example: Calibration of a spectrofluorometer with quinine, followed by the analysis of unknown concentrations of quinine in tonic water.

Analytical Method Considerations

COST

SPEED NEEDS

• COST: Training, equipment...

• SPEED: When is it required...

• NEEDS: E.g., required detection limit...• Overall: An equal balance among the three.

Basic TermsChapter 1 Page 11, Table 1-3

(Skoog Holler Nieman)

• Precision (Reproducibility): Absolute or relative standard deviation, coefficient of variation, variance

• Bias: Absolute or relative systematic error

• Sensitivity: Calibration or analytical...

• Response Factor: Response as a function of analyte concentration

• Limit of Quantitation (LOQ): A numerical value usually directly related to LOD (>)

• Detection Limit (LOD or MDL): Different methods (statistical) to establish LOD

• Not Detected: Signal not discernible from noise using the given analytical system

• Concentration Range: From LOQ to LOL

• Selectivity: Coefficient of selectivity (%)

• Standard Deviation (s)

for a small set of data.

• Rel. Standard Deviation (using arithmetic mean)

• %RSD or CV (Coefficient of variation)

• Standard Deviation of the Mean

• Variance

• Overall standard deviation

Precision

223

22

21 ... Nsssss

2s

N

ssm

1

)(1

2

N

xxs

N

ii

%100 xsCV

xsRSD

N

xN

ii

1

2)(

Small set of data

Population

F-Test - Comparison of Variances

• Variance (s1)2 of TEST METHOD #1 (sample)

• Variance (s2)2 of TEST METHOD #2 (sample)

• The ratio of variances (where s22>s1

2)

21s

22s

21

22

s

sF

Gaussian (Normal)

Distributions

t-Test - Comparison of Means

• Mean of TEST METHOD #1 (sample)

• Mean of TEST METHOD #2 (sample)

• Are the means equal? The t-test we use is dependent on whether the variances of the two test methods are the same or different. Again, assume Gaussian Distributions. One-tailed t-test for equal variances, and two-tailed for unequal variances. The two-tailed presents more relaxed constraints on establishing equivalence.

1X

2X

A Look At Calibration Curves: A Primer

Do I use R² to indicate linearity, the appearance of the curve...what then?

• Linear Least Square Regression, Unweighted

• Ideal Calibration Curve

• Y-Int=0

Curve 1

y = 5.00000x

R2 = 1.00000

0

2000

4000

6000

8000

10000

12000

14000

16000

18000

0 500 1000 1500 2000 2500 3000 3500

[ ] ppm

Res

pons

e

ppm Arb Units1 55 2525 125

125 625625 31253125 15625

Curve 1

Curve 2

• Linear Least Square Regression, Unweighted

• With an extreme example, Points 1, 2, and 3 are each increased by 100%

ppm Arb Units1 105 50

25 250125 625625 3125

3125 15625

Curve 2

y = 4.9872x + 34.165

R2 = 0.9999

0

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6000

8000

10000

12000

14000

16000

18000

0 500 1000 1500 2000 2500 3000 3500

[ ] ppm

Res

pons

e

Curve 3

• Point 6 is increased by only 10%

y = 5.5067x - 69.415

R2 = 0.9997

0

2000

4000

6000

8000

10000

12000

14000

16000

18000

20000

0 500 1000 1500 2000 2500 3000 3500

[ ] ppm

Res

pons

e ppm Arb Units1 55 2525 125

125 625625 31253125 17188

Curve 3

Summary of Curve Comparisons

Curve 1 Curve 2 Curve 3Arb Units Arb Units Arb Units ppm

5 10 5 125 50 25 5

125 250 125 25625 625 625 1253125 3125 3125 625

15625 15625 17188 3125

Slope 5.00000 4.98720 5.50665Y-Int 0.00000 34.16524 -69.41482

R² 1.00000 0.99994 0.99967

Er% slope -0.26% 10.13%

Curve 1 Curve 2 Curve 3ppm ppm Er% Curve 1 ppm Er% Curve 2 ppm Er% Curve 3

1 1 0% -5 -585% 14 1251%5 5 0% 3 -36% 17 243%25 25 0% 43 73% 35 41%

125 125 0% 118 -5% 126 1%625 625 0% 620 -1% 580 -7%3125 3125 0% 3126 0% 3134 0%

• Interpolation of concentrations using the estimated slopes and Y-intercepts

Use of Response Factors to Understand Curve Linearity

• Consider the following calibration curve. We have a value of 5 Units of “Bias” for all measurements.

• There is also detector saturation at the highest target analyte concentration, resulting in ~10% loss of expected response.

• Is this curve linear, and usable for quantitation at the low concentration range?

y = 4.4932x + 74.4370

R2 = 0.9995

0

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6000

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10000

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16000

0 500 1000 1500 2000 2500 3000 3500

[ ] ppm

Res

pons

e

ppm Arb Units1 105 3025 130

125 630625 31303125 14067

Analysis of Data Using Response Factor

• U.S.EPA indicates that a RF <15% CV is acceptable, with up to 25% CV on RF as acceptable for some methods.

ppm Arb Units RF1 10 105 30 625 130 5.2

125 630 5.04625 3130 5.0083125 14067 4.501

slope 4.5 Ave. RF 6.0Y-int 74.4 Std. Dev. 2.0RSQ 0.9995 CV 34%

• What strategies can be employed to establish a clear understanding of the calibration curve?

• What about the Dixon Test to check the RF’s. Remember, a larger Confidence is actually a less reliable result.

• The RF data provide strong clues with respect to the detection limit (see later).

Calibration Curve Example

NXX

XXQ

1

21explowest highest to fromRank

...,, 321 NXXXX

Six Approaches to Quantification

1. External Standard: A calibration curve is first developed. No consideration for matrix here. The samples are run, as close to the same conditions as the standards. Sample responses are interpolated from the calibration curve.

2. Bracketing standards: Here, two standards around a sample response are used to quantify the target analyte concentration. Is just as accurate as external standard calibration. Not usually accepted by regulatory bodies, but works well particularly for research applications, surveys.

3. Average Response Factors: Simple, but powerful. Here, provided the y-int is small, the Ave. RF can be quite effectively used, and is as accurate as external standard calibration.

4. Internal Standard: A more elegant technique involving introducing a compound into the sample extract just prior to analysis. The ratio of responses of the internal standard to the known concentrations of target analyte is used to develop a calibration curve. Then, in the sample, the ratio of IS response to target analyte response of unknown concentration is interpolated from the curve. Good correction for instrumental factors affecting quanitification.

Six Approaches to Quantification

5. Standard Addition: Here, the actual background matrix is spiked with various (known) concentrations of the target analyte.

6. Isotope Dilution: The most elegant of all quanitification procedures. It involves adding an isotope of the target analyte to the sample prior to extraction. It therefore acts as a surrogate compound and internal standard, but most importantly, will elute at almost exactly the same retention volume as the unlabelled analyte.

There are various versions of isotope dilution, such as radio immunoassay and neutron activation.

The ultimate isotope dilution would be to present the radioactive isotope of the target analyte to the test system prior to incorporation into plant and animal tissue. This would therefore quite thoroughly mimic the actual physical-chemical environment of the target analyte. This strategy is actually used with new pesticide registrations. Here, radioactive isotopes of the pesticides are presented to plants. The fate of the pesticide can be easily tracked.