Calibration-Curve-Based Analysis - American Chemical Society

8 Calibration-Curve-Based Analysis Use of Multiple-Curve and Weighted Least-Squares Procedures with Confidence Band Statistics DOUGLAS G. MITCHELL

Downloaded by CORNELL UNIV on October 6, 2016 | http://pubs.acs.org Publication Date: July 15, 1985 | doi: 10.1021/bk-1985-0284.ch008

Center for Laboratories and Research, New York State Department of Health, Albany, NY 12201

Two procedures for improving precision i n c a l i b r a t i o n curve-based-analysis are described. A multiple curve procedure i s used to compensate for poor mathematical models. A weighted least squares procedure i s used to compensate for non-constant variance. Confidence band s t a t i s t i c s are used to choose between alternative c a l i b r a t i o n strategies and to measure precision and dynamic range. T h i s paper d e s c r i b e s the use o f s t a t i s t i c a l techniques to improve p r e c i s i o n i n r o u t i n e chemical a n a l y s i s at a modest e x t r a cost and to measure the p r e c i s i o n of such analyses. Mote the key words i n t h i s aim: P r e c i s i o n , not accuracy. Accuracy i s mainly a chemical problem, whereas p r e c i s i o n i s a chemical, instrumental and s t a t i s t i c a l problem. Routine chemical a n a l y s i s . T h i s i m p l i e s a n a l y s i s o f many samples, and use of c a l i b r a t i o n curves i s an economic n e c e s s i t y . I n g e n e r a l , the two-standard method, w i t h standards b r a c k e t i n g each sample analyzed, i s economical f o r the a n a l y s i s of up to about 10 samples. Conventional l e a s t squares curve o f best f i t procedures are economical f o r a n a l y s i s of 10 to 500 samples. The procedures d e s c r i b e d here are cost e f f e c t i v e f o r the a n a l y s i s of 500 samples or more. Cost. There i s always a t r a d e - o f f of cost versus data q u a l i t y . Data q u a l i t y can be improved by f u r t h e r method development, more extensive c a l i b r a t i o n , r e p l i c a t e a n a l y s i s or b e t t e r statistics. Measurement of p r e c i s i o n . Measurement of data q u a l i t y i s v a l u a b l e f o r both the a n a l y s t and the data user. L e a s t squares c u r v e - o f - b e s t - f i t s t a t i s t i c a l programs u s u a l l y provide some i n f o r m a t i o n on p r e c i s i o n ( c o r r e l a t i o n c o e f f i c i e n t , standard e r r o r of e s t i m a t e ) . However, these are not s u f f i c i e n t l y q u a n t i t a t i v e and o f t e n o v e r s t a t e the q u a l i t y parameters of the data. To provide optimum data i n r o u t i n e chemical a n a l y s i s , the a n a l y t i c a l method must meet four c r i t e r i a . The f i r s t two are 0097-6156/85/0284-0115S06.00/0 © 1985 American Chemical Society

Kurtz; Trace Residue Analysis ACS Symposium Series; American Chemical Society: Washington, DC, 1985.

TRACE RESIDUE ANALYSIS


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fundamental and determine the upper limit of method performance. The last two concern calibration. 1. The method must be precise. (Without precision we cannot get accuracy, unless we carry out many replicate analyses.) 2. The method must be accurate, or at least free from unpredictable bias. 3. The calibration process must not excessively degrade precision. (Use of a calibration curve w i l l usually result in less precise analyses, compared with bracketing each sample measurement with standard measurements.) 4. The calibration curve must be stable, i . e . free from drift. This paper is concerned with the effects of the calibration process on data quality. Risks to Data Quality in the Calibration Process Over a limited dynamic range, say 1 to 10, use of least squares procedures has l i t t l e adverse effect on data quality. With calibration over wide dynamic ranges and with non linear curves, precision is lost because: The mathematical model may not closely f i t the data. For example, Figure 1 shows calibration data for the determination of iron in water by atomic absorption spectrometry (AAS). At low concentrations the curve is first-order, at high concentrations i t is approximately second-order. Neither model adequately f i t s the whole range. Figure 2 shows the effects of blindly f i t t i n g inappropriate mathematical models to such data. In this case, a manually plotted curve would be better than either a f i r s t - or second-order model. Calibration curves yield the best precision at the mean concentration of the standards. For example, a curve based on standard with concentrations of 1, 4 and 10 yields best precision at 5 (assuming constant variance). To achieve maximum precision the standards should be selected so that their mean concentration is equal to the most important sample concentration, such as an action level. The curve w i l l yield increasingly poor precision with increasing distance from this mean. The least-squares curve-of-best-fit procedure implicitly assumes the same variance (standard deviation) at a l l concentrations. This assumption is rarely correct. Figure 3a shows hypothetical replicate standard analysis data with constant variance. This pattern is almost never seen in routine chemical analyses. Figure 3b shows a much more r e a l i s t i c pattern in which the variance increases with concentration. Proposed Solutions to Calibration Problems There are several approaches to minimizing the loss of precision inherent in calibration-curve-based analysis. We have chosen a



8. MITCHELL

Calibration-Curve-Based Analysis

117

CONCENTRATION (^g/ml)

Fig. 1. Calibration data for determination of iron in water by AAS (Reprinted with permission from D. G. Mitchell and J . S. Garden, Talanta, 1982, 29, 921-929.)

Iron Concentration Fig. 2. Inappropriate use of first-and second-order least squares-curves-of- best-fit procedures with part of the data shown in Figure 1.




118

F i g . 3. H y p o t h e t i c a l c a l i b r a t i o n data showing r e p l i c a t e w i t h (a) c o n s t a n t and (b) n o n c o n s t a n t v a r i a n c e .

standard


analyses

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8. MITCHELL


hybrid calibration-curve, two-standard method for calibration over wide dynamic ranges and a weighted least-squares procedure to compensate for nonconstant variance. Commonly used measures of precision such as correlation coefficients, standard errors of estimate and relative standard deviations (RSD) cannot reliably evaluate alternative calibration strategies. This is beacause they do not measure precision in terms of predicted sample concentration. For example, a correlation coefficient (r) of 0.98 is better than a value of 0.97 providing the same standards are used. If different standards are used, the high r value curve w i l l not necessarily yield best precision for sample analysis. Before proceeding further, i t is necessary to choose appropriate s t a t i s t i c a l techniques. Confidence Band Statistics. The confidence-band s t a t i s t i c a l approach is described in texts by Natrella (1) and Miller 2) and in three papers from our laboratory (3-5). A computer program, REGRES, (See Appendix) was used to carry out a l l the computations described in this paper. The confidence band approach is illustrated in Figure 4. An appropriate mathematical model is chosen. This is usually a f i r s t order (signal = b + b x concentration) or second order (signal = b + b x concentration + b x concentration ) linear equation. A calibration curve is then calculated using a least-squares curve-of-best f i t procedure (6). Next a confidence band is calculated around the curve using the regression band equation from Table I . This band encloses the curve with a, say, 90% level of probability. A confidence band is then calculated around the signal using the appropriate signal band equation (Table I). The two bands are combined as shown in Figure 4 to yield a confidence band around the predicted concentration. The resulting band around the predicted concentration gives a conservative estimate of the precision of the analysis, including the effects of error in both sample and curve. There is a debate among statisticians concerning the best procedures for this application, and our approach may be too conservative. To some extent a conservative bias is probably an advantage, because i t could approximately compensate for (uncalculable) errors due to minor inaccuracies inherent in many methods. Even i f i t is too conservative, the bands w i l l be selfconsistent and should provide accurate estimates of relative precision. Improved mathematical models. F i r s t or second order linear equations adequately f i t much calibration data. If neither model is appropriate, the following semi-empirical multiple curve procedure may be used. Standards covering the proposed dynamic range are analyzed, and the resulting calibration data entered into program REGRES. Each sample is analyzed, and REGRES chooses the combination of contiguous standards enclosing the sample which yields 0

x

2

0

±

2



120


X,

x

2

x

3

0

x

4

x

5

CONCENTRATION, X Fig. 4. Calibration curve with confidence bands around the curve, sample signal, and predicted concentration. (Reproduced with permission from D. G. Mitchell, W. N. M i l l s , J . S. Garden, and M. Zdeb, Anal. Chem, 1977, 49, 1655-1660, copyright 1977, American Chemical Society)



0

1

2

x

i

2

(2F)

1

1 / 2

s

.m

| l _ + (Xo - X ) Z(D Za)(x. - X )

n-2

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So

1/2

2 2

2

(Zw.ZwXY - Za)X.Zo)Y)/[Za).Za)X

2

1

1/2

2

2

- (Zu)X) ]

- (ZwX) ]

ZU)X.Z(DXY)/[ZU).ZU>X

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X+e e~N(0,aVa))

2

- ZX.ZXY)/[n.ZX

x

2

2

- (ZX)
Y - b^wXY, ZY - b ZY - b ZXY. Band around p r e d i c t e d c o n c e n t r a t i o n s s u b s t i t u t e Y values (mean s i g n a l ± AY) i n Y = b + b 8 + band around r e g r e s s i o n . Solve f o r X . a = 0.05; = 1.96. ° ° (Reprinted w i t h permission from Ref* 4.)

Regression band

S i g n a l band, AY

Slope, b

2

+ b

(ZwY.ZwX -

Zu)X/Zu)

Mean, X

Intercept, b

Y = b

0

V a r i a b l e - v a r i a n c e data

Model

Parameter

Table I . A l g e b r a i c Equations f o r F i r s t - O r d e r Regression C a l c u l a t i o n s




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the narrowest band around the p r e d i c t e d c o n c e n t r a t i o n . F i g u r e 5 shows t y p i c a l curves s e l e c t e d by t h i s procedure. At low concent r a t i o n s a f i r s t - o r d e r equation based on, say, the three lowest standards i s chosen. At high concentrations a second-order equation y i e l d s the narrowest band. Note that t h i s procedure o f t e n does not use a l l a v a i l a b l e data, an omission which seeirs i n t u i t i v e l y i n c o r r e c t . The procedure w i l l improve p r e c i s i o n when the b e n e f i t s from b e t t e r mathematical modeling exceed the l o s s e s from not using some data. In general, the m u l t i p l e - c u r v e procedure produces maximum b e n e f i t s at the low-concentration end of a long, n o n l i n e a r curve. F o r example, i n t y p i c a l data f o r the determination of f e n v a l e r a t e by gas chromatography (Table I I ) , use of the m u l t i p l e - c u r v e procedure improved the p r e c i s i o n of the a n a l y s i s by a f a c t o r of two at the 1 meg l e v e l , and a f a c t o r of three a t the 5 meg l e v e l . C o r r e c t i o n f o r nonconstant v a r i a n c e . To c o r r e c t f o r nonconstant v a r i a n c e , i t i s necessary t o weight standard measurements according t o t h e i r l o c a l v a r i a n c e , S . For each standard conc e n t r a t i o n the variance i s determined by r e p e t i t i v e a n a l y s i s at that l e v e l , and a weighting f a c t o r , w = 1/s , i s c a l c u l a t e d . These f a c t o r s are used i n the equations given i n Table I . The computation r e q u i r e s o n l y that the variance r a t i o s be a c c u r a t e l y known. The absolute p r e c i s i o n of the method may change from day t o day without a f f e c t i n g the v a l i d i t y of e i t h e r the l e a s t - s q u a r e s curve-of-best f i t procedure or the confidence band c a l c u l a t i o n s . ( I t i s not p r a c t i c a l t o r e g u l a r l y monitor l o c a l v a r i a n c e s , and e r r o r s may develop i n v a r i a n c e r a t i o s . Eowever, the e r r o r due t o i n c o r r e c t r a t i o s w i l l almost always be much l e s s than the e r r o r due t o assuming constant v a r i a n c e . Even guessed values of, say, S a c o n c e n t r a t i o n are l i k e l y t o y i e l d more p r e c i s e data.) An unweighted l e a s t squares procedure i s o f t e n adversely a f f e c t e d by high c o n c e n t r a t i o n standards, w i t h h i g h (absolute) v a r i a n c e s . These may cause l a r g e e r r o r s i n the slope of f i r s t order equations. The l i n e i s ' r o t a t e d ' , causing large r e l a t i v e e r r o r s at low c o n c e n t r a t i o n s . The weighting proceprocedure deemphasiz.es these p o i n t s , thus reducing t h i s e f f e c t . Figure 6 shows data f o r the determination of lead i n blood by Delves cup AAS. The f i r s t - o r d e r curve i s known t o pass through zero. The weighted least-squares l i n e i s c l o s e , w i t h an 4 i n t e r c e p t of 1.5, but the unweighted l i n e has been ' ' r o t a t e d ' ' by a s i n g l e low value (not an o u t l i e r ) a t 65 ug/dL, g i v i n g an i n c o r r e c t i n t e r c e p t of 3.3. A sample y i e l d i n g a s i g n a l of 5.3 has a c a l c u l a t e d lead c o n c e n t r a t i o n of 10 meg Pb/dl using the weighted l e a s t squares l i n e and 6 mcg/dl using the unweighted l i n e - a 40% e r r o r . S i m i l a r l y , f i g u r e 7 shows part of the weighted and unweighted l e a s t - s q u a r e s curves f o r symposium Dataset B. Standards over the range 0.05 to 20 r i g f e n v a l e r a t e were analyzed, and the f i g u r e shows a range of only 0-1. The v a r i a n c e at each amount l e v e l was known, so both weighted and 2

2

2



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Concentration F i g . 5. Use of m u l t i p l e - c u r v e procedure. Subsets of c a l i b r a t i o n data, each comprising s e v e r a l standards b r a c k e t i n g the samples, are used to c a l c u l a t e p r e d i c t e d c o n c e n t r a t i o n s f o r high-and low-concentration samples.

Concentration (pig PB/dl)

F i g . 6. Determination of lead i n blood by Delves-cup AAS. and unweighted curves of best f i t are shown.

Both weighted


124



Table

II.

Use o f m u l t i p l e c u r v e p r o c e d u r e t o i m p r o v e p r e c i s i o n o f f e n v a l e r a t e a n a l y s i s by g a s c h r o m a t o g r a p h y

F e n v a l e r a t e Amount (meg)

Single-Curve C a l i b r a t i o n RCB

Multiple-Wave C a l i b r a t i o n C a l i b r a t i o n Range (%) Low High RCB%

0.05

1.00

24

33

0.05

1.00

20

1.00

33

0.05

5.00

17

5.00

26

1.00

20.00

7

20.00

20

1.00

20.00

7

0.05 0.25

Note:

40

a = 0.05; Z, = l-a/2 / 0

1.96



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125

F i g . 7. Determination of f e n v a l e r a t e by gas chromatography w i t h DATASET B showing weighted and unweighted second-order curves.



126

unweighted curves could be c a l c u l a t e d . These data are very p r e c i s e , w i t h f i v e r e p l i c a t e measurements at 20 ng f e n v a l e r a t e having a range of + 2.6%. Use of the unweighted procedure caused s i g n i f i c a n t e r r o r s only at amount l e v e l s below 1 ng fenvalerate.


Improved Measurement of P r e c i s i o n C a l i b r a t i o n curve q u a l i t y . C a l i b r a t i o n curve q u a l i t y i s u s u a l l y evaluated by s t a t i s t i c a l parameters, such as the c o r r e l a t i o n c o e f f i c i e n t and standard e r r o r of estimate, and by e m p i r i c a l indexes, such as the length of the l i n e a r range. Using c o n f i dence band s t a t i s t i c s , curve q u a l i t y can be b e t t e r described i n terms of confidence band widths at s e v e r a l key c o n c e n t r a t i o n s . Other s e m i - q u a n t i t a t i v e indexes become redundant. A l t e r n a t i v e l y , the e f f e c t s of curve q u a l i t y can be incorporated i n t o statements of sample a n a l y s i s data q u a l i t y . Sample a n a l y s i s data q u a l i t y . P r e c i s i o n of sample a n a l y s i s i s almost always measured by determining the RSD at two o r more concentrations without using a c a l i b r a t i o n curve. Such data do not include the e f f e c t s of the c a l i b r a t i o n process on p r e c i s i o n , flluch b e t t e r i n f o r m a t i o n i s given by the r e l a t i v e confidence bandwidth (RCB) defined as: RCB(%) =

band - lower band) x 100 2 x P r e d i c t e d Concentration

^PPer

For example, Figure 8 shows both RSD and RCB data f o r determinat i o n of c h l o r i d e and lead i n water. I n Figure 8a, the l e a s t squares curve of best f i t c l o s e l y f i t s the lead standard data, and the c a l i b r a t i o n process has l i t t l e adverse e f f e c t on p r e c i s i o n . RSD's and RCB's are almost equal. On the other hand, c h l o r i d e standard data i n F i g u r e 8b does not c l o s e l y f i t the mathematical model, and the RSD data o v e r s t a t e s the p r e c i s i o n of the a n a l y s i s by a f a c t o r of about two. Minimum r e p o r t a b l e c o n c e n t r a t i o n . The lower c o n c e n t r a t i o n l i m i t f o r a method i s u s u a l l y measured by determining the d e t e c t i o n l i m i t . This i s b a s i c a l l y an instrument s i g n a l t o noise r a t i o , and i t does not include c a l i b r a t i o n e f f e c t s . At low concentrations the c a l i b r a t i o n process o f t e n has a major adverse e f f e c t on p r e c i s i o n . D e t e c t i o n l i m i t s are u s e f u l f o r comparing the inherent s e n s i t i v i t y of methods, but they are not r e a l i s t i c indexes of measurable concentrations i n r o u t i n e a n a l y s i s . We suggest using a new parameter, the minimum r e p o r t a b l e c o n c e n t r a t i o n , defined as the c o n c e n t r a t i o n whose confidence band j u s t i n c l u d e s zero (5.). This parameter i s obtained by reducing the value of s i g n a l Yo, f i g u r e 4, u n t i l the band around p r e d i c t e d c o n c e n t r a t i o n , Xo, j u s t touches zero. For example, f o r the determination of i r o n i n water by AAS, (data given i n Table I I I ) the d e t e c t i o n l i m i t , d e f i n e d as the c o n c e n t r a t i o n a t which the


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8. MITCHELL

C XI

•u

W

T3 C •H O £ -H

C

OJ

CO «H PQ >

1.0

on on

2.5

5

Concentration

(yg/ml)

30 c

(0

X u T)

•H

c

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20

10

1

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4->

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20 Concentration

40 (pg/ml)

Fig. 8. Comparison of RCB (-) and RSD ( ) for determination of (a) chloride and (b) lead in water. (Reprinted with permission from D. G. Mitchell and J . S. Garden, Talanta 1982, 29, 921-929, copyright 1982, Pergamon Press Ltd.)



128


Table

I I I . Determination by AAS

Standard (vig/ml)

0.05 0.10 0.25 0.50 1.00 1.5 2.5 5.0 10.0 15.0 18.0 20.0 22.0 25.0 40.0 50.0 75.0 100.0

Absorbance a t nm

0. 004 0. 008 0. 022 0. 045 0. 093 0. 142 0. 222 0. 430 0. 750 0. 961 1. 054 1. 086 1. 145 1. 191 1. 268 1. 300 1. 360 1. 405

o f Maximum C o n c e n t r a t i o n o f I r o n i n Water

Single-Curve Calibration: RCB ( % ) -

40 26 13 9 8 7 7 5 6 11 60 60 60 60 60 60 60 60

Multiple-Curve Calibration Calibration r a n g e (pg/ml) Low High RCB (%)

0 .05 0 .05 0 .05 0 .1 0 .1 0 .1 0 .1 0 .1 0 .1 0 .1 15 .0 15 .0 15 .0 15 .0 25 .0 0 .05 40 .0 40 .0

1.5 1.5 1.5 1.5 1.5 20.0 20.0 20.0 20.0 20.0 40.0 40.0 40.0 40.0 40.0 100.0 100.0 100.0

N o t e : a = 0.05; Z, = 1.96 l-a/2 —A w e i g h t e d l e a s t - s q u a r e s t e c h n i q u e was u s e d : c a l i b r a t i o n 0.05-18 ug/ml.

33 19 6 5 4 6 6 4 5 8 11 12 16 25 38 60 60 60

/ 0

Source:

Reproduced w i t h

permission

from

R e f . 5.


range


8.

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RSD i s 50%, i s 0.015 ug/ml. The minimum r e p o r t a b l e c o n c e n t r a t i o n i s a f a c t o r of 2 higher when the method i s c a l i b r a t e d over a narrow, low c o n c e n t r a t i o n range (0.05 to 0.1 ug/ml). I t i s a f a c t o r of 20 higher when the method i s ( i n a p p r o p r i a t e l y ) c a l i b r a t e d over a dynamic range of 100 (0.05 to 5 ug/ml). Maximum r e p o r t a b l e c o n c e n t r a t i o n . The upper l i m i t of measurement f o r a method i s u s u a l l y defined as the c o n c e n t r a t i o n at which the curve shows a c e r t a i n d e v i a t i o n from l i n e a r i t y . This i s a v a l i d e m p i r i c a l c r i t e r i o n , since s e n s i t i v i t y and hence p r e c i s i o n decreases as the curve f l a t t e n s . However, l i n e a r i t y does not d i r e c t l y measure the performance parameter of i n t e r e s t : p r e c i s i o n . In p r a c t i c e an a n a l y s t would accept curves at h i g h c o n c e n t r a t i o n s , p r o v i d i n g the p r e c i s i o n i s s t i l l adequate and p r o v i d i n g the method does not have accuracy problems at h i g h concentrations e.g., because of l i g h t s c a t t e r i n g i n a b s o r p t i o n methods. Confidence bands are d i r e c t p r e c i s i o n data, and the maximum r e p o r t a b l e c o n c e n t r a t i o n can be defined as the maximum c o n c e n t r a t i o n at which the method y i e l d s adequate p r e c i s i o n (5.) (excluding measurements near the minimum r e p o r t a b l e c o n c e n t r a t i o n , where poor p r e c i s i o n i s unavoidable). Table I I I shows RCB f o r the determination of i r o n i n water by AAS. The analyst may consider a RCB of say, 15% to be adequate. The maximum r e p o r t a b l e c o n c e n t r a t i o n would be 15 ug/ml from a s i n g l e , weighted least-squares curve, and 20 ug/ml by the m u l t i p l e - c u r v e method. Samples c o n t a i n i n g > 20 ug/ml should be d i l u t e d to 1-10 ug/ml and analyzed using standards c o n t a i n i n g 0.05 - 15 pg/roL. (Mote that i t i s always b e t t e r to include a standard above the maximum d e s i r e d c o n c e n t r a t i o n . The p r e c i s i o n of t h i s standard measurement w i l l be poor, but poor data at t h i s l e v e l are b e t t e r than none.) I m p l i c a t i o n s For Method Development. The e f f e c t s of the c a l i b r a t i o n process on p r e c i s i o n suggest the need f o r an a d d i t i o n a l step i n the development of an a n a l y t i c a l method. A suggested flow chart i s shown i n F i g u r e 9. The a n a l y s t should f i r s t develop a method of adequate accuracy and p r e c i s i o n without using c a l i b r a t i o n curves. The c a l i b r a t i o n step i s then added, and the p r e c i s i o n i s rechecked. I f p r e c i s i o n has been e x c e s s i v e l y degraded, the a n a l y s t can choose among a l t e r n a t i v e c a l i b r a t i o n s t r a t e g i e s , such as use of more standard measurements and use of the m u l t i p l e - c u r v e procedure. Conclusion I have described a reasonably complete set of mathematical techniques f o r improving the p r e c i s i o n of c a l i b r a t i o n - c u r v e - b a s e d analyses and measuring t h e i r p r e c i s i o n . Each technique may not be the optimum s o l u t i o n to each problem, but the o v e r a l l philosophy should be c o r r e c t . We should develop s t a t i s t i c a l techniques to measure p r e c i s i o n which are s e l f - c o n s i s t e n t and



130

SELECT AND

SAMPLE

PREPARATION

MEASUREMENT

DEVELOP

TECHNIQUES

FOR ROUTINE

ANALYSIS


YES

YES

YES

METHOD

SATISFACTORY

FOR ROUTINE

ANALYSIS

Method development procedure for calibration-curve-based analysis. (Reproduced with permission from D. G. Mitchell S. Garden, Talanta, 1982, 29, 921-929, copyright 1982, Pergamon Press Ltd.)


8.

MITCHELL


131

which account f o r a l l the f a c t o r s a f f e c t i n g p r e c i s i o n . We should use them t o choose optimum c a l i b r a t i o n s t r a t e g i e s and t o measure the p r e c i s i o n of the r e s u l t i n g data.


The computer program REGRES was w r i t t e n by John S. Gorden, New York State Department of Health. I t can be obtained by sending a check f o r f 3 0 . 0 0 , made out t o Health Research I n c . , and a 9-track magnetic tape t o John S. Gorden, New York State Department of H e a l t h , CSMDP, Concourse L e v e l , Empire State P l a z a , Albany, New York 12237.

Literature Cited

1. Natrella, M.G., "Experimental Statistics", National Bureau of Standards Handbook 91 (1963) 2. Miller, R.G., "Simultaneous Statistical Inference", McGraw-Hill, New York (1966) 3. Mitchell, D. G., Mills, W., N. Garden, J.G., and Zdeb, M., Anal. Chem., 1977, 49, 1655-1660. 4. Garden, J . S., Mitchell, D. G., and Mills, W. M., Anal. Chem., 1980, 52, 2310-2315. 5. Mitchell, D. G., and Garden, J. S., Talanta, 1982, 29, 921-929. 6. Draper, N.R., and Smith, H., "Applied Regression Analysis", J . Wiley and Sons, New York, (1966) RECEIVED March 25, 1985


Calibration-Curve-Based Analysis - American Chemical Society

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