Trace Residue Analysis - ACS Publications - American Chemical Society

5 The Many Dimensions of Detection in Chemical Analysis Downloaded by CORNELL UNIV on October 19, 2016 | http://pubs.acs.org Publication Date: July 15, 1985 | doi: 10.1021/bk-1985-0284.ch005

with Special Emphasis on the One-Dimensional Calibration Curve LLOYD A. CURRIE Center for Analytical Chemistry, National Bureau of Standards, Washington, DC 20234

Simple detection decisions generally involve the comparison of scalar quantities (gross s i g n a l , blank). Conventional chromatography and spectrometry, on the other hand, involve one-dimensional variables (time, mass, wavelength, energy) where signal and baseline traces may be examined to decide whether a peak i s present at a given location. Linked techniques, such as GC-MS or two-parameter nuclear spectroscopy, raise the question of detection i n two dimensions. F i n a l l y , problems wherein a set of samples i s characterized by many independent chemical and physical observations raise the issue of multidimensional detection. A unified approach for all such problems i s given by the statistical theory of hypothesis testing. Following a brief review of underlying assumptions and techniques for applying the theory to detection decisions and detection l i m i t s , primary attention i s given to a one-dimensional (reduced from two) problem involving the calibration curve and the pesticide, Fenvalerate. Other topics addressed include information-loss through faulty reporting (at trace levels) and its impact on regulatory issues, and chemometric quality assurance through standard interlaboratory test data sets. One of the fundamental performance c h a r a c t e r i s t i c s of any anal y t i c a l procedure i s the L i m i t of D e t e c t i o n . Just as with the imprecision (standard d e v i a t i o n ) , with which i t i s i n t i m a t e l y connected, the Detection L i m i t ( L ) i s undefined unless there D

This chapter not subject to U.S. copyright. Published 1985, American Chemical Society

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

Downloaded by CORNELL UNIV on October 19, 2016 | http://pubs.acs.org Publication Date: July 15, 1985 | doi: 10.1021/bk-1985-0284.ch005

50

TRACE RESIDUE ANALYSIS

e x i s t s a f u l l y - s p e c i f i e d Chemical Measurement Process (CMP) i n a s t a t e of complete c o n t r o l . When these requirements are met, i t i s convenient t o define Lp i n accordance w i t h the s t a t i s t i c a l theory o f Hypothesis T e s t i n g (1,2). Although t h i s theory i s w e l l e s t a b l i s h e d there continues t o be a great d i v e r s i t y of terminology and formulations which generate needless confusion i n our d i s c i p l i n e . S t i l l more s e r i o u s i s i n t e r d i s c i p l i n a r y confusion, when a n a l y s t s are c a l l e d upon t o provide v a l i d methods and c r i t i c a l data f o r r e g u l a t o r y , c l i n i c a l , or environmental decision-making ( 3 ) . The o b j e c t i v e s of t h i s review w i l l be t o summarize the b a s i c concepts of d e t e c t i o n i n A n a l y t i c a l Chemistry, w i t h the development f o l l o w i n g a stepwise increase i n d i m e n s i o n a l i t y . Prime emphasis i s given t o the assumptions which must be met, and t o i l l u s t r a t i o n s having d i f f e r i n g dimensions. I n keeping w i t h the Symposium t i t l e and i n response to the i n v i t a t i o n of the Symposium o r g a n i z e r , a d e t a i l e d e x p o s i t i o n i s presented f o r the t r a c e d e t e c t i o n of a p e s t i c i d e (Fenvalerate) by gas chromatography — an e x e r c i s e which h i g h l i g h t s the r e l a t i o n s h i p of the c a l i b r a t i o n process t o the d e t e c t i o n c h a r a c t e r i s t i c , and which exposed a s u r p r i s i n g (and unnecessary) l i m i t a t i o n t o the d e t e c t i o n c a p a b i l i t y . Treatment of a r e a l , imperfect c a l i b r a t i o n data s e t revealed the f u l l complexity and breadth of the c a l i b r a t i o n curve - d e t e c t i o n l i m i t problem, ranging from v a r y i n g s t a t i s t i c a l weights t o an u n c e r t a i n model and data c o n t a i n i n g p o s s i b l e blunders t o an a r t i f i c i a l l y imposed response t h r e s h o l d . Attempts t o s i m p l i f y an a c t u a l l y complicated s i t u a t i o n were r e j e c t e d i n favor of a f u l l e x p o s i t i o n i n c l u d i n g an Appendix c o n t a i n i n g worked-out numerical examples. SIMPLE HYPOTHESIS TESTING - SCALAR SIGNALS The b a s i c d e t e c t i o n concepts can be presented f o r the "zerodimensional" case where d e t e c t i o n d e c i s i o n s and d e t e c t i o n l i m i t s are e s t a b l i s h e d simply from the c h a r a c t e r i s t i c s of the chemical s i g n a l (instrument response), without g i v i n g d e t a i l e d a t t e n t i o n t o other dimensions such as time, wavelength, analyte concentrat i o n , e t c . A c t u a l l y , higher dimensional s i t u a t i o n s (multiparameter separations or detector responses) reduce t o t h i s case e i t h e r through s e q u e n t i a l c l a s s i f i c a t i o n schemes or v i a a l g o rithms which operate d i r e c t l y on the multidimensional data. Our b a s i c task i s t o d i s t i n g u i s h the blank or background ( H , n u l l h y p o t h e s i s ) , from a s i g n a l a t the d e t e c t i o n l i m i t (H-|, alternative hypothesis). A straightforward p r o b a b i l i s t i c f o r m u l a t i o n can be given provided that the observed s i g n a l s ( a r i s i n g from an u n d e r l y i n g " t r u e " s i g n a l ) are random, independent and s t a t i o n a r y . To completely s p e c i f y the f a l s e p o s i t i v e (a) and f a l s e negative (3) r i s k s , we must know the form of the d i s t r i b u t i o n and i t s parameters. For most a n a l y t i c a l s i t u a t i o n s Q


5.

51

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Dimensions of Detection in Chemical Analysis

we assume the d i s t r i b u t i o n t o be normal (Gaussian), and the d i s p e r s i o n parameter i s simply the i m p r e c i s i o n (standard deviation, o). As shown i n Reference ( 2 ) , i t i s s u f f i c i e n t t o have an estimate o f the blank (B) and i t s standard d e v i a t i o n ( 0 5 ) p l u s the v a r i a t i o n of o w i t h the s i g n a l magnitude (y) t o s p e c i f y a d e c i s i o n c r i t e r i o n o r l e v e l (LQ) given a, and a d e t e c t i o n l i m i t ( L D ) given LQ and 3 . (See e s p e c i a l l y F i g u r e 2 i n Reference (2j). I f oy i s independent of s i g n a l magnitude (at and below the d e t e c t i o n l i m i t ) , and i f y i s normally d i s t r i b u t e d , one concludes that y


L

z

1a

C - 1-a°o

L

= LQ

D

< ) (1b)

+ Z1-300

where z 0, then the d i s p e r s i o n about the f i t t e d values ( y i ) y i e l d s an estimate f o r o . In every case, independent experiments and r e p l i c a t e s are v i t a l f o r e x t e r n a l v a l i d a t i o n of the presumed OJ'S — e.g., w i t h the a i d of the A n a l y s i s of Variance. f


2

D e v i a t i o n s from the I d e a l Model. In a l l r e a l s i t u a t i o n s , the e r r o r terms have a s t r u c t u r e q u a l i t a t i v e l y represented by Equation 4, e « BA i s n e c e s s a r i l y zero; so K = 1. Second, i f « 1 ( p r e c i s e A or A-known), both K and 1 - 1 ; so x = 2 XQ. T h i r d , s i n c e PBA i s negative and | P B A I » OB £ o , the r a t i o K/I may be w r i t t e n as [1-e(z )]/[l-(z )(z )] where 0 £ e < 1. Thus, there can be no a n a l y t e d e t e c t i o n l i m i t ( x ») i f A

< 1

A

N

D

D

0

A

A

A

D

5 1/z = 1/1 .645 =* 0.608 A

A l s o x = 2 x i f e = | P B A I ° B O + H> l i m i t , ->0. I f e < z, XD > 2XQ and the converse. The minimum i n the r a t i o xp/xc occurs when the design {x^} i s such t h a t x - xp (see Table I I and Equation 1 0 ) . More g e n e r a l l y , the c a l i b r a t i o n curve can be represented as a m a t r i x equation /O

D

Z

a

n

d

c

i n

t n e

A

A

y - M8 + e

(8)

whose weighted l e a s t - s q u a r e s s o l u t i o n i s

8 = (MTwM)"" M Wy 1

T

(9a)


60


V

T

1

= (M WM)-

e

T

1

* (M M)" V

(9b)

y

s

The approximation f o r the variance o f 6 i n (9b) h o l d i n g when V. i s approximately constant, A summary of the a p p l i c a t i o n of Equation 9 t o the l i n e a r c a l i b r a t i o n curve d e r i v e d from known a n a l y t e concentrations {x} - (x-j, X 2 ... x ) and corresponding s t a t i s t i c a l weights ( i n v e r s e v a r i a n c e s ) i s given i n Table I I . n

Table I I . D e c i s i o n and D e t e c t i o n (weighted l e a s t

squares)


(y = M0 = B + Ax) eT = (B A)

(

Zw

Zwx \

where W£ = 1 / V ( x i ) [k/V , i f k-replicates]

) , Zwx Zwx2y

y

y

Then, 1 T , 1 *^ V (x) + — A L Decision: x = 0

V

x x

v

2

Z

w

(x - x ) 2

1

w

+

— :

Z w ( x

w

a

" w) J D e t e c t i o n : x » x^ x

n

2

Expressions f o r the v a r i a n c e s and covariance o f B and A f o l l o w from the i n v e r s e m a t r i x (MTWM)*' . See the d i s c u s s i o n o f "casef" from Table V i n the Appendix, f o r e x p l i c i t formulas. 1

(Note t h a t x represents the weighted mean o f the {x}.) Given the d e f i n i n g expressions f o r d e c i s i o n and d e t e c t i o n l i m i t s together w i t h the c a l i b r a t i o n design {x}, the equation f o r V i n Table I I immediately y i e l d s the d e s i r e d q u a n t i t i e s f o r the l i n e a r c a l i b r a t i o n curve. For equal weights (Vy - const.) and t a k i n g r o o t s , the e x p r e s s i o n s i m p l i f i e s t o r . -|1/2 w

x


5.

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61 Dimensions of Detection in Chemical Analysis

Non-linear curves may be t r e a t e d using Equation 9 d i r e c t l y , using the techniques of n o n - l i n e a r l e a s t squares, when a p p r o p r i a t e . (Note that a n o n - l i n e a r c a l i b r a t i o n curve does not n e c e s s a r i l y imply n o n - l i n e a r l e a s t squares. The l a t t e r i s necessary only i f the problem i s n o n - l i n e a r i n the estimated parameters (16). For example, y = a+bx+cx and y = a+bx a r e both n o n - l i n e a r f u n c t i o n s , but o n l y the l a t t e r i s n o n - l i n e a r i n the parameters.)


2

c

Fenvalerate Data. C a l i b r a t i o n data f o r the GC measurement of Fenvalerate were f u r n i s h e d by D. K u r t z (17). Average responses f o r f i v e r e p l i c a t e s a t each o f f i v e standard concentrations a r e given i n Table I I I . I t should be noted t h a t the s t a t e d r e sponses are not raw o b s e r v a t i o n s , but r a t h e r o n - l i n e computer generated peak area estimates (cm ). (Had we s t a r t e d w i t h the raw data [chromatograms], the problem would a c t u a l l y have been two-dimensional, i n c l u d i n g as v a r i a b l e s r e t e n t i o n time and concentration.) The s t a t e d u n c e r t a i n t i e s i n the peak areas are based on a l i n e a r f i t (o * a+bx) o f the r e p l i c a t i o n standard d e v i a t i o n s t o c o n c e n t r a t i o n ; and the " l o c a l s l o p e s " [ f i r s t d i f f e r e n c e s ] i n the l a s t column of Table I I I a r e presented 2

Table I I I . Fenvalerate (GC) Data - Set B

(averages of 5 r e p l i c a t e s ) 2

Response (y, c m )

a

Amount (x, ng)

Ay/Ax

[0.023]

0.05

23.6

7.08 ± 0.06s C0.365]

0.25

29.5

[0.18]

1.00

30.1

1.18 ± 0 . 0 2

29.68 ± 0.23

4

209.0

± l.l-i

[1.87]

5.00

44.8

920.6

± 4.4

[4.32]

20.00

47.4

0

U n c e r t a i n t i e s represent standard e r r o r s , based on the f i t t e d equation o(y) - (0.028 + 0.49 x)//5~. Q u a n t i t i e s i n brackets a r e the observed standard e r r o r s .



62

simply t o i n d i c a t e the extent of n o n - l i n e a r i t y i n the c a l i b r a t i o n curve, ( T h i s i s not so easy t o grasp from a p l o t , because of the very wide dynamic range.) In order t o c a l c u l a t e V , and t h e r e f o r e the d e t e c t i o n l i m i t , i t i s necessary f i r s t t o estimate V as a f u n c t i o n o f c o n c e n t r a t i o n and then t o use t h i s i n f o r m a t i o n t o estimate the parameters of the c a l i b r a t i o n curve u s i n g weighted l e a s t squares (WLS) f i t t i n g . Rigorous a p p l i c a t i o n o f WLS r e q u i r e s knowledge of r e l a t i v e weights, but the technique i s already considered adequate when n £ 5 (18). In Table IV we present the r e s u l t s of f i t t i n g a l t e r n a t i v e models t o the p a t t e r n of weights and the c a l i b r a t i o n curve. Before using the r e s u l t s i n Tables I I I and IV t o c a l c u l a t e detection l i m i t s , x


y

Table IV. A l t e r n a t i v e C a l i b r a t i o n Models

Model (1)

y - B + Ax(a)

(2)

y = B + Ax

(3)

y = B + Aq(b)

t

^ 0.042

j^A/dff

22.8

-1.04 ± 0.02 0.042 ± 0.024

38.81 ± 0.12

32.7

32.46 ± 0.10

9.64

a

( ^ T h i s model i s taken t o be exact — i t uses B from model-3 together w i t h the i n i t i a l p o i n t , (x,y) = (0.05, 1.18), t o d e r i v e A. take q t o be e x a c t l y x - t o account f o r the nonl i n e a r i t y i n the curve; the two parameters (B, A) are then estimated by l i n e a r l e a s t squares, u s i n g weights as i n d i c a t e d i n Table I I I . 1

1 2

a number of observations should be made: (a) The observed SE's (Table I I I ) a r e g e n e r a l l y monotonic ( c e r t a i n l y not constant) w i t h i n c r e a s i n g c o n c e n t r a t i o n and c o n s i s t e n t w i t h the l i n e a r model, w i t h the exception o f the value a t x = 0.25 ng. (b) The i n i t i a l o b s e r v a t i o n (at x = 0.05) has a response already > f o r t y times the z e r o - p o i n t standard d e v i a t i o n


5.

CURRIE

63


(1,18/0.028); thus, i t i s c l e a r l y way i n excess of the detect i o n l i m i t . A very l a r g e e x t r a p o l a t i o n i s t h e r e f o r e necessary to estimate both the background (B) and standard d e v i a t i o n ( o ) i n the r e g i o n of the d e t e c t i o n l i m i t . T h i s i s the b a s i s of i n t r o d u c i n g model-1, f o r i l l u s t r a t i v e purposes (Table I V ) . (c) Model-2 (Table IV) o b v i o u s l y i s inadequate; the s i g n i f i c a n t l y negative i n t e r c e p t and poor f i t r u l e i t out (over the e n t i r e data range). Not shown are simple polynomial f i t s , which are a l s o inadequate. (d) Model-3 i s b e t t e r . The i n t e r c e p t i s c o n s i s t e n t w i t h zero (to be expected from the technique of c a l c u l a t i n g net GC peak a r e a ) . The f i t , however, i m p l i e s an a d d i t i o n a l (nonr e p l i c a t i o n ) e r r o r source. Again, f o r i l l u s t r a t i v e purposes, the f u n c t i o n q(x) has been taken exact i n order t o avoid the d i s t r i b u t i o n a l p e r t u r b a t i o n s of n o n - l i n e a r l e a s t squares (not j u s t i f i e d i n view of the foregoing l i m i t a t i o n s of the d a t a ) . Before t u r n i n g t o the question of d e t e c t i o n , i t i s i l l u m i n a t i n g t o examine a p l o t of the data, and the r e s i d u a l s from the f i t of model-3. These are shown i n Figure 1. The p r i n c i p a l observations which d e r i v e from the r e s i d u a l p l o t are that the assumed shape of the curve and v a r i a t i o n of s t a t i s t i c a l weight w i t h c o n c e n t r a t i o n are g e n e r a l l y a c c e p t a b l e . The magnitude of the r e s i d u a l s and d i s p e r s i o n f o r c e r t a i n r e p l i c a t e s and concent r a t i o n s are not. That i s , there i s a d d i t i o n a l s c a t t e r about the f i t t e d curve, unaccounted f o r by the r e p l i c a t i o n e r r o r ; and c e r t a i n r e p l i c a t e s , e s p e c i a l l y (• a n d • ) i n the 0.25 ng and 5 ng samples are more widely separated than the o t h e r s . Queries which f o l l o w e d these observations l e d to suggestions that some untoward d i l u t i o n e r r o r s may have been i n v o l v e d i n preparing two of the standards, and random e r r o r s i n "x (concentrations of standards) may not be n e g l i g i b l e . Thus, a d e t a i l e d e v a l u a t i o n of the c a l i b r a t i o n process would r e q u i r e s c r u t i n y (or restanda r d i z a t i o n ) of standard s o l u t i o n s f o r p o s s i b l e blunders (outl i e r s ) , and the d i f f i c u l t task of f i t t i n g the c a l i b r a t i o n data t a k i n g i n t o account e r r o r s i n both v a r i a b l e s (19).


y

ft

Fenvalerate D e t e c t i o n L i m i t s . To the extent that d e t e c t i o n l i m i t s r e q u i r e knowledge of the c a l i b r a t i o n curve and random e r r o r ( f o r x) as a f u n c t i o n of c o n c e n t r a t i o n , a l l of the f o r e g o i n g d i s c u s s i o n i s r e l e v a n t — both f o r d e t e c t i o n and e s t i m a t i o n . However, curve shape and e r r o r s where x >> x , are r e l a t i v e l y unimportant at the d e t e c t i o n l i m i t , i n c o n t r a s t to d i r e c t observations of the i n i t i a l slope and the blank and i t s variability. ( I t w i l l be seen t h a t the i n i t i a l o b s e r v a t i o n i n the current data set exceeded the u l t i m a t e d e t e c t i o n l i m i t by more than an order of magnitude!) To give some p e r s p e c t i v e t o the above remarks a set of a l t e r n a t i v e d e c i s i o n and d e t e c t i o n l i m i t s are given i n Table V, d e r i v e d from a p p r o p r i a t e i n f o r m a t i o n i n the preceding three t a b l e s . F i r s t , we observe t h a t there are two broad c l a s s e s of D


64



1000.

0.03

0.1

1.

10. 30.

Amount (ng) Fenvalerate—Set B

CO D "35

o> cc "O (B/S ) + (B/S ) A BA A B D B D y

d

(11)

f

where the s are r e l a t i v e standard d e v i a t i o n s , and Sj) i s the net s i g n a l ( y - B ) a t the d e t e c t i o n l i m i t . (See the Appendix, Case-f, f o r the a p p l i c a t i o n o f Equation t1.) D

Within each o f the two c l a s s e s i n Table V, the f i r s t two s e t s o f l i m i t s ( ( a ) , ( b ) , ( d ) , (e)) use the constant and v a r i a b l e weights, r e s p e c t i v e l y , and assume B and A are e x a c t l y known (model-1 i n Table I V ) . The remaining l i m i t s i n v o l v e estimated parameters, based on the design {x} and the equations of Table I I I . Method (c) u t i l i z e s the parameters o f Model-1 and constant weight; method ( f ) uses Model-3 and v a r i a b l e y-err ors (we i g h t ) . P r i n c i p a l conclusions t o be drawn from t h i s e x e r c i s e , d i s p l a y e d g r a p h i c a l l y i n Figure 2, are t h a t : © The "black-box" t h r e s h o l d imposes a l a r g e and unnecessary increase i n d e t e c t i o n l i m i t . © I n the r e g i o n o f the d e t e c t i o n l i m i t , f o r t h i s data s e t , the a l t e r n a t i v e weighting scheme or model s e l e c t e d has l i t t l e effect. © The a d d i t i o n a l , n o n - r e p l i c a t i o n , s c a t t e r about the f i t t e d c a l i b r a t i o n curve — perhaps due t o random e r r o r i n the x - v a r i a b l e — does show a s u b s t a n t i a l e f f e c t . (See l a s t paragraph, Append!x.) © Optimal assessment of the minimum d e t e c t i o n l i m i t would r e q u i r e a design {x} w e l l below the c u r r e n t standard conc e n t r a t i o n s and i n c l u d i n g the blank. The scope of t h i s a r t i c l e does not permit the c o n s i d e r a t i o n of p h y s i c a l v s . e m p i r i c a l models f o r the c a l i b r a t i o n curve, nor the e f f e c t of new designs on the d e t e c t i o n l i m i t , but these a r e extremely important issues i n c a l i b r a t i o n . For example, i t can be shown t h a t with an inadequate design the d e t e c t i o n l i m i t ( f o r a » 3 = 0.05) may not even e x i s t ! (XD «>.)


5.

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67 Dimensions of Detection in Chemical Analysis


[y(cm2)l

Figure 2. C a l i b r a t i o n curve f o r Fenvalerate i n the Regions of the D e t e c t i o n L i m i t s . Numerical values of d e c i s i o n (C) and d e t e c t i o n l i m i t s (D) are shown f o r the "No Threshold", case (b) and "Threshold", case (e) from Table V. For the former, a- and 3-errors are i n d i c a t e d q u a l i t a t i v e l y . The f i r s t (lowest c o n c e n t r a t i o n ) data p o i n t i s shown at (x,y) = (50, 1.18). (Though t o p o l o g i c a l l y c o r r e c t , the s c a l e s have been d e l i b e r a t e l y d i s t o r t e d t o encompass both cases, and e s p e c i a l l y near the o r i g i n t o dramatize the e f f e c t of some designs on the uncert a i n t y of the i n t e r c e p t and the r a t i o X\)/XQ.)



68

HIGHER DIMENSIONS:

EXPLORATION AND VALIDATION


Space remains f o r o n l y a b r i e f glance a t d e t e c t i o n i n higher dimensions. The b a s i c concept of hypothesis t e s t i n g and the c e n t r a l s i g n i f i c a n c e of measurement e r r o r s and c e r t a i n model a s sumptions, however, can be c a r r i e d over d i r e c t l y from the lower dimensional d i s c u s s i o n s . I n the f o l l o w i n g t e x t we f i r s t examine the nature o f d i m e n s i o n a l i t y (and i t s r e d u c t i o n t o a s c a l a r f o r d e t e c t i o n d e c i s i o n s ) , and then address the c r i t i c a l i s s u e of d e t e c t i o n l i m i t v a l i d a t i o n i n complex measurement s i t u a t i o n s . Physicochemical A n a l y s i s v s . Chemometric R e s o l u t i o n . Once we pass beyond the one-dimensional c a l i b r a t i o n of a pure substance, we enter the realm of mixtures, and the a n a l y z i n g dimensions of chromatography, spectrometry, r e l a x a t i o n times, morphology, chemical " f i n g e r p r i n t i n g " , e t c . When one has s u f f i c i e n t r e s o l v i n g power, whether by means of a simple dimension of extreme r e s o l u t i o n or a l i n k e d ("hyphenated") s e r i e s of independent dimensions y i e l d i n g the product of t h e i r i n d i v i d u a l r e s o l v i n g powers, then the problem reduces t o the zerodimensional case. That i s , one simply measures the s i g n a l i n the a p p r o p r i a t e hypercube i n multidimensional space which marks the l o c a t i o n of the species of i n t e r e s t . An outstanding example of such m u l t i s p e c t r a l s o r t i n g i s the new technique o f A c c e l e r a t o r Mass Spectrometry (20), which has l e d t o a r e v o l u t i o n i n measurements f o r radiocarbon d a t i n g , isotope geophysics, nuclear geology, e t c . Here, f o r example, 1**C atoms and c l u s t e r s a r e i n i t i a l l y mass analyzed as high energy (-2 MeV) negative i o n s , a f t e r which a l l molecular fragments are destroyed and most e l e c t r o n s removed; then a d d i t i o n a l a c c e l e r a t i o n and mass a n a l y s i s occurs w i t h i o n s , and f i n a l d i s c r i m i n a t i o n takes place w i t h 8 MeV i o n s on the b a s i s of i o n i z a t i o n d e n s i t y (dE/dx) and energy (E) or range. The r e s o l v i n g power i s so enormous that one can i s o l a t e the s i g n a l of one 1^0 atom from the a s s o c i a t e d 1 0 -10 * C atoms. A s u b t l e dimension i n t h i s spectroscopy i s time, i n t h a t the overwhelming background o f % i s e l i m i n a t e d by the decay o f N~ during the i n i t i a l a c c e l e r a t i o n phase. F i n a l q u a n t i t a t i v e e s t i m a t i o n comes from i n t e g r a t i n g counts i n the a p p r o p r i a t e r e g i o n o f the dE/dx, E - plane. More commonly, we are faced w i t h the need f o r mathematical r e s o l u t i o n o f components, u s i n g t h e i r d i f f e r e n t patterns (or s p e c t r a ) i n the v a r i o u s dimensions. That i s , l i t e r a l l y , mathematical a n a l y s i s must supplement the chemical or p h y s i c a l a n a l y s i s . I n t h i s case, we very o f t e n i n i t i a l l y l a c k s u f f i c i e n t model i n f o r m a t i o n f o r a r i g o r o u s a n a l y s i s , and a number of methods have evolved t o "explore the data", such as p r i n c i p a l components and " s e l f - m o d e l i n g " a n a l y s i s (21), c r o s s c o r r e l a t i o n (22). F o u r i e r and d i s c r e t e (Hadamard, . . .) transforms (23), d i g i t a l f i l t e r i n g (24), rank a n n i h i l a t i o n (25), f a c t o r a n a l y s i s (26), and data matrix r a t i o i n g (27). 1

1

1 2

11

1 2

1


5.

69


CURRIE

Under the best of circumstances we can express the m u l t i dimensional s i g n a l (y) as a l i n e a r f u n c t i o n of the unknown concentrations ( x ) , such as decaying nuclear or o p t i c a l spectra: k

v

ij

=

1

v

A

ijk - U Ui)

ijk

=

1

A

i j k *k + eij

(12)

where e-tj/^k

k

(13)

(The f i r s t f a c t o r U ( A i ) i s the spectrum of species-k vs wavelength (A^); the second i s the decay curve v£ time~Ttj) w i t h mean l i f e x . ) I f the e^j are normally d i s t r i b u t e d w i t h known ( r e l a t i v e ) v a r i a n c e s , and we know the s p e c t r a and l i f e t i m e s f o r a l l components, then weighted, l i n e a r l e a s t squares w i l l provide estimates f o r x and °x (28). Since each x i s a l i n e a r sum of the normally d i s t r i b u t e d o b s e r v a t i o n s , i t too i s normal, and i t i s (almost) s t r a i g h t f o r w a r d t o compute the d e c i s i o n l e v e l (XQ) and d e t e c t i o n l i m i t ( x ) f o r s p e c i e s - k . I n p r i n c i p l e , the q u a n t i t i e s would r e q u i r e the e v a l u a t i o n o f o as x i n c r e a s e s from zero t o i t s d e t e c t i o n l i m i t . I f the s i g n a l i s r e l a t i v e l y weak (x

2

(3b) A

In the present case of = 0.00724, so the i n c l u s i o n of J would i n c r e a s e o and t h e r e f o r e xc by l e s s than 1 p a r t i n 10 *. The e x i s t e n c e of such a f a c t o r i s important i n p r i n c i p l e , however, f o r i t s i g n i f i e s the c o n t r i b u t i o n of e to the variance of the estimated net a n a l y t e c o n c e n t r a t i o n (or amount) even when that c o n c e n t r a t i o n i s zero. A

1

0

A

[ i i ] S t r i c t l y speaking, XQ and xn as c a l c u l a t e d above must be viewed as approximations (though extremely good ones) s i n c e A i s


76


not e x a c t l y known (denominator of Eq. 1 0 ) . A u s e f u l viewpoint i s to consider x as e x a c t l y Z-J-QJ o /A, where A i s a s i n g l e , s e l e c t e d outcome 1ST! 1

c

z

0

1-a'/A -

z = + 2pAA*B + i|; q A B

(11')

B

where

*B = < J > B i ——V

B|

yvD-B J

—V

n d

ys J

*

X D

=

*q

/

K

1

2

D

D

Using the previous r e s u l t s , we f i n d PBA = -0.292 SD - YD-B = B

- yc-B+zi-3°VD * 1-.042+1.645(.028+0.49[0.0464])

(jTJ

=

°

•A = °A/A

B / S d

024

* °- 3/1 .041 = 0.0233

* 0.103/32.46 = 0.00317

Thus, q . 0.0226; so c|> =* 0.0202 The ("1o") confidence i n t e r v a l f o r the d e t e c t i o n l i m i t i s thus, D

XD

x : 46.4 ± 0.9l| pg D

A symmetric and normal confidence i n t e r v a l i s a good approximation, s i n c e the u n c e r t a i n t y i s dominated by 03 (numerator of Eq 3b). F i n a l l y , _ i f the poor f i t simply r e f l e c t e d p r o p o r t i o n a t e l y e x t r a y - v a r i a n c e , then: *i

2

+ W i / ( 9 . 6 4 ) and Oy + o

y

• (9.64)//5


80


and , would be increased by a f a c t o r of A

B

9.64

The r e s u l t i n g estimate f o r xp would be x : D

59.1 ± 1 1 . 5 pg

T h i s r e s u l t , however, should not be taken too s e r i o u s l y , because the poor f i t may not be simply r e l a t e d to e x t r a y - v a r i a n c e . Literature Cited


1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20.

Kaiser, H. Two papers on the Limit of Detection of a Complete Analytical Procedure. London:Hilger, 1968. Currie, L. A. Anal. Chem. 1968, 40, 586. Rogers, L. B. Subcommittee dealing with the s c i e n t i f i c aspects of Regulatory Measurements, American Chemical Society, 1982. Ingle, J . D . , J r . J . Chem. Educ. 1974, 51, 100-5. Currie, L. A. Pure & Appl. Chem. 1982, 54, 715-754. Currie, L. A. Nucl. Instr. Meth. 1972, 100, 387. Currie, L. A. i n "Modern Trends i n Activation Analysis"; DeVoe, J. R.; LaFleur, P. D . , Eds.; Nat. Bur. Stand (U.S.) Spec. Publ. 312, 1968; p. 1215. Currie, L. A. i n "Treatise on Analytical Chemistry"; Elving, P.; Kolthoff, I . M., Eds.; J. Wiley & Son:New York, 1978; Vol 1, Chap. 4. Ingle, J . D., Jr.; Wilson, R. L. Anal. Chem. 1976, 48, 1641. Patterson, C. C . ; S e t t l e , D. M. 7th Materials Res. Symposium, Nat. Bur. Stand (U.S.) Spec. Publ. 422, 1976; p. 321. Scales, B. Anal. Biochem. 1963, 5, 489. Horwitz, W. (FDA); Meinke, W. W. (NRC), personal communica tions, 1982. See also Reference (3). Horwitz, W.; Kamps, L. R.; Boyer, K. W. J. Assoc. Off. Anal. Chem. 1980, 63, 1344. Hubaux, A . ; Vos, G. Anal. Chem. 1970, 42, 849. Ku, H. H. J. Res. N a t l . Bur. Stand. 1966, 70c, 263. Brownlee, K. A. i n " S t a t i s t i c a l Theory and Methodology i n Science and Engineering"; J . Wiley & Son:New York, 1960. Kurtz, D. A . , personal communication, 1982, 1983. Jacquez, J . A . ; Mather, F. J . and Crawford, C. R. i n "Linear Regression with Non-Constant, Unknown Error Variances"; Biometrics 1968, 24, 607. Golub, G. H . ; van Loan, C. F. J . Numer. Anal. 1980, 17, 883. Purser, K.; Russo, C . ; Gove, H . ; Elmore, R.; Ferraro, R.; Beukens, K.; Chang, L.; K i l i u s , L.; Lee, H . ; Litherland, A. Chapt. 3 i n Symposium on Nuclear and Chemical Dating Techniques, Currie, L. A., Ed.; American Chemical Society: Symposium Series No. 176, Washington, D.C., 1982.


5. CURRIE Dimensions of Detection in Chemical Analysis

21. 22. 23. 24. 25. 26.


27. 28. 29. 30. 31.

81

Lawton, W. H.; Sylvestre, E. A.; Maggio, M. S. Technometrics 1972, 14, 3. Horlick, G. Anal. Chem. 1973, 45, 319. Marshall, A. G . , Ed.; "Fourier, Hadamard, and Hilbert Transforms i n Chemistry", Plenum Press:New York, 1982. Savitzky, A . ; Golay, M.J.E. Anal. Chem. 1964, 36, 1627. Ho, C . - N . ; Christian, G. D . ; Davidson, E. R. Anal. Chem. 1981, 53, 92. Malinowski, E. R.; Howery, D. G. i n "Factor Analysis i n Chemistry"; J . Wiley & Son:New York, 1980. Fogarty, M. P . ; Warner, I . M. Anal. Chem. 1981, 53, 259. Nicholson, W. L.; Schlosser, J . E . ; Brauer, F. P. Nucl. Instr. Meth. 1963, 25, 45. Parr, R. M . ; Houtermans, H . ; Schaerf, K. Computers i n A c t i vation Analysis and Gamma-ray Spectroscopy Ed., Conf. -780421 1979, p. 544. Currie, L. A . ; Gerlach, R. W.; Lewis, C. W. Atm. Environ ment 1984, 18, 1517. Liggett, W. ASTM Conf. on Quality Assurance for Environ mental Measurements, STP (in press) 1984, Boulder, CO.

RECEIVED March 25, 1985


Trace Residue Analysis - ACS Publications - American Chemical Society

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