Detection: Overview of Historical, Societal, and Technical Issues - ACS

Chapter 1

Detection: Overview of Historical, Societal, and Technical Issues Lloyd A . Currie

Downloaded by UNIV OF ARIZONA on January 6, 2013 | http://pubs.acs.org Publication Date: December 9, 1987 | doi: 10.1021/bk-1988-0361.ch001

Center for Analytical Chemistry, National Bureau of Standards, Gaithersburg, M D 20899

Practical s o c i e t a l needs and basic s c i e n t i f i c advances frequently r e l y on Measurement Processes possessing s p e c i f i e d detection c a p a b i l i t i e s with acceptable p r o b a b i l i t i e s of false p o s i t i v e s and false negatives. The first part of t h i s overview introduces the basic concept of (chemical) detection, together with i t s applicability to selected s o c i e t a l problems such as the detection of natural hazards and the implementation of c e r t a i n regulations. Basic scientific measurement issues concerning assumptions and t h e i r validity, plus hypothesis t e s t i n g and decision theory as r e l a t e d to analyte detection are next introduced. Part two comprises a b r i e f historical review, h i g h l i g h t i n g major contributions to the concept and r e a l i z a t i o n of detection i n chemical applications. The current state of the art i s then considered. Part three i s the most extensive, as i t seeks to expose most of the technical issues involved i n deriving meaningful detection decisions and detection limits, considering the o v e r a l l Chemical Measurement Process. Those concerned p r i m a r i l y with s o c i e t a l or historical matters may wish to pass over t h i s part. Among the topics discussed are: systematic and model error; non-normal random error; the s p e c i a l problem of the blank; r e p l i c a t i o n vs Poisson variance; issues concerning complex data evaluation, c a l i b r a t i o n , and reporting -- including pitfalls associated with "black box" algorithms; OC curves; power of the t - t e s t ; and q u a l i t y . The section concludes with some new material on d i s c r i m i n a t i o n limits, lower and upper regulatory l i m i t s , multiple detection decisions, and univariate and m u l t i variate identification. A brief summary follows, bringing together historical, s o c i e t a l , and technical highlights. A concluding observation i s that a meaningful approach to p r a c t i c a l s o c i e t a l needs i s at hand, but that order must be brought out of the extant d i v e r s i t y of technical views on detection. T h i s chapter not subject to U . S . copyright Published 1988 A m e r i c a n C h e m i c a l Society

In Detection in Analytical Chemistry; Currie, L.; ACS Symposium Series; American Chemical Society: Washington, DC, 1987.


2

DETECTION IN ANALYTICAL CHEMISTRY The DETECTION LIMIT (Lp) i s one o f t h e most important c h a r a c t e r i s t i c s o f any Measurement P r o c e s s . Recognizing the e x i s t e n c e o f such l i m i t s i s c r u c i a l b o t h f o r s t r i c t l y s c i e n t i f i c endeavors, such as t h e s e a r c h f o r a new fundamental p a r t i c l e ( i l , and f o r v i t a l s o c i e t a l a p p l i c a t i o n s o f s c i e n t i f i c measurements, such as t h e d e t e c t i o n o f a p a t h o l o g i c a l s t a t e o r a hazardous l e v e l o f a heavy m e t a l . I n t h i s l a t t e r r e g a r d , i m p o r t a n t p r o g r e s s has been made i n c o n v e y i n g t o t h e p u b l i c and t h e i r p o l i c y makers t h a t i t i s a l a w o f measurement s c i e n c e t h a t t h e d e t e c t i o n c a p a b i l i t y o f a l l Measurement P r o c e s s e s must s t o p s h o r t o f z e r o , i n c l o s e analogy w i t h t h e T h i r d Law o f Thermodynamics. R e c o g n i t i o n t h a t Lq may n o t be z e r o , has a l l e v i a t e d e a r l i e r l e g i s l a t i v e problems, such as t h e d i c t u m t h a t no r e s i d u e o f p r o v e n a n i m a l c a r c i n o g e n s may be p r e s e n t i n c e r t a i n f o o d p r o d u c t s ( 2 ) . The f a c t , however, t h a t d e t e c t i o n l i m i t s c a n , a t a c o s t and w i t h t e c h n o l o g i c a l advances, be made e v e r s m a l l e r h a s f o r c e d reexamin a t i o n o f r e g u l a t o r y i s s u e s i n t h e l i g h t o f e x t a n t and even potential detection capabilities. The consequence has been t h e c o n s i d e r a t i o n o f c o s t / b e n e f i t o r "acceptable r i s k " a l t e r n a t i v e s t o "no d e t e c t a b l e r e s i d u e " r e g u l a t o r y p o l i c y ( 3 ) . Such a l t e r n a t i v e s are mandatory i n l i g h t o f t h e fundamental p r i n c i p l e s o f d e t e c t i o n . D e f i n i n g a c c e p t a b l e l e v e l s o f r i s k ( 4 ) , whether i n a r e g u l a t o r y s e t t i n g o r w i t h r e s p e c t t o m e d i c a l d e c i s i o n s o r even i n terms o f governmental actions i n connection with potential natural d i s a s t e r s , i s p r i m a r i l y a s o c i o p o l i t i c a l matter. Although t h i s i s s u e i s o f c e n t r a l importance, i t t r a n s c e n d s t h e theme o f t h i s c h a p t e r , w h i c h i s t o examine t h e h i s t o r i c a l e v o l u t i o n and c u r r e n t s t a t e o f t h e a r t o f d e t e c t i o n from t h e p e r s p e c t i v e o f c h e m i c a l measurement s c i e n c e . I n o r d e r t o h i g h l i g h t t h e importance o f D e t e c t i o n D e c i s i o n s and Detection L i m i t s , and t o u n d e r l i n e the f a c t that the p r o b a b i l i t y o f d e t e c t i o n does n o t i m m e d i a t e l y pass from z e r o t o u n i t y a t t h e D e t e c t i o n L i m i t , we have p r e s e n t e d i n F i g . 1 s e v e r a l s i t u a t i o n s where v a l i d d e t e c t i o n d e c i s i o n s and adequate d e t e c t i o n l i m i t s a r e o f c o n s i d e r a b l e p r a c t i c a l importance. (The p r e s e n c e o f a f i n i t e r i s k o f e r r o r ( f a l s e negative) a t the detection l i m i t i . e . , t h e absence o f " c e r t a i n t y " i s t h e second a s p e c t o f t h e problem t h a t i s somewhat f o r e i g n t o t h e common u n d e r s t a n d i n g , t h e f i r s t being the f a c t that zero d e t e c t i o n l i m i t s a r e unattainable.) This f i g u r e introduces the Hypothesis Testing foundation f o r D e t e c t i o n , and i t demonstrates t h a t i t i s e s s e n t i a l f o r those o f us i n v o l v e d i n measurement s c i e n c e t o d e v e l o p a sound, common, and q u a n t i t a t i v e approach t o t h e f o r m u l a t i o n o f D e t e c t i o n L i m i t s . I n a d d i t i o n , t h i s f o r m u l a t i o n must be communicated i n an e f f e c t i v e manner b o t h w i t h i n t h e s c i e n t i f i c community and w i t h t h o s e who depend on o u r measurements f o r s o c i e t a l d e c i s i o n s and p o l i c y making. As a f i n a l i n t r o d u c t o r y n o t e , i t s h o u l d be o b s e r v e d t h a t from the p e r s p e c t i v e s o f b a s i c d i s c o v e r i e s i n S c i e n c e and t h e e a r l y d i s c e r n m e n t o f fundamental changes i n t h e G l o b a l Environment (e.g.,


1. CUtfRIE

Overview of Historical, Societal, and Technical Issues

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CO 60) t h e e x p r e s s i o n y i e l d s a r e a s o n a b l y good e s t i m a t e f o r V , as good as w o u l d r e s u l t from 100 o r more r e p l i c a t e s . If V i s t a k e n t o be z e r o , t h e c r i t i c a l l e v e l may t h e n be c a l c u l a t e d d i r e c t l y from V w h i c h h e r e i s e s t i m a t e d as t h e number o f counts fNote 51. If V i s n o t known, we have t h r e e a l t e r n a t i v e s . One o f extreme c o n s e r v a t i s m would be t o use t h e lower and upper l i m i t s f o r V , b a s e d on r e p l i c a t i o n [ s - e s t ( V ) ] and knowledge o f V . I n t e r m e d i a t e , and most common, i s s i m p l y t o c a l c u l a t e as t * s , where t i s b a s e d on t h e e x t e r n a l number o f d f (number o f r e p l i c a t e s minus o n e ) . More i n t e r e s t i n g i s t h e use d f o u r knowledge t h a t V >0, and a v a r i a n c e w e i g h t e d t . (A f o u r t h a l t e r n a t i v e , i g n o r i n g the p o s s i b l e e x i s t e n c e o f V i s a l l t o o common; t h e unsupported assumption t h a t counting s t a t i s t i c s o r other i n t e r n a l i n s t r u m e n t a l v a r i a n c e f i x e s t h e o v e r a l l i m p r e c i s i o n c a n generate T-^'s and C I ' s t h a t a r e t o o s m a l l and c o r r e s p o n d i n g l y l a r g e f a l s e positive probabilities.) The m e r i t o f t h e v a r i a n c e w e i g h t e d t t e c h n i q u e i s t h a t i t p e r m i t s us t o use o u r e x c e l l e n t knowledge o f together w i t h the f a c t t h a t V cannot be n e g a t i v e t o o b t a i n a s i g n i f i c a n t l y s m a l l e r o r CI than we would u s i n g s d i r e c t l y . I t provides p r o t e c t i o n a g a i n s t u n a n t i c i p a t e d e x t e r n a l random e r r o r w i t h l i t t l e p e n a l t y i f t h a t e r r o r component i s i n f a c t n e g l i g i b l e . The t e c h n i q u e sugg e s t e d h e r e i s t e n t a t i v e and approximate, b u t i t appears t o be c o n s e r v a t i v e and a s y m p t o t i c a l l y c o r r e c t fNote 61. To i l l u s t r a t e , l e t us c o n s i d e r t r i p l i c a t e measurements o f a sample u s i n g a c o u n t i n g t e c h n i q u e , such as i o n c o u n t i n g mass s p e c t r o m e t r y o r photon c o u n t i n g i n o p t i c a l s p e c t r o m e t r y , X - r a y f l u o r e s c e n c e a n a l y s i s o r gamma r a y s p e c t r o m e t r y . Internal variance d e r i v e s from P o i s s o n c o u n t i n g s t a t i s t i c s [V ] where t h e a p p r o p r i a t e v a l u e o f t e q u a l s i t s normal l i m i t z o r 1.645 f o r a - 0.05. T o t a l v a r i a n c e [ V ] f o r t h e 3 r e p l i c a t e s i s e s t i m a t e d as s , where t i s 2 . 9 2 f o r 2 d f . Excess v a r i a n c e [V ] i s V -V , and e s t i m a t e d as s - V , w i t h t h e c o n s t r a i n t t h a t V may n o t be n e g a t i v e . 1^,' ( o r CI) i s c a l c u l a t e d as t's'/Jri where: fc

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26

DETECTION IN ANALYTICAL CHEMISTRY 2

F o r the example a t hand, we e s t i m a t e V as s - V . (Eq. 1) thus y i e l d s s'=a i f s < , o r s'=s i f s > a . (Eq. 2) becomes t'=1.645 (k) + 2.92 (1-k) where k - a . / s . For example, i f a. i s e q u i v a l e n t t o 1.75 ng-Ca, and s, t o 3.04 ng-Ca, k would e q u a l ( 1 . 7 5 / 3 . 0 4 ) = 0.331. As a r e s u l t , s' - 3.04 ng-Ca, t ' - 2.50, and 1^' = 4.39 ng-Ca. I n v e s t i g a t i o n o f the p r o p e r t i e s o f ' f o r V =0 to V » V shows i t t o be c o n s e r v a t i v e [a' < 0.05] w i t h a l i m i t i n g v a l u e when V=«=0 o f a p p r o x i m a t e l y 0.03. A l s o i n t h i s l i m i t i n g case, 1^, ' on the average i s o n l y s l i g h t l y g r e a t e r (< 10%) t h a n i t would be i f one assumed V was i d e n t i c a l l y z e r o , whereas f o r the convent i o n a l approach [1^ - 2.92(s/7n)) i t would be 78% l a r g e r . When the s t a k e s a r e h i g h e r -- e.g., CI's o r L^'s f o r a=0.01 -- the c o n t r a s t becomes even g r e a t e r . I n e f f e c t , we have used our knowledge o f V to e x c l u d e v e r y s m a l l v a l u e s f o r e s t i m a t e d t o t a l a, and g a i n e d smaller CI's, ' s , and d e t e c t i o n l i m i t s i n r e t u r n . 3.1.10 e f f e c t s o f r o u n d i n g and truncation. Premature rounding of experimental data d i s t o r t s i t s e r r o r d i s t r i b u t i o n , r e s u l t i n g i n erroneous c o n c l u s i o n s r e g a r d i n g the shape o f the distribution, i t s parameters [mean, v a r i a n c e ] , and r e s u l t s o f s t a t i s t i c a l t e s t s ( e . g . , d e t e c t i o n d e c i s i o n s , q u a l i t y o f f i t ) and confidence i n t e r v a l s . The most o b v i o u s d i s t o r t i o n i s t h a t an i n h e r e n t l y c o n t i n u o u s d i s t r i b u t i o n i s made d i s c r e t e ; the e f f e c t i s analogous t o " d i s c r e t i z a t i o n n o i s e " w h i c h i s o f t e n found w i t h m u l t i c h a n n e l and m u l t i d e t e c t o r a r r a y t e c h n i q u e s i n v o l v i n g windows i n t i m e , space, energy, w a v e l e n g t h , e t c . ( 5 6 ) . The tolerable degree o f r o u n d i n g depends on the d i s t r i b u t i o n . For n o r m a l l y d i s t r i b u t e d d a t a , t h e r e i s about a 10% chance o f f i n d i n g r e s u l t s w i t h i n a/8 o f the mean. S c a l e d i v i s i o n s much s m a l l e r t h a n a/4 a r e t h e r e f o r e r e q u i r e d i f one i s t o a v o i d f a l s e c o i n c i d e n c e s , and f i t s t h a t a r e "too good", e t c . I n f a c t , c l u e s t o e x c e s s i v e r o u n d i n g o r t r u n c a t i o n may be found i n x F s t a t i s t i c s which are u n u s u a l l y s m a l l , o r i n p d f ' s e x h i b i t i n g unexpected d e v i a t i o n s from n o r m a l i t y (57) . A b n o r m a l i t y i s n o t e d a l s o by Cheeseman and W i l s o n f o r c o n s t r a i n e d b a l a n c e - p o i n t measurements, such as the galvanometer needle which i s p h y s i c a l l y c o n f i n e d to non-negative s c a l e readings (36). The importance of these c o n s i d e r a t i o n s f o r databases i n c o r p o r a t i n g l o w - l e v e l r e s u l t s i s discussed i n (34). x

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3.1.11 the e v a l u a t i o n p r o c e s s [ d a t a r e d u c t i o n : fitting]. The d a t a e v a l u a t i o n p r o c e s s [EP] i s an i n t e g r a l p a r t o f the CMP, and as such i t h e l p s d e f i n e o and the c r i t i c a l l e v e l . It is perhaps o b v i o u s t h e n t h a t 1^, , C I ' s , and the d e t e c t i o n l i m i t w i l l d i f f e r f o r the v e r y same e x p e r i m e n t a l d a t a , depending on the EP a p p l i e d . A s i m p l e i l l u s t r a t i o n i s found i n the f i t t i n g o f s p e c t r a l or c h r o m a t o g r a p h i c peaks. One may use the peak h e i g h t as the q u a n t i t a t i v e s i g n a l measure, o r a model-independent peak a r e a may be used, o r a more s o p h i s t i c a t e d t e c h n i q u e such as l i n e a r o r n o n - l i n e a r l e a s t squares may be employed t o e s t i m a t e the peak s i z e a c c o r d i n g t o a s e l e c t e d f u n c t i o n a l model such as a G a u s s i a n o r skewed G a u s s i a n (58). The point i s that without explicit s p e c i f i c a t i o n o f the e n t i r e CMP, i n c l u d i n g t h e EP employed, the d e t e c t i o n c h a r a c t e r i s t i c s o f the measurement p r o c e s s a r e u n d e f i n e d . Because o f t h i s , a s l i g h t problem o c c u r s when the EP i s g i v e n as a " b l a c k box", o r a l g o r i t h m whose c h a r a c t e r i s t i c s a r e u n c l e a r . ( T h i s i s s u e , i n c l u d i n g the common a v a i l a b i l i t y o f e x e c u t a b l e s o f t w a r e Q



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Overview of Historical, Societal, and Technical Issues 27

CURRIE

w i t h o u t source code, w i l l be t r e a t e d f u r t h e r i n the d i s c u s s i o n o f d e t e c t i o n l i m i t s i n the n e x t s e c t i o n . ) When the EP comprises l i n e a r computations ( l i n e a r i n the o b s e r v a t i o n s ) such as s i m p l e d i f f e r e n c e s , y - B, or l i n e a r l e a s t squares or l i n e a r m u l t i v a r i a t e c o m p u t a t i o n s , i n i t i a l n o r m a l i t y ( o f the o b s e r v a t i o n s y) i s p r e s e r v e d f o r the e s t i m a t e d q u a n t i t i e s . N o n - l i n e a r c o m p u t a t i o n s , such as a r i s e commonly i n i t e r a t i v e model s e l e c t i o n and peak s e a r c h r o u t i n e s , produce e s t i m a t e d parameters h a v i n g non-normal d i s t r i b u t i o n s (.59). Caution i s i n order, i n t h o s e c a s e s , i n a p p l y i n g "normal" v a l u e s o f t e s t s t a t i s t i c s t o c a l c u l a t e 1^-. and C I ' s . (Other f a c t o r s t o c o n s i d e r are the e x t e n t o f n o n - l i n e a r i t y , the l e v e l o f c o n f i d e n c e or s i g n i f i c a n c e [ 1 - a ] , and the r o b u s t n e s s o f the s t a t i s t i c i n q u e s t i o n . ) F i n a l l y , i t s h o u l d be n o t e d t h a t an erroneous model w i l l g i v e erroneous r e s u l t s . T h i s seeming t r u i s m i s i m p o r t a n t because models w h i c h pass s t a t i s t i c a l t e s t s [e.g., x t e s t o f f i t ] are c o n s i s t e n t w i t h the d a t a b u t not n e c e s s a r i l y c o r r e c t . Because o f m u l t i c o l linearity, model error may go undetected, while producing s i g n i f i c a n t b i a s i n the r e s u l t s (48). 3.1.12 c a l i b r a t i o n error. A number o f d i f f e r e n t approaches may be t a k e n t o i n c o r p o r a t e the u n c e r t a i n t y i n the c a l i b r a t i o n f a c t o r A i n t o the c r i t i c a l l e v e l . To i l l u s t r a t e , l e t us c o n s i d e r the s i m p l e s t f u n c t i o n a l r e l a t i o n f o r the E v a l u a t i o n P r o c e s s : 2

x

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S/A

(3)

U n f o r t u n a t e l y , t h i s i s a l r e a d y a n o n - l i n e a r r e l a t i o n , so we cannot e x p e c t x t o be n o r m a l l y d i s t r i b u t e d . I f the r e l a t i v e e r r o r i n A i s s m a l l (e.g., < 10%) i t s i n f l u e n c e on 1^, i s l i k e w i s e s m a l l , and d e v i a t i o n s from n o r m a l i t y are m i n i m a l . I f the r e l a t i v e u n c e r t a i n t y i n A i s not n e c e s s a r i l y s m a l l , or i f i t i n c l u d e s possible s y s t e m a t i c e r r o r , a s t r a i g h t f o r w a r d approach i s t o use the lower bound f o r A t o c a l c u l a t e an upper bound f o r L^. (here x ) w h i c h can be u s e d t o make c o n s e r v a t i v e detection decisions [a < 0.05]. ( I n c o r p o r a t i o n o f bounds f o r s y s t e m a t i c e r r o r i s d i s c u s s e d more f u l l y i n the s e c t i o n on d e t e c t i o n l i m i t s . ) E r r o r p r o p a g a t i o n from the f i t t i n g o f a c a l i b r a t i o n c u r v e can be used to t r e a t d e t e c t i o n and interval estimation (almost) r i g o r o u s l y p r o v i d e d the model i s c o r r e c t -- a c a u t i o n b e i n g t h a t the i n t e r c e p t - B may not represent the b l a n k - B (60.61). An i n t e r e s t i n g a l t e r n a t i v e i s to estimate a and the c o r r e s p o n d i n g detection characteristics directly for x [i.e., i n units of c o n c e n t r a t i o n ] by f u l l r e p l i c a t i o n o f the CMP a t the l e v e l s o f c o n c e r n , o b s e r v i n g y, B, and A f o r each r e p l i c a t e . ( T h i s i s the " p a i r e d comparison" concept extended t o c a l i b r a t i o n , where a b l a n k and s t a n d a r d i s r u n f o r e v e r y sample.) The s t a t i s t i c a l p r o p e r t i e s o f the o b s e r v e d x d i s t r i b u t i o n can t h e n be used t o d i r e c t l y calculate x [as t s , i f A - v a r i a t i o n i s not too g r e a t ] and e s t i m a t e the d e t e c t i o n l i m i t . An added b e n e f i t o f t h i s scheme i s t h a t d i r e c t o b s e r v a t i o n o f the b l a n k decouples i t from the c a l i b r a t i o n curve f i t t i n g process, so t h a t an assumed s t r a i g h t l i n e model [ c o n s t a n t s e n s i t i v i t y A] can be t e s t e d by f i t t i n g a l i n e [S=Ax+e] t h r o u g h the o r i g i n (62.). For v a l i d c o n c l u s i o n s , o f c o u r s e , due a t t e n t i o n must be g i v e n to i n t e r f e r e n c e and m a t r i x e f f e c t s on b o t h c

Q

c

Q


28

DETECTION IN ANALYTICAL CHEMISTRY

parameters, B and A. An i l l u s t r a t i o n of an observed x d i s t r i b u t i o n for the n u l l hypothesis [x - 0] i s shown f o r I i n (44). 3.1.13 reporting of l o w l e v e l data. Problems associated with data rounding and truncation extend to the reporting of f i n a l r e s u l t s . Also, j u s t as i n the case of the data evaluation step of the CMP, reporting must be treated as an i n t e g r a l part of the o v e r a l l CMP. Bias and information-loss are the prime considerations. At the lower extreme, where x - 0 [null hypothesis] suppression of negative estimates forces a p o s i t i v e bias, on the average. Other biases a r i s e when a l l non-detected r e s u l t s are reported as zero or as equal to (or less than) the detection l i m i t . The d i f f i c u l t i e s are evident as soon as one attempts to: develop a database comprising large amounts of low-level data (14); to compute temporal or s p a t i a l averages for higher order detection decisions (2£); or to compute average concentrations across d i f f e r e n t materials as i n the USDIET-1 exercise (62). This l a s t example i l l u s t r a t e s the point: composite samples of the U.S d i e t were prepared for measurement of a broad range of e s s e n t i a l and toxic chemical constituents, including the trace element Se. Comparison of the r e s u l t for Se i n the composite sample [128 /ig/g] with the weighted average from the large number of i n d i v i d u a l contributing foods [100 Mg/g]» showed a s i g n i f i c a n t negative bias for the l a t t e r . This was a r e s u l t of s e t t i n g a l l "trace" observations (defined as those below a q u a n t i f i c a t i o n l i m i t , L^) to zero. Adjusting these upward to L^/2 led to an improvement [110 Mg/g] » but negative bias was s t i l l apparent. These kinds of problems can be completely circumvented i f concentration estimates, even i f negative, are always reported together with t h e i r uncertainties (64. 65) . Detection decisions can be made by comparison with , and upper l i m i t s may be given as x + Zs/Jri.


1 3 1

3.1.14 thresholds. The threshold for discriminating "real signals" from blanks may be set i n various ways. The only way that i s consistent with the r e l a t i o n between confidence intervals and significance tests i s the one described, Lg - t s . Other techniques include the use of a constant m u l t i p l i e r ks or ka, with k - 3 a popular choice; and use of a f i x e d threshold signal or concentration, such as 1 mV or 2 ng. A drawback of these alternative techniques i s that they seldom recognize the existence or magnitude of the a-error, which, however, does e x i s t , and which w i l l take on varying values depending on the number of degrees of freedom or the magnitude of the f i x e d threshold i n comparison to a . For confidence intervals a i s conventionally taken as 0.05, so there seems l i t t l e j u s t i f i c a t i o n for depressing i t by a factor of f o r t y to 0.0013 (corresponding to 3a) for detection decisions [a-known], or by more than a factor of ten (corresponding to 3s) for 20 degrees of freedom. Instruments having hardware or software discriminators may have r e s u l t i n g dead zones which correspond to vanishingly small a's. In one case recently, the threshold was set so high that a was beyond the range of any of the s t a t i s t i c a l tables, 1^/^ « 34; i n fact the threshold was so high that the c r i t i c a l l e v e l exceeded even the conventional l i m i t of q u a n t i f i c a t i o n -- i . e . , the RSD at Lg was but 5% (61, p. 76, case-e) ! Such high and varying thresholds for detection decisions lead both to needless confusion and to measurement processes operating f a r short of t h e i r inherent capabilities. Q


1.

CURRIE


3.2 M a t t e r s C o n c e r n i n g the E r r o r o f the Second K i n d , and the A n a l y t e D e t e c t i o n L i m i t . The c o n c e r n i n the p r e c e d i n g s e c t i o n was the v a l i d i t y o f d e t e c t i o n d e c i s i o n s , based on comparisons o f e x p e r i m e n t a l outcomes [ e s t i m a t e d s i g n a l s o r c o n c e n t r a t i o n s ] w i t h a p p r o p r i a t e c r i t i c a l l e v e l s o r d e c i s i o n t h r e s h o l d s . Here, we t u r n to i s s u e s c o n c e r n i n g the i n h e r e n t d e t e c t i o n c a p a b i l i t y o f the CMP i n question t h a t i s , the t r u e s i g n a l s o r c o n c e n t r a t i o n s w h i c h can be d e t e c t e d w i t h , f o r example, a 95% p r o b a b i l i t y [£=0.05], g i v e n the c r i t i c a l l e v e l ( o r e q u i v a l e n t l y a) t o b e used f o r t e s t i n g o b s e r v e d r e s u l t s . L^ i s thus t i e d i n t i m a t e l y t o and t o a o r . A l t h o u g h a s i g n i f i c a n c e t e s t may be p e r f o r m e d w i t h no c o n s i d e r a t i o n o f H o r the d e t e c t i o n l i m i t , Lp i s ambiguous w i t h o u t the s p e c i f i c a t i o n o f p and a ( o r ). That i s , f o r a g i v e n Lp t h e r e i s an i n f i n i t e s e t o f p o s s i b l e a, p p a i r s . Passing a s i g n i f i c a n c e test -- e.g., x < x -- i s commonly s a i d t o mean "acceptance" o f the n u l l h y p o t h e s i s -- i . e . , x - 0. This i s unfortunate terminology, for only consistency with the null hypothesis has been demonstrated. " P r o o f " o f the n u l l h y p o t h e s i s ( w i t h i n c e r t a i n f u z z y bounds) demands a t t e n t i o n t o a l l p o s s i b l e a l t e r n a t i v e hypotheses H; t h a t i s the t e s t i n use must be s u f f i c i e n t l y p o w e r f u l t o " d e t e c t " [0 < 0.05, g i v e n a - 0.05] H. A major r e a s o n f o r i n t e r e s t i n d e t e c t i o n l i m i t s i s thus t o a l l o w us t o s e l e c t o r d e s i g n a measurement p r o c e s s h a v i n g the c a p a c i t y t o d e t e c t s i g n a l s o r a n a l y t e s a t p r e s c r i b e d l e v e l s o f importance. An o v e r v i e w o f selected technical issues follows. 3.2.1 i g n o r a n c e o f the e r r o r o f the second k i n d (B). False n e g a t i v e s o c c u r whether t h e i r e x i s t e n c e i s r e c o g n i z e d o r n o t . The common p r a c t i c e o f making d e t e c t i o n d e c i s i o n s a t the s o - c a l l e d d e t e c t i o n l i m i t , o r LOD, e t c . , has the e f f e c t o f s e t t i n g - 1^ , w i t h the r e s u l t t h a t p - 50% -- e q u i v a l e n t t o the p r o v e r b i a l f l i p o f the c o i n . W i t h a a - c o e f f i c i e n t o f 3, a may be as s m a l l as 0.0013, r e s u l t i n g i n an imbalance [P/a] o f a f a c t o r o f n e a r l y 400! Ignorance o f t h i s m a t t e r makes p o s s i b l e i n a d v e r t e n t o r even i n t e n t i o n a l m i s r e p r e s e n t a t i o n o f d e t e c t i o n c a p a b i l i t y . For example, the s u b t l e t r a d e - o f f between a and p c o u l d be employed t o a v o i d p e n a l t i e s f o r f a l s e p o s i t i v e s a s s o c i a t e d w i t h an inadequately c o n t r o l l e d blank. 3.2.2 lower and upper d e t e c t i o n l i m i t s . For c e r t a i n types o f chemical measurements there are dual null hypotheses and c o n s e q u e n t l y d u a l 1^'s and Lp's f o r c o n c e n t r a t i o n s d i f f e r i n g from these n u l l l e v e l s . Examples are found where a lower l i m i t i s s e t by background n o i s e , and an upper l i m i t , by some type o f maximum s i g n a l l i m i t a t i o n such as i n s t r u m e n t a l d e t e c t o r s a t u r a t i o n . A d u a l i l l u s t r a t i o n i s shown i n F i g . 7 f o r two e x p o n e n t i a l phenomena, r a d i o a c t i v e decay and r a d i a t i o n a b s o r p t i o n . I n each case the lower Lj3 i s g i v e n by the s m a l l e s t d e t e c t a b l e d i f f e r e n c e from a comparator ( z e r o age s t a n d a r d o r b l a n k s o l u t i o n ) , and the upper Lp i s g i v e n by the s m a l l e s t d e t e c t a b l e d i f f e r e n c e from an i n f i n i t e l y o l d sample (no n e t e m i t t e d r a d i o a c t i v i t y ) o r an i n f i n i t e l y a b s o r b i n g sample (no n e t t r a n s m i t t e d r a d i a t i o n . 3.2.3 the a - 8 c o n n e c t i o n : PC and ROC c u r v e s , and d e t e c t i o n power. A c o n v e n i e n t way t o v i s u a l i z e the r e l a t i o n s h i p between f a l s e p o s i t i v e [a] and f a l s e n e g a t i v e [p] e r r o r s and the n o r m a l i z e d d i f f e r e n c e [d] between the means o f two p o p u l a t i o n s f o r a g i v e n s t a t i s t i c a l t e s t has come t o us from s i g n a l d e t e c t i o n t h e o r y [ 9 ] .


A

c

A

A



30


F i g . 7. Lower and Upper D e t e c t i o n L i m i t s . When a measurement p r o c e s s has b o t h minimum and maximum s i g n a l bounds, as i n r a d i o a c t i v e decay and o p t i c a l a b s o r p t i o n s p e c t r o m e t r y , LLD and ULD must b o t h be c o n s i d e r e d . Dashed l i n e s i g n a l lower and upper d e t e c t i o n l i m i t s map onto the age and c o n c e n t r a t i o n lower and • upper l i m i t s (arrows) v i a the e x p o n e n t i a l f u n c t i o n .


1.

CURRIE


I n t h i s t h e o r y the R e c e i v e r O p e r a t i n g C h a r a c t e r i s t i c [ROC] curve f o r a g i v e n t e s t t r a c e s the r e l a t i o n s h i p between the t r u e p o s i t i v e p r o b a b i l i t y [1-0] and the f a l s e p o s i t i v e p r o b a b i l i t y [a] f o r a g i v e n mean n o r m a l i z e d s i g n a l d i f f e r e n c e d. Another r e p r e s e n t a t i o n o f the r e l a t i o n s h i p , denoted the Power Curve i n s t a t i s t i c s , i s the c u r v e w h i c h t r a c e s the r e l a t i o n between D e t e c t i o n Power -- w h i c h i s synonymous w h i c h the t r u e p o s i t i v e p r o b a b i l i t y [1-/3] -- and the d i f f e r e n c e d, f o r a g i v e n v a l u e f o r a. (The complementary r e l a t i o n : f} vs d, g i v e n a, i s d e s c r i b e d i n s t a t i s t i c s as the O p e r a t i n g C h a r a c t e r i s t i c [OC] c u r v e . ) F i g . 8 shows the normal ROC c u r v e f o r d - 3.29 i n u n i t s o f o [ i . e . , the d e t e c t i o n l i m i t ] , and the power c u r v e f o r a - 0.05. The former [ROC] r e p r e s e n t a t i o n i s the more c o n v e n i e n t f o r the comparison o f t e s t s and the s e l e c t i o n o f a l t e r n a t i v e a, £ p a i r s , f o r a g i v e n d i f f e r e n c e i n p o p u l a t i o n means. For t h i s reason, i t i s used t o compare the d i a g n o s t i c power o f a l t e r n a t i v e t e s t s i n c l i n i c a l c h e m i s t r y , where t h e r e are two d i s c r e t e p o p u l a t i o n s [d - f i x e d ] (66)I t may be u s e f u l a l s o f o r e x a m i n i n g the v a l u e o f a t e s t o r the s e l e c t i o n o f an " o p t i m a l " a, /3 p a i r i n a r e g u l a t o r y s e t t i n g where, f o r example, a s p e c i f i e d d i f f e r e n c e [6^] i s o f c o n c e r n . A l s o , i f the s o u r c e s o r i d e n t i t i e s o f c h e m i c a l s p e c i e s are c h a r a c t e r i z e d by unique element or i s o t o p e r a t i o s , an ROC c u r v e c o u l d be used t o r e p r e s e n t the d i s c r i m i n a t i n g power o f s e l e c t e d measurement t e c h n i q u e s . The second [Power c u r v e ] r e p r e s e n t a t i o n i s more a p p r o p r i a t e when one i s i n t e r e s t e d i n the d e t e c t i o n power as a f u n c t i o n o f ( n e t ) s i g n a l l e v e l o r c o n c e n t r a tion. Thus, i t i s c l e a r from the c u r v e t h a t the power i s b u t 50% when d — 1.645. A second s c a l e on the a b s c i s s a makes i t c o n v e n i e n t in this representation to see the relation i n units of concentration. OC and power c u r v e s are r e g u l a r l y used i n the e v a l u a t i o n o f statistical t e s t s (67.68). S i m i l a r l y , one f i n d s ROC curves employed i n m e d i c i n e and p s y c h o l o g y [12, d i s c u s s i o n & r e f e r e n c e s i n Chapt. 2 5 ] . They appear t o be l i t t l e used i n A n a l y t i c a l C h e m i s t r y , though L i t e a n u and R i c a have proposed the use o f d i f f e r e n t two d i m e n s i o n a l p r o j e c t i o n s o f the t h r e e d i m e n s i o n a l r e l a t i o n s h i p [a, ft, d] as r e p r e s e n t a t i o n s o f the " d e t e c t i o n c h a r a c t e r i s t i c " ( 8 ) . 3.2.4 power of the t-test. The three dimensional r e l a t i o n s h i p d e s c r i b e d above i s expanded t o f o u r f o r S t u d e n t ' s t , w i t h the a d d i t i o n o f the number o f degrees o f freedom. I f we r e s t r i c t our a t t e n t i o n t o the d e t e c t i o n l i m i t , by f i x i n g a and /? b o t h t o 0.05, the r e m a i n i n g two dimensions can be v i e w e d as a c u r v e , d vs d f -- i . e . , the d e t e c t i o n l i m i t ( i n u n i t s o f a ) as a f u n c t i o n o f the number o f degrees o f freedom, where t _ i s used f o r making d e t e c t i o n d e c i s i o n s . I n t h i s c a s e , the v a l u e o f d i s d e t e r m i n e d by r e q u i r i n g a 95% p r o b a b i l i t y (1-/9) t h a t the e s t i m a t e d n e t s i g n a l d i v i d e d by i t s e s t i m a t e d s t a n d a r d d e v i a t i o n [ ( y - B ) / s ] w i l l exceed the c r i t i c a l l e v e l f o r S t u d e n t ' s t . T h i s r a t i o i s c a l l e d the n o n - c e n t r a l t , w i t h n o n - c e n t r a l i t y parameter d, because i t i s d i s p l a c e d from z e r o by t h i s amount. The n e t s i g n a l d e t e c t i o n l i m i t i s g i v e n by da . Another i m p o r t a n t a p p l i c a t i o n o f the non-central t d i s t r i b u t i o n has been t o t e s t the v a l i d i t y o f presumed d e t e c t i o n l i m i t s , f o r example, i n c o n n e c t i o n w i t h m e d i c a l d i a g n o s t i c d e v i c e s (69.) .


Q

Q

x

a

Q

Q


32


1.00


I

0.50 0.05

0.10

0.20

0.30

False Positive (a) 1.00 —0.95

0.80

-

[ 8. F o r 4 7 degrees o f freedom t h e c o r r e c t v a l u e s a r e 4.07, 3.87, 3.75, and 3.68. To i l l u s t r a t e , l e t us suppose t h a t 5 p a i r e d y, B o b s e r v a t i o n s were made and t h e mean d i f f e r e n c e and e s t i m a t e d s t a n d a r d e r r o r were 1.8 ± 1.2 mV. The c r i t i c a l l e v e l f o r 4 degrees o f freedom would be t s - (2.13) (1.2) - 2.6 mV, so t h e c o n c l u s i o n would be "not d e t e c t e d . " The d e t e c t i o n l i m i t would be da 4.07a . U s i n g s as an e s t i m a t e f o r a, we would e s t i m a t e LQ as (4.07)(1.2)=4.9 mV. 3.2.5 uncertainties i n detection limits. The p r e v i o u s example r a i s e s an e x t r e m e l y i m p o r t a n t p o i n t , namely, t h a t u n l e s s a i s known w i t h o u t e r r o r , t h e d e t e c t i o n l i m i t cannot be e x a c t l y known. This i s i n c o n t r a s t w i t h the c r i t i c a l l e v e l , which can always be e x p l i c i t l y c a l c u l a t e d from Student's t and t h e e s t i m a t e d standard error. We c a n , however, d e r i v e a c o n f i d e n c e i n t e r v a l f o r Lp from t h e bounds f o r a, g i v e n s and d f . F o r n o r m a l l y d i s t r i b u t e d e r r o r s these bounds c a n be d e r i v e d from t h e x d i s t r i b u t i o n . ( s / a i s d i s t r i b u t e d as x / d f . ) One f i n d s , f o r example, t h a t a t l e a s t 13 r e p l i c a t e s a r e n e c e s s a r y t o o b t a i n s w i t h i n 50% o f t h e t r u e a (90% confidence l e v e l ) . For p r a c t i c a l a p p l i c a t i o n o f d e t e c t i o n l i m i t s -- e.g., i n m e e t i n g a r e s e a r c h o r r e g u l a t o r y r e q u i r e m e n t - - a " s a f e r " procedure i s t o quote t h e upper l i m i t f o r Lp . This i n e f f e c t casts the u n c e r t a i n t y onto i n t h a t a s p e c i f i c v a l u e ( r a t h e r t h a n a range) can be g i v e n f o r t h e d e t e c t i o n l i m i t , b u t w i t h t h e p r o v i s o t h a t £b> ) a

BvsS Al Concentration steel-B steel-A

(mg/g) 10 13

Si 12 8

IV Ca 45 70

Cr 3.2 3.3

c

A vs 6 Fe

Mn 22 16

160 120

1. 1

1.0

5.8

0.32

1.9

14

window [ I ±]

4. ,00

3.63

21.1

1.16

6.91

50.9

distance

3.,0

25.

0.10

a

fi

[A]

-4.0

-6.0

-40.

0..74

0.40

0.32

0.98

0.63

0.71

1,.93

-2.83

3.05

0.22

-2.23

-2.02

(a) Based on d a t a from Ref. 72. (b) V a l u e s o f I and fi are g i v e n f o r n=5. (c) F i g . 13 d e p i c t s the windows [ I ] and v a r i a b l e s e p a r a t i o n s


[A].


1. CURRIE

Concentrations f o r s i x elements c h a r a c t e r i z i n g two s t e e l a e r o s o l samples (72.) a r e g i v e n i n t h e f i r s t two rows. S t e e l - B i s t a k e n as t h e c o n t r o l , and s t e e l - A as t h e a l t e r n a t i v e s o u r c e . H i s r e p r e s e n t e d by t h e v e c t o r o r p a t t e r n d i f f e r e n c e , (x - x ) ; H , by (x - x ) . The l a s t f i v e rows o f t h e t a b l e i n d i c a t e , r e s p e c t i v e l y : the s t a n d a r d d e v i a t i o n s [a] f o r t h e elements i n q u e s t i o n , t h e m a t c h i n g i n t e r v a l s [ I ] , t h e c o n c e n t r a t i o n d i f f e r e n c e s [A] under H , the p r o b a b i l i t y o f f a l s e matches [fi], and t h e r a t i o s o f c o n c e n t r a t i o n d i f f e r e n c e s [A] t o t h e p a i r e d measurement s t a n d a r d d e v i a t i o n s [o ] . 1-0 and A / a b o t h s e r v e as measures o f i n d i v i d u a l element d i s c r i m i n a t i n g power. The q u a n t i t y I i s computed by r e q u i r i n g 1-a' t o be 0.95; f o r n=5, t h i s means a - 0.0102 o r z ( 2 - s i d e d ) = 2.57. (For 6-member p a t t e r n s , z i n c r e a s e s t o 2.63.) Then I = ± z c r , where a = ajl. P a t t e r n d i f f e r e n c e s [A] , i n d i c a t e d by t h e open c i r c l e s , a r e shown i n comparison w i t h m a t c h i n g i n t e r v a l s i n F i g . 13. For t h i s example, p a t t e r n i d e n t i f i a b i l i t y ( H " p r o v a b i l i t y " ) has been approached i n two ways. F i r s t , p' has been c a l c u l a t e d as the p r o d u c t o f t h e i n d i v i d u a l 's, r e f l e c t i n g the series of i n d i v i d u a l element matching d e c i s i o n s . ( F o r n = 5, o m i t t i n g Ca, t h i s p r o d u c t e q u a l s 0.13.) Second, t h e v e c t o r d i f f e r e n c e r e p r e s e n t e d by H i s examined t h r o u g h t h e use o f t h e n o n - c e n t r a l x s t a t i s t i c , where S ( A / a ) i s t h e n o n - c e n t r a l i t y parameter (7J3) . * t h i s second case t h e t e s t o f t h e v e c t o r match ( i . e . , H t e s t ) i s c a r r i e d o u t by comparing t h e sum o f squares o f t h e n observed normalized differences with the c r i t i c a l l e v e l f o r the c e n t r a l x f o r n - degrees o f freedom. The rms v a l u e from t h e sum o f squares -- ( A / a ) -- r e p r e s e n t s t h e m u l t i v a r i a b l e g e n e r a l i z a t i o n o f t h e u n i v a r i a t e normalized differences. I t i s a convenient s i n g l e parameter measure ( i n d e x ) f o r t h e v e c t o r d i s c r i m i n a t i o n power (1-/?'), as f$' i s u n i q u e l y d e t e r m i n e d by t h i s q u a n t i t y , g i v e n a' and the number o f degrees o f freedom. T a b l e IV g i v e s r e s u l t s f o r t h e two t y p e s o f t e s t and s e v e r a l c h o i c e s o f element p a t t e r n s . Important d u a l p a t t e r n i d e n t i f i c a t i o n conclusions follow: (a) D i s c r i m i n a t i o n power ( i d e n t i f i a b i l i t y ) d i f f e r s a c c o r d i n g t o t h e type o f t e s t , x being significantly b e t t e r and becoming more so w i t h i n c r e a s e d d i m e n s i o n a l i t y . (b) O p t i m a l f e a t u r e s e l e c t i o n (e.g., f o r n=5) g i v e s o p t i m a l d i s c r i m i n a t i n g power f o r t h e number o f v a r i a b l e s s e l e c t e d . ( c ) There e x i s t s an o p t i m a l number o f dimensions ( v a r i a b l e s ) . The most p o w e r f u l v a r i a b l e (here, Ca) i s used f o r n—1; a second d i s c r i m i n a t i n g v a r i a b l e y i e l d s i n c r e a s e d power w i t h n=2; b u t e v e n t u a l l y a d d i t i o n o f poorly d i s c r i m i n a t i n g v a r i a b l e s " d i l u t e s " the d i s c r i m i n a t i o n power -- e.g., n=6 compared t o t h e b e s t s e t o f 5. (d) I n c r e a s e d d i m e n s i o n a l i t y g i v e s enormous l e v e r a g e t o modest improvements i n p r e c i s i o n , through t h e p r o d u c t ILp . (See bottom l i n e , T a b l e IV.) These f o u r c o n c l u s i o n s d i r e c t l y i n d i c a t e t h e way toward improved d i s c r i m i n a t i o n power, t h e l a s t b e i n g t h e most i n f l u e n t i a l . (X i n t h e t a b l e denotes t h e n o n - c e n t r a l x •) 3.3.6 generalization. The f o r e g o i n g considerations of h y p o t h e s i s t e s t i n g and p a t t e r n i d e n t i f i c a t i o n l i m i t s were necess a r i l y s i m p l i f i e d , an extended d i s c u s s i o n b e i n g beyond t h e scope o f 0

B

A

B

A

B

A

0

Q

c

c

c

Q


Q

Q

2

A

2

n

Q

Q

2

Q

r i n s

2

i

2

2

American Chemical Society Library 115516th st, N.w.Currie, L.; In Detection in Analytical Chemistry; ACS Symposium Series; Washington, American Chemical Society: Washington, DC, 1987. D.C. 20036


48


+6 O)

E

+4

+2

5

0

S

-2

Al

Si

Cr

Mn

1

Fe (1/10)

I I I

Ca (1/5)

Element

1

F i g . 13. M u l t i v a r i a b l e I d e n t i f i c a t i o n . H windows [ I ] and H concentration differences [open c i r c l e s ] f o r the m u l t i - v a r i a b l e (element) p a t t e r n s c h a r a c t e r i s t i c o f p a r t i c l e e m i s s i o n s from two s t e e l p l a n t s . [See T a b l e I I I . ] Q


A

1. CURRIE

Overview of Historical, Societal, and Technical Issues

T a b l e IV.

P a t t e r n D i s c r i m i n a t i o n Power [1 - P']

(Steel-A p a r t i c l e s vs Steel-B p a r t i c l e s

Sequential m a t c h i n g ^ X - test< ) 2

b

*8% i n c r . p r e c i s i o n ( o 's) i

[ c o n t r o l ] ; a' - 0.05)

n-1 (Ca)

(-Ca)

n=5

Detection: Overview of Historical, Societal, and Technical Issues - ACS

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