Systematic Error in Chemical Analysis - ACS Symposium Series (ACS

3 Systematic Error in Chemical Analysis L . A . C U R R I E and J. R. D E V O E

Downloaded by UNIV OF AUCKLAND on May 3, 2015 | http://pubs.acs.org Publication Date: June 1, 1977 | doi: 10.1021/bk-1977-0063.ch003

Analytical Chemistry Division, Institute for Materials Research, National Bureau of Standards, Washington, DC 20234

The fundamental limitation to accuracy in chemical analysis is systematic error. Unfortunately, systematic error--which comprises all nonrandom deviations of analytical results from the truth--is the rule in analytical chemistry. Systematic error comes about whenever the actual nature of the analytical process differs from that assumed. It results from invalid sampling, operator or equipment instability and blunders, unrecognized sample loss or contamination, poor instrument calibration, inadequate physical (mathematical) or random error distribution models, and faulty reporting of data. These problems, which will be covered in some detail below, are not exceptional. It is only through exhaustive, quantitative evaluation of the individual and collective effects of such violations in assumption that the analyst can hope to provide meaningful bounds for systematic error. The impact of erroneous analytical measurements can be considerable. A recent New York Times article (1) entitled, "Medical Labs May Not Be All That Accurate" pointed up the fact that in a survey of the clinical laboratories involved in Interstate Commerce (and consequently under the monitoring of the Federal Center for Disease Control, USPHS) 31 percent were unable to identify sickle-cell anemia from blood smears. Additional tests such as hemoglobin and electrolyte content in blood were unsatisfactory in a similar fraction of laboratories. Naturally, this situation has resulted in some lack of confidence on the part of the physician; and confidence-erosion can be dangerous. Instances have occurred where a test result deviated from the norm to such an extent that the physician who ignored the result (assuming laboratory error, when there was none) made an improper diagnosis with serious consequences to the patient. Another example of somewhat less immediate severity but greater long term importance is the measurement of ozone in the atmosphere. Figure 1 shows the deviations from the true concen-*Contribution of the National Bureau of Standards. Not subject to copyright. 114

In Validation of the Measurement Process; DeVoe, J.; ACS Symposium Series; American Chemical Society: Washington, DC, 1977.

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tration (2) o f experimental results from a number o f laboratories. C o l l e c t i v e l y , one sees t h a t the l a b o r a t o r i e s produced r e s u l t s whose (negative) bias exceeds the i m p r e c i s i o n bound. As a r e s u l t o f both systematic and random e r r o r components reported 0 concentrations were too low by 20 percent to 60 percent ( a t the A i r Q u a l i t y Standard l e v e l ) . In t h i s case the "true" concentration was provided t o the t e s t i n g l a b o r a t o r i e s i n the form o f an accurately-prepared gaseous reference sample. This example r a i s e s an important p o i n t r e l a ted t o the r o l e o f reference m a t e r i a l s f o r t r a n s f e r r i n g accura cy from one l a b o r a t o r y t o another. Though reference m a t e r i a l s are exceedingly useful f o r d i s c l o s i n g l a b o r a t o r y e r r o r , they do not e l i m i n a t e the need f o r the q u a n t i t a t i v e assessment o f a l l p o t e n t i a l sources o f b i a s .


3

The overwhelming importance o f the systematic component o f e r r o r may be grasped from eq. ( 1 ) : total error: random e r r o r :

e= δ +Δ

(la)

δ = z^SE = ζ(σ/\[η)

(lb)

Where e represents the t o t a l e r r o r i n χ and δ and Δ represent the random and systematic components, r e s p e c t i v e l y . * * I f normality i s assumed (Gaussian random e r r o r d i s t r i b u t i o n ) , the random e r r o r i s simply the product o f the random normal deviate ( z ) and the standard e r r o r (SE). The standard e r r o r i n χ depends upon the p r e c i s i o n parameter σ (standard d e v i a t i o n ) and the number of r e p l i c a t i o n s n. With increased r e p l i c a t i o n the standard e r r o r tends toward zero, w i t h the r e s u l t t h a t the t o t a l e r r o r a s y m p t o t i c a l l y approaches the b i a s - - i . e . , Θ-»Δ. The u l t i m a t e c a p a b i l i t y o f any a n a l y t i c a l procedure thus r e s t s upon the magnitude o f the b i a s . The problem i s compounded by the f a c t t h a t only the p r e c i s i o n may be d i r e c t l y estimated through experiment ( r e p l i c a t i o n ) . The two examples c i t e d above simply i l l u s t r a t e the consequences o f i g n o r i n g o r g i v i n g inadequate a t t e n t i o n t o t h i s extremely important, but more d i f f i c u l t t o estimate,systematic component of e r r o r . When adequate care i s given to e s t i m a t i n g bounds f o r Δ , the r e s u l t s may appear s u r p r i s i n g . For example, i n the most recent t a b u l a t i o n o f Eu-152 γ-ray decay p r o b a b i l i t i e s , the estimated l i m i t s f o r systematic e r r o r exceed the standard e r r o r by f a c t o r s of 2.5 t o 40 03). As s t a t e d bounds f o r systematic e r r o r a r e h i g h l y dependent upon the s c i e n t i f i c judgment and philosophy o f the experimenter, under- and over-estimation o f such bounds can completely cloud the meaning of a n a l y t i c a l r e s u l t s . An i n c i s i v e d i s c u s s i o n of t h i s p a r t i c u l a r problem, as r e l a t e d to the funda mental p h y s i c a l constants, has been given by T a y l o r et a]_. ( 4 ) . **A

l i s t of terms and symbols i s given a t the end o f the


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OF T H E

M E A S U R E M E N T PROCESS

In the d i s c u s s i o n which f o l l o w s we s h a l l f i r s t examine the means and l i m i t a t i o n s of nonrandom e r r o r d e t e c t i o n . A systematic a n a l y s i s of the i n d i v i d u a l steps of the Chemical Measurement Process (CMP) w i l l then be undertaken i n order t o expose the sources and methods f o r c o n t r o l l i n g t h i s component of e r r o r . F i n a l l y , some simple, y e t powerful d i a g n o s t i c techniques w i l l be presented f o r the i d e n t i f i c a t i o n of b i a s and blunders a f f e c t i n g experimental r e s u l t s .


SYSTEMATIC ERROR BOUNDS L i m i t s f o r systematic e r r o r may be a r r i v e d a t i n two d i f f e r e n t ways. (1) They may be e s t i m a t e d ^ ( i n the s t a t i s t i c a l sense) by comparing an experimental r e s u l t χ w i t h the t r u e value τ ( i f known) or w i t h values obtained by independent, r e l t a b l e methods or l a b o r a t o r i e s . * * * As t r u e values i n the s t r i c t e s t sense e x i s t only by d e f i n i t i o n , the f i r s t type of comparison g e n e r a l l y i m p l i e s the a v a i l a b i l i t y of "accepted" or " c e r t i f i e d " v a l u e s , such as those which accompany reference m a t e r i a l s d i s t r i b u t e d by n a t i o n a l s t a n d a r d i z i n g l a b o r a t o r i e s . (2) The second approach to systematic e r r o r e v a l u a t i o n i s through d e t a i l e d a n a l y s i s of the s t r u c t u r e of the CMP i n order t o i n f e r bounds f o r o v e r a l l propagated systematic e r r o r . This approach, i n c o n t r a s t to the former, r e l i e s wholly upon sound, s c i e n t i f i c judgment. The f i r s t , e m p i r i c a l approach thus y i e l d s e = χ - τ

(2a)

f o r the estimated b i a s , and £

±

= e ± δ

(2b)

Μ

f o r i t s upper and lower l i m i t s . In eq. (2b), δ represents the absolute value of the maximum l i k e l y random error—commonly taken to be two t o three times the standard e r r o r . The second, " t h e o r e t i c a l " approach y i e l d s i n f e r r e d bounds M

Δ

±

= Ρ(Δ ).

(3)

±

where ( Δ ) . represents the c o n t r i b u t i o n of s t e p - i of the CMP, and Ρ symbolizes the a p p r o p r i a t e propagation o p e r a t i o n which i n the s i m p l e s t case i s merely summation--e.g., Δ = Σ (Δ ).. +

+

+

I t i s e s s e n t i a l t h a t both types of a n a l y s i s take place. V e r i f i c a t i o n of measurement accuracy can only come through intercomparison. ( $ method-! must cover zero f o r an unbiased + 9

***Intercomparisons l a c k i n g e i t h e r independence or r e l i a b i l i t y are f r u i t l e s s . A dramatic i l l u s t r a t i o n has been given by Yolken ( 5 ) , who c o n t r a s t s r e s u l t s obtained by expert l a b o r a t o r i e s w i t h those obtained using " c e r t i f i c a t i o n by concensus" o f nonexperts.


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measurement process.) However, meaningful u n c e r t a i n t y bounds f o r any giveji experiment u l t i m a t e l y depend upon c a r e f u l e v a l u a t i o n of Δ (method-2). Furthermore, having both types o f estimate makes" p o s s i b l e an extremely valuable check f o r consistency: the overlap of ^ and Xj.. (Further d i s c u s s i o n o f bounds f o r systematic and t o t a l e r r o r w i l l be given i n the s e c t i o n on r e p o r t i n g of data.) ^ Although s c i e n t i f i c e v a l u a t i o n o f systematic e r r o r bounds (Δ ) i s q u i t e d i f f i c u l t , adequate e s t i m a t i o n v i a intercomparison C^v) i s perhaps even more d i f f i c u l t . This i s because o f random e r r o r . I t i s evident from eq. (1) t h a t any observed d i f f e r e n c e or e r r o r (e) w i l l have a random component (δ) which l i m i t s our a b i l i t y t o estimate Δ. J u s t t o ( r e l i a b l y ) detect systematic e r r o r , i t can be shown f o r n o r m a l l y - d i s t r i b u t e d random e r r o r s , t h a t Δ must exceed SE by a f a c t o r of 4 or more (28). In order t o detect a systematic e r r o r which i s comparable t o the standard d e v i a t i o n (σ), one t h e r e f o r e needs a t l e a s t 15 observations. Figure 2 i n d i c a t e s i n a d i f f e r e n t way the d i f f i c u l t y i n d e t e c t i n g sources o f e r r o r . The s o l i d curve shows the d e t e c t i o n l i m i t f o r bias (Δ) r e l a t i v e t o the standard d e v i a t i o n (σ) as a f u n c t i o n o f the number o f observations. The dotted curve gives the same type o f information f o r another common problem: extraneous random e r r o r (σ^) a d d i t i o n a l to the Poisson component i n counting experiments C6). In t h i s case, i f the a d d i t i o n a l random e r r o r i s twice the Poisson component one must have t e n observations t o demonstrate i t s existence. I f the two are comparable, 47 observations s u f f i c e ; and i f the a d d i t i o n a l e r r o r i s h a l f the Poisson e r r o r , several hundred observations are required. I n c i d e n t a l l y , the same (dotted) curve a p p l i e s t o the d e t e c t i o n o f the i n t e r ! a b o r a t o r y e r r o r (corresponding to σ ) f o r a group o f l a b o r a t o r i e s having comparable intra!aboratory i m p r e c i s i o n (corresponding t o σ). C l e a r l y , i n the absence o f a very large number o f measure ments and long term s t a b i l i t y , one cannot e m p i r i c a l l y (through intercomparison) e s t a b l i s h e r r o r bounds (Δ o r σ ) much smaller than the standard d e v i a t i o n o f a s i n g l e measurement (σ). There i s no s u b s t i t u t e , however, f o r intercomparison and r e p l i c a t i o n f o r the d e t e c t i o n o f u n a n t i c i p a t e d blunders o r bias o r l a c k o f c o n t r o l which i s r e l a t i v e l y Targe compared t o the standard deviation. +


+

SOURCES OF SYSTEMATIC ERROR The most e f f e c t i v e way t o i d e n t i f y and c o n t r o l sources of bias i n chemical a n a l y s i s i s t o t r e a t the CMP as a c a r e f u l l y defined system. C r i t i c a l a n a l y s i s o f the i n d i v i d u a l steps and t h e i r linkage w i l l then make i t p o s s i b l e t o estimate i n d i v i d u a l bias components as w e l l as on o v e r a l l propagated e r r o r s f o r systematic e r r o r . For t h i s purpose a g e n e r a l i z e d flow diagram i s given i n f i g u r e 3. In the f o l l o w i n g paragraphs we s h a l l examine each of the steps f o r p o s s i b l e systematic e r r o r c o n t r i b u t i o n s . A


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Figure 1.

PROCESS

Results of a collaborative test of the EPA reference method for ambient ozone (2). Dashed line indicates the true value.

Figure 2. Detection limits vs number of observations for extraneous random error (a , dashed curve) and systematic error (Δ, solid curve) e


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measurement process cannot be s a i d t o e x i s t i n the absence o f c o n t r o l ( 7 ) . When c o n t r o l i s achieved both the c e n t r a l value and the v a r i a b i l i t y are s t a b l e . Under such circumstances the e r r o r can be completely defined by a f i x e d systematic component (the b i a s ) and a random component having constant standard d e v i a t i o n . Such b i a s may be the r e s u l t o f f i x e d mistakes o r blunders i n experiment o r theory, o r i t may a r i s e from converted random e r r o r s — i . e . , e r r o r s which occur i n a random f a s h i o n but which remain f i x e d because one o r more steps o f the CMP a r e not repeated. I f the systematic e r r o r i s not constant, i t becomes impossible t o generate meaningful u n c e r t a i n t y bounds f o r experimental data. Lack o f c o n t r o l may a r i s e through carelessness o r e r r a t i c blunders, such as t r a n s c r i p t i o n e r r o r s . These may be exposed v i a r e p l i c a t i o n . A l t e r n a t e l y , nonrandom v a r i a t i o n s may come about from the e f f e c t s o f systematic trends i n u n c o n t r o l l e d v a r i a b l e s (such as barometric p r e s s u r e ) , o r from u n a n t i c i p a t e d e f f e c t s o f seemingly remote f a c t o r s . (Such e f f e c t s are not n e c e s s a r i l y a nuisance. They may provide an opportunity f o r discovery--as i n the case o f v a r i a t i o n s i n the c a l i b r a t i o n curve f o r radiocarbon d a t i n g induced by the a c t i v i t i e s o f man and v a r i o u s geophysical and c l i m a t i c phenomena (8)·) The assessment o f whether a measurement process i s i n c o n t r o l i s f r e q u e n t l y accomplished through the use o f c o n t r o l c h a r t s - - a technique which has been thoroughly discussed above. The c o n t r o l c h a r t , o f course, merely s i g n a l s i n s t a b i l i t y ; i t does not g e n e r a l l y compensate f o r i t . In order t o achieve c o n t r o l , the experimenter must i d e n t i f y and e i t h e r s t a b i l i z e o r c o r r e c t f o r sources o f e r r a t i c behavior. When v a r i a b l e s cannot be held constant, i t i s o f t e n e f f e c t i v e t o c o r r e c t f o r changes by means o f an i n t e r n a l o r e x t e r n a l standard. Figure 4 gives one such example (39). Because o f the extremely low concentrations present i t was necessary t o measure a sample o f r a d i o a c t i v e A r over a p e r i o d o f a month, during which time there was about a 10 percent d r i f t i n gain. Though i t was not p o s s i b l e t o prevent the d r i f t , which came from s l o w l y changing p r o p o r t i o n a l counter gas composition, i t was p o s s i b l e t o c o r r e c t f o r i t . This was accomplished w i t h an e x t e r n a l m o n i t o r — a n x-ray source which simulated the response o f the d e t e c t o r to the sample r a d i a t i o n . 3 7

Sample V a l i d i t y Among the more s e r i o u s problems a f f e c t i n g the sample a r e contamination, heterogeneity and i n s t a b i l i t y . Contamination w i l l be discussed below. The most l i k e l y consequence of heterogeneity i s a nonrepresentative sample. Quite o f t e n one can observe a major d i f f e r e n c e i n sample composition w i t h amount taken f o r a n a l y s i s . For example, the appearance o f severe heterogeneity among t r a c e elements i n Orchard Leaves (SRM #1571)


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PROCESS

CRITICAL ASPECTS Existence of CMP (definition, control) Sampling

Homogeneity, contamination, stability

Separation

Recovery, contamination

(sample prep.)

Calibration, resolution


Model, error

structure

Adequate reporting Meaningful error bounds

Figure 3. The chemical measurement proc ess—flow diagram

σ

c c σ

I

29

Am-241 (Cu-ka)

Ar - Sample (SAWM-7)

28 27 26

25 220

225

230

235

240

245

1974 Days Figure 4.

Quenching of Ar; external standard control (39) 37



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was demonstrated when 10 mg r a t h e r than the recommended 250 mg samples were taken ( 9 ) . Such heterogeneity depends, o f course, upon sample type. (The authors o f r e f . (10) noted the "extreme homogeneity" o f t r a c e elements i n fresh l i v e r . ) Also, homogeneity o f some elements cannot assure the same f o r others. For c e r t a i n methods of a n a l y s i s , sample homogeneity requirements are indeed s t r i n g e n t . E l e c t r o n probe m i c r o a n a l y s i s , f o r example, r e q u i r e s standards o f e x p e r i m e n t a l l y demonstrated microhomogeneity (10). Another major source o f systematic e r r o r r e l a t e s t o the change o f sample composition w i t h time. A e r o s o l s , f o r example, are known t o be s u s c e p t i b l e t o moisture and gaseous contaminants. Spurious s u l f a t e r e s u l t s have been obtained from the gradual o x i d a t i o n o f S 0 on a i r f i l t e r s (11) and H S 0 aerosol r e s u l t s have been f a l s e l y low because o f p a r t i a l n e u t r a l i z a t i o n by t r a c e s o f NH i n l a b o r a t o r y a i r (12). Loss o f trace species from aqueous samples a t c o n t a i n e r w a l l s (adsorption or d i f f u s i o n ) i s another common source o f instability. This i s p a r t i c u l a r l y marked f o r heavy, v o l a t i l e elements such as mercury (13). 2

2

4

3

The Blank Figure 5 i l l u s t r a t e s a systematic e r r o r t h a t i s troublesome i n the f i r s t two steps o f the CMP: t h a t i s the occurrence o f an unmeasured blank. A s i g n i f i c a n t d i f f e r e n c e i s shown between a simple s o l u t i o n o f l e a d and the apparent lead content i n whole blood (14). When t h e d e v i a t i o n s a t these low l e v e l s o f c o n c e n t r a t i o n are a l l p o s i t i v e , a good s u p p o s i t i o n i s a blank problem from contamination o f reagents used t o prepare the sample of whole blood but not used f o r the aqueous s o l u t i o n . A common p i t f a l l i n trace analysis i s i n s u f f i c i e n t a t t e n t i o n t o the v a r i a b i l i t y o f the blank. I f v a r i a b i l i t y due t o contamination i s such t h a t i t may p l a y an important p a r t i n the s e t t i n g o f u n c e r t a i n t y bounds o r d e t e c t i o n l i m i t s , some c a u t i o n i s necessary i n i n t e r p r e t i n g the r e s u l t o f j u s t one o r a few blank observations (19). Results such as those quoted above show the danger o f b l i n d l y assuming t h a t the r e l a t i v e range o f the blank i s no more than 10 percent, 100 percent, a f a c t o r o f 2 o r even a f a c t o r o f 1 0 (15). Thus, even i f the blank i s ten times s m a l l e r than the s i g n a l o f i n t e r e s t , i t s v a r i a b i l i t y must be measured. I f t h i s i s accomplished, f o r example, by examining the d i f f e r e n c e o f j u s t two experimental blanks, there i s a s i g n i f i c a n t chance t h a t the actual range o f the blanks w i l l exceed t h a t measured d i f f e r e n c e by a f a c t o r o f 25, under the best of circumstances ( n o r m a l l y - d i s t r i b u t e d blanks; 95 percent tolerance interval). G i v i n g up the assumption of n o r m a l i t y , but r e q u i r i n g the blank to be under ( s t a t i s t i c a l ) c o n t r o l , one can be f a i r l y (95 percent) c e r t a i n t h a t h a l f o f the blanks w i l l f a l l w i t h i n the range o f 8 o b s e r v a t i o n s , o r 90 percent o f them w i t h i n the range of 47 observations! 6

1


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VALIDATION O F T H E

Aqueous Solution 140 ng Pb/g 75

PROCESS

Porcine Blood 30 ng Pb/g

60

456

109 801

+ 40 r Downloaded by UNIV OF AUCKLAND on May 3, 2015 | http://pubs.acs.org Publication Date: June 1, 1977 | doi: 10.1021/bk-1977-0063.ch003

MEASUREMENT

453

90

62T

+ 20h

•20h

-40

L

1

2

3

4

5

6

7

1 LABORATORY

2

3

4

5

6

7

NUMBER National Bureau of Standards Special Publication

Figure 5. Comparison of interlaboratory Pb results for an aqueous standard vs. whole blood (14)


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Sample P r e p a r a t i o n


Besides problems w i t h the blank, great care must be taken when performing p h y s i c a l o r chemical separations o f components i n a sample. I f the recovery o f the d e s i r e d component i s not q u a n t i t a t i v e , - - e . g . , l o s s o f v o l a t i l e components during sample d i s s o l u t i o n — s e r i o u s systematic e r r o r may r e s u l t (16). The recovery f a c t o r presents the same o p p o r t u n i t y to e r r as does the instrument c a l i b r a t i o n f a c t o r — n a m e l y , the assumption t h a t the (average) y i e l d i s q u a n t i t a t i v e or constant and t h a t i t s v a r i a b i l i t y ( r e l a t i v e standard d e v i a t i o n ) i s f i x e d . Such f i x e d values may be deduced by assumption, a t h e o r e t i c a l model ( s o l u b i l i t y product, p a r t i t i o n c o e f f i c i e n t , ...) o r b e t t e r s t i l l , by a few measurements on pure s o l u t i o n s . The t r a p i s l a i d : as soon as v a r y i n g concentrations and complex samples are encountered low and f l u c t u a t i n g y i e l d s w i l l occur. One o f the most r e l i a b l e means f o r e l i m i n a t i n g bias due t o n o n q u a n t i t a t i v e s e p a r a t i o n i s isotope d i l u t i o n . Provided t h a t the d i l u t i n g isotope i s added at the e a r l i e s t p o s s i b l e stage and t h a t complete i s o t o p i c mixing takes p l a c e , t h i s technique i s capable of very high accuracy. The a r t has perhaps reached i t s u l t i m a t e l e v e l a t the hands o f s k i l l e d chemical mass spectrom e t r i s t s , who have succeeded i n measuring isotope r a t i o s w i t h u n c e r t a i n t i e s of only 0.03 percent (17,18). Measurement The measurement step provides many chances f o r e r r o r . Operator b i a s , f o r example, commonly occurs i n the making o r r e c o r d i n g o f o b s e r v a t i o n s , as shown i n f i g u r e s 6 and 7 (19,20). Results of 1,000 weighings ( f i g . 6) show t h a t operators favor the values o f 0 and 5 f o r the l a s t d i g i t , and t h a t even numbers tend to be favored over odd numbers. From 1,510 buret readings ( f i g . 7 ) , on the other hand, one can observe t h a t small numbers are favored over l a r g e numbers. The p o s s i b i l i t y of operator b i a s i s , perhaps, s u f f i c i e n t j u s t i f i c a t i o n f o r c o n s i d e r i n g computer c o n t r o l f o r such types o f processes. (Since even computers are programmed and run by the operator o f the instruments, however, the t h r e a t of e r r o r s (blunders) of t h i s type i s only reduced, not eliminated.) Two o f the most important c h a r a c t e r i s t i c s o f a n a l y t i c a l measurements a r e the c a l i b r a t i o n f u n c t i o n and instrumental r e s o l u t i o n . To assume t h a t the c a l i b r a t i o n f a c t o r i s constant, independent o f the nature ( m a t r i x ) o r c o n c e n t r a t i o n o f the sample, i s t o i n v i t e b i a s . I t i s i n the c a l i b r a t i o n f a c t o r , together w i t h recovery f a c t o r s , t h a t " r e a l " samples d i f f e r most s t r i k i n g l y from pure s o l u t i o n s . Aside from the use o f sound t h e o r e t i c a l o r semi-empirical c o r r e c t i o n formulas, the most r e l i a b l e method t o assure a c o r r e c t c a l i b r a t i o n i s the use of an




ANALYTICAL BALANCE

TERMINAL DIGIT

Figure 6. Operator bias—analytical balance. Histogram depicts observed terminal (estimated) digit distribution for 1000 student weighings. Dashed line indicates expected distribution. (Data from Ref. 20).

BURET

300

u Ζ

Ο Ul ÛÈ

200 Η

100

0 1

2 3

4

5 6 7 8 9

TERMINAL DIGIT

Figure 7. Operator bias—buret reading. Histogram depicts terminal digit distribu tion for 1510 student observations. Dashed lines delimit the 95% confidence interval for a uniform distribution. (Data from Ref. 20).


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i n t e r n a l standard. By adding t o the sample a known a l i q u o t o f the substance being measured, one can t r a n s l a t e the d i f f e r e n t i a l response i n t o an e f f e c t i v e c a l i b r a t i o n f a c t o r f o r the actual sample a t hand (21 ). Instrumental r e s o l u t i o n , j u s t l i k e chemical o r p h y s i c a l "résolution"--!.e. , s e p a r a t i o n — i s one o f the most important means of p r e v e n t i n g systematic e r r o r from u n a n t i c i p a t e d components o r i l l - d e f i n e d s p e c t r a l f e a t u r e s . Some o f the p e n a l t i e s from inadequate r e s o l u t i o n w i l l be examined below i n our d i s c u s s i o n o f data e v a l u a t i o n . When d e a l i n g w i t h complex m a t e r i a l s c o n t a i n i n g p o t e n t i a l l y i n t e r f e r i n g s p e c i e s , however, a small investment i n increased chemical r e s o l u t i o n w i l l be w e l l r e p a i d i n decreased b i a s . Data E v a l u a t i o n Although the l a s t two steps o f the CMP do not n e c e s s a r i l y i n v o l v e any work i n the chemical l a b o r a t o r y , they are nevert h e l e s s an i n t e g r a l p a r t o f the o v e r a l l measurement system and thus must be recognized as p o t e n t i a l c o n t r i b u t o r s o f systematic e r r o r . In f a c t , p e r f e c t l y v a l i d sampling, chemistry and i n s t r u mental measurement can be rendered meaningless by f a u l t y e v a l uation o r r e p o r t i n g . This p o t e n t i a l f o r i n a c c u r a t e data e v a l uation has been recognized r e c e n t l y i n an i n t e r n a t i o n a l comparison devoted s t r i c t l y t o the e v a l u a t i o n (and r e p o r t i n g ) phase o f gamma-ray spectroscopy (22). E r r o r s which are due t o the d i f f e r e n c e s between pure s o l u t i o n s and complex samples o f t e n remain l a t e n t u n t i l the e v a l u a t i o n stage. Systematic e r r o r can be minimized provided t h a t such d i f f e r e n c e s — c o n n e c t e d w i t h the blank, matrix e f f e c t s , component i n t e r f e r e n c e — a r e adequately recognized i n the evaluation process. (The p o s s i b i l i t y o f making proper c o r r e c t i o n s may, o f course, depend upon the p r i o r i n t r o d u c t i o n of a recovery t r a c e r o r use o f a high r e s o l u t i o n measuring device.) For s i n g l e component measurements a common source o f e v a l u a t i o n b i a s i s the assumed c a l i b r a t i o n "constant." M a t r i x c o r r e c t i o n s represent one area where the a n a l y s t must c o r r e c t l y a d j u s t t h i s f a c t o r (10,23). The other r e l a t e s t o the f u n c t i o n a l r e l a t i o n s h i p assumed between the q u a n t i t a t i v e response o f an a n a l y t i c a l chemistry measurement system and the composition o f standards. Many times the r e l a t i o n s h i p i s l i n e a r o r a t l e a s t i t appears t o be so. However, one soon l e a r n s t h a t he can d e f i n e a f i t t o a mathematical model i n a v a r i e t y o f ways. I t i s i n t h i s process o f determining whether the model adequately represents the experimental data, t h a t systematic e r r o r s can a r i s e . A common but p o t e n t i a l l y misleading c a l i b r a t i o n procedure i s f i t t i n g a s t r a i g h t l i n e t o the data and the subsequent examination o f a t e s t s t a t i s t i c t o assess the goodness o f f i t . In Validation of the Measurement Process; DeVoe, J.; ACS Symposium Series; American Chemical Society: Washington, DC, 1977.

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An example i s the c a l i b r a t i o n of l i n e a r i t y of the energy s c a l e of a G e ( L i ) γ-ray detector. Figure 8 shows a l i n e a r f i t where the r e s i d u a l r e l a t i v e standard d e v i a t i o n ( a measure o f f i t ) was l e s s than 0.1 percent, and the c o r r e l a t i o n c o e f f i c i e n t ( a measure of l i n e a r i t y ) was 0.9999. However, through more d e t a i l e d examination we found t h a t the f i t was not r e a l l y adequate when compared w i t h t h a t expected on the b a s i s o f Poisson counting s t a t i s t i c s . A very i n f o r m a t i v e way t o evaluate the f i t i s t o observe the p l o t of r e s i d u a l s ( a l g e b r a i c d i f f e r e n c e between the experimental data p o i n t s and the f i t t e d mathematical model vs γ-ray energy). One can see i n the f i g u r e t h a t the X's are not d i s t r i b u t e d randomly about zero. In f a c t , e r r o r s i n both the c a l i b r a t i o n f u n c t i o n and the t a b u l a t e d standard energies were detected; the c o r r e c t e d r e s u l t s are represented by the dots. Although great improvement was obtained, one can see by i n s p e c t i o n t h a t there i s a s l i g h t decrease i n the spread of the r e s i d u a l s a t higher channel numbers. This i n d i c a t e s a p o s s i b l e a d d i t i o n a l problem t h a t might warrant f u r t h e r study. Multicomponent methods o f a n a l y s i s o f t e n s u f f e r b i a s from inadequate r e s o l u t i o n . The problem o f a c c u r a t e l y r e s o l v i n g o b v i o u s l y overlapping peaks, such as those shown i n f i g u r e 9, has r e c e i v e d c o n s i d e r a b l e a t t e n t i o n i n the s p e c t r o s c o p i c and chromatographic l i t e r a t u r e (24). Not so w e l l a p p r e c i a t e d , however, i s the f a c t t h a t s i g n i f i c a n t systematic e r r o r may be introduced when an i n t e r f e r i n g peak i s present but not apparent, and hence excluded from the data r e d u c t i o n model (25). The magnitude o f the r e s u l t i n g b i a s , when an undetected peak l i e s b u r i e d w i t h i n the peak of i n t e r e s t i s shown i n f i g u r e 10. I t i s a s u r p r i s i n g r e s u l t t h a t the l e v e l o f e r r o r can be so l a r g e and s t i l l go undetected. P l o t t e d i n the f i g u r e i s the r a t i o o f the systematic e r r o r t o the standard d e v i a t i o n o f the estimated area of a (Guassian) peak as a f u n c t i o n o f i t s separation from a neighboring (undetected) peak. I t can be seen t h a t i f the overlap i s equal t o o r l e s s than the h a l f w i d t h , very l a r g e systematic e r r o r can r e s u l t (26). Improved instrumental r e s o l u t i o n may e l i m i n a t e the above pitfall. In f a c t , advanced instrumentation may reveal q u i t e a s u r p r i s i n g degree o f complexity; f i g u r e 11 shows the s t r u c t u r e a c t u a l l y contained i n the apparent γ-ray doublet o f f i g u r e 9. Reporting Results and U n c e r t a i n t i e s Among the r e s u l t s reported i n a recent t r a c e a n a l y s i s l a b o r a t o r y intercomparison o f an NBS Standard Reference M a t e r i a l (SRM 1577, bovine l i v e r ) , one f i n d s the f o l l o w i n g :


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2 1000 h /

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+1.0

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Figure 8. a. Gamma-ray calibra tion curve: energy (keV) vs. chan nel number; b. residuals (AkeV) from gamma-ray calibration curve vs. channel number, (x = linear function; · = cubic function; Ο = bad physical input data [tabulated y-energy ].)

Figure 9. Gamma-ray spectrum from Bremsstrahlung-activated gold: NaI(Tl)detector


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PROCESS

Figure 10. Model error bias. Curve shows the (maximum) bias in the estimated area of the major peak in a spectral doublet when an undetected minor peak is omitted from the mathematical model. The minor peak is not detectable if it lies below the solid curve.

Figure 11.

Gamma-ray spectrum from Bremsstrahlung - activated gold: Ge(Li)detector


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Systematic Error in Chemical Analysis - ACS Symposium Series (ACS

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