Sampling for Chemical Analysis of the Environment: Statistical

2 Sampling for Chemical Analysis of the Environment: Statistical Considerations B. KRATOCHVIL

Downloaded by UNIV LAVAL on July 13, 2016 | http://pubs.acs.org Publication Date: July 15, 1985 | doi: 10.1021/bk-1985-0284.ch002

Department of Chemistry, University of Alberta, Edmonton, Alberta, Canada T6G 2G2

A statistically valid sampling plan requires careful design and execution so that generalizations based on mathematical probability can be drawn from a small number of test portions. Guidelines are given for estimation of the minimum number and size of sample increments needed to achieve a given level of confidence i n chemical analyses. Accurate sampling f o r p e s t i c i d e s and p e s t i c i d e residues i n the environment presents formidable problems. The p o p u l a t i o n o f i n t e r e s t i s l i k e l y t o be c o m p l e x . I t may c o n s i s t o f s u c h d i v e r s e m a t r i c e s as a i r , water, v e g e t a t i o n , s o i l , sediment, f i s h , or w i l d l i f e . Furthermore, concentrations of the soughtf o r s u b s t a n c e may b e l o w a n d u n e v e n l y d i s t r i b u t e d . The n e c e s s i t y f o r a sound s a m p l i n g program i n any study o f p e s t i c i d e d i s t r i b u t i o n i n the environment i s g e n e r a l l y recognized. Y e t programs a r e o f t e n s o d e s i g n e d as t o be s t a t i s t i c a l l y unsound, or a v a l i d , w e l l designed p l a n i s compromised by expediency o r c a r e l e s s n e s s . The e f f o r t expended on e v a l u a t i o n o f s a m p l i n g d e s i g n s f o r p e s t i c i d e m o n i t o r i n g i s u s u a l l y e x c e e d i n g l y s m a l l compared w i t h t h a t expended on the a n a l y t i c a l measurements. I n o n l y a few c a s e s have g e n e r a l c o n s i d e r a t i o n s f o r s t a t i s t i c a l l y sound e n v i r o n m e n t a l sampling p l a n s b e e n d i s c u s s e d (J_#_2) • An e x a m p l e o f a t h o r o u g h s a m p l i n g s t u d y i s t h e i n v e s t i g a t i o n of fungicide persistence i n s o i l by a randomized s a m p l i n g p l a n (3)• O t h e r a u t h o r s h a v e p r e s e n t e d general c r i t e r i a f o r sampling m a t r i c e s such as s o i l s ( 4 ) , p l a n t s and s o i l s {5), a n d a i r ( 6 ) ; a m o r e g e n e r a l r e v i e w o n s a m p l i n g f o r c h e m i c a l a n a l y s i s i s a v a i l a b l e (J7). A u s e f u l d i s c u s s i o n c o n t a i n i n g much p r a c t i c a l i n f o r m a t i o n h a s b e e n p r o v i d e d b y t h e m o n i t o r i n g panel o f the F e d e r a l Working Group on Pest Management i n t h e U . S . A . ( 8 ) . T h i s g r o u p o b s e r v e d t h a t most

0097-6156/85/0284-0005$06.00/0 © 1985 American Chemical Society

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


6

T R A C E RESIDUE ANALYSIS

recorded data on d e l e t e r i o u s substances i n the environment have not come from programs designed according to s t a t i s t i c a l p r i n c i p l e s , and so the r e l i a b i l i t y of e x t r a p o l a t i o n s from the r e s u l t s cannot be assessed. The r e l i a b i l i t y of any environmental a n a l y t i c a l data depends upon the r e l i a b i l i t y of sample q u a l i t y . To g e n e r a l i z e from a n a l y t i c a l r e s u l t s on a small p o r t i o n of m a t e r i a l to a l a r g e r population r e q u i r e s c a r e f u l planning and execution i f b i a s i s to be avoided. This a r t i c l e considers the general problems involved i n sampling heterogeneous bulk populations such as s o i l , a i r , and n a t u r a l waters; s p e c i f i c d e t a i l s f o r p a r t i c u l a r types of m a t e r i a l s are not i n c l u d e d . These problems i n c l u d e the heterogeneity of most environmental m a t e r i a l s ; the costs i n time, manpower, and e f f o r t required f o r c o l l e c t i o n of r e a l samples; and the need to avoid contamination or decomposition of samples a f t e r c o l l e c t i o n . A s e t of d e f i n i t i o n s of terms f r e q u e n t l y used i n sampling i s provided because usage sometimes d i f f e r s among s t a t i s t i c i a n s , chemists, and o t h e r s . The d e f i n i t i o n s have been s e l e c t e d a f t e r c o n s i d e r a t i o n of the recommendations of various standards o r g a n i z a t i o n s • Background Sources of e r r o r i n an a n a l y s i s may be c l a s s i f i e d as random or systematic. Systematic e r r o r s g e n e r a l l y bias a r e s u l t i n one d i r e c t i o n i n a r e l a t i v e l y reproducible way and are not u s u a l l y amenable to s t a t i s t i c a l treatment. Random e r r o r s vary i n a nonreproducible way around the true value and can be t r e a t e d s t a t i s t i c a l l y by the laws of p r o b a b i l i t y . Therefore i n t h i s d i s c u s s i o n we s h a l l deal only with random e r r o r s , keeping i n mind that most e r r o r s are p a r t l y random and p a r t l y systematic and that systematic e r r o r s i n the a n a l y t i c a l operations can be c o n t r o l l e d by proper use of blanks, standards, and reference samples. Because poor samples are not i d e n t i f i a b l e by such checks, sampling u n c e r t a i n t y i s often t r e a t e d s e p a r a t e l y . For random e r r o r s the o v e r a l l variance s i s the sum of the 2

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.

sampling variance

2

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.

.

s_ and the variance of the remaining —s 2 2 2 2 2 a n a l y t i c a l operations : -f-a * s may be obtained by s u b t r a c t i o n of s (known i f a measurement —a o process i s i n s t a t i s t i c a l c o n t r o l ) from (obtained by a n a l y s i s of the samples). A l t e r n a t e l y , a s e r i e s of r e p l i c a t e measurements or samples can be designed to evaluate both standard d e v i a t i o n s . Reduction i n the o v e r a l l u n c e r t a i n t y r e q u i r e s , t h e r e f o r e , a t t e n t i o n to both sampling and a n a l y t i c a l operations. Once the a n a l y t i c a l standard d e v i a t i o n s i s one t h i r d or l e s s of the sampling standard d e v i a t i o n s^, f u r t h e r reduction i n has l i t t l e e f f e c t on (9). An example of the importance of sampling i s i n the determination of a f l a t o x i n s (a c l a s s of h i g h l y t o x i c compounds =

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e

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2.

KRATOCHVIL

Sampling for

Chemical Analysis of the Environment

produced by molds) i n peanuts (10 11) • Because the d i s t r i b u t i o n of contaminated kernels i s t y p i c a l l y patchy and uneven, and because the t o l e r a b l e l e v e l of contamination i s so low (about 25 ppb), sampling i s the major source of a n a l y t i c a l u n c e r t a i n t y even with samples of over 20 kg. The o v e r a l l a n a l y t i c a l process can be d i v i d e d i n t o f i v e s t e p s — c o n s t r u c t a model, design a plan, take samples, perform analyses, and evaluate r e s u l t s (12)• The model defines the population to be studied, the substances to be measured ( i n c l u d i n g s p e c i a t i o n ) , the extent of d i s t r i b u t i o n w i t h i n the population, and the l e v e l of p r e c i s i o n r e q u i r e d . The sampling plan s p e c i f i e s the number, s i z e , and l o c a t i o n of the sample increments, the extent of combining of increments (compositing), and the procedures f o r reduction of the bulk or gross sample to a l a b o r a t o r y sample and to t e s t p o r t i o n s (subsampling). The plan should be w r i t t e n as a d e t a i l e d p r o t o c o l before work begins and r e v i s e d as warranted by new i n f o r m a t i o n . I t should include o n - s i t e c r i t e r i a f o r c o l l e c t i o n of a v a l i d sample, such as whether a substance should be considered f o r e i g n to the population and r e j e c t e d . A discarded piece of metal or p l a s t i c i n a f i e l d , f o r example, might be considered f o r e i g n f o r a s o i l a n a l y s i s and t h e r e f o r e l e g i t i m a t e l y r e j e c t e d . I t should a l s o include information on procedures f o r p r o t e c t i o n of the sample from contamination before and a f t e r c o l l e c t i o n , f o r p r e s e r v a t i o n , and f o r l a b e l i n g and recording of a l l appropriate i n f o r m a t i o n . F i e l d sampling operations are o f t e n c o s t l y i n time and manpower. Those c o l l e c t i n g samples should be aware of the p o s s i b i l i t y of b i a s and contamination.


r

Random and

Systematic

Sampling

In d e v i s i n g a model f o r an a n a l y t i c a l operation, we i d e n t i f y a t a r g e t population to which we want our conclusions to apply. This w i l l d i f f e r from the parent population from which the samples are a c t u a l l y taken. The d i f f e r e n c e may be reduced by random s e l e c t i o n of i n d i v i d u a l p o r t i o n s (increments) f o r a n a l y s i s so that each part of the population has an equal chance of s e l e c t i o n . Genuinely random sampling i s d i f f i c u l t because b i a s , unconscious or d e l i b e r a t e , i s r e a d i l y i n t r o d u c e d . Untrained i n d i v i d u a l s o f t e n have d i f f i c u l t y i n accepting that an apparently unsystematic sampling pattern must be followed to be v a l i d . For s i m p l i c i t y and convenience, sampling at evenly spaced i n t e r v a l s over a population i s often used i n place of random sampling. For example, a f i e l d may be d i v i d e d i n t o uniform segments, and a sample taken from the center of each segment. This procedure i s g e n e r a l l y subject to more b i a s than random sampling• Should p e r i o d i c i t y i n the population be present or suspected, segments to be sampled should be s e l e c t e d with the


1


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a i d of a t a b l e of random numbers (13) • The sampling s i t e w i t h i n each segment should then be s e l e c t e d by f u r t h e r d i v i s i o n i n t o imaginary subsegments, each assigned a number, and the one to be sampled s e l e c t e d from a table of random numbers. Sometimes random sampling i s d i f f i c u l t to execute, as when a stream i s being monitored with a t i m e - a c t i v a t e d automatic remote sample c o l l e c t i o n d e v i c e . Under such c o n d i t i o n s a random s t a r t or other superimposed random time element may be s u b s t i t u t e d . The e f f i c i e n c y of systematic sampling improves as the population becomes b e t t e r understood. Both t h e o r e t i c a l and experimental s t u d i e s of t h i s p o i n t have been made (14)• When the component of i n t e r e s t i s d i s t r i b u t e d i n a segregated way, s p e c i a l sampling precautions may be needed. Thus, a p e s t i c i d e may have been d i s t r i b u t e d i n higher c o n c e n t r a t i o n i n one p a r t of the area under study or may have undergone more r a p i d degradation i n a low wet p o r t i o n of a f i e l d . To o b t a i n a v a l i d sample of a s t r a t i f i e d m a t e r i a l , the procedure recommended (15) i s to ( i ) d i v i d e the population i n t o segments ( s t r a t a ) based on the known or suspected p a t t e r n of segregation, ( i i ) f u r t h e r d i v i d e the major s t r a t a i n t o subsections and s e l e c t the r e q u i r e d number of subsections to be sampled by use of a t a b l e of random numbers, and ( i i i ) c o l l e c t samples p r o p o r t i o n a l i n number to the r e l a t i v e s i z e of the major s t r a t a . S t r a t i f i e d random sampling i s p r e f e r a b l e to u n r e s t r i c t e d random sampling, provided the number of major s t r a t a i s kept s u f f i c i e n t l y small that s e v e r a l increments can be taken from each. Composite Samples When only the average p r o p e r t i e s of a population, and not the v a r i a b i l i t y or d i s t r i b u t i o n of the sought-for component, are of i n t e r e s t , a composite sample may be prepared and analyzed. D i s t i n c t i o n should be made between composite and r e p r e s e n t a t i v e samples. A r e p r e s e n t a t i v e sample i s f r e q u e n t l y defined as one that possesses the average p r o p e r t i e s of a population; a composite sample i s u s u a l l y produced by homogenizing i n any of s e v e r a l ways one or more sample increments, and i t c o n s t i t u t e s one approach to producing r e p r e s e n t a t i v e samples. Compositing u s u a l l y means fewer analyses are required, and sample storage, r e c o r d i n g , and handling are s i m p l i f i e d once compositing i s completed. But much u s e f u l information may be l o s t i n preparing a composite sample. A n a l y s i s of i n d i v i d u a l samples c o l l e c t e d by a p r o p e r l y designed and executed sampling plan permits determination of the between-sample and within-sample v a r i a b i l i t y as w e l l as the average composition. This i n f o r m a t i o n helps to e s t a b l i s h the heterogeneity of the p o p u l a t i o n , i d e n t i f y anomalous samples, and evaluate d i f f e r e n c e s w i t h i n and between l a b o r a t o r i e s . Thus composite samples provide l i m i t e d information and should be employed only a f t e r c a r e f u l c o n s i d e r a t i o n of the disadvantages i n v o l v e d .


2.

KRATOCHVIL

Sampling for Chemical Analysis of the Environment


Subsampling I f the sample increment i s l a r g e r than the amount ( t e s t portion) needed per measurement, subsampling i s necessary• This operation may be simple, as with many l i q u i d or gaseous m a t e r i a l s , or complex, as with c e r t a i n bulk s o l i d s . The work r e q u i r e d to produce a uniform subsample depends on the heterogeneity of the o r i g i n a l m a t e r i a l . Subsampling of s o l i d s may require s e v e r a l steps of p a r t i c l e s i z e r e d u c t i o n and mixing; much has been w r i t t e n on t h i s t o p i c . P a r t i c l e s i z e r e d u c t i o n i s important when the p a r t i c l e s d i f f e r a p p r e c i a b l y i n composition because sampling e r r o r may occur even i n a w e l l mixed sample i f too few p a r t i c l e s are taken f o r a n a l y s i s . One approach to determining the extent of the r e d u c t i o n needed i s to t r e a t the sample as a two-component mixture, with each component c o n t a i n i n g a d i f f e r e n t amount of the substance of i n t e r e s t (16,17)• This treatment i s based on a binomial d i s t r i b u t i o n of the two kinds of p a r t i c l e s . Because i t has been covered i n d e t a i l elsewhere, i t w i l l not be considered here. D i s t r i b u t i o n s Found i n Nature For the purpose of sampling f o r chemical a n a l y s i s three types of d i s t r i b u t i o n s can be c o n s i d e r e d . These are the Gaussian ( a l s o known as the normal, Laplace, or DeMoivre), the Poisson, and the negative b i n o m i a l . Knowledge of the type of d i s t r i b u t i o n i s u s e f u l i n d e v i s i n g the most e f f i c i e n t sampling d e s i g n . Gaussian and Poisson d i s t r i b u t i o n s are both c l o s e l y r e l a t e d to the binomial d i s t r i b u t i o n , which a p p l i e s to the p r o b a b i l i t y of whether or not an event w i l l be observed i n a s e r i e s of independent o b s e r v a t i o n s . [The binomial d i s t r i b u t i o n i s based on the p r o b a b i l i t y of an event or property being observed _p, or not observed 1-p_, i n a s e r i e s of _n independent o b s e r v a t i o n s . The d i s t r i b u t i o n of the number of times the event i s observed, x, i n j i t r i a l s i s given by

For f u r t h e r i n f o r m a t i o n see Reference 18.] The event might be the presence of any p a r t i c u l a r a t t r i b u t e i n a sample, such as the d e t e c t i o n of a p e s t i c i d e . Only two l e v e l s of the a t t r i b u t e are p o s s i b l e , present or not present. If many a t t r i b u t e s c o n t r i b u t e to the r e s u l t of an observation, the binomial p r o b a b i l i t y d i s t r i b u t i o n approaches a l i m i t i n g curve whose equation i s given by _y = (1/a v2%) exp[-(x-ji) / 2 a ] . As a p p l i e d to an a n a l y t i c a l measurement of a substance, y_ i s the p r o b a b i l i t y of a measurement value being observed, j i i s the 2


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true value f o r the substance, and o_ i s the standard d e v i a t i o n i n jx. This equation d e s c r i b e s the Gaussian d i s t r i b u t i o n . This d i s t r i b u t i o n i s observed f o r a large f r a c t i o n of the systems encountered i n chemical a n a l y s i s ; a c h a r a c t e r i s t i c i s that j i i s greater than JJ. The Poisson d i s t r i b u t i o n i s c l o s e l y r e l a t e d to the binomial, and i s l i k e w i s e d e r i v e d from c o n s i d e r a t i o n of discrete properties. [The Poisson d i s t r i b u t i o n i s given by pOO = e~ _^ /x! where \ = N£_ when N_ i s large and p_ i s s m a l l . Thus \ i s the expected number of events o c c u r r i n g on any given o b s e r v a t i o n , x = \. The Poisson d i s t r i b u t i o n i s a l i m i t i n g form of the binomial d i s t r i b u t i o n (18).] I t a p p l i e s when the p o s s i b l e number of values N_ i s large but the p r o b a b i l i t y p^ of the a t t r i b u t e of i n t e r e s t being observed i s s m a l l . One example i s the measurement of r a d i o a c t i v e decay, where the p r o b a b i l i t y of any one of a large number of atoms undergoing decay a t a given time may be s m a l l . Another example might be the l o c a t i o n of a weed s e e d l i n g or a l i v e i n s e c t i n a f i e l d a f t e r spraying with a p e s t i c i d e . In the f i e l d there are a large and u n s p e c i f i e d number of p o i n t s where a weed p l a n t or i n s e c t might be found, but the p r o b a b i l i t y of f i n d i n g one a t a given p o i n t w i l l be small i f the a p p l i c a t i o n of p e s t i c i d e has been s u c c e s s f u l . The Poisson d i s t r i b u t i o n i s c h a r a c t e r i z e d by j i , the mean or average, being equal to the variance . Thus the standard d e v i a t i o n s_ f o r a s e t of measurements i n a Poisson d i s t r i b u t i o n i s e a s i l y obtained by taking the square root of the average, s_ = /x. Each observed event must be independent f o r the Poisson d i s t r i b u t i o n to h o l d . A t h i r d type of p r o b a b i l i t y d i s t r i b u t i o n f r e q u e n t l y encountered i n nature i s where the occurence of one event at some l o c a t i o n increases the p r o b a b i l i t y of other events being observed nearby. This leads to clumping or patchiness, c h a r a c t e r i s t i c of many b i o l o g i c a l systems such as weed or i n s e c t i n f e s t a t i o n s , and mold growth i n stored g r a i n s . Although a v a r i e t y of p r o b a b i l i t y d i s t r i b u t i o n s have been considered f o r contagious systems, the most s u c c e s s f u l appears to be the negative b i n o m i a l . Here a d i s t i n g u i s h i n g c h a r a c t e r i s t i c i s that o* i s greater than j i . Major considerations" i n any sampling plan are the s i z e and number as w e l l as the l o c a t i o n of the sampling increments. The f o l l o w i n g s e c t i o n s consider aspects of these p o i n t s . 2


X

2

g

E s t i m a t i o n of Minimum Size of Sample

Increments

For the determination of a chemical or a p e s t i c i d e i n a f i e l d the sampling increment may be a bulk q u a n t i t y such as a core of s o i l , a volume of a i r passed through a p a r t i c u l a t e s c o l l e c t o r , or a q u a n t i t y of vegetation gathered from a s i n g l e s i t e . A u s e f u l method f o r r e l a t i n g the amount of sample i n an increment to the sampling u n c e r t a i n t y , developed by Ingamells (19,20)


2.

KRATOCHVIL


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f o r mining e x p l o r a t i o n , can be a p p l i e d e f f e c t i v e l y to unsegregated Gaussian d i s t r i b u t i o n s . In t h i s approach a sampling constant _Kg, corresponding to the weight of sample required to l i m i t the sampling u n c e r t a i n t y to 1% r e l a t i v e with 68% confidence, i s defined by 2


Kg = WR

(1)

where W_ represents the weight of sample taken and R_ i s the r e l a t i v e standard d e v i a t i o n i n sample composition. For a given population, Kg i s evaluated by performing a s e r i e s of analyses on sets of samples of d i f f e r i n g s i z e e i t h e r by c a l c u l a t i o n or with the a i d of a sampling diagram. An example i s a study of human l i v e r homogenate prepared by cryogenic g r i n d i n g at the N a t i o n a l Bureau of Standards (21)• The e f f e c t i v e n e s s of the homogenization step was assessed by withdrawing a small p o r t i o n of t i s s u e , i r r a d i a t i n g i t , adding i t to the remainder of the sample, performing the homogenization operation, and measuring the sodium-24 a c t i v i t y i n ten samples each of about 0.1, 1, and 5.5 g. For the f i r s t s e t of ten a value of 13.1 was obtained for the percent r e l a t i v e standard d e v i a t i o n R_, f o r the second set a value of 5.5%, and f o r the t h i r d 2.53. From Equation 1 values for Kg are 17, 30, and 35. Thus the value f o r K approaches 35, and t h i s i s the best estimate of the sampling constant. From Equation 1, then, we f i n d that the weight of subsample i n grams r e q u i r e d to hold the sampling u n c e r t a i n t y to 1% r e l a t i v e i s 35 g. Equation 1 can be used to estimate the sampling u n c e r t a i n t y for subsamples of other s i z e s . In the above example, a subsample of 0.5 g would be expected to give a sampling u n c e r t a i n t y of about 8% r e l a t i v e . Note that p r e l i m i n a r y measurements are necessary to e s t a b l i s h the degree of heterogeneity of the i n d i v i d u a l sample increments whenever the p r o p e r t i e s of the p o p u l a t i o n are unknown. Under such c o n d i t i o n s e s t i m a t i o n of K„ should not be —s based on a s i n g l e increment, but on r e s u l t s from s e v e r a l . I t i s always sound p r a c t i c e whenever p o s s i b l e to perform a p r e l i m i n a r y assessment of an unknown population by c o l l e c t i n g a few samples and a n a l y z i n g f o r the component of i n t e r e s t . These samples can be s e l e c t e d on the b a s i s of experience and judgment. Then on the b a s i s of the p r e l i m i n a r y r e s u l t s a r e f i n e d sampling plan can be designed. E s t i m a t i o n of Minimum Number of Sample Increments c

A second f a c t o r to consider i n a v a l i d sampling plan i s the c o l l e c t i o n of enough i n d i v i d u a l sample increments to ensure t h a t heterogeneity on a large scale does not b i a s the r e s u l t s . Estimation of t h i s number can be made s t r a i g h t f o r w a r d l y i f the component of i n t e r e s t i s d i s t r i b u t e d


12

TRACE RESIDUE ANALYSIS throughout the population according to a known s t a t i s t i c a l relation. I f the d i s t r i b u t i o n i s Gaussian or b i n o m i a l , the minimum number of increments can be estimated from 4

n = ^ = f - x () R x_ where t_ i s the Student's _t-table value f o r the l e v e l of confidence d e s i r e d , and are estimated from p r e l i m i n a r y measurements on or previous knowledge of the population, and _R i s the percent r e l a t i v e standard d e v i a t i o n acceptable as sampling u n c e r t a i n t y . I n i t i a l l y , t_ can be set at the value f o r 95% confidence l i m i t s , 1.95, and an i n i t i a l estimate of _n c a l c u l a t e d . The t_ value f o r t h i s n_ can then be s u b s t i t u t e d , and the system i t e r a t e d to constant _n. If the d i s t r i b u t i o n i s Poisson, s = x, and Equation 2 s i m p l i f i e s to


1

0

2

2

* 4 n = — x 10 R £

(3)

For a negative binomial d i s t r i b u t i o n an index of clumping k_ must be incorporated, and Equation 2 becomes 4

n - ^ ^ H l O ) (4) R x ii Both k_and x^are estimated from p r e l i m i n a r y measurements. E s t i m a t i o n of Number and Size of Increments f o r a Population

Segregated

When the p o p u l a t i o n i s segregated, a number of samples should be taken from each stratum or segment. A guide to the number of samples to c o l l e c t under these circumstances has been developed by Visman (22,23)• Through an e m p i r i c a l study, subsequently put on a t h e o r e t i c a l f o o t i n g by Duncan (24,25), Visman derived the r e l a t i o n ±Q

2

= A/w

n_ + JB/n_

(5)

2 where i s the variance of the average of n_ samples of i n d i v i d u a l weight w, and _A and _B are constants f o r a given p o p u l a t i o n . The magnitude of A^ depends on the degree of homogeneity a t the l o c a l l e v e l , and may be c a l c u l a t e d from Ingamell's subsampling constant jC and the average c o n c e n t r a t i o n of sought-for component x^ by A = 10~ — Once

4

x —

2

K —s

and x have been estimated by n p r e l i m i n a r y measurements



2.

KRATOCHVIL


on a given m a t e r i a l , B_ can be estimated by c a l c u l a t i n g for the same p r e l i m i n a r y measurements and s u b s t i t u t i o n i n t o Equation 5. The magnitude of _B depends on the extent of segregation or s t r a t i f i c a t i o n i n the m a t e r i a l . Once A_and J5 are known, and an acceptable l e v e l f o r the standard d e v i a t i o n of sampling decided on, various combinations of w_ and n_ can be chosen to hold s„ w i t h i n the s e l e c t e d value. —6 Two other methods of o b t a i n i n g values f o r A_and B_ have been developed. In the f i r s t , two sets of samples, one of r e l a t i v e l y large and the other of r e l a t i v e l y small increments, are c o l l e c t e d ; the constant i s obtained from the measurements on the small samples, and the constant from the l a r g e samples. Small samples make the f i r s t terms on the r i g h t side of Equation 5 l a r g e r than the second by emphasizing the e f f e c t s of l o c a l heterogeneity and by making the value of smaller. Large samples have the reverse e f f e c t , and when w^ i s of such a s i z e that the second term swamps the f i r s t , a value f o r _B can be c a l c u l a t e d . I f the m a t e r i a l being sampled c o n s i s t s of d i s c r e t e p a r t i c l e s such that an average p a r t i c l e mass can be c a l c u l a t e d , then s t i l l another method i s u s e f u l . In t h i s procedure the constants _A and B_ of Equation 5 are obtained from the i n t r a c l a s s c o r r e l a t i o n c o e f f i c i e n t r_ between p a i r s of s m a l l , single-increment samples of equal mass, the increments of each p a i r being c o l l e c t e d near each other and the p a i r s d i s t r i b u t e d over the population under study. The value of r_ can be estimated from the r e l a t i o n 2Z(x-x)(x'-x) Z(x-x) + E(x'-x) where the sums are over a l l p a i r s x_ and x_' and i s the mean of a l l measurements (26). From t h i s p i l o t study of 10 to 20 p a i r s the constants A_ and B_ are obtained by A_ = s^/( rm + 1/w) and B_ = rAm. Here jn equals 1/(average p a r t i c l e mass), the mass of the i n d i v i d u a l sample increments, and js_ the pooled standard d e v i a t i o n f o r the measurements. An a t t r a c t i v e aspect of t h i s approach i s that i t a l s o allows c a l c u l a t i o n of a minimum d e t e c t a b l e bias (MDB) i n the sampling o p e r a t i o n f o r any s p e c i f i e d confidence l e v e l and number of samples from the relation MDB

= ts/2/n

The value f o r t_ i s obtained from a table of student's t_ values (see, f o r example, Table T-5 i n Ref. 13, or Table A-4 i n Ref. 26) f o r the d e s i r e d confidence l e v e l and number n_ of samples taken. The need to estimate the average p a r t i c l e mass l i m i t s t h i s method to granular m a t e r i a l s . An example of a c a l c u l a t i o n of Tj _A, B, and MDB i s given i n the Appendix.


13

14



Estimation of Sample Size when Form of Population D i s t r i b u t i o n i s Unknown In the preceding s e c t i o n s the Gaussian, Poisson, and clumped d i s t r i b u t i o n s have been discussed, and methods of c a l c u l a t i n g the number of samples i n each case have been g i v e n . When no information i s a v a i l a b l e about a p o p u l a t i o n , however, the q u e s t i o n a r i s e s as to the best approach to use. If sufficient samples can be c o l l e c t e d and analyzed to e s t a b l i s h the d i s t r i b u t i o n as one of the three, the problem i s s o l v e d . I f the d i s t r i b u t i o n does not f i t one of the above, i t should be checked to see whether i t can be converted to Gaussian by taking the logarithm of the v a l u e s . Transformations using f u n c t i o n s other than l o g a r i t h m i c may be considered, but are not e a s i l y r e l a t e d to most r e a l systems. For unknown d i s t r i b u t i o n forms where only l i m i t e d data i s a v a i l a b l e i t i s p o s s i b l e to draw u s e f u l conclusions without knowledge of the d i s t r i b u t i o n . For example, a confidence i n t e r v a l can be e s t a b l i s h e d f o r a set of a n a l y t i c a l values by p l o t t i n g cumulative percent of the number of analyses on the v e r t i c a l a x i s against the i n d i v i d u a l a n a l y t i c a l values on the h o r i z o n t a l a x i s . Then draw l i n e s p a r a l l e l to t h i s p l o t a t a distance of 100 d ,the values f o r &^_ being read from a t a b l e for various numbers of samples and confidence i n t e r v a l s (see, f o r example, Table A-21 i n Ref. 18). Tables are a l s o a v a i l a b l e to determine the number of samples r e q u i r e d to be able to s t a t e that the population cumulative d i s t r i b u t i o n i s w i t h i n a defined band at a s e l e c t e d confidence l e v e l (Ref. 18, Table A-21b)• The numbers tend to be l a r g e . For example, to be 95% sure of c o n t a i n i n g the d i s t r i b u t i o n w i t h i n an i n t e r v a l of ±10% r e l a t i v e 740 samples would be r e q u i r e d . C l e a r l y the p r i c e r e q u i r e d f o r not knowing the form of the population d i s t r i b u t i o n i s more d a t a . 1 - a

a

Conclusions A general theory f o r sampling a heterogeneous system such as the environment f o r trace l e v e l s of substances such as p e s t i c i d e s i s not l i k e l y to become a v a i l a b l e f o r some time. Although a v a r i e t y of models have been proposed to d e s c r i b e s p e c i f i c d i s t r i b u t i o n s , each r e q u i r e s p r i o r knowledge of the system under study. The best approach appears to be to c a r r y out a set of p r e l i m i n a r y sampling and a n a l y s i s operations based on knowledge of s i m i l a r systems from past experience. The extent of the p r e l i m i n a r y work depends on the time and resources a v a i l a b l e ; the more care and e f f o r t expended, the b e t t e r i s the q u a l i t y of the data u l t i m a t e l y c o l l e c t e d . On the b a s i s of t h i s i n i t i a l information a model and sampling plan can


2.

KRATOCHVIL


be developed. I t must be borne i n mind that the p l a n may need to be a l t e r e d as a r e s u l t of data being c o l l e c t e d i n the course of the work. Such a l t e r a t i o n i s v a l i d i f s t a t i s t i c a l p r i n c i p l e s are c a r e f u l l y adhered to throughout. Acknowledgments


The a s s i s t a n c e of Ram Thapa with the c a l c u l a t i o n s and of Annabelle Wiseman with p r e p a r a t i o n of the manuscript i s g r a t e f u l l y acknowledged. This work was supported by the N a t u r a l Sciences and Engineering Research C o u n c i l of Canada and the U n i v e r s i t y of A l b e r t a . Appendix Example of A p p l i c a t i o n of Sampling Theory to P e s t i c i d e A n a l y s i s T a y l o r , Freeman, and Edwards (27) performed a study of the pathways and rate of loss of the p e s t i c i d e d i e l d r i n from a grass-meadow s o i l . The large number and v a r i e t y of samples c o l l e c t e d and analyzed allow e v a l u a t i o n of the u n c e r t a i n t y a s s o c i a t e d with the sampling o p e r a t i o n . B r i e f l y , i n one p a r t of t h e i r i n v e s t i g a t i o n a s e t of three s o i l cores of d i f f e r i n g diameters was c o l l e c t e d i n a diagonal p a t t e r n w i t h i n each square meter of a 6 m x 6 m square p o r t i o n of a f i e l d (Figure 1). Core depth was 17.7 cm; core diameters were 21, 24, and 44 mm. The cores were each e x t r a c t e d with 1:1 hexane:2-propanol. The e x t r a c t was washed with water and the r e s i d u a l hexane i n j e c t e d i n t o a gas chromatograph. The p r e c i s i o n of the e x t r a c t i o n and measurement operations can be estimated to be of the order of a few per cent. R e l a t i v e to the v a r i a b i l i t y observed i n the o v e r a l l r e s u l t s , these u n c e r t a i n t i e s can be considered n e g l i g i b l e . The r e s u l t s , c a l c u l a t e d on an area b a s i s to f a c i l i t a t e comparison, are reproduced i n Table I . The data show a wide range; the values i n Columns B and C are r e l a t i v e l y high, while those i n Column E are r e l a t i v e l y low. The authors suggested that these v a r i a t i o n s may r e f l e c t i r r e g u l a r i t i e s i n the spray a p p l i c a t i o n of the p e s t i c i d e . A second, more l o c a l , v a r i a b i l i t y was a t t r i b u t e d to incomplete mixing of the s o i l a f t e r the spray a p p l i c a t i o n . The r e s u l t i s a large value f o r the o v e r a l l standard d e v i a t i o n , 166 mg per square meter. Given these data, what statements can we make about the number and s i z e of samples that would have to be taken to hold the sampling standard d e v i a t i o n to some p r e s c r i b e d l e v e l ? C a l c u l a t i o n of I n g e i ^ H ^ subsampling constant i s not appropriate since segregation i s present. The minimum number of sample increments r e q u i r e d can be c a l c u l a t e d from e i t h e r Equation 2 or 5. From Equation 2, assuming that a 50% l e v e l of confidence i s d e s i r e d and that an acceptable percent r e l a t i v e standard d e v i a t i o n R i s 50, j i


15

TRACE RESIDUE ANALYSIS

• Downloaded by UNIV LAVAL on July 13, 2016 | http://pubs.acs.org Publication Date: July 15, 1985 | doi: 10.1021/bk-1985-0284.ch002

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s

/

•

/ / / / /

*

s

7 /

/

/

/ /

m

Contour

Spray track B Figure 1. Sampling g r i d (6>

Sampling for Chemical Analysis of the Environment: Statistical

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