3 Determining Uncertainty in Physical Parameter Measurements by Monte Carlo Simulation 1
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David W. Coy , Gregory A. Kew , Michael E. Mullins , and Phillip V. Piserchia Evaluation of Pesticides in Ground Water Downloaded from pubs.acs.org by FUDAN UNIV on 12/18/16. For personal use only.
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Research Triangle Institute, Research Triangle Park, NC 27709 Office of Health and Environmental Assessment (RD-689), U.S. Environmental Protection Agency, Washington, D C 20460
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A statistical approach, often called Monte Carlo simulation, has been used to examine propagation of error and to better characterize the uncertainty associated with measurement of several parameters important in predicting environmental transport of chemicals. These parameters are vapor pressure, water solubility, octanolwater partition coefficient, and "volatilization from water" (based on the ratio of laboratory-measured volatilization rate constant to oxygen reaeration rate constant for a specific system). Column chromatographic and high pressure liquid chromatographic (HPLC) methods are replacing more traditional equilibrium methods (e.g. shake flask, isoteniscope) for measuring the first three parameters. The newer methods tend to underpredict aqueous solubility and vapor pressure and overpredict octanol-water partition coefficient, although deviations for both the equilibrium and dynamic systems are similar. Measurement error proves not to be normally distributed, with differing bias for each parameter. For "volatilization from water", determinations of the ratio of rate constants for compounds whose Henry's Law constant equals or exceeds 1,000 torr/mole/liter typically report 95% percent confidence limits equal to 5 to 10 percent of the ratio. Analysis of a regression approach often used to determine the ratio suggests underestimation of both the ratio and its variance. Monte Carlo simulation did not confirm underestimation of the ratio but suggests variances may be underestimated by a factor of 2.3. Using this statistical approach in other cases might allow an investigator to choose levels of a parameter (e.g. a drinking water standard) knowing the uncertainty associated with the choice, or the converse. 0097-6156/ 86/ 0315-0039506.50/ 0 © 1986 A m e r i c a n C h e m i c a l Society
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Evaluation of Pesticides in Ground Water Downloaded from pubs.acs.org by FUDAN UNIV on 12/18/16. For personal use only.
The purpose of t h i s study i s t o i l l u s t r a t e a s t a t i s t i c a l approach f o r d e t e r m i n i n g o v e r a l l u n c e r t a i n t y i n a parameter c a l c u l a t e d from more than one i n p u t , when d i s t r i b u t i o n of e r r o r f o r the i n p u t s can be e s t i m a t e d . The approach i n c o r p o r a t e s t h i s i n f o r m a t i o n i n t o a c o n c i s e statement of o v e r a l l u n c e r t a i n t y . In p a r t i c u l a r , i t p e r m i t s d e t e r m i n i n g the p r o b a b i l i t y of e x c e e d i n g a specific level. The r e s u l t s then a r e compared w i t h e r r o r e s t i m a t e s f o r s p e c i f i c e x p e r i m e n t a l procedures i n the l i t e r a t u r e . T h i s approach seems a p p l i c a b l e t o b r o a d e r i n v e s t i g a t i o n s than a r e e x p l o r e d h e r e , such as p r e d i c t i n g e n v i r o n m e n t a l t r a n s p o r t and f a t e of c h e m i c a l s where p r o p e r t i e s of v a r i o u s compounds a r e used i n models. Even more g e n e r a l l y , i t might be a p p l i e d t o exposure assessment where, u s u a l l y , too few d a t a a r e a v a i l a b l e to c h a r a c t e r i z e d i s t r i b u t i o n s of exposure t o p a r t i c u l a r c h e m i c a l s . G e n e r a l approaches f o r e x p r e s s i n g u n c e r t a i n t y most e v i d e n t i n the l i t e r a t u r e may be viewed as v a r i o u s forms of " p r o p a g a t i o n of e r r o r . " V a r i a n t s range from " s e n s i t i v i t y a n a l y s i s " where the e f f e c t s of i n d i v i d u a l i n p u t v a r i a b l e s a r e examined by h o l d i n g a l l o t h e r v a r i a b l e s a t midrange w h i l e the one under s t u d y i s v a r i e d from minimum t o maximum, t o the use of f i x e d formulae d e r i v e d f o r v a r i o u s mathematical r e l a t i o n s h i p s i n s p e c i f y i n g a b s o l u t e maxima and minima f o r o u t p u t s . None of these methods can s u p p l y a p r o b a b i l i t y of o c c u r r e n c e f o r a g i v e n v a l u e w i t h i n the range of p o s s i b l e e r r o r . What appears t o be y e t a n o t h e r v a r i a n t , a s t a t i s t i c a l approach o f t e n c a l l e d Monte C a r l o s i m u l a t i o n , r e c e n t l y has been demonstrated t o a p p l y by Walentowicz and F a l c o i f the d i s t r i b u t i o n of v a l u e s f o r each i n p u t v a r i a b l e i s known o r can be approximated (J^). T h e r e a f t e r , t h i s i n f o r m a t i o n can be manipul a t e d to y i e l d a d i s t r i b u t i o n of outputs e x p r e s s i n g p r o b a b i l i t y of o c c u r r e n c e f o r any g i v e n v a l u e . A l s o , a r e c e n t s t u d y by Whitmore G O d e v e l o p s the c o n c e p t s of Walentowicz and F a l c o and o f f e r s c o n c l u s i o n s on the form of f i n a l l i m i t i n g d i s t r i b u t i o n s ( e . g . , s i m u l a t e d o u t p u t s f o r product f u n c t i o n s approach l o g normal) i f s p e c i f i c m a t h e m a t i c a l c o n d i t i o n s a r e met. The p r e c i s e degree t o which the c o n d i t i o n s must be f u l f i l l e d f o r the t e c h n i q u e t o y i e l d a u s e f u l r e s u l t i n the p r e s e n t a p p l i c a t i o n have not y e t been f u l l y e x p l o r e d . While the examples c o n s i d e r e d h e r e a r e e x p e r i m e n t a l d e t e r m i n a t i o n of (1) water s o l u b i l i t y , (2) o c t a n o l - w a t e r p a r t i t i o n c o e f f i c i e n t , (3) v a p o r p r e s s u r e , and (4) v o l a t i l i z a t i o n from water, the r e s u l t i n g d i s t r i b u t i o n s presumably then c o u l d become i n p u t s t o l a r g e r problems such as d e t e r m i n i n g medias p e c i f i c e n v i r o n m e n t a l c o n c e n t r a t i o n , i n t u r n e x p r e s s e d as a well-characterized distribution. S i m u l a t i o n of Method
Error
The p r o c e s s used f o r d e t e r m i n i n g a d i s t r i b u t i o n of aqueous s o l u b i l i t y v a l u e s i n v o l v e s assuming a " t r u e " s o l u b i l i t y v a l u e e q u a l t o 1 . 0 0 a r b i t r a r y u n i t s , a s s i g n i n g an e r r o r d i s t r i b u t i o n f o r each r e c o g n i z e d e r r o r s o u r c e , and c a l c u l a t i n g the e f f e c t on t r u e v a l u e of the s e v e r a l s o u r c e s of e r r o r . After assigning
Evaluation of Pesticides in Ground Water Downloaded from pubs.acs.org by FUDAN UNIV on 12/18/16. For personal use only.
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COY E T A L .
Determining Uncertainty in Physical Parameter Measurements
the " t r u e " v a l u e , random numbers g e n e r a t e d by computer program were used t o p i c k i n p u t v a l u e s w i t h i n t h e s o l u b i l i t y range of concern. The c o r r e s p o n d i n g e r r o r was then c a l c u l a t e d f o r t h e s o u r c e f i r s t a f f e c t i n g t h e s o l u b i l i t y d e t e r m i n a t i o n . The r e s u l t i n g d i s c r e t e e r r o r was added t o o r s u b t r a c t e d from t h e t r u e v a l u e and t h e r e s u l t a n t used as i n p u t t o t h e d i s t r i b u t i o n for the second-occurring e r r o r source. C o n t r i b u t i o n s from s o u r c e s o p e r a t i n g s i m u l t a n e o u s l y , r a t h e r than s e q u e n t i a l l y , were c a l c u l a t e d and summed s i m u l t a n e o u s l y . This process i s r e p e a t e d t o i n c l u d e a l l r e c o g n i z e d s o u r c e s and r e s u l t s i n a s i n g l e s o l u b i l i t y v a l u e c o r r e s p o n d i n g t o t h e s i n g l e random number o r i g i n a l l y chosen. The p r o c e s s o f c h o o s i n g a random number and c a l c u l a t i o n o f c a s c a d i n g e r r o r i s r e p e a t e d 10,000 t i m e s , g e n e r a t i n g a d i s t r i b u t i o n o f s o l u b i l i t y v a l u e s r e l a t i v e t o t h e " t r u e " v a l u e which c a n be c h a r a c t e r i z e d m a t h e m a t i c a l l y , p l o t t e d , and used t o f u r n i s h the l i k e l i h o o d o r p r o b a b i l i t y o f any p a r t i c u l a r s o l u b i l i t y v a l u e i n the range b e i n g exceeded. E s s e n t i a l l y t h e same s t e p s were f o l l o w e d i n s i m u l a t i o n s f o r o t h e r parameters i n v e s t i g a t e d , even f o r t h o s e on v a p o r p r e s s u r e , where e f f e c t s of temperature v a r i a t i o n are compound-specific. The c a l c u l a t i o n procedures used a r e a v a i l a b l e i n s o f t w a r e of the S t a t i s t i c a l A n a l y s i s System (SAS), v e r s i o n 82-3. Central p r o c e s s i n g u n i t time p e r output d i s t r i b u t i o n on a main frame computer (IBM Model 3280) was around 20 seconds and c o s t per o u t p u t d i s t r i b u t i o n was l e s s than $6. D i s t r i b u t i o n s g e n e r a t e d by t h e p r o c e s s d e s c r i b e d above may n o t be easy t o l a b e l as "normal" o r "skewed l o g - n o r m a l " , f o r example, b u t i n g e n e r a l ( f o r P e a r s o n d i s t r i b u t i o n s ) they a r e c o m p l e t e l y c h a r a c t e r i z e d by s p e c i f y i n g t h e f i r s t f o u r s t a t i s t i c a l moments, i . e . mean, v a r i a n c e , skew, and k u r t o s i s ( 3 ) . C o n s e q u e n t l y , r a t h e r than s u p p l y i n g l e s s q u a n t i t a t i v e l a b e l s , t h e s e moments a r e p r o v i d e d f o r a l l output d i s t r i b u t i o n s which follow. Comparisons w i t h e x p e r i m e n t a l r e s u l t s i n t h e t e x t a r e l i m i t e d t o d i s c u s s i o n of t h e mean and t h e square r o o t o f v a r i a n c e or s t a n d a r d d e v i a t i o n , because few e x p e r i m e n t s (and none l o c a t e d i n t h i s s t u d y ) a r e r e p e a t e d o f t e n enough t o g e n e r a t e d a t a s e t s a l l o w i n g m e a n i n g f u l c a l c u l a t i o n o f t h i r d and f o u r t h moments. "Beta f u n c t i o n s " have been used t o r e p r e s e n t s e v e r a l s o u r c e s of e r r o r . The b e t a f u n c t i o n i s e x t r e m e l y f l e x i b l e i n t h e sense t h a t i t can assume a g r e a t v a r i e t y o f shapes depending on v a l u e s chosen f o r exponents, as seen i n F i g u r e 1, and seems p a r t i c u l a r l y useful i n representing negative bias. E x p e r i m e n t a l Methods f o r S o l u b i l i t y
Determination
S o l u b i l i t y d a t a f o r o r g a n i c compounds i n water a r e o b t a i n e d r e a d i l y f o r h i g h l y soluble m a t e r i a l s , but f o r s p a r i n g l y soluble compounds measurements can be d i f f i c u l t due t o p o t e n t i a l l o s s e s to a i r o r t o t h e c o n t a i n e r w a l l s , l o n g e q u i l i b r a t i o n t i m e s , need f o r extreme p u r i t y i n s t a r t i n g m a t e r i a l s , and p o s s i b l y u n a n t i c i p a t e d minor r e a c t i o n s i n s o l u t i o n . As a r e s u l t , e r r o r s of an o r d e r o f magnitude f o r some compounds have been r e p o r t e d by MacKay e t a l . ( 4_).
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E V A L U A T I O N O F PESTICIDES IN G R O U N D WATER
For s l i g h t l y s o l u b l e compounds, two methods dominate the c u r r e n t l i t e r a t u r e , t h e s h a k e - f l a s k method, and t h e g e n e r a t o r column method. I n t h e s h a k e - f l a s k ( o r s t i r - f l a s k ) method, a known amount o f s o l u t e e x c e e d i n g t h e a n t i c i p a t e d s o l u b i l i t y l i m i t i s added t o a measured w a t e r sample i n a s e a l e d c o n t a i n e r . This m i x t u r e i s shaken o r s t i r r e d t h o r o u g h l y and t h e n a l l o w e d t o s i t f o r e i g h t hours t o s e v e r a l days t o e n s u r e t h a t s o l u t e - s o l v e n t e q u i l i b r i u m i s a t t a i n e d . The m i x t u r e i s then f i l t e r e d o r c e n t r i fuged t o remove suspended p a r t i c l e s , and t h e s o l u t i o n c o n c e n t r a t i o n i s determined u s i n g customary chromatographic o r s p e c t r a s c o p i c t e c h n i q u e s . Changing range w i t h i n a method may a f f e c t e r r o r . The column g e n e r a t o r (and r e l a t e d HPLC) method o f d e t e r m i n i n g aqueous s o l u b i l i t y has become t h e predominant method i n the l i t e r a t u r e o v e r t h e past 5 y e a r s . I n t h i s method, a column i s packed w i t h an i n e r t s o l i d s u p p o r t w h i c h p r o v i d e s a h i g h s u r f a c e a r e a f o r t h e s o l u t e t o i n s u r e q u i c k e q u i l i b r a t i o n between i t and t h e aqueous phase. A f t e r t h e s u p p o r t i s impregnated w i t h the s o l u t e of i n t e r e s t and excess s o l u t e has been d i s p l a c e d , t h e column i s b r o u g h t t o t h e d e s i r e d t e m p e r a t u r e and t h e aqueous c o n c e n t r a t i o n i s determined as a f u n c t i o n of f l o w r a t e . The e q u i l i b r i u m c o n c e n t r a t i o n corresponds to the h i g h e s t e l u t i n g f l o w r a t e where s o l u t e c o n c e n t r a t i o n remains flow-independent. Sources o f E r r o r i n S o l u b i l i t y
Determination
The s h a k e - f l a s k method f o r d e t e r m i n i n g s o l u b i l i t y has f o u r major sources of determinate e r r o r a s s o c i a t e d w i t h i t . In the f i r s t c a t e g o r y a r e l o s s e s from a d s o r p t i o n o f t h e s o l u t e on t h e f l a s k walls or to evaporation. T h i s q u a n t i t y i s somewhat dependent on the v a p o r p r e s s u r e of the s o l u t e i n q u e s t i o n , b u t i f t h e q u a n t i t y of s o l u t i o n used i s f a i r l y l a r g e ( i . e . , 250 m l ) , t h e s e l o s s e s may r e p r e s e n t a maximum o f 5 p e r c e n t o f t h e t o t a l s o l u t e . A b e t a d i s t r i b u t i o n was chosen f o r t h i s e r r o r s o u r c e . The presence o f u n d i s s o l v e d s o l u t e suspended i n t h e water phase may cause a sample t o r e g i s t e r anomalously h i g h s o l u b i l i t y . Centrifuging or f i l t e r i n g reduces t h i s c o n t r i b u t i o n , a l t h o u g h t h e problem may p e r s i s t f o r more n e u t r a l l y bouyant s o l u t e s . T h i s e r r o r i s r e p r e s e n t e d by an e x p o n e n t i a l d i s t r i b u t i o n w i t h t h e v e r y low s t a n d a r d d e v i a t i o n expected i f t h e t e c h n i q u e d e s c r i b e d by K a r i c k h o f f and Brown (_5) i s p r o p e r l y f o l l o w e d (jf2 p e r c e n t w i t h 95-percent confidence). The t h i r d s o u r c e a r i s e s from t h e e x t r a c t i o n e f f i c i e n c y f o r the removal of s o l u t e from water, a t e c h n i q u e common t o most methods. P e t e r s ( 6 ) , has shown t h i s t o i n t r o d u c e a n o t h e r negat i v e b i a s w i t h i n 8 percent; t h e r e f o r e a beta d i s t r i b u t i o n with a 95-percent c o n f i d e n c e l e v e l o f -8 p e r c e n t was s e l e c t e d . F i n a l l y , the a c c u r a c y o f t h e d e t e c t i o n method was modeled by a normal d i s t r i b u t i o n w i t h a 95-percent c o n f i d e n c e l e v e l of +2 p e r c e n t as recommended by M a l l o n and H a r r i s o n (7^)· Note t h a t t h e e r r o r i n t r o d u c e d by each s t e p i s based upon the e x i t c o n c e n t r a t i o n of s o l u t e from t h e p r e v i o u s s t e p and, therefore, i s m u l t i p l i c a t i v e i n nature. In l i n e with the c o n c l u s i o n s o f Whitmore (2), t h e r e s u l t i n g output d i s t r i b u t i o n
COY ET AL.
Evaluation of Pesticides in Ground Water Downloaded from pubs.acs.org by FUDAN UNIV on 12/18/16. For personal use only.
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Determining Uncertainty in Physical Parameter Measurements
might then be expected t o approximate log-normality· However, the r e s u l t s o f t h e computer s i m u l a t i o n o f e r r o r f o r aqueous s o l u b i l i t y by t h e s h a k e - f l a s k method shown i n F i g u r e 2 c l e a r l y e x h i b i t a n e g a t i v e b i a s . T h i s b i a s was i n t r o d u c e d by p o t e n t i a l l o s s e s and by e x t r a c t i o n e f f i c i e n c y , a l t h o u g h the s t a n d a r d d e v i a t i o n o f 3,9 p e r c e n t compares q u i t e f a v o r a b l y t o t h a t c i t e d by s e v e r a l r e s e a r c h e r s . The mean s i m u l a t e d v a l u e u n d e r e s t i m a t e s the t r u e v a l u e by s l i g h t l y over 2 p e r c e n t . The column method a l s o has s e v e r a l s t e p s which may r e s u l t i n a n e g a t i v e b i a s on measured aqueous s o l u b i l i t y . The l a c k o f w a t e r - s o l u t e e q u i l i b r i u m i n t h e g e n e r a t o r column i t s e l f may produce an o u t l e t c o n c e n t r a t i o n l o w e r than t h e " t r u e " s o l u b i l i t y a c c o r d i n g t o S t o l z e n b u r g and Andren ( 8 ) . T h i s r e s u l t a l s o has been modeled w i t h a b e t a d i s t r i b u t i o n ( 9 5 - p e r c e n t c o n f i d e n c e l e v e l o f -3 p e r c e n t ) . The e f f i c i e n c i e s f o r a d s o r p t i o n and e x t r a c t i o n s t e p s a g a i n a r e e s t i m a t e d t o be s i m i l a r t o t h o s e suggested by P e t e r s ( 6 ) , t h e r e f o r e , t h e same b e t a d i s t r i b u t i o n s as b e f o r e a r e employed. However, note t h a t f o r t h e column g e n e r a t o r , two-step o r g a n i c removal t y p i c a l l y i s u s e d . F i n a l l y , one may a s c r i b e t h e same d e t e c t i o n l i m i t s as those o b t a i n e d f o r t h e s h a k e - f l a s k method. The column method r e s u l t s ( F i g u r e 3) e x h i b i t a mean s i m u l a t e d v a l u e 7 percent under t h e t r u e v a l u e , o r 5 percent l e s s than t h a t o b t a i n e d s i m u l a t i n g t h e s h a k e - f l a s k method. The d i f f e r e n c e may be a t t r i b u t e d t o t h e a d d i t i o n a l e x t r a c t i o n s t e p and s o l u t e - e q u i l i b r a t i o n s t e p i n t h e column method. The tendency of t h i s method t o u n d e r p r e d i c t r e a f f i r m s the need f o r c a l i b r a t i o n of t h e column system v i a s h a k e - f l a s k s t a n d a r d s p r i o r t o u s e , although the p r e d i c t e d standard d e v i a t i o n of 4 percent i s q u i t e good. Methods and E r r o r i n O c t a n o l - W a t e r P a r t i t i o n Determinations
Coefficient
The o c t a n o l - w a t e r - p a r t i t i o n c o e f f i c i e n t ( K ) i s the most f r e q u e n t l y c i t e d measure of e n v i r o n m e n t a l p a r t i t i o n i n g b e h a v i o r . P a r t i t L o n c o e f f i c i e n t s t r a d i t i o n a l l y have been determined by some v a r i a t i o n on t h e s h a k e - f l a s k method, however, o v e r t h e past s e v e r a l y e a r s h i g h - p r e s s u r e l i q u i d chromatography has been shown to measure o c t a n o l - w a t e r p a r t i t i o n c o e f f i c i e n t s a c c u r a t e l y over a wide range o f v a l u e s w i t h g r e a t e r e a s e . S i n c e t h e column/HPLC and s h a k e - f l a s k methods a r e a l s o t h e main s o u r c e o f s o l u b i l i t y v a l u e s , t h e e v a l u a t i o n o f e r r o r f o r K i s very s i m i l a r . Loss o f s o l u t e t o w a l l s o r t h e atmosphere a r e s t i l l o f c o n c e r n , as a r e t h e e x t r a c t i o n e f f i c i e n c i e s f o r s o l u t e removal from water. E q u i l i b r a t i o n o f each of the t h r e e phases i n v o l v e d ( i n s t e a d o f two f o r s o l u b i l i t y ) i s o f c o n c e r n . The s h a k e - f l a s k method f o r K d i f f e r s from t h e s o l u b i l i t y d e t e r m i n a t i o n o n l y i n the measurement of s o l u t e c o n c e n t r a t i o n i n the o c t a n o l phase. Since t h i s t y p i c a l l y i n v o l v e s standard chromatographic o r s p e c t r o s c o p i c t e c h n i q u e s , e r r o r a g a i n was assumed t o be n o r m a l l y d i s t r i b u t e d about t h e t r u e c o n c e n t r a t i o n w i t h a 95-percent c o n f i d e n c e l e v e l o f +2 p e r c e n t . o w
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