Sampling and Analysis of Environmental Contaminants
Sampling and Analysis of Environmental Contaminants ENVS 541
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This 34 page Class Notes was uploaded by Gerry Spinka on Friday October 23, 2015. The Class Notes belongs to ENVS 541 at University of Idaho taught by Maxine Dakins in Fall. Since its upload, it has received 31 views. For similar materials see /class/227750/envs-541-university-of-idaho in Environmental Science at University of Idaho.
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Date Created: 10/23/15
4122002 Module 2 2 Module 2 kiron mental 22 Simp Sampling The Sampling DI ution of the Mean Confidence Intervals on the Mean quotl 6 Simple Random Sampling o A simple random sample SRS is one that gives each sample unit an equal chance of being selected to be in the sample A 39A t Using SRS the sample statistics are the same as shown in Modules 12 and 14 Other sampling designs such as stratified and systematic sampling result in different equations for calculating the values ofthe E 5 sample statistics 4122002 0 Module 22 A 9 9 l A r Simple Random Sampling o Sample statistics under SRS I l i1 i1 X 1 S 9 4122002 x Ear ff 77 1 i1 Module 22 6 Simple Random Sampling 6 o Another useful measure is the coef cient of variation CVy which is the standard deviation divided by the mean and gives the dispersion of the data as a proportion ofthe average When multiplied by 100 it gives the dispersion as a percentage ofthe mean a it s CV2 X Module 22 V l 39 9 4122002 r V o Just like data having underlying distributions Of Q o The sampling distribution ofthe sample mean I 7 9 4122002 The Sampling Distribution of the Mean that describe their probability characteristics sample statistics have distributions as well Distributions of statistics are called sampling distributions is a distribution of particular importance Module 22 5 v31 459 v 19 The Sampling Distribution of the Mean o lmagine taking repeated samples of size n from a population and calculating the sample means Those means could then be complied together into a histogram to display their probability distribution With some thought you can see that this distribution should be narrower than the distribution ofthe data since really high or low data values would be to some extent moderated by other data points when calculating the means 0 4122002 Module 22 6 amp 91 amp Zvc y A The Sampling Distribution of the Mean o For example ifthe data follows a normal distribution with mean 50 and standard deviation of 10 then most ofthe data are in the range between 20 and 80 Means of samples of size 10 would tend to be close to 50 and would rarely be as low as in the 20s or in the 70s 4122002 Module 22 amp 39 9i 6i amp 3 59 r l 9 The Sampling Distribution of the Mean o In fact the central limit theorem gives us a theoretical way to determine the sampling distribution ofthe mean when data are drawn from a normal distribution It is approximately true fordata from any distribution although it is less and less true as the underlying data distribution becomes more asymmetric 4122002 Module 22 r The Sampling Distribution of the Mean o If X is Normal no and N is relatively small then X is Normal with mean it and standard error 2 2 0y Vary n 1 N Q 7 on Q 1 4 f N 9 4122002 Module 22 9 a 2 j 0ySEy1 ij 6 The Sampling Distribution of the Mean 6 o Note that the dispersion of a statistic is called its standard error A Y o These are estimated of course by replacing owith s V 39 9 4122002 Module 22 10 h A r The Sampling Distribution of the Mean A o When N is infinite or even vary large then the nN term is zero or close to it so it disappears from the equations Then X 1 Q o GgtSEGgt t 9 4122002 Module 22 11 Q l 39 Con dence Intervals on the Mean I 7 o The sample mean estimates the population mean but would rarely be expected to be Q a equal to it l Q o A confidence interval is an interval calculated from the sample mean and its standard error 7 that has a specified probability of containing Q the true parameter E 9 4122002 Module 22 12 Q i 39 79quot Confidence Intervals on the Mean 7 o Fordata from a normal distribution at least 39 approximately and large samples ngt25 a Q N 1x confidence interval on p is 1 039 s Q inaZSEyinaZWinaZ j X 9 4122002 Module 22 13 6 5 Con dence Intervals on the Mean 39 o For small sample sizes the sample standard deviation is not a very good r estimate of the population standard deviation Use the Student s t distribution instead of the normal s y ta2n 1 y La2W1 V r39 9 4122002 Module 22 14 3 A Vwb A r Conclusion A 9 4122002 X 95 N o Similarly population totals and proportions can be calculated They have sampling distributions and standard errors and confidence intervals can be calculated o These topics are covered more thoroughly in the text Module 22 113 Usine 1 Decision 7 aking H i To know one s ignorance is the best part of knowledgequot Lao Tzu The Tao i 39 8152003 Module 113 Q to 0 Outline 4 o Understanding Uncertainty O Probabilistic Assessment Q quot o Using Uncertainty Introduction to Decision Analysis o Example of Understanding and Using Uncertainty in Risk Based Decision l Making under CERCLA Q7 I o Conclusions l 0 8152113 Module 113 r Q t by Understanding Uncertainty q o Traditional calculations and models use single value estimates for input variables and Q result in a single value risk models l O engineering calculations o If conservatism is needed a variety of methods are used to provide it O use worst case estimates rather than best guess a W apply a safety factor to the final calculations l 0 8152113 Module 113 9 Understanding Uncertainty o This approach provides little information on the range of values that may exist and their relative probabilities The degree of conservatism is unknown and controversial Usually involves an ad hoc discussion of O O 999 9 v uncertainty o Doesn t distinguish between variability and uncertainty 8152003 Module 113 5 A l Understanding Uncertainty A Oquot o It s important to distinguish between variability and uncertainty Variability refers to the differences that come from heterogeneity or diversity in a population Variability is usually not reducible by further study Uncertainty refers to the lack of knowledge about specific factors parameters or models Uncertainty can potentially be reduced by additional study l O 8152CO3 Module 113 99 9 Understanding Uncertainty Q go Probabilistic Assessment replaces single point estimates with probability distributions Q i Distributions re ect variability andor uncertainty 4 K Distributions are created from Data 4 Information from the scientific literature Q Expert Judgement experience l 0 8152113 Module 113 a Understanding Uncertainty Q Q 0 4 f o Either mathematical techniques calculus or empirical techniques Monte Carlo Analysis are used to propagate the uncertainty through the risk model o In this way a distribution on the model output variable is created l 0 8152113 Module 113 vafvftt Understanding Uncertainty 0 Monte Carlo Analysis Monte Carlo methods involve a random selection of values from each input probability distribution These values are then used in the model to get an output value This procedure is repeated a large number of times N100 500 1000 10000 Together the output values define a probability distribution on the model output Ens2mm Module ll 3 a Q v a vf l Understanding Uncertainty Model Inputs Model Output E uation h odel Ens2mm Module ll 3 RISK Functions 3152003 Module 113 winning J 611quot E F uaw Mmm skmg Lam an a r I gpaag stng UK um 39 Tm wan 3152003 Module 113 vimu 4 im a mum rnangm 75x muss FEB MadelmleLElUNV Nmmaiiuluun FEB MadamxlFL UM V lellmml m mm u minmin mu m 41 Numaiiasugm P uvmsusu 3m mai F mm mm mm 1 Tunanwiadaslmtwinw um swam1 El nmI quot 35mm 65mm 7125111211 EMMA XMlmux Emmi gems dr0 am9gQ 59 125PMquot 3152003 Module 113 13 E5 Histogram and CDF of Output Distribution W D lwl 5mm 7 can m BEE Distribution forTota PCBCT Distributipn forTotai PCBC7 FROEAEiLTY EuvventVa a es1nu1puL5mput Settings Simuat inns1 Heva ans amquot am Emu va may mapF Kt wamfmqg 3152003 Module 113 14 4 i 9 Q If we begin with certainties we shall 39 end in doubts but if we begin with Q doubts and are patient in them we i 91 shall end in certainties t I l 0 Francis Bacon oi l O 8152C03 Module 113 15 39 Using Uncertainty in Scienti c Decision Making 7 quot o Once the assessment is complete a single value 0 from the distribution could be used 95th percentile 99th percentile O o But this ignores most of the information contained in 2 the distribution And distributions with differing Q shapes have the same upper percentile o Could there be important characteristic of the distribution shape central tendency which should be taken into account i 0 8152113 Module 113 16 l 991c v v Using Uncertainty in Scienti c Decision Making o Two types of decision errors can occur in risk management decision making o Type False Positive Conclude that remediation is warranted at a site when it isn t Do more remediation than is necessary to meet management goals 8152603 Module 113 l q 039 Q Q a 0 6 on i O 8152CO3 Using Uncertainty in Scienti c Decision Making o Type II False Negative Conclude that remediation is not warranted at a site when it is Do too little remediation to meet risk management goals o A loss function makes the losses for each of these types of decision errors explicit Module 113 Q 939 Using Uncertainty in quot Scienti c Decision Making o Once the set or space of alternative decisions is defined and the loss function is determined Decision analysis provides a framework for calculating the expected loss for each altemative decision using the probability distribution from the PA The best decision is the one that minimizes the expected loss This determination can be used along with all other information to help determine the best course of action 8152003 Module 113 19 9 9 vo yc v vn 9 A 39 Using Uncertainty in Scienti c Decision Making o Note Loss functions can be difficult to define and get agreement on There may be multiple decision makers and they may have different loss functions 1 It is dif cult and sometimes impossible to turn values into numerical quantities Q 4 39 s I 8152603 Module 113 20 939 Using Uncertainty in quot Scienti c Decision Making Q However there may be value in having the discussion Q Exposes areas of agreement and disagreement Educates everyone about the other s point of r view q Increases transparency of decision making I Can repeat the analysis with different loss l 8152C03 Module 113 l functions 0 j i 2 Q The man who insists upon l 04 seeing with perfect clearness before he decides never 39 9i decides k Frederic Am iel f O 8152003 Module 113 t my Example of Understanding and Using 1 Uncertainty in Risk Based Decision Making 439 under CERCLA p 39 o Exposure analysis involving a food web model of l9 uptake of PCBs into flounder in New Bedford Harbor in 39 Eleven model input variables had significant uncertainty 4 39 Sensitivity analysis showed that six of the eleven were important in controlling output uncertainty If 39 Using literature values and expert judgement W uncertainty distributions were created for the six A I t variables is 3152003 Module 113 23 quot kample of Understanding and Using Uncertainty in Risk Based Decision Making A der CERCLA mils winaaw elD mu duva WEXplau x EEvavhi mainly lt1l EBHE Q EJ u 123w it 3152003 Module 113 24 Q v v vev Example of Understanding and Using Uncertainty in Risk Based Decision Making under CERCLA TotalPCBBodyBurdenforFlomder Management Question How many cubic meters of sediment in inner New Bedford Harbor must be dredged to meet a management criteria of 2 micrograms of PCB per gram of flounder Probabilin 46810121416182022 Flounder Body Burden micrograms per gram 8152603 Module 113 25 w 4f 9 r9 4 t 0 8152113 9 9 Example of Understanding and Using Uncertainty in Risk Based Decision Making under CERCLA The model and some assumptions were used to relate a reduction in PCB concentration in the sediment to a reduction in ounder body burden A loss function was created that estimated the losses associated with underremediation and with over remediation assuming 1000 per cubic meter dredged and incinerated 50 million penalty for underremediation resulting from a need to keep the fishery closed for five additional years while undergoing reremediation Module 113 26 l v91 319319 9 Example of Understanding and Using Uncertainty in Risk Based Decision Making under CERCLA quot Loss Function LA Bi 1000 A for A gt Bi 1000 Bi 50 million for A lt Bi where A is the area to be dredged m3 under a particular decision scenario Bi is the correct but unknown area necessary to dredge to just meet the management criterion ifthe ith replication of the Monte Carlo was true and LABi is the loss associated with making a less than optimal decision 8152603 Module 113 27 V v 96 Example of Understanding and Using ncertainty in Risk Based Decision Making nder CERCLA rt Expected Loss where ELA is the expected loss for V decision A and N is the number of replications from the Monte Carlo 8152603 Module 113 28 l l a 39r 3 0 Example of Understanding and Using Uncertainty in Risk Based Decision Making under CERCLA 0 Decision Calculate the expected loss for different management decisions and select the one that minimizes the expected loss ElliZEUS Mudule ll 3 29 r 1 Conclusions j o Monte Carlo techniques allow the uncertainty in a r 3 lo Q scientific decision making situation to be determined by propagating the uncertainties in the inputs through the equation or model to get a probability distribution on the output o Decision analysis can be useful in scientific decision making especially where significant uncertainties exist because it makes explicit the losses associated l39 with different actions and helps the decision maker r 9 Ell EZEIEIS select an alternative that minimizes the expected loss i Mudule ll 3 3D 4 T we demand rigidly defined Q areas of doubt and uncertainty 9 lt0 Q The Hitchhiker s Guide to the Galaxy l 8152603 Module 113 31 M il l i l Probability Distributions o Often it is important to know something about the underlying probability distribution that data come from c There is a branch of statistics that does not require any assumptions about distributions These are non parametric methods and will be discussed later 4122002 Module 31 A rt Oil 1 66 X t 95 Probability Distributions o Information about the underlying distribution can come from Knowledge ofthe data generating process Past data of a similar type Theoretical considerations 4122002 Module 31 3 0 s a w 75 Q i L39 Q l t 7 399 Discrete Probability Distributions o Discrete probability distributions are used when the data can only take on specific values a o Examples Contaminated or Noncontaminated0 or 1 Count data 0 12 3 and so on Value ofa rolled die 1 2 6 4122002 Module 31 4 9 9 V v Variance VarX r Discrete Probability a Distributions Notation Px Probability Xx PX1 PX2 PX3 PXn 1 P09 gt 0 Expected Value EX let PoCl xi Pom 4122002 Module 31 v yfyfv v Discrete Probability Distributions Binomial IV Applies when there are n independent trials Two outcomes possible success and failure p probability of success is constant Probability of X successes in n trials is Prx quotCxplt1 p Expected Value EX p Variance VarX npUD 4122002 Module 31 6 Note on Notation 24 n v1 quot0 xn x vfy 4122002 Module 31 7 Discrete Probability Distributions Poisson quot Applies when there are n independent events Events occur at a constant rate p 7 Probability of X events in a given period oftime 0 area of space is i x Px e X v4 rt Q yfvsv of Expected Value EX p Variance VarX p 4122002 Module 31 8 9fv zfi 9 r Discrete Probability a Distributions Hypergeometric Applies when a population of N units contains R successes Probability that a sample of size n contains X successesis RCLXNRC Px Wd Expected Value EX nRN Variance VarX RNRnN2 Module 31 4122002 vfyfv v 9 VarX 62 Continuous Probability Distributions Normal Bell Shaped Curve Occurs in nature heights weights etc when the variable is the sum of other variables distribution of sample mean total proportion EX u 4122002 Module 31 10 95915i 9 r Continuous Probability a Distributions Lognormal Right skewed has long tail to the right Occurs when the variable is the product of other variables often a good model for environmental contamination EX expp 622 VarX exp2p 62 exp621 4122002 Module 31 v y v 9 v Continuous Probability Distributions Exponential Applies when the time until an event occurs or between events is of interest EX l1 t VarX p2 4122002 Module 31 25 Other Sampling Designs m I Cluster Sampling o Useful when population units are found in groups or clusters Often useful in sampling plants and animals o Randomly select a cluster or find one and then sample all of the units in that cluster 4122002 Module 25 l A r Cluster Sampling 6 o Advantages May be cheaper and easierthan simple random sampling in some situations X o Disadvantages Population units in a cluster may be more similar than across the population as a X whole so the variance may be underestimated 9 4122002 amp Module 25 39 Multi Stage Sampling Q o Population units fall within a staged structure v o First sample stage 1 then stage 2 and so on o When sampling for lead contamination in homes first randomly select a Q neighborhood then a block then a v home 9 4122002 I 9 of l Module 25 Multi Stage Sampling 6 F1 O VA X 6quot 39 v Lay 4122002 Module 25 Composite Sampling p o Useful when the cost of taking samples 7 i is lower than the cost of measuring Q 7 them o This is often true when samples must Q x be chemically analyzed for contaminants V t 9 4122002 Module 25 Q Q 9 Q l 91 0 i l i Q Qx 79quot Composite Sampling o Compositing involves taking several physical samples from an area mixing them together and taking a subsample from the mixture o It is a type of physical averaging as opposed to mathematical averaging o It s appropriate when information on the mean is sought and NOT appropriate when information on high values andor 41va iability is neededlezs 7 amp I bi vi amp b 43vquot y l 9 Other Sampling Designs o There are other sampling designs as well Ranked set sampling Ratio estimation Double sampling 4122002 Module 25
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