Statistics for Environmental Science and Management - Chapter 11 ppsx

CHAPTER 11 Monte Carlo Risk Assessment 11.1 Introduction Monte Carlo simulation for risk assessment is a relatively new idea, made possible by the increased computer power that has become available to environmental scientists in recent years. The essential idea is to take a situation where there is a risk associated with a certain variable, such as an increased incidence of cancer when there are high levels of a chemical in the environment. The level of the chemical is then modelled as a function of other variables, some of which are random variables, and the distribution of the variable of interest is generated through a computer simulation. It is then possible, for example, to determine the probability of the variable of interest exceeding an unacceptable level. The description 'Monte Carlo' comes from the analogy between a computer simulation and repeated gambling in a casino. The basic approach for Monte Carlo methods involves five steps: A model is set up to describe the situation of interest. Probability distributions are assumed for input variables, such as chemical concentrations in the environment, ingestion rates, exposure frequency, etc. Output variables of interest are defined, such as the amounts of exposure from different sources, the total exposure, etc.). Random values from the input distributions are generated for the input variables, and the resulting output distributions are derived. The output distributions are summarised by statistics such as the mean, the value exceeded 5% of the time, etc. There are three main reasons for using Monte Carlo methods: (1) The alternative is often to assume the worse possible case for each of the input variables contributing to an output variable of © 2001 by Chapman & Hall/CRC interest. This can then lead to absurd results, such as the Record of Decision for a US Superfund site at Oroville, California, which specifies a clean-up goal of 5.3 x 10 -7 µg/litre for dioxin in groundwater, which is about 100 times lower than the drinking water standard and 20 times lower than current limits of detection (United States Environmental Protection Agency, 1989b). Thus there may be unreasonable estimates of risk, and unreasonable demands for action associated with those risks, leading to the questioning of the whole process of risk assessment. (2) Properly conducted, a probabilistic assessment of risk gives more information than a deterministic assessment. For example, there may generally be quite low exposure to a toxic chemical, but occasionally individuals may get extreme levels. It is important to know this, and in any case the world is stochastic rather than deterministic, so deterministic assessments are inherently unsatisfactory. (3) Given that a probability-based assessment is to be carried out, the Monte Carlo approach is usually the easiest way to do this. On the other hand, Monte Carlo methods are only really needed when the 'worse case' deterministic scenario suggests that there may be a problem. This is because making a scientifically defensible Monte Carlo analysis, properly justifying assumptions, is liable to take a great deal of time. 11.2 Principles for Monte Carlo Risk Assessment The United States Environmental Protection Agency has put considerable effort into the development of reasonable approaches for using Monte Carlo simulation. Their website on this topic (United States Environmental Protection Agency, 1999) is full of useful information, as is their policy document (United States Environmental Protection Agency, 1997) that can be obtained from the same source. In the policy document the following guiding principles are stated for Monte Carlo studies: The purpose and scope should be clearly explained in a ‘problem formulation’. © 2001 by Chapman & Hall/CRC The methods used (models, data, assumptions) should be documented and easily located with sufficient detail for all results to be reproduced. Sensitivity analyses should be presented and discussed. Correlations between input variables should be discussed and accounted for. Tabular and graphical representation of input and output distributions should be provided. The means and upper tails of output distributions should be presented and discussed. Deterministic and probabilistic estimates should be presented and discussed. The results from output distributions should be related to reference doses, reference concentrations, etc. It is stressed that this is a minimum set of principles that are not intended to restrict the use of new scientifically defensible methods. 11.3 Risk Analysis Using a Spreadsheet Add-On For many applications, the simplest way to carry out a Monte Carlo risk analysis is using a spreadsheet add-on. Two such add-ons are @Risk (Palisade Corporation, 2000), and Crystal Ball (Decisioneering Inc., 2000). In both cases these products use the spreadsheet as a basis for calculations, adding extra facilities for simulation. Typically, what is done is to set up the spreadsheet with one or more random input variables and one or more output variables that are functions of the input variables. Each recalculation of the spreadsheet yields new random values for the input variables, and consequently new random values for the output variables. What @Risk and Crystal Ball do is to allow the recalculation of the spreadsheet hundreds or thousands of times, followed by the automatic generation of tables and graphs that summarise the characteristics of the output distributions. The following example illustrates the general procedure. © 2001 by Chapman & Hall/CRC Example 11.1 Contaminant Uptake Via Tapwater Ingestion This example concerns cancer risks associated with tapwater ingestion of Maximum Contaminant Levels (MCL) of tetrachloroethylene in high risk living areas. It is a simplified version of a case study considered by Finley et al. (1993). A crucial equation gives the dose of tetrachloroethylene received by an individual (mg/kg-day) as a function of other variables. This equation is Dose = (C x IR x EF x ED)/(BW x AT) (11.1) where C is the chemical concentration in the tapwater (mg/litre), IR is the ingestion rate of water (litres/day), EF is the exposure frequency (days/year), ED is the exposure duration (years), BW is the body weight (kg), and AT is the averaging time (days). The numerator is the total dose received in EF x ED exposure days, while the denominator is the total number of days in the period considered. Dose is therefore the average daily dose of tetrachloroethylene per kilogram of body weight. The aim in this example is to determine the distribution of this variable over the population of adults living in a high risk area. The variables on the right-hand side of equation (11.1) are the input variables for the study. These are assumed to have the following characteristics: C, the chemical concentration, is assumed to be constant at the MCL for the chemical of 5 µg/litre; IR, the ingestion rate of tapwater, is assumed to have a mean of 1.1 and a range of 0.5-5.5 litres per day, based on survey data; EF, the exposure frequency, is set at the United States Environmental Protection Agency upper point estimate of 350 days per year; ED, the exposure duration, is set at 12.9 years based on the average residency tenure in a household in the United States; BW, the body weight is assumed to have a uniform distribution between 46.8 (5th percentile female in the United States) and 101.7 kg (95th percentile male in the United States); and AT, the averaging time, is set at 25,550 days (70 years). © 2001 by Chapman & Hall/CRC Thus C, EF, ED and AT are taken to be constants, while IR and BW are random variables. It is, of course, always possible to argue with the assumptions made with a model like this. Here it suffices to say that the constants appear to be reasonable values, while the distributions for the random variables were based on survey results. For IR the empirical distribution shown in Table 11.1 is used because this gives the correct mean and range. Table 11.1 Distribution used for the ingestion rate of tapwater for the individuals living in high risk areas Ingestion rate (l/day) Probability 0.50 0.2857 0.75 0.2571 1.00 0.2286 1.50 0.0857 2.00 0.0571 2.50 0.0286 3.00 0.0143 3.50 0.0143 4.00 0.0114 4.50 0.0086 5.00 0.0057 5.50 0.0029 Total 1.0000 There are two output variables: Dose, the dose received (mg/kg-day) as defined before; and ICR, the increased cancer risk (the increase in the probability of a person getting cancer) which is set at Dose x CPF(oral), where CPF(oral) is the cancer potency factor for the chemical taken orally. For the purpose of the example CPF(oral) was set at the United States Environmental Protection Agency's upper limit of 0.051. A spreadsheet was set up containing dose and ICR as functions of the other variables, with the @Risk add-on activated. Each recalculation of the spreadsheet then produced new random values for IR and BW, and consequently for dose and ICR, to simulate the situation for a random individual from the population at risk. The number of simulated sets of data was set at 10,000. Table 11.2 © 2001 by Chapman & Hall/CRC shows some of the summary output obtained (minimums, maximums, means, etc.), while Figure 11.1 shows the distribution obtained for the ICR. (The dose distribution is the same but with the horizontal axis divided by 0.051.) The 50th and 95th percentiles for the ICR distribution are 0.05x10 -5 and 0.20 x 10 -5 , respectively. Finlay et al. (1993) note that the 'worse case' scenario gives an ICR of 0.53 x 10 -5 , but a value this high was never seen with the 10,000 simulated random individuals from the population at risk. Hence the 'worse case' scenario actually represents an extremely unlikely event. At least, this is the case based on the assumed model. Figure 11.1 Simulated distribution for the increased cancer risk as obtained from the output of @RISK. © 2001 by Chapman & Hall/CRC Table 11.2 Summary of output from @Risk based on simulating 10,000 random individuals from the population living in a high contamination area Name Dose*10 -5 ICR*10 -5 Description Output Output Cell A35 C35 Minimum = 0.4344 0.0222 Maximum = 10.2119 0.5208 Mean = 1.3778 0.0703 Std Deviation = 1.1539 0.0588 Variance = 1.3314 0.0035 Skewness = 2.7076 2.7076 Kurtosis = 11.9364 11.9364 Errors Calculated = 0 0 Mode = 1.1234 0.0573 5% Perc = 0.4795 0.0245 10% Perc = 0.5348 0.0273 15% Perc = 0.6042 0.0308 20% Perc = 0.6671 0.0340 25% Perc = 0.7069 0.0361 30% Perc = 0.7560 0.0386 35% Perc = 0.8141 0.0415 40% Perc = 0.8748 0.0446 45% Perc = 0.9161 0.0467 50% Perc = 0.9746 0.0497 55% Perc = 1.0509 0.0536 60% Perc = 1.1452 0.0584 65% Perc = 1.2516 0.0638 70% Perc = 1.3708 0.0699 75% Perc = 1.5357 0.0783 80% Perc = 1.7690 0.0902 85% Perc = 2.1374 0.1090 90% Perc = 2.6868 0.1370 95% Perc = 3.8249 0.1951 11.4 Further Information A good starting point for more information is the Risk Assessment Forum home page (United States Environmental Protection Agency, 2000). For examples of a range of applications of Monte Carlo methods, a special 400 page issue of the journal Human and Ecological Risk Assessment will be useful (Association for the Environmental Health of Soils, 2000). For more information about @Risk, see the book by Winston (1996). © 2001 by Chapman & Hall/CRC 11.5 Chapter Summary The Monte Carlo method uses a model to generate distributions for output variables from assumed distributions for input variables. These methods are useful because 'worse case' deterministic scenarios may have a very low probability of ever occurring, stochastic models are usually more realistic, and Monte Carlo is the easiest way to use stochastic models. The guiding principles of the United States Environmental Protection Agency for Monte Carlo analysis are summarised. An example is provided to show how Monte Carlo simulation can be done with the @RISK add-on for spreadsheets. Sources of further information are noted. © 2001 by Chapman & Hall/CRC . dose and ICR as functions of the other variables, with the @Risk add-on activated. Each recalculation of the spreadsheet then produced new random values for IR and BW, and consequently for dose and. 3.8249 0.1951 11. 4 Further Information A good starting point for more information is the Risk Assessment Forum home page (United States Environmental Protection Agency, 2000). For examples of. for a US Superfund site at Oroville, California, which specifies a clean-up goal of 5.3 x 10 -7 µg/litre for dioxin in groundwater, which is about 100 times lower than the drinking water standard