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 becom

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

interest This can then lead to absurd results, such as the Record

of Decision for a US Superfund site at Oroville, California, which

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’

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

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

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:

MCL for the chemical of 5 µg/litre;

1.1 and a range of 0.5-5.5 litres per day, based on survey data;

Environmental Protection Agency upper point estimate of 350 days per year;

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

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

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

There are two output variables:

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

shows some of the summary output obtained (minimums, maximums,

ICR (The dose distribution is the same but with the horizontal axis divided by 0.051.)

and 0.20 x 10-5, respectively Finlay et al (1993) note that the 'worse

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

Table 11.2 Summary of output from @Risk

based on simulating 10,000 random individuals from the population living in a high contamination area

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)

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