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Appendixes

Appendix A Lexicon INTRODUCTION ally called âanalysisâ (indeed, employees assigned to information gathering are called intelligence analysts), The lexicon in this appendix, prepared by the Commit- and the second step is called an âassessmentâ of the tee on Methodological Improvements to the Department of situation. Homeland Securityâs Biological Agent Risk Analysis, is in- â¢ The risk and decision community reverses these tended to be an exemplar of what might be used in any public definitions: the first step of gathering information (in presentation and discussion of a probabilistic risk analysis particular, obtaining information about the uncertainty and presented as a supplement to this report. Without a clear of events and their possible consequences) is usu- and consistent use of language in this technical arena, there ally called âassessment,â while the second stepâthe will be a tendency for conclusions to be misinterpreted and process of using this information and combining it in for policy recommendations based on these conclusions to such a way that a decision maker can make better deci- be misguided. sionsâis usually called âanalysis.â Because many of the terms in this lexicon (Table A.1) are found in everyday usage, often with implications or mean- For this reason, in the lexicon the committee has taken pains ings different from those presented here, it was suggested to break out the various components of ârisk analysisâ as that the committee also include âlay definitionsâ in order used in its report. to provide a comparison and to help in interpreting various loosely written documents and statements made available to the committee (and the public). However, the commit- ALTERNATIVE DEFINITIONS FOR âRELATIVE RISKâ tee has intentionally not done this, in order to avoid giving The term ârelative riskâ has a well-accepted definition credence to analyses that might be flawed by improper use in the biomedical community: âThe risk of harm among a or interpretation of various technical terms. The committee population exposed to a potentially damaging substance, recommends that any governmental agency issuing a report compared to the risk amongst an unexposed population.â on or engaging in a discussion of risk analysis consider using The term may also be used to describe the ratio: {cumula- terms as defined in this lexicon, or establish from the begin- tive incidence rate in the exposed population}/{cumulative ning reasons for using alternative definitions. incidence rate in the unexposed population}. However, the Department of Homeland Security (DHS) has chosen to ANALYSIS AND ASSESSMENT use the term for a completely different concept. In particu- lar, ârelative riskâ for a particular agent is determined as There is an unfortunate (but readily dealt with) incon- follows: sistency in usage between two communities importantly involved in understanding the risk of terrorist events: intel- â¢ For each agent i an expected consequence E(Ci) is ligence analysts and risk analysts. calculated (by Monte Carlo simulation), â¢ In the intelligence community it is customary first to â R.M. Anderson and R.M. May. 1991. Infectious Diseases of Humans. gather information about an opponentâs intentions and Oxford, United Kingdom: Oxford University Press. capabilities and then to use this information to present a â Department of Homeland Security. 2006. Bioterrorism Risk Assess- statement of the current situation. The first step is usu- ment. Biological Threat Characterization Center of the National Biodefense Analysis and Countermeasures Center, Fort Detrick, Md., p. C-95. 63

64 DEPARTMENT OF HOMELAND SECURITY BIOTERRORISM RISK ASSESSMENT â¢ Probability pi is assigned to the event {agent i will be available on the World Wide Web but developed for used}, promoting commercial software packages, consulting â¢ An overall âtotalâ expected consequence (or âriskâ) is services, and such. These sites are, for the most part, computed R = â piE(Ci), poorly conceived and, more problematic, have not been â¢ The relative risk for agent i is Ri = piE(Ci)/R. vetted by any professional independent set of experts, academics, practitioners, or professional societies. That is, ârelative riskâ is the proportion of the total expected â¢ The main portion of the lexicon (Part A.1.A), although risk contributed by a particular agent. Since this definition developed for biological risks, can also be appropri- is quite different from that used by the biomedical com- ately applied to nonbiological (chemical, radioactive, munity, it presents a major source of potential confusion agricultural, and other) threats. The second part of the and misinterpretation, particularly among readers who are lexicon (Part A.1.B), specifically, the terms used in knowledgeable in epidemiology. susceptible, exposed, infected, and recovered (SEIR) modeling, applies only to biological risk analysis. â¢ Although the committee recognizes the long philo- COMMENTS ON THE CONSTRUCTION sophical history of the controversy surrounding the AND USE OF THE LEXICON nature of uncertainty, it takes the position that, for the â¢ Since the committeeâs primary objective is to provide purposes of policy development and decision making consistency among the various terms, the terms are (the eventual goal of DHSâs risk analysis), all uncer- cross-referenced as needed. Column 1 provides syn- tainty (subjective, frequency-derived, and so on) must onyms and cross-references for the terms defined. It eventually be encoded into probabilities. also gives quoted definitions from the DHS document â¢ The entry â[None]â in second column, âCommitteeâs entitled âA Lexicon of Risk Terminology and Meth- Recommended Definition,â indicates a conclusion by odological Description of the DHS Bioterrorism Risk the committee that it is not necessary (or it is poten- Assessmentâ (DHS, 2007). tially confusing) to provide a definition. Indeed, the â¢ References are given in footnotes to the table. Rather committee recommends that such terms not be used in than using highly theoretical sources, the commit- any formal discussion of methods, results, and so on, tee chose to rely on widely accepted introductory or unless they are used as exemplars of what not to say. basic texts or more contemporary but focused refer- â¢ Due to the (committee) process by which the lexicon ences (e.g., Meyer and Booker). Where appropriate, was developed, it may not include terms that some selected Web sites from well-regarded sources have readers might find important; further, choices among also been used. However, the committee has intention- alternative accepted definitions were made where ally avoided the use of glossaries and lexicons readily necessary. â For example: W. Feller, 1968, An Introduction to Probability Theory and Its Applications, New York: Wiley; D.V. Lindley, 1965, Introduction to REFERENCE Probability and Statistics from a Bayesian Viewpoint; Part 1: Probability, DHS. 2007. âA Lexicon of Risk Terminology and Methodological De- Cambridge, U.K.: Cambridge University Press; and B. deFinetti, 1974, scription of the DHS Bioterrorism Risk Assessment.â Written com- Theory of Probability, Hoboken, N.J.:Wiley. munication to the Committee on Methodological Improvements to the â For example: M.S. Meyer and J.M. Booker, 2001, Eliciting and Analyz- Department of Homeland Securityâs Biological Agent Risk Analysis. ing Expert Judgment: A Practical Guide, Philadelphia, Pa.: American Statis- April 14, 2007. tical Association and the Society for Industrial and Applied Mathematics.

APPENDIX A 65 TABLE A.1â Lexicon of Probabilistic Risk Assessment Terms PART A.1.Aâ TERMS APPLICABLE TO BIOLOGICAL RISKS AND TO OTHER, NONBIOLOGICAL THREATS Term, with Synonyms, Cross- References, and DHS Lexicona Definitions Committeeâs Recommended Definition Notes, Comments, and References accuracy A measure of agreement between the estimated See note under precision value of some quantity and its true value. See also precision. (Adapted from Society for Risk Analysis [SRA] Glossary.b) agent-conditional expected risk The conditional expected risk computed using probabilities conditional upon the use of a See also conditional expected risk. particular agent. agent-conditional relative risk The conditional relative risk using probabilities conditional upon the use of a particular agent. See also agent-conditional expected risk. aleatory probability âA measure of the uncertainty of an unknown event whose occurrence is governed by some Synonym: aleatory uncertainty random physical phenomena that are either (1) predictable, in principle, with sufficient See also probability, epistemic information (e.g., tossing a die) or (2) essentially probability. unpredictable (radioactive decay).âc approximation âThe result of a computation or assessment that may not be exactly correct, but that is adequate See also estimation. for a particular purpose.âd arc (directed arc) In an event tree: an outcome from a preceding event to a subsequent event; in a decision tree: Synonym: branch either an action or an outcome from a preceding event to a subsequent event. See also split fraction. arithmetic average The sum of n numbers divided by n.e,f,g The average is a simple arithmetic operation, requiring a set of n numbers. It is often confused with the mean (or expected Synonyms: arithmetic mean, sample value), which is a property of a probability distribution. mean One reason for this confusion is that the average of a set of realizations of a random variable is often a good estimator of See also mean. the mean of the random variableâs distribution. conditional expected risk Expected risk computed using conditional The conditioning event is typically the choice of agent; probabilities. however, it could be other events such as good weather, See also agent-conditional successful manufacture, or ineffective countermeasures. expected risk. conditional probability The probability of an event supposing (i.e., It is important to note that subjectively assessed probabilities âconditioned uponâ) the occurrence of other are based on the state of knowledge that holds at the time of See also probability. specified events. In the aleatory theory of the probability assessment. probability, the conditional probability of event A given event B is equal to the probability of the joint occurrence of events A and B divided by the probability of event B, if the probability of event B is not zero. (After Feller [1968],g DeFinetti [1974],h and Lindley [1965].i) conditional relative risk The proportion of the total expected risk If pi = P{conditioning event i} and Ci = expected consequence contributed by a particular conditioning event. associated with event i, then total expected risk is R = Î£ piCi and the total conditional relative risk associated with event i is piCi/R. continued

66 DEPARTMENT OF HOMELAND SECURITY BIOTERRORISM RISK ASSESSMENT TABLE A.1â Continued Term, with Synonyms, Cross- References, and DHS Lexicona Definitions Committeeâs Recommended Definition Notes, Comments, and References conditional risk Risk computed using conditional probabilities The expected risk associated with a particular agent, as (follows from the definition of conditional measured by the expected number of deaths, may be See also risk, conditional probability). conditioned upon (for example) the direction of the wind. probability. confidence interval A range of values [a,b] determined from a Confidence intervals should not be interpreted as implying sample, using a predetermined rule chosen that the parameter itself has a range of values; it has only one See also uncertainty range. such that, in repeated random samples from the value. The confidence limits a and b, being computed from same population, the fraction Î± of computed a sample, are random variables, the values of which (for a ranges will include the true value of an unknown particular sample) either do or do not include the true value parameter. The values a and b are called a of the parameter. However, in repeated samples, a certain confidence limits; Î± is called the confidence fraction of these intervals will include the parameter, provided coefficient (commonly chosen to be .95 or .99); that the actual population satisfies the initial hypothesis. and 1 â Î± is called the confidence level. (Adapted from SRA.b) consequence A description of a scenario, in terms of For DHS risk analyses, typical and important consequence measurable factors, that decision makers may measures are lives lost, morbidities, direct and indirect dollar Synonym: outcome consider in assessing preferences over different losses, and others. scenarios; these factors are often random variables. (Adapted from McCormick [1981],j with âdamageâ replaced by consequences.â) continuous random variable A random variable that has an absolutely continuous cumulative distribution function.i,e See also cumulative distribution function, probability density function. cost-benefit analysis âA formal quantitative procedure comparing SRA also includes in its definition: âTo determine a rank costs and benefits of a proposed act or policy.âb ordering of projects to maximize rate of return when available funds are unlimited, the quotient of benefits divided by costs is the appropriate form; to maximize absolute return given limited resources, benefits minus costs is the appropriate form.â This method of rank-ordering is inappropriate for risk analysis in that it implies specific (and presumably known) trade-offs between noncommensurable benefits and costs. A better procedure is to plot the costs and benefits associated with each possible decision and then to present the results to decision makers to assess the trade-offs, which may (or may not) result in the linear or multiplicative functions inherent in the cost-benefit computations. cumulative distribution function The function Æ(x) whose value is the probability The cumulative distribution function always exists for any (CDF) that a random variable, X, will be less than or random variable; it is monotonic and nondecreasing in x, equal to a value x; written as P{X â¤ x}.e,g,k and (being a probability) 0 â¤ P{X â¤ x} â¤ 1. If P{X â¤ x} is Synonyms: cumulative distribution, absolutely continuous in x, then X is called a âcontinuousâ distribution function random variable; if it is discontinuous at a finite or countably infinite number of values of x, and constant otherwise, X is See also probability distribution, called a âdiscreteâ random variable. probability density function, probability mass function.

APPENDIX A 67 TABLE A.1â Continued Term, with Synonyms, Cross- References, and DHS Lexicona Definitions Committeeâs Recommended Definition Notes, Comments, and References decision tree A tree with event nodes that are random The operations used in a decision tree are elementary: variables or decision nodes that represent expectation over consequences at event nodes and See also event tree, fault-tree decisions of an active agent. Each branch (path maximization (or minimization) at decision nodes. analysis. of event and decision nodes leading to a terminal node) may have consequences (e.g., in dollars, Decision trees can be infinite (with no terminal nodes, lives, utility) associated with its terminal node. as in recursive game trees) and/or can have intermediate consequences at nonterminal nodes. directed arc In an event tree: an ordered pair of nodes, representing a preceding event, followed by a Synonym: branch subsequent event. It is usual to interpret an arc as the outcome of an event. See also split fraction. In a decision tree or game tree: an ordered pair of nodes representing either an action or a preceding event, followed by a subsequent action or event or terminal (âpayoffâ) node. discrete random variable âA random variable that has a non-zero A probability mass function is used to represent the set of probability for only a finite, or countably infinite, probabilities for all values of a discrete random variable. See also cumulative distribution set of values.âc function, probability mass function. epistemic probability âA representation of uncertainty about Some examples of epistemic probability are (1) the assigning propositions due to incomplete knowledge. Such of a probability to the proposition that a proposed law of Synonym: epistemic uncertainty propositions may be about either past or future physics is true; (2) determination of the probability that events.âc a terrorist will use a particular agent, based on evidence See also aleatory probability, presented. uncertainty. DHS Lexicon: âarising from limited state of knowledgeâa estimation (of parameters in âA procedure by which sample data are used to Estimation procedures are usually based on statistical analyses probability models) assess the value of an unknown quantity.âf that address their efficiency, effectiveness, limiting behaviors, degrees of bias, etc. The most common methods of parameter Also see approximation. estimation are maximum likelihood and the method of moments. Bayesian methods tend to avoid producing estimates and instead treat parameters as unknown quantities, with associated probability distributions. event A subset of the sample space.f,g In a decision or Events are the basic building blocks of a probabilistic risk event tree, a random variable whose values are assessment; they are the entities for which probabilities See also random variable, event possible outcomes. are assessed and/or computed. Event descriptions must be space. carefully and unambiguously articulated. The terminal event â100 people dieââwithout making explicit the time frame within which they die, their geographical distribution, their demographics, etc.âis quite different from â100 people dieâ within the first 48 hours of the attack, all of whom are within 5 km of the city center, 60% of whom are age 65 and older, and so on. The important thing to consider here is that the granularity of probability risk assessment events should be only as fine as needed to capture the consequences of the scenarios that include the events. continued

68 DEPARTMENT OF HOMELAND SECURITY BIOTERRORISM RISK ASSESSMENT TABLE A.1â Continued Term, with Synonyms, Cross- References, and DHS Lexicona Definitions Committeeâs Recommended Definition Notes, Comments, and References event space The set of all possible outcomes of an experiment or of some (unknown) phenomenon. Synonym: sample space (After Feller [1968]g and Statistical Education Through Problem-Solving [STEP] Consortium.f) See also event. event tree A tree formed of a sequence of random An event tree is essentially a decision tree with the decisions variables, called events. The branching point at removed or replaced by nodes representing events that are Synonyms: probability tree, chance which a new variable is introduced in the tree the result of probabilistic decisions (made either by the tree is called a node. Each node is followed by the decision maker or some other agency). If a node in an event possible random variable realizations, called tree represents a decision taken by an adversary, then the See also tree, decision tree, fault- outcomes, and their probability distributions (conditional) probabilities of the resulting events must be tree analysis. conditional on outcomes of previous random assessed or computed just as those for any other event nodes. variables in the tree. The outcomes are Note that some computations (perhaps based on game- DHS Lexicon: âa logic diagram represented as arcs leading from one event to the theoretic approaches) might produce event probabilities of 0 consisting of both decisions and next. The joint probability of the intersection of or 1, associated with âknowingâ with certainty what action the physical events in which the events that constitute a sequence (or scenario) adversary will take. potential outcomes are represented is found by multiplication. A natural way to by a finite, complete, discretized set construct an event tree is to place events in the There is no need to disallow infinite or continuous outcomes, of outcomes (branches). The events chronological order in which they occur, if this as the DHS definition would imply. are not necessarily consecutive order is known.l in time and are, in general, not independent.âa expected risk A summary measure of risk for an event, The committee strongly recommends that, wherever possible, scenario, etc., as expressed by the expected the term âexpected riskâ be replaced by the specific measure Synonym: expected consequences value of any one of the measurable of consequences, such as âexpected deaths,â âexpected loss of consequences associated with the risk. (Adapted income,â âexpected illnesses.â If these measures are combined Although âexpected riskâ is not from McCormick [1981]j with âdamageâ in some functional way, for example via a utility function, in the DHS Lexicon, DHS reports replaced by âconsequence.â) then âexpected riskâ should be replaced by âexpected utility.â and presentations seem to imply One difficulty with defining âexpected riskâ is the historical synonymy among the terms âriskâ reality that the discipline of probabilistic risk assessment (as related to a specific set of events arose from an understanding of the risks associated with or scenarios), âexpected risk,â and nuclear reactors, chemical plants, and such. In these situations, âtotal risk.â expected risk is defined to be [expected frequency of occurrence of an event] times [expected consequences of that event]. expected value The first moment of the probability distribution The arithmetic average of random samples taken from the of a random variable X; often denoted as E(X) distribution converges to the mean for all sufficiently large Synonym: expectation and defined as â xi p(xi) if X is a discrete random sample sizes, under certain conditions. variable and as â«xÆ(x)dx if X is a continuous See also mean. random variable.g,e Ironically, in many cases the expected value of a random variable is a numerical value that the random variable can never take on. For example, if a random variable X has P{X = 0} = .5 and P{X = 100} = .5, then E(X) = 50, even though X can only take on values of 0 or 100. There is also a common confusion between expected value and average, due to the fact that, in the limit, as the sample size becomes very large, the average of a set of observations of a random variable will approach the mean of the random variableâs probability distribution. (A curious linguistic note: in French the expectation is called lâesperance, which in rough translation means âhoped for.â Being simply the result of a mathematical operation, it is neither hoped for nor truly âexpected.â)

APPENDIX A 69 TABLE A.1â Continued Term, with Synonyms, Cross- References, and DHS Lexicona Definitions Committeeâs Recommended Definition Notes, Comments, and References fault-tree analysis âA technique by which events that interact to produce other events can be related using simple logical relationships permitting a methodical building of a structure that represents the system.âb frequency âThe fraction of events that satisfy some The two DHS definitions confound four different ideas: prespecified criterion; a record of how often expected value, rate, fraction of past events that satisfy some DHS Lexicon: â1. The number of each value (or set of values) of the variable in criterion, fraction of future events that satisfy some criterion. events that would be expected to question occurs.âf occur in a time period.âa â2. A rate (with units, #/time).âa in-degree The number of arcs resulting in an event. In a tree, the in-degree is one for all events, except the initial event, which has an in-degree of 0. initial event The first node in an event tree. Synonym: initial node initiating event An event with the potential to initiate a sequence of other events leading to undesirable Synonym: initial event consequences. DHS Lexicon: âAn action taken by a terrorist organization to begin the process that may culminate in an act of terrorism.âa likelihood The likelihood, L(A | D), of an event A, given In informal usage, âlikelihoodâ is often a qualitative the data D and a specific model, is often taken description of probability or frequency. However, equally See also likelihood function, to be proportional to P(D | A), the constant of often these descriptions do not satisfy the axioms of probability, uncertainty. proportionality being arbitrary.m probability. For example, âlikelihoodâ has been used by DHS as a âweightâ when informally assessing uncertainties, even though the collection of these weights do not add to 1. likelihood function A weighting function interpreted as a function of The maximum (with respect to the parameter value) of parameters with the random variable(s) replaced the likelihood function often produces an estimator of the See also likelihood. by its (their) observed values.h,n parameter with desirable properties. mean The first moment of a probability distribution, See note under expected value. with the same mathematical definition as See also expected value, expected value. The mean is a parameter arithmetic average. that represents the central tendency of the distribution. (After Glossary of Statistics Terms,e STEP Consortium,f Ross [2000],o Devore [2000].k) measurement error âThe unexplainable discrepancy between Measurement error is often decomposed into two components: a measurement and the quality that the (1) random variation of measurements on objects of identical measurement instrument is intended to quality; (2) a systematic error in measurement (e.g., a measure.âp measurement device may be out of adjustment). continued

70 DEPARTMENT OF HOMELAND SECURITY BIOTERRORISM RISK ASSESSMENT TABLE A.1â Continued Term, with Synonyms, Cross- References, and DHS Lexicona Definitions Committeeâs Recommended Definition Notes, Comments, and References model A representation of some portion of the world Mathematical models are used to aid our understanding of in a readily manipulable form. A mathematical some aspects of the real world and to aid in decision making. See also simulation. model is an abstraction that uses mathematical They are also valuable rhetorical tools for presenting the language to describe the behavior of system. rationale supporting various decisions, since they arguably (Adapted from Wikipedia.q) allow for transparency and reproduction of results by others. However, models are only as good as their (validated) relationship to the real world and within the context for which they are designed. It is wise to remember the advice of George E.P. Box: âAll models are wrong, but some may be useful.â node A representation of an event or decision in a decision tree. A representation of an event in an See also event. event tree. node-to-node branch An ordered pair of nodes; a course of action leading from a preceding event to a subsequent See also course of action. one. normal distribution A symmetric âbell-shapedâ probability density The normal distribution commonly used, since (1) it is (with function, (1/(Ïâ2Ï))(-exp((x - Î¼)2/(2Ï2))), certain conditions) the limiting distribution of the sum of Synonym: Gaussian distribution completely characterized by two parameters: random variables, (2) it has a certain degree of mathematical mean Î¼ and standard deviation Ï.g,k,o tractability, (3) there exist many well-known methods for estimating its parameters, and (4) it represents a reasonable fit to data obtained for a wide variety of situations. normalized risk The proportion of the total expected risk contributed by a particular agent. See also conditional relative risk, relative risk. out-degree The number of directed arcs leaving a node. path A sequence of arcs. Synonym: scenario Poisson distribution A commonly used probability mass function The Poisson distribution is often used to reflect ârandomnessâ associated with a random variable X = number of events over timeâP{time between consecutive occurring of events that occur in a given period of time. events â¥ t} = eâÂµt, which does not depend on the time of any The formula is P{X = x} = ÂµxeâÂµ / x!, for previous event. x = 0, 1, . . ., where the parameter Âµ = E(X) is the mean of the distribution.k precision The implied degree of certainty with which a Consider two statements assessing âW = Bill Gatesâs value is stated, as reflected in the number of net worthâ: A precise but inaccurate assessment is âW is See also accuracy. significant digits used to express the valueâthe $123,472.89â; an imprecise but accurate assessment is: âW is more digits, the more precision. (Adapted from more than $8 billion.â SRA.b) probabilistic risk assessment An analytical tool that (1) identifies and delineates logical combinations of basic (not Synonym: risk assessment analyzed further) events that, if they occur, could lead to an accident (or other undesired event, called the top event); (2) assesses or approximates the probability of the top event from the probabilities of logical combinations of basic events; and (3) assesses the probable consequences associated with occurrence of the top event.

APPENDIX A 71 TABLE A.1â Continued Term, with Synonyms, Cross- References, and DHS Lexicona Definitions Committeeâs Recommended Definition Notes, Comments, and References probability One of a set of numerical values between 0 and The definition holds for all quantification of uncertainty: 1 assigned to a collection of random events subjective or frequentist. See also likelihood, conditional (which are subsets of a sample space) in such a probability. way that the assigned numbers obey two axioms: DHS Lexicon: â1. A probability 1. 0 â¤ P{A} â¤ 1 for any A, and assignment is a numerical encoding 2. P{A} + P{B} = P{A âª B} for two mutually of the relative state of knowledge exclusive events A and B.o (Society for Risk Analysis). 2. The subjectivist viewpoint of probability: the analystâs state of knowledge or degree of belief.âa probability density function (PDF) The derivative of an absolutely continuous The PDF is the common way to represent the probability cumulative distribution function.p distribution of a continuous random variable, because its shape often displays the central tendency (mean) and variability For a scalar random variable X, a function f such (standard deviation). From its definition, P{a â¤ X â¤ b} is the that, for any two numbers, a and b, with a â¤ b, integral of the PDF between a and b. P{a â¤ X â¤ b} = â« ab f(x)dx. probability distribution See cumulative distribution function. probability elicitation âA process of gathering, structuring, and coding There are many approaches for probability elicitation, expert judgment (about uncertain events or the most common of which are those used for obtaining a Synonym: probability assessment quantities).âr priori subjective probabilities. However, in some sense all probabilities, even those that result from statistical analysis of large data sets, are subjective and therefore require elicitation. This is because the conditions under which the data have been collected, and the relevance of these conditions to future events for which probabilities are desired, are a matter of expert and subjective judgment. Note that the results of probability elicitations are sometimes called probability âassessmentsâ or âassignments.â probability mass function (PMF) A function that gives the probability that a discrete random variable takes on a particular See also discrete random variable. value.k,o random error [None] This term is meaningful only in the context of analyzing the results of a particular experiment and therefore should not be See also measurement error. used. random variable âA real valued function defined on a sample (or The random variables of interest to a PRA are those that event) space.âg describe the consequences of a particular event. For example, See also event, probability suppose that the event space consists of only three events: distribution, continuous random A = â100 deaths, 500 illnessesâ; B = â0 deaths, 0 illnessesâ; variable, discrete random C = â75 deaths, 375 illnessesâ; and their respective probabilities variable. are P{A} = .3, P{B} = .2, P{C} = .5. Then, if the random variables are defined to be X = âthe number of deaths associated with the event space,â and Y = âthe number of illnesses associated with the event space,â this implies P{X = 100} = P{A} = .3; P{X = 12} = 0 (there are no events with X = 0); P{Y = 375} = .5; P{Y/X = .5} = P{A} + P{C} = .8, etc. A probability distribution, constructed on the range of the random variable, can then be used to assign probabilities to events in the event space. continued

72 DEPARTMENT OF HOMELAND SECURITY BIOTERRORISM RISK ASSESSMENT TABLE A.1â Continued Term, with Synonyms, Cross- References, and DHS Lexicona Definitions Committeeâs Recommended Definition Notes, Comments, and References relative risk (in an epidemiological See health terms in Part A.1.B of this table. context Synonyms: risk ratio; odds ratio reload capacity A measure of the ability to introduce a pathogen (Formulated by former Navy Secretary Danzig, according to into more than one country and/or on more than Marc Lipsitch of the Harvard School of Public Health.) one occasion. risk âThe potential for unwanted, adverse It is important to distinguish between the term risk, which consequences.âb involves uncertainties, consequences and conditioning See also expected risk. statements, and expected risk, which combines these factors using the linear additive expectation operation. It is essential DHS Lexicon: âwhen used in a to be absolutely clear when using these two these terms. general sense: The potential for Unfortunately, even SRAâs Glossaryb intermixes them, since realization of unwanted, adverse after giving the definition in Column 2, it goes on to say, consequences to human life, health, âestimation of risk is usually based on the expected value of property or the environmentâ the conditional probability of the event occurring times the [American Heritage Dictionary]; consequence of the event given that it has occurredâbâwhich â(âtechnical meaningâ): The set of is technically incorrect as well as misleading. triplets of frequency, scenario and consequences, for all scenarios <f, To make things even more confusing, Appendix C3 (âRisk s, c>; (âas the output of quantitative Integrationâ) of the DHSâs 2006 report Bioterrorism Risk risk assessmentâ): First moment Assessments defines âriskâ as âthe probability or frequency of of the risk probability density an event multiplied by the consequences of the event,â which function.âa is both inconsistent and technically meaningless. risk analysis An overall process that involves risk assessment, risk perception, risk communication, and risk DHS Lexicon: âA detailed management. The hazards to be analyzed (e.g., examination including risk physical, chemical, radioactive, and biological assessment, risk evaluation and risk agents) may result from natural events (e.g., management alternatives, performed earthquakes and hurricanes), technological to understand the nature of events (e.g., chemical accidents), and human unwanted negative consequences to activity (e.g., design and operation of engineered human life, health, property or the systems or attack by a terrorist). (Adapted from environment; an analytical process SRA.b) to provide information regarding undesirable events; the process of quantification of the probabilities and expected consequences for identified risks (from SRA).âa risk assessment The systematic process of identifying hazards and quantifying their potential adverse See also risk analysis. consequences (magnitude, spatial scale, duration, and intensity) and associated probabilities, DHS Lexicon: âThe process of including the uncertainties surrounding these establishing information regarding estimates. It may include a description of the acceptable levels of a risk and/or cause-and-effect links between hazards, the levels of risk for an individual, nature of the interdependencies, vulnerabilities, group, society, or the environment. and consequences. (Adapted and expanded from (From SRA).âa SRA.b)

APPENDIX A 73 TABLE A.1â Continued Term, with Synonyms, Cross- References, and DHS Lexicona Definitions Committeeâs Recommended Definition Notes, Comments, and References risk communication The process used by risk analysts, decision makers, policy makers, and intelligent adversaries to provide data, information, and knowledge to change the risk perceptions of individuals and organizations and enable them to assess the risk more accurately than they otherwise might. risk curve A graph describing frequency of events as a function of consequences. Alternatively, a curve describing frequency of events with consequences greater than or equal to some level as a function of that level. risk estimation âThe determination of the characteristics of risks Although SRA provides a definition, the committee sees no such as the magnitude, spatial scale, duration, need to include this term in a formal lexicon, since the term See also risk analysis. and intensity of adverse consequences and their âriskâ by itself has many connotations and in any event is a associated probabilities of the cause-and-effect random variable which, by definition, cannot be âestimated.â DHS Lexicon: âThe scientific links.âb There are also many overlaps with ârisk assessment.â determination of the characteristics of risks, usually in as quantitative a way as possible. These include the magnitude, spatial scale, duration and intensity of associated probabilities as well as adverse consequences and their description of the cause and effect links. (from SRA)âa risk management The process of constructing, evaluating, Taken from the definition in the committeeâs interim report: implementing, monitoring, and revising âIn the case of an individual, private sector or public sector See also risk analysis. strategies for reducing (or distributing) losses organization, these strategies enable them to transfer, mitigate, from future hazards and dealing with the or accept their perceived risks. Risk management strategies DHS Lexicon: âThe process recovery process should a hazard occur. Risk can be evaluated by undertaking cost-benefit analyses to of constructing and evaluating management strategies include a combination determine the tradeoff between the reduction of risk and strategies for reducing losses from of options such as providing information (i.e., the costs of undertaking such measures. In evaluating a risk future hazards and dealing with the risk communication), economic incentives (e.g., management strategy one needs to be concerned with the way recovery process should a disaster subsidies, fines), insurance, compensation, resources are allocated (i.e. efficiency considerations) as well occur.âa regulations, and standards. (Adapted and as the impact of these measures on different stakeholders (i.e. expanded from SRA.b) distribution or equity considerations).ât risk perception Beliefs, attitudes, judgments, and perceptions held by individuals, communities, societies, See also risk analysis. groups, or organizations about the risks of a hazard. Risk perception is concerned with DHS Lexicon: âBeliefs held the psychological and emotional factors. Risk by individuals or organizations perceptions can be influenced by personal about the risks of a hazard. Risk knowledge, experience, and beliefs; they can be perception is concerned with affected by changing perceptions of the threat, the psychological and emotional the vulnerabilities, and/or the consequences; factors, which have been shown they may be influenced by information about to have an enormous impact on hazards, risk assessments, risk policies, and risk behavior.âa management decisions. (Adapted and expanded from SRA.b) continued

74 DEPARTMENT OF HOMELAND SECURITY BIOTERRORISM RISK ASSESSMENT TABLE A.1â Continued Term, with Synonyms, Cross- References, and DHS Lexicona Definitions Committeeâs Recommended Definition Notes, Comments, and References scenario A complete enumeration of one path on a tree, from the initial event to the terminal node (if DHS Lexicon: âOne of a possible any). combination of approaches leading to the execution of an act of terrorism. An end of an event tree.âa simulation âA model constructed so that the input of a large By its inherent nature, each set of ârunsâ of a simulation number of random variables drawn from defined represents the outcomes of a series of experiments. Analysis Synonym: Monte Carlo simulation probability distributions will generate outputs of simulation output data therefore requires a proper that are representative of the random behavior of experimental design, followed by the use of statistical See also model. a particular system, phenomenon, consequences, techniques to estimate parameters, test hypotheses, etc. etc., of a series of events.âu split fraction [None] Presumably this term has been used by DHS to be synonymous with âconditional probability.â However, the See also conditional probability. DHS definition is not consistent with the DHS definition of âfrequency,â and ârelative frequencyâ is not defined by DHS. DHS Lexicon: âFor an event, the relative frequency of a branch.âa standard deviation âThe square root of the variance of a distribution.âo See also variance. terminal event An event in an event tree or a decision tree with out-degree 0. Synonym: terminal node total expected risk The probability-weighted sum of expected risks It is preferable that âtotal riskâ should depend on the specific associated with all agents. (Implied by DHS context: the consequence (deaths, utility, etc.) and the events Synonym: total risk usage). over which the sum is taken (e.g., agents, other conditioning events, etc.). For example, if pi = P{conditioning event i}, and Ci is the expected consequence associated with event i, then total expected risk is R = Î£ piCi. tree A connected acyclic directed graph with exactly one distinguished (root) node with in-degree 0, See also event tree, decision tree. and every other node with in-degree 1. uncertainty The condition of being unsure about something; The formal definition of âuncertaintyâ is really not important a lack of assurance or conviction.d to the understanding of any PRA method. However, having See also probability. a clear and agreed on definition of the uses to which any quantification of âuncertaintyâ is put is crucial. The DHS DHS Lexicon: âTwo types of Lexicon definitions are neither clear nor agreed on, and in uncertainty are considered and fact they confuse the notion of âuncertaintyâ with the various treated differently: aleatory methods used to quantify it in a useful way. uncertaintyâarising from variability (e.g., weather variability); epistemic uncertaintyâ arising from limited state of knowledge.âa

APPENDIX A 75 TABLE A.1â Continued Term, with Synonyms, Cross- References, and DHS Lexicona Definitions Committeeâs Recommended Definition Notes, Comments, and References uncertainty range [None] Depending on the context, DHS apparently means either (1) a range of probabilities associated with a particular event, DHS Lexicon: âTypically, a scenario, etc.âpossibly due to disagreements among subject- confidence interval. For the matter experts, outputs of a simulation or analytical model, common definition of risk given or results of an experiment, etc.; or (2) the range of uncertain above [presumably âexpected outcomes associated with a particular event, scenario, etc. consequences per unit time or within a time interval,â but not shown in this table since the committee does not display âlay definitionsâ], âthe confidence interval associated with the epistemic uncertaintyâ.âa utility A real valued function of a consequence. In economics, âutilityâ captures ârelative happinessâ or satisfaction gained by goods and services. Synonym: utility function In decision analysis, âutilityâ captures returns to scale and risk DHS Lexicon: âfunction preference. that transforms measures of consequences into a number.âa In both cases, the assessment of utility values (and hence utility functions) for consequences is an inherently subjective exercise and so depends on the individual (or organization) confronting the possible consequences. Formally, let A be the most-preferred possible outcome of a risky prospect, B be the least-preferred, and C be any other outcome. If a decision maker is indifferent between C and a prospect having probability u of getting A and probability (1 - u) of getting B, then u is defined as the (von Neumann- Morgenstern) utility of C. variance The second moment of a probability distribution, The variance is a common measure of variability around the defined as E(X - Âµ)2, where Âµ is the first moment mean of a distribution. Its square root, the standard deviation, See also standard deviation. of the random variable X. having dimensional units of the random variable, is a more intuitively meaningful measure of dispersion from the mean. weight of evidence The logarithm of K = P{x | A} / P {x | B}, where A nonstandard and nonstatistical definition, used by some x is a realization of a random variable, and A and analysts but not recommended, is as follows: âAn elicitation See also probability risk B are alternative hypotheses. (K is also called the of uncertainty that results in a non-normalized set of numbers assessment. likelihood ratio.)q which can be normalized (by dividing by the sum over all possible events) to produce probabilities.â In some statistical usage, the âweight of evidenceâ is defined to be 10 times the log-likelihood ratio. More generally, a loosely defined or undefined term indicating the extent to which studies are judged to support a conclusion. continued

76 DEPARTMENT OF HOMELAND SECURITY BIOTERRORISM RISK ASSESSMENT TABLE A.1â Continued PART A.1.Bâ TERMS USED IN SUSCEPTIBLE, EXPOSED, INFECTED, AND RECOVERED (SEIR) MODELING AND APPLICABLE ONLY TO BIOLOGICAL RISK ANALYSIS Term, with Synonyms, Cross- References, and DHS Lexicona Definitions Committeeâs Recommended Definition Notes, Comments, and References Bioshield A federal program authorized in 2004 to For more information see http://www.whitehouse.gov/infocus/ improve medical countermeasures protecting bioshield/. Americans against a chemical, biological, radiological, or nuclear (CBRN) attack. contagious A person who is infected and capable of This can be used as an adjective or noun, but most often, in transmitting an infectious agent to another host. the modeling context, as a noun. DHS Lexicon: âinfected and (Adapted from Thomas and Weber [2001]v.) capable of spreading disease.âa dose The amount (or concentration) of desired matter or energy deposited at the site of effect. (Adapted from SRA.b) exposed A person or population that came in contact with For SEIR modeling, but generally not other usage, âexposedâ the infectious agent or toxin. includes only those who received an infectious dose. This See also infected. can be used as an adjective or noun, but most often, in the modeling context, as a noun. DHS Lexicon: âpopulation who came in contact with the infectious agent or toxin and received an infectious dose.âa ill Infected or intoxicated population showing This can be used as an adjective or noun, but most often, in clinical signs and symptoms. the modeling context, as a noun. DHS Lexicon: âinfected or intoxicated population showing symptoms.âa infected An individual or population that has an This can be used as an adjective or noun, but most often, in infectious agent enter and multiply in its tissues.v the modeling context, as a noun. DHS Lexicon: âpopulation that has been exposed and received an infectious dose.âa infectious dose X (IDX) A dose that is expected to lead to the infection of Typically X = 50, but it is sometimes set to 10, 90, or other X percent of individuals exposed. values, depending on the intent of the analysis. intoxicated Population that has been exposed to a threshold amount of toxin and will become ill in the DHS Lexicon: âpopulation that has absence of intervention. been exposed and received a toxic dose of a toxin.âa lethal concentration X (LCX) A concentration that is calculated to kill X percent of a population. (Adapted from SRA.b) lethal dose X (LDX) A dose that is expected to kill X percent Typically X = 50, but it is sometimes set to 10, 90, or other of a population in the absence of medical values, depending on the intent of the analysis. intervention(s).b R0 The mean number of secondary cases of R0 is a property of the pathogen. R0 is a theoretical number infection to which one primary case gives rise and does not hold if the population is not entirely susceptible, Synonym: basic reproduction throughout its infectious period, if introduced or even in the case where there is more than 1 contagious number into a population consisting solely of susceptible person (since the entire population is not susceptible). individuals. (Adapted from Anderson and May See also R. [1991].w)

APPENDIX A 77 TABLE A.1â Continued Term, with Synonyms, Cross- References, and DHS Lexicona Definitions Committeeâs Recommended Definition Notes, Comments, and References R The number of secondary cases of infection R is a property of both the pathogen and the populationâs to which a single contagious case gives rise relative susceptibility. Under conditions of stable endemic Synonym: effective reproduction throughout its infectious period, in a host infection, R = 1. Note that the R value can and does number population where not all persons are susceptible. change as the outbreak progresses. The change in R may be due to reduction in the susceptible population, through See also R0. natural infections, changes in social behavior, or medical interventions. relative risk (Biomedical context) The ratio of the risk of disease or death among the exposed to the risk Synonyms: risk ratio; odds ratio among the unexposed. removed Population that has recovered, has been âRemovedâ may also include vaccinated individuals in some successfully immunized, or has died. models. DHS Lexicon: âpopulation that has recovered or died.âa susceptible Individual or population who, if exposed to an This can be used as an adjective or noun, but most often, in infectious agent, could become infected. the modeling context, as a noun. DHS Lexicon: âpopulation who [sic] is at risk of becoming infected if exposed to an infectious agent.âa aDepartment of Homeland Security. 2007. âA Lexicon of Risk Terminology and Methodological Description of DHS Bioterrorism Risk Assessment.â April 14. bSociety for Risk Analysis (SRA), Glossary of Risk Analysis Terms. Available at sra.org/resources_glossary.php. Accessed Feb. 22, 2008. cCornell LCS Statistics Laboratory. See http://instruct1.cit.cornell.edu:8000/courses/statslab/Stuff/index.php. Accessed Feb. 22, 2008. dAmerican Heritage Dictionary. 2000. Boston: Houghton, Mifflin. eGlossary of Statistics Terms. Available at www.stat.berkeley.edu/users/stark/SticiGui/Text/gloss.htm. Accessed Feb. 22, 2008. fStatistical Education Through Problem Solving [STEP] Consortium. Available at www.stats.gla.ac.uk/steps/index.html. Accessed Feb. 22, 2008. gW. Feller. 1968. An Introduction to Probability Theory and Its Applications. New York, N.Y.: Wiley. hB. DeFinetti. 1974. Theory of Probability. Hoboken, N.J.: Wiley. iD.V. Lindley. 1965. Introduction to Probability and Statistics from a Bayesian Viewpoint; Part 1: Probability. Cambridge. U.K.: Cambridge University Press. jN.J. McCormick. 1981. Reliability and Risk Analysis. San Diego, Calif.: Academic Press. kJ.L. Devore. 2000. Probability and Statistics for Engineering and the Sciences. Pacific Grove, Calif.: Duxbury Press. lE. PatÃ©-Cornell. 1983. âFault Trees vs. Event Trees in Reliability Analysis.â Risk Analysis 4(3):177-186. mA.F.W. Edwards. 1992. Likelihood. Baltimore, Md.: Johns Hopkins University Press. nThe White House. 2004. Available at www.whitehouse.gov/infocus/bioshield. Accessed Feb. 22, 2008. oS.M. Ross. 2000. Introduction to Probability Models. New York, N.Y.: Academic Press. pDuke University. 1998. Statistical and Data Analysis for Biological Sciences. Available at isds.duke.edu/courses/Fall98/sta210b/terms.html. Accessed Feb. 22, 2008. qWikipedia: âStatistics.â Available at en.wikipedia.org/wiki/Statisics. Accessed Feb. 22, 2008. rM.S. Meyer and J.M. Booker. 1991. Eliciting and Analyzing Expert Judgment. Los Alamos, N.M.: Los Alamos National Laboratory. sDHS (Department of Homeland Security). 2006. Bioterrorism Risk Assessment. Biological Threat Characterization Center of the National Biodefense Analysis and Countermeasures Center. Fort Detrick, Md. tNational Research Council. 2006. Interim Report on Methodological Improvements to the Department of homeland Securityâs Biological Agent Risk Analysis. Washington, D.C.: The National Academies Press. uE.J. Henley and H. Kumamoto. 1981. Reliability Engineering and Risk Assessment. Upper Saddle River, N.J.: Prentice-Hall. vJ.C. Thomas and D.J. Weber. 2001. Epidemiologic Methods for the Study of Infectious Diseases. Oxford, U.K.: Oxford University Press. wR.M. Anderson and R.M. May. 1991. Infectious Diseases of Humans. Oxford, U.K.: Oxford University Press.