11
Risk Assessment Models and Methods

RISK ASSESSMENT METHODOLOGY

The occurrence of cancers is known to be related to a number of factors, including age, sex, time, and ethnicity, as well as exposure to environmental agents such as ionizing radiation. Understanding the role of exposure in the occurrence of cancer in the presence of modifying effects is a difficult problem. Contributing to the difficulty are the stochastic nature of cancer occurrence, both background and exposure related, and the fact that radiogenic cancers are indistinguishable from nonradiogenic cancers.

This section summarizes the theory, principles, and methods of risk assessment epidemiology for studying exposure-disease relationships. The two essential components of risk assessment are a measure of exposure and a measure of disease occurrence. Measuring exposure to radiation is a challenging problem, and dosimetry issues are discussed in detail elsewhere in this report; the common epidemiologic measures of disease occurrence are reviewed in this section. Evaluation of the association between exposure and disease occurrence is aided by the use of statistical models, and the types of models commonly used in radiation epidemiology are described below, as are the methods for fitting the models to data. This section ends with a description of the use of fitted models for estimating probabilities of causation and certain measures of lifetime detriment associated with exposure to ionizing radiation.

Rates, Risks, and Probability Models

Some individuals exposed to environmental carcinogens (e.g., ionizing radiation) develop cancer and some do not; the same is true of unexposed individuals. Thus, cancer is not a necessary consequence of exposure, and exposure is not necessary for cancer. However, the greater incidence of cancer in individuals exposed to known carcinogens indicates that the probability or risk of developing cancer is increased by exposure. Compared to unexposed individuals, the elevated risks of exposed individuals are manifest by increased cancer rates in the latter group. Risks and rates are the basic measures used to compare disease occurrence in exposed and unexposed individuals. This section describes rates and risks and their relationship to one another as a prelude to the sections on modeling and model fitting.

Incidence Rate

A common measure of disease occurrence used in cancer epidemiology is the incidence rate. Incidence refers to new cases of disease occurring among previously unaffected individuals. The population incidence rate is the number of new cases of the disease occurring in the population in a specified time interval divided by the sum of observation times, in that interval, on all individuals who were disease free at the beginning of the time interval. In general an incidence rate is time dependent and depends on both the starting point and the length of the interval.

With data from studies in which subjects are followed over time, incidence rates can be estimated by partitioning the following period into intervals of lengths Lj having midpoints tj for j = 1,…,J, and estimating a rate for each interval. Let nj denote the number of individuals who are disease free and still under observation at time tj, and dj the number of new diagnoses during the jth interval. An estimate of the incidence rate at time tj is obtained by dividing dj by the product of nj and Lj:

The denominator in is an approximation to the sum of observation times on the nj population members in the jth interval and in practice is usually replaced by the actual observation time, which accounts for the fact that the dj diagnoses of disease did not occur exactly at time tj.



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11 Risk Assessment Models and Methods RISK ASSESSMENT METHODOLOGY creased by exposure. Compared to unexposed individuals, the elevated risks of exposed individuals are manifest by The occurrence of cancers is known to be related to a increased cancer rates in the latter group. Risks and rates are number of factors, including age, sex, time, and ethnicity, as the basic measures used to compare disease occurrence in well as exposure to environmental agents such as ionizing exposed and unexposed individuals. This section describes radiation. Understanding the role of exposure in the occur- rates and risks and their relationship to one another as a pre- rence of cancer in the presence of modifying effects is a lude to the sections on modeling and model fitting. difficult problem. Contributing to the difficulty are the sto- chastic nature of cancer occurrence, both background and exposure related, and the fact that radiogenic cancers are Incidence Rate indistinguishable from nonradiogenic cancers. A common measure of disease occurrence used in cancer This section summarizes the theory, principles, and meth- epidemiology is the incidence rate. Incidence refers to new ods of risk assessment epidemiology for studying exposure- cases of disease occurring among previously unaffected in- disease relationships. The two essential components of risk dividuals. The population incidence rate is the number of assessment are a measure of exposure and a measure of dis- new cases of the disease occurring in the population in a ease occurrence. Measuring exposure to radiation is a chal- specified time interval divided by the sum of observation lenging problem, and dosimetry issues are discussed in de- times, in that interval, on all individuals who were disease tail elsewhere in this report; the common epidemiologic free at the beginning of the time interval. In general an inci- measures of disease occurrence are reviewed in this section. dence rate is time dependent and depends on both the start- Evaluation of the association between exposure and disease ing point and the length of the interval. occurrence is aided by the use of statistical models, and the With data from studies in which subjects are followed types of models commonly used in radiation epidemiology over time, incidence rates can be estimated by partitioning are described below, as are the methods for fitting the mod- the following period into intervals of lengths Lj having mid- els to data. This section ends with a description of the use of points tj for j = 1,…,J, and estimating a rate for each interval. fitted models for estimating probabilities of causation and Let nj denote the number of individuals who are disease free certain measures of lifetime detriment associated with expo- and still under observation at time tj, and dj the number of sure to ionizing radiation. new diagnoses during the jth interval. An estimate of the incidence rate at time tj is obtained by dividing dj by the Rates, Risks, and Probability Models product of nj and Lj: Some individuals exposed to environmental carcinogens ˆ dj λ (t j ) = . (e.g., ionizing radiation) develop cancer and some do not; n j Lj the same is true of unexposed individuals. Thus, cancer is ˆ The denominator in λ(t j ) is an approximation to the sum of not a necessary consequence of exposure, and exposure is observation times on the nj population members in the jth not necessary for cancer. However, the greater incidence of interval and in practice is usually replaced by the actual ob- cancer in individuals exposed to known carcinogens indi- servation time, which accounts for the fact that the dj diag- cates that the probability or risk of developing cancer is in- noses of disease did not occur exactly at time tj. 259

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260 BEIR VII Risk The cumulative incidence rate and the distribution function satisfy the relationship Risk is defined as the probability that an individual devel- ops a specified disease over a specified interval of time, F(t) = 1 – exp{–Λ(t)}, (11-1) given that the individual is alive and disease free at the start of the time period. As with the incidence rate, risk is time from which it follows that the instantaneous incidence rate dependent and depends on both the starting point and the completely determines the first-occurrence distribution F(t). length of the interval. In a longitudinal follow-up study as The risk of first disease occurrence in the interval (t, t + h), described above, the proportion of new occurrences dj among given no previous occurrence, is the conditional probability nj disease-free individuals still under observation at time tj, dj p(t j ) = ˆ , F (t + h ) − F (t ) nj p(t, t + h) = . 1 − F (t ) is an estimate of the risk or probability of disease occurrence in the jth time interval. When h is not too large, so that the difference quotient {F(t Incidence rates and risks are related via the general for- + h) – F(t)} / h approximates f(t) = dF(t) / dt, mula, risk = rate × time. For the longitudinal follow-up study estimates defined above, the relationship is manifest by the F (t + h ) − F (t ) h f (t ) equation p(t, t + h) = ≈ h = λ(t )h. h 1 − F (t ) 1 − F (t ) ˆ p(t ) = λ(t ) L . ˆ j j j Thus, among individuals who are disease free at time t, the Probability Models risk of disease in the interval (t, t + h) is approximately λ(t)h. This approximation is the theoretical counterpart of the re- The description of rates and risks in terms of estimates lationship between risks and rates described in the discus- from a longitudinal follow-up study is informative and sion of risk. In the remainder of this chapter, incidence rate clearly indicates the relevance of these numerical quantities means instantaneous incidence rate unless explicitly noted to the study of disease. However, the development of a gen- otherwise. eral theory of risk and risk estimation requires definitions of rates and risks that are not tied to particular types of studies or methods of estimation. Probability models provide a math- Incidence Rates and Excess Risks ematical framework for studying incidence rates and risks It is clear that the incidence rate plays an important role in and also are used in defining statistical methods of estima- the stochastic modeling of disease occurrence. Conse- tion depending on the type of study and the data available. quently, models and methods for studying the dependence of Models for studying the relationship between disease and disease occurrence on exposure are generally formulated in exposure are usually formulated in terms of the instanta- terms of incidence rates. In the following it is assumed that neous incidence rate, which is the theoretical counterpart of individuals have been stratified on the basis of age, sex, cal- the incidence rate estimate defined below. The instantaneous endar time, and possibly other factors related to disease oc- incident rate is defined in terms of the probability distribu- currence, and that incidence rates are stratum specific. In the tion function F(t) of the time to disease occurrence. That is, simple case of two exposure categories, exposed and unex- F(t) represents the probability that an individual develops posed, let λE(t) and λU(t) denote the incidence rates of the the disease of interest in the interval of time (0, t). Two func- exposed and unexposed groups, respectively. If disease oc- tions derived from F(t) are used to define the instantaneous currence is unrelated to exposure, one expects that λE(t) = incidence rate. One is the survivor function, which is the λU(t), whereas lack of equality between these two incidence probability of being disease free throughout the interval (0, rates indicates an association between disease occurrence t) and is equal to 1 – F(t). The second is the probability and exposure. density function, which is the derivative of F(t) with respect A common measure of discrepancy between incidence to t, that is, f(t) = (d / dt)F(t), and measures the rate of in- rates is the difference crease in F(t). The instantaneous incidence rate, also known as the hazard function, is the ratio EAR(t) = λE(t) – λU(t), f (t ) λ(t ) = . which by convention is called the excess absolute risk (EAR) 1 − F (t ) even though it is, technically, a difference in rates. Rear- Integrating the instantaneous incident rate yields the cumu- ranging terms results in lative incidence rate t Λ(t ) = ∫ λ(u)du. λE(t) = λU(t) + EAR(t), 0

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RISK ASSESSMENT MODELS AND METHODS 261 showing that EAR(t) describes the additive increase in inci- detect (using statistical methods) in the presence of back- dence rate associated with exposure. For example, if the EAR ground risks. is constant, EAR(t) = b, then the effect of exposure is to The difficulties can be seen by considering the estimates increase the incidence rate by the constant amount b for all of risk from the longitudinal follow-up study described in time periods. Note that b = 0 corresponds to the case of no “Rates, Risks, and Probability Models.” For a time period Lj, association. let nj,E, dj,E and nj,U, dj,U denote the number of individuals at A second common measure of discrepancy is the relative risk at the start of the interval and the number of occurrences risk (RR), defined as of disease during the interval for the exposed and unexposed λ E (t ) subgroups, respectively. A direct estimate of the excess risk RR(t ) = . for the jth time period is the difference between two propor- λ U (t ) tions (dj,E / nj,E) – (dj,U / nj,U). Even in the favorable situation Rearranging terms shows that in which the baseline risk is relatively well estimated com- pared to the risk of the exposed group (when nj,U is large λE(t) = RR(t)λU(t), relative to nj,E), the ability to reliably detect small increases in risk associated with exposure requires a large number of so that RR(t) describes the multiplicative increase in inci- exposed individuals at risk. For example, using the usual dence rate associated with exposure. When the RR is con- criterion for statistical testing in order to detect with prob- stant, RR(t) = r, the effect of exposure is to alter incidence ability .80 a 5% increase in risk when the baseline risk is rate by the factor r. If exposure increases risk, then r > 1; if 0.10, the number of individuals at risk in the exposed group exposure decreases risk, then r < 1, and r = 1 corresponds to would have to be approximately nj,E = 30,000. the case of no association. The excess relative risk ERR(t) is A key objective of this report is the calculation of quanti- tative estimates of human health risks (e.g., cancer) associ- ERR(t) = RR(t) – 1. ated with exposure to ionizing radiation for specific sub- populations defined by stratification on variables such as sex, The ERR of the exposed and unexposed incidence rates are age, exposure profile, and smoking history. In theory, such related via the equation estimates could be derived by identifying a large group of individuals having common exposure profiles within each λE(t) = λU(t) {1 + ERR(t)}. stratum and following the groups over a long period of time. As described above, the proportion of individuals in each group who develop cancer in specific time periods provides RISK MODELS the desired estimates of risk. However, this approach is not feasible because sufficient data are not available. At low lev- Direct Estimates of Risk els of exposure, cancer risks associated with exposure are The previous section defined the fundamental quantities small relative to baseline or background risks. The increases used in risk estimation: risks, rates, EAR, RR, and ERR, and in observed cancer rates associated with exposure are small established their relevance to the study of environmental relative to the natural random fluctuations in baseline cancer carcinogens. These measures enable the study of differences rates. Thus, very large groups of individuals would have to in disease occurrence in relationship to time, by studying be followed for very long periods of time to provide suffi- either EAR(t) or ERR(t) between unexposed and exposed ciently precise estimates of risk associated with exposure. groups. For most carcinogens, exposure is not a simple Consequently, direct estimates of risk are not possible for dichotomy (unexposed, exposed) but occurs on a continuum. stratified subpopulations. The alternative is to use math- That is, the exposure or dose d can vary from no exposure ematical models for risk as functions of dose and stratifying (d = 0) upward. In such cases the relationship between risk— variables such as sex and age. or EAR(t) or ERR(t)—and dose is of fundamental impor- tance. For all carcinogens it is generally agreed that suffi- Estimation via Mathematical Models for Risk ciently large doses increase the risk of cancer. By definition there is no increase in risk in the absence of exposure (d = 0). Model-based estimation provides a feasible alternative to That is, when d = 0, both EAR(t) = 0 and ERR(t) = 0. Thus, direct estimation. Model-based estimates efficiently exploit for many carcinogens the only open or unresolved issue is the information in the available data and provide a means of the dependence of risk on small or low doses. Low-dose deriving estimates for strata and dose profile combinations ranges are often the most relevant in terms of numbers of for which data are sparse. This is accomplished by exploit- exposed individuals. They are also the most difficult ranges ing assumptions about the functional form of a risk model. for which to obtain unequivocal evidence of increased risk. Of course, the validity of estimates derived from models These difficulties result from the fact that small increases in depends on the appropriateness of the model; thus model risk associated with low levels of exposure are difficult to choice is important. The accepted approach in radiation epi-

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262 BEIR VII demiology is to base models on radiobiological principles The number of initiated cells arising from the normal cell and theories of carcinogenesis to the fullest extent possible, pool is described by a Poisson process with a rate of vX. The keeping in mind statistical limitations imposed by the quan- initiated cells then divide either symmetrically or non- tity and quality of data available for model fitting. Biologi- symmetrically. Symmetrical division results in two initiated cally based and empirically derived mathematical models cells, while nonsymmetrical division results in an initiated for risk are discussed in the next two sections. cell and a differentiated cell. The rate of symmetrical division is designated by α(t), and the death differentiation rate by β(t). The difference α – β is the net proliferation rate for initiated Biologically Based Risk Models cells. The rate of division into one initiated cell and one malig- Biologically based risk models are designed to describe nant cell is designated by µ(t) (Hazleton and others 2001). the fundamental biological processes involved in the trans- TSCE models for radiation carcinogenesis have now been formation of somatic cells into malignant cancer cells. The applied successfully to a number of important data sets, in- use of biologically based risk models in epidemiologic analy- cluding atomic bomb survivors (Kai and others 1997) and ses can result in a greater understanding of the mechanisms occupational groups such as nuclear power plant workers of carcinogenesis. These models can also help to expose the and miners (Moolgavkar and others 1993; Luebeck and oth- complex interrelationships between different time- and age- ers 1999; Sont and others 2001). A study of atomic bomb dependent exposure patterns and cancer risk. Biologically survivors illustrates the usefulness of the two-stage model in based risk models provide an analytical method that is radiation epidemiology (Kai and others 1997). Findings from complementary to the traditional, well-established, empiri- this analysis include the observation of a high excess risk cal approaches. among children that may not be explained by enhanced tis- Armitage and Doll (1954) observed that for many human sue sensitivity to radiation exposure. The temporal patterns cancers the log-log plot of age-specific incidence rates ver- in cancer risk can be explained in part by a radiation-induced sus age is nearly linear, up to moderately old ages. This ob- increase in the pool of initiated cells, resulting in a direct servation has led to the development of models for carcino- dose-rate effect (Kai and others 1997). Exact solutions of the genesis. In brief, Armitage and Doll’s theory postulates that two-stage model (Heidenreich and others 1997) and multi- malignant transformation occurs following the kth stage of a stage models (Heidenreich and others 2002b) have been ap- series of spontaneous and irreversible changes (Armitage plied to atomic bomb survivors’ data. 1985). The corresponding hazard function is of the form Another data set to which application of the TSCE has λ(t) = atk–1, where t denotes time and a is a constant reflect- been useful is the National Dose Registry (NDR) of Canada. ing the dependence of the hazard on the number of stages, k. This database contains personal dosimetry records for work- These models have been fit to various data sets, leading to ers exposed to ionizing radiation since 1951, with current the observation that most cancers arise after the occurrence records for more than 500,000 Canadians (Ashmore and oth- of five to seven stages. Comprehensive reviews of the math- ers 1998). Application of the TSCE model to the NDR sug- ematical theory of carcinogenesis have been given by gests an explanation of the apparently high excess relative Armitage and Doll (1961), Whittemore (1978), and Armitage risk observed, relative to the A-bomb data (Sont and others (1985). 2001). The TCSE model reveals that the dose-response for In response to the multiplicity of parameters produced by the NDR cohort is consistent with the lung cancer incidence their earlier models, Armitage and Doll proposed a simpler in the A-bomb survivors’ cohort, provided that proper ad- two-stage model designed to avoid parameters not readily justments are made for the duration of exposure and differ- estimable from available data. A major limitation of these ences in the background rate parameters. early two-stage models is their failure to address the multi- In addition to the TSCE model, the Armitage-Doll model plication and death of normal cells, which was known to of carcinogenesis has evolved into several other analytic occur in tissues undergoing malignant change (Moolgavkar methods, including the general mutagen model (Pierce and Knudson 1981). A revised two-stage model was later 2002). The basic assumption of this model is that a malig- proposed by Moolgavkar and colleagues, which allowed for nant cell results from the accumulation of mutations, with k the growth of normal tissue and the clonal expansion of in- mutations required for malignancy. The effect of exposure is termediate cells (Moolgavkar and Knudson 1981). Numer- that an increment of dose at age a, at rate d(a), results in a ous two-stage models have since been described in the lit- multiplicative increase λr[1 + βd(a)] in the rate of all k mu- erature (Fisher 1985; Moolgavkar 1991; Sielken and others tations. Although this model applies to both recessive and 1994; Luebeck and others 1996; Heidenreich and others dominant mutations, it does not explicitly allow for selective 1999, 2002a, 2002b; Moolgavkar and others 1999; Heiden- proliferation of cells having only some of the required muta- reich and Paretzke 2001; Moolgavkar and Luebeck 2003). tions. The general mutagen model has been applied success- The two-stage clonal expansion (TSCE) model assumes a fully to A-bomb survivor data (Pierce and Mendelsohn 1999; normal stem cell population of fixed size X and a rate of first Pierce and Preston 2000) and to underground miners exposed mutation of v(d), depending on the dose d of the carcinogen. to radon (Lubin and others 1995).

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RISK ASSESSMENT MODELS AND METHODS 263 Whereas empirical approaches to risk modeling rely on females, 0 for males) and other study population-specific statistical models to describe data, biologically based mod- factors generically represented by p. For example, the study els depend on fundamental assumptions regarding the population-specific parameters for A-bomb survivor data mechanisms of radiation carcinogenesis. The parameters cre- models are city c and calendar year y, that is, p = (c, y). The ated by modern biologically based risk models have direct incidence rate is, in general, a function λ(a, e, d, s, p) of all of biological interpretation, provide insight into cancer mecha- these factors. By definition, the background incidence rate nisms, and generate substantive questions about the path- does not depend on either d or e, so the EAR formulation of ways by which exposure to ionizing radiation can increase the exposed incidence rate has the form cancer risk. These models also provide a way of describing temporal patterns of exposure and risk. λ(a, e, d, s, p) = λ(a, s, p) + EAR(a, e, d, s, p), Although biologically based risk models have many and the ERR formulation is strengths, some general limitations are associated with their use. Such models can only approximate biological reality λ(a, e, d, s, p) = λ(a, s, p) {1 + ERR (a, e, d, s, p)}, and require an understanding of the complex mechanisms of radiation carcinogenesis for interpretation. In addition, it is where EAR (a, e, d, s, p) and ERR (a, e, d, s, p) are the EAR difficult to distinguish among alternative models that yield and ERR, respectively. When the excess risk functions are similar dose-response curves without direct information on dependent on the study population—that is, when they the fundamental biological processes represented by the depend on the factor p—estimates of risk derived from the model, which are often unknown. Biologically based risk models are specific to the study population and therefore of models are generally more complex than empirical models limited utility for estimating risks in other populations. Thus, and may require richer databases to develop properly. De- it is desirable to find suitable models in which either the spite these limitations, biologically based models have found excess risk or the excess relative risk does not depend on many applications for important epidemiologic data sets, and population-specific parameters. Consequently, models used the successes achieved to date afford support for the con- in radiation risk estimation are often of the form tinual development of such models for future analyses that will directly inform the association between radiation expo- λ(a, e, d, s, p) = λ(a, s, p) + EAR(a, e, d, s) sure and human cancer risk. or Biologically based models have not been employed as the λ(a, e, d, s, p) = λ(a, s, p) {1 + ERR (a, e, d, s)}. primary method of analysis in this report for several reasons. The mechanisms of radiation carcinogenesis are not fully That is, the excess risk functions depend only on a, e, d, and understood, which makes the development of a fully bio- s, but not p. Note that if t represents time after exposure, then logically based model difficult. The data required for a bio- because t = a – e, any two of the variables t, a, and e deter- logically based model, such as rates of cell proliferation and mine the third, so at the current level of generality, the ex- mutation, are also generally not available. The availability cess risk functions could also be written as functions of t, e, of empirical risk models that provide a good description of d, and s. Also, because there is no excess risk at ages prior to the available data on radiation and cancer permits the prepa- exposure (a < e), ER(a, e, d, s) = 0 (a < e), EAR(a, e, d, s) = ration of useful risk projection. 0 and ERR(a, e, d, s) = 0 for a < e and thus, λ(a, e, d, s, p) = λ(a, s, p) for a < e. The formulas and equations in the re- mainder of this chapter are described only for the relevant Empirically Based Risk Models case a e. The following symbols are used to describe the variables Radiobiological considerations suggest that for low-dose, that enter into risk models based on the Japanese A-bomb low-LET (linear energy transfer) radiation, the risk of dis- survivor data: ease for an individual exposed to dose d depends on a linear or quadratic function of d. That is, risk depends on dose d a: attained age of an individual through a function of the form e: age at exposure to radiation d: dose of radiation received f(d) = α1d + α2d2, s: code for sex (1 if the individual is a female and 0 if male) where α1 and α2 are parameters to be estimated from the p:study population-specific factors data. At higher doses of radiation, cell sterilization and cell Models also sometimes include time since exposure (t). death compete with the process of malignant transformation, Since t = a – e, models that include a and e implicitly thereby attenuating the risk of cancer at higher doses. A more include t. general model applicable to a broader dose range and used Models for the incidence rate for individuals of age a, extensively in radiation research is exposed to dose d, at age e, generally depend on sex s (1 for

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264 BEIR VII f(d) = α1d + a2d2) exp(–α3d – α4d2). modifying factors depend on parameters that must be esti- mated from data. The most common method of fitting risk The models for dependence on dose are generally incorpo- model data (i.e., estimating the unknown parameters in the rated into risk models by assuming that the excess risk func- model) is the method of maximum likelihood reference. tions are proportional to f(d), where the multiplicative con- Given a model for the probability density of the observed stant (in dose) depends on a, e, and s. data, a likelihood is obtained by evaluating the density at the observed data. The likelihood is a function of the data and VARIABLES THAT MODIFY THE DOSE-RESPONSE the unknown parameters in the probability density model. RELATIONSHIP The parameters are estimated by those values in the param- eter space (the set of all allowable parameter values) that In general, cancer rates vary considerably as functions of maximize the likelihood for the given data values. attained age, and there is strong evidence indicating that can- There are several approaches for the numerical calcula- cer risks associated with radiation exposure also vary as tions of likelihood analysis. Estimation based on grouped functions of attained age and age at exposure. For example, data using a Poisson form of the likelihood (Clayton and it has been observed that after instantaneous exposure to ra- Hills 1993) has been used for the analyses of atomic bomb diation, leukemia and bone cancer rates rise for a short pe- survivors and other major epidemiologic studies of radiation riod of time (≈ years) and then decrease to baseline rates health risks. over a longer period of time (≈ years). In contrast, the avail- This analysis is facilitated by forming a table so that indi- able evidence suggests, and it is generally believed, that rates viduals contributing information to each cell of the table have for most other cancers increase after exposure to radiation equal, or approximately equal, background rates. In particu- and possibly remain at elevated levels at all ages. lar, the table is formed by the cross-classification of indi- Models for the dependence of risk on variables such as viduals into categories of age at exposure, time period, expo- age at exposure, attained age, and time since exposure are sure dose, and all other variables that appear in the model. often empirical and are justified more by epidemiologic and The key summary variables required for each cell are the statistical principles than by radiobiological theory. A useful total person-years (PY) of observation in the cell, the num- class of models that includes the modifying effects on radia- ber of new cases of cancer, the mean dose, the mean age at tion dose-response of attained age, age at exposure, and gen- exposure, and the mean age or mean time since exposure. der has the form For an RR model, the contribution to the likelihood from λ(a, e, d, s, p) = λ(a, s, p) + f (d)g(a, e, s); the data in each cell of the table has the same form as a Poisson likelihood (thus permitting well-understood and for EAR models, and straightforward computations), with the mean equal to the λ(a, e, d, s, p) = λ(a, s, p) {1 + f (d)g(a, e, s)}; product of PY; a parameter for the common, cell-specific background rate; and the RR 1 + fg, where f and g are func- for ERR models, where g(a, e, s) is a function of attained tions of dose and of age, age at exposure, and sex, described age, age at exposure, and gender. Because time since expo- previously. sure is equal to the difference t = a – e, this class of models The full likelihood is the product of the cell-specific Pois- includes models defined as functions of time since exposure. son likelihoods. Numerical optimization is required to maxi- Often g depends on e and t via exponential and power mize the likelihood, and statistical inference generally is functions. based on large-sample approximations for maximum likeli- For example, the committee’s preferred model for solid hood estimation. cancer uses g(a, e, s) = exp (γ˜ + η1n(a) + θs), e Using the Estimated Model where e is e – 30 years if e is less than 30, and 0 if e is greater ˜ The models developed as described above can be used to than or equal to 30; and γ, η, and θ are unknown parameters, estimate both lifetime risks and probabilities of causation, which must be estimated from the data. both of which are discussed below. Following this, several limitations in the use of these models, which lead to uncer- tainties in estimated risks, are discussed. Further discussion Model Parameter Estimation of uncertainties and the committee’s approach to quantify- Models describe the mathematical form of a risk func- ing them can be found in Chapter 12. tion, but the parameters in the model must be estimated from data. For example, a linear dose model presupposes that risk Estimating Lifetime Risks increases linearly with dose but the slope of the line, which measures the increase in risk for a unit increase in dose, must To calculate the lifetime risk for a particular age at expo- be estimated from data. Similarly, models for the effect of sure and a particular gender, one essentially follows a sub-

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RISK ASSESSMENT MODELS AND METHODS 265 ject forward in time and calculates the risk of developing a Modeling Caveats radiation-induced cancer at each age subsequent to age at The theory of risk assessment, modeling, and estimation exposure. This requires probabilities of survival to each sub- and the computational software for deriving statistically sequent age, which are obtained from life tables for the popu- sound parameter estimates from data provide a powerful set lation of interest. ERR models are expressed in terms of a of tools for calculating risk estimates. Risk models provide relative increase in the sex- and age-specific background the general form of the dependence of risk on dose and risk- rates for the cancer of interest; these rates are usually ob- modifying factors. Specific risk estimates are obtained by tained from cancer mortality vital statistics for the popula- fitting the models (estimating unknown parameters) to data. tion of interest (or incidence rates if cancer incidence is to be The role of data in the process of risk estimation cannot be estimated). overemphasized. Neither theory, models, nor model-fitting An important issue in estimating lifetime risks is the ex- software can overcome limitations in the data from which trapolation of risks beyond the period for which follow-up risk estimates are derived. In human epidemiologic studies data are available. No population has been followed for more of radiation, both the quality and the quantity of the data than 40 or 50 years; thus, it is not possible to model the EAR available for risk modeling are limiting factors in the estima- or ERR directly for the period after follow-up has ended, a tion of human cancer risks. The quality of data, or lack limitation that is primarily important for those exposed early thereof, and its impact on risk modeling are discussed below in life. Estimating lifetime risks for this group thus requires under three broad headings. The primary consequence of assumptions that are usually based on the observed pattern less-than-ideal data is uncertainty in estimates derived from of risk over the period for which data are available. For ex- such data. ample, if the ERR appears to be a constant function of time since exposure, it may be reasonable to assume that it re- mains constant. Alternatively, if the EAR or ERR has de- Incomplete Covariate Information clined to nearly zero by the end of the follow-up period, it The specificity of risk models is limited by the informa- may be reasonable to assume that the risk remains at zero. tion available in the data. Even the most extensive data sets Another important issue is how to apply risks estimated contain, in addition to measurements of exposure, informa- from studying a particular exposed population to another tion on only a handful of predictor variables such as dose, population that may have different characteristics and dif- age, age at exposure, and sex. Consequently, models fit to ferent background risks. Specifically, the application of esti- such data predict the same risk of cancer for individuals hav- mates based on Japanese atomic bomb survivors to a U.S. ing the same values of these predictor variables, regardless population is a concern, since background rates for some of other differences between the two individuals. For ex- specific cancers (including stomach, colon, liver, lung, and ample, two individuals who differ with respect to overall breast) differ substantially between the two populations. The health status, family history of cancer (genetic disposition to BEIR V (NRC 1990) committee calculations were based on cancer), exposure to other carcinogens, and so on, will be the assumption that relative risks (ERR) were comparable assigned the same estimated risk provided they were exposed for different populations; however, the BEIR III (NRC 1980) to the same dose of radiation, are of the same age, and have committee modified its ERR models based on the assump- the same age at exposure and the same gender. tion that absolute risks were comparable. Some recent ef- Consequently, among a group of individuals having the forts have used intermediate approaches with allowance for same values of the predictor variables in the model, some considerable uncertainty (NIH 1985, 2003). will have a higher personal risk than that predicted by the model and some will have a lower personal risk. However, Estimating Probabilities of Causation on average, the group risk will be predicted reasonably well by the model. The situation is similar to the assessment of The probability of causation (PC; NIH 1985, 2003) is insurance risk. Not all teenage males have the same personal defined as the ratio of ERR to RR: risk of having an automobile accident (some are better driv- ERR ers than others), yet as a group they are recognized as having PC = 1 + ERR a greater-than-average risk of accidents, and premiums are where for brevity the dependence of ERR on dose, time vari- set accordingly. From the insurance company’s perspective, ables, and possibly other individual characteristics is sup- the premiums are set fairly in the sense that their risk models pressed. For the RR models described previously, ERR = fg, adequately predict the claims experience of the group. where f = f(d) and g = g(a, e, s), in which case Radiation risk models are similar in that they adequately fg predict the disease experience of a group of individuals shar- PC = . ing common values of predictor variables in the model. How- 1 + fg ever, such estimated risks need not be representative of indi- Thus, the ERR model provides immediate PC estimates. vidual personal risks.

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266 BEIR VII Estimated Doses The estimation of risk models from atomic bomb survi- vors has been carried out with a statistical technique that The standard theory and methods of risk modeling and accounts for the random uncertainties in nominal doses estimation are appropriate under the assumption that dose (Pierce and others 1990). To the extent that it is based on is measured accurately. Estimated radiation dose is a com- correct assumptions about the forms and sizes of dose uncer- mon characteristic of human epidemiologic data, and ques- tainties, it removes the bias due to random dose measure- tions naturally arise regarding the adequacy of dose esti- ment errors. mates for the estimation of risk parameters and the calculation of risk estimates. These are different problems and are discussed separately. Data from Select Populations First, consider the problem of calculating risk estimates Ideally, risk models would be developed from data gath- from a given risk equation. Suppose that the risk equation ered on individuals selected at random from the population has been estimated without bias and with sufficient preci- for which risk estimates are desired. For example, in esti- sion to justify its use in the calculation of risks. Assume mating risks for medical workers exposed to radiation on the also that risk increases with dose: that is, the risk equation job, the ideal data set would consist of exposure and health yields higher risks for higher doses. Suppose that an esti- information from a random sample of the population of such mate of lifetime risk is desired for an individual whose workers. However, data on specific populations of interest dose is estimated to be d. If d overestimates the in- are generally not available in sufficient quantity or with ex- dividual’s true dose, the lifetime risk will be overesti- posures over a wide enough range to support meaningful mated; if d underestimates the true dose, the risk will be statistical modeling. Radiation epidemiology is by necessity underestimated. This is intuitive and is a consequence of opportunistic with regard to the availability of data capable the fact that risk is an increasing function of dose. of supporting risk modeling, as indicated by the intense study The problem of estimating risk equation parameters of A-bomb survivors and victims of the Chernobyl accident. from data with estimated doses is a little more complicated. A consequence of much significance and concern is the Errors in estimated doses can arise in a number of different fact that risk models are often estimated using data from one ways, not all of which have the same impact on risk param- population (often not even a random sample) for the purpose eter estimation. For example, flaws in a dosimetry system of estimating risks in some other population(s). Cross-popu- have the potential to affect all (or many) dose estimates in lation extrapolation of this type is referred to as “transport- the same manner, leading to systematic errors for which all ing” the model from one population to another. The poten- (or many) dose estimates are too high or too low. Errors or tial problem it creates is the obvious one—namely, that a incomplete records in data from which dose estimates are risk equation valid for one population need not be appropri- constructed (e.g., badge data from nuclear industry work- ate for another. Just as there are differences in the risk of ers) are likely to result in more or less random errors in cancer among males and females and among different age dose estimates (i.e., some individuals will have dose esti- groups, there are differences in cancer risks among different mates that are too high and others will have estimates that populations. For example, the disparity between baseline are too low). Systematic errors can result in biased esti- rates for certain cancers (e.g., stomach cancer) in Japanese mates of risk equation parameters. The type of bias de- and U.S. populations suggests the possibility of differences pends on the nature of the systematic error. For example, in the risks due to radiation exposure. risk equations derived from data with doses that are overes- Transporting models is generally regarded as a necessity, timated by a constant factor (>1) will result in an underesti- and much thought and effort are expended to ensure that mation of risk at a particular given dose d; doses that are problems of model transportation are minimized. The deci- underestimated by a constant factor (<1) will result in an sion to use EAR models or ERR models is sometimes influ- overestimation of risk. Random errors in dose estimates enced by concerns of model transport. Problems of trans- also have the potential to bias estimated risk equations. porting models from one population to another can never be Random error-induced bias generally results in the under- eliminated completely. However, to avoid doing so would estimation of risk. That is, random errors tend to have the mean that risk estimates would have to be based on data so same qualitative effect as systematic overestimation of sparse as to render estimated risks statistically unreliable. doses.